VIBRATION ANALYSIS FOR ELECTRONIC EQUIPMENT

VIBRATION ANALYSIS FOR ELECTRONIC EQUIPMENT THIRD EDITION

Dave S. Steinberg Steinberg & Associates and University of California, Los Angeles

A WILEY-INTERSCIENCE PUBLICATION

JOHN WILEY & SONS, INC. New York

Chichester

Weinheim

Brisbane

Singapore

Toronto

This book is printed on acid-free paper. @ Copyright G2000 by John Wiley & Sons, Inc. All rights reserved. Published simultaneously in Canada. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4744. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 605 Third Avenue, New York, NY 10158-0012, (212) 850-6011, fax (212) 850-6008, E-Mail: [email protected]. Library of Congress Cataloging in Publication Data:

Steinberg, Dave S. Vibration analysis for electronic equipment/Dave S. Steinberg.--3rd ed. p. cm “ A Wiley-Interscience publication.” ISBN 0-471-37685-X 1. Electronic apparatus and appliances-Vibration. I. Title. TK7870S8218 2000 621.381--dc21 Printed in the United States of America 109876

99-056617

To my wife, Annette, and to my two daughters, Con and Stacie

Electronic equipment continues its relentless expansion into virtually every area associated with commercial, industrial, and military applications throughout the entire world. Exotic technology has become commonplace in medicine, entertainment, communication, travel, transportation, manufacturing, education, and commerce. The internet has dramatically and forever changed the way many companies run their businesses, without the need for large offices and a large staff of people to perform clerical duties, largely due to the extensive use of the fast, small, powerful personal computers. This has resulted in improved profit margins in companies that embraced the new technologies, and disasters for companies that resisted the tidal wave of changes. Extensive changes were made in this third edition to reflect the changes that are taking place in the world of electronics. One of the big changes is the dramatic reduction in the costs of manufacturing the electronic equipment due to improved automation. This has resulted in the use of more electronics with improved performance over mechanical functions in a wide variety of products such as cameras, automobile controls for braking, ignition. air conditioners, transmission shifting, and combustion control, computers, computer printers, FAX machines, and televisions. One of the most dramatic changes has occurred in the military area, where high costs have forced the Department of Defense to scrap many of the military specifications in favor of best commercial practice electronic hardware for their new sophisticated equipment. Lower costs often lead to lower quality, which can result in a reduced reliability. This may only be an inconvenience when your television set fails or your automobile will not start. This can be a disaster in a military vehicle if the guns do not fire, or if the navigation system fails, or if the communications system fails. Therefore a new chapter was added that investigates and evaluates the effects of manufacturing methods and tolerances on the reliability of the electronic hardware. Another new chapter was added to show methods for improving the ruggedness of commercial hardware for improved reliability in harsh military environments. An additional chapter was added that relates to a better understanding of the bending distortion of the circuit board at its resonant condition, and how it effects the forces, stresses, and fatigue life in the component lead wires and solder joints. A new chapter was also added in the area of environmental stress screening, to provide a better understanding of how various combinations of xvii

XViii

PREFACE

thermal cycling and vibration can affect the accumulated damage and the amount of fatigue life used up by various screens. New equations are derived that show how to evaluate the effectiveness of a lead wire strain relief to reduce the dynamic forces and stresses in the lead wires and solder joints, and the resulting increase in the fatigue life. A simpler method for evaluating the fatigue life in random vibration is shown, which has the same accuracy as the three-band technique with much less work. A quick way of estimating the effectiveness of stiffening ribs on plug-in PCBs is shown, which is very simple to use and easy to apply. A new added section discusses a quick and easy method for evaluating the first three natural frequencies for flat plates and PCBs that are rectangular, octagonal, triangular, and round, with 16 different edge and point supports for each shape. A new and more accurate method for calculating the expected transmissibilty Q values for beams, circuit boards, and chassis is presented that relates to the natural frequency and the input acceleration G level. The data for this method are based on extensive vibration tests which showed that very low input acceleration G levels produced very low stresses and damping, which resulted in very high Q values. At the opposite end, high input G levels produced high stresses and damping, which resulted in low Q values. Many detailed sample problems are shown throughout this textbook, to demonstrate methods of analysis and evaluation that are based on 40 years of analysis and testing many different types of electronic systems. A section on vibration testing case histories was added to provide additional knowledge and insights for improving the testing results. Field failures in electronic equipment hardware compiled by the U.S. Air Force over a period of about 20 years show that about 40% of these failures are related to connectors, 30% to interconnects, and 20% to component parts. Failures in these areas can be due to handling, vibration, shock, and thermal cycling. Field failures related to operating environments show that about 55% of the failures are due to high temperatures and temperature cycling, 20% of the failures are related to vibration and shock, and 20% are due to humidity. Since high temperatures and temperature cycling events appear to be the major environmental causes of electronic failures, more information in these areas is very important. These subjects are not addressed in this book in great detail. More information is available on high temperatures and temperature cycling forces, stresses, and fatigue life effects [54]. Details on electronic components, electrical lead wires, solder joints, and plated through-holes for surface mounted and through-hole components, due to differences in the thermal coefficients of expansion (TCE), can be found in my book entitled Cooling Techniques for Electronic Equipment, Second Edition published by John Wiley & Sons. DAVES. STEINBERG Westlake Village, California September, 1999

DLIST OF SYMBOLS

A

Area (in.2), amplification g (dimensionless ratio) a length (in.) H ASIC Application-specific integrated circuit h Length (in.) Hz B Fatigue exponent, width (in.) I b Dynamic constant C C Length (in.) Lri C G Center of Gravity Distance from neutral axis J C K to outer fiber, length (in.), damping coefficient (lb . s/in.) Critical damping (lb .s/in.) cc D Dissipation energy KIN (lb .in./s) diameter (in.), plate bending stiffness Kt factor (lb . in.) Decibel dB KO D,, Plate torsional stiffness factor (Ib * in.) d L Diameter (in.), length (in.) DIP Dual inline package LCCC E Modulus of elasticity (lb/in.2) A4 e MT Bolted efficiency factor (%) ESS Environmental stress screen M , F Force (lb) Frequency (Hz) f f" Natural frequency (Hz) Rotational natural Mo fr MS frequency (Hz) FEM Finite element method Shear modulus (lb/in.2) G rn N with subscript: acceleration in gravity units (dimensionless)

Acceleration of gravity, 386 in./s2 Horizontal force Ob), drop heightr (in. or ft) Height (in.), thickness (in.) Frequency (cycles/s) Moment of inertia (area) (in.4) Mass moment of inertia (Ib in: s2) Torsional form factor (in.4> Linear spring rate (Ib in.), stiffness ratio (dimensionless), buckling form factor dimensionless) Kinetic energy Theoretical stressconcentration factor (dimensionless) Angular spring rate (Ib .in./rad) Length (in.) Leadless ceramic chip carrier Bending moment (lb . in.) Total moment (lb in.) Bending moment per unit length along the X axis (in: lb/inJ, bending moment at point X(lb. in.) Momentum (lb .s) Margin of safety (dimensionless) mass (lb s2/in.) Number of fatigue cycles to fail xix

XX

LIST OF SYMBOLS

Number of positive zero crossings, (Hz) n Actual number of fatigue cycles NS Number of sweeps through a resonance P Force (lb) P. PSD Power spectral density

N,+

wd W

X X X

x

z

Greek Symbols

(G*/HZ)

PCB d'

P

Q

4

R

Rn

RC

R, RMS RSS r sb

se

S SMD TCE T t

U V

W

Printed circuit board Dynamic force Unit load (lb/in.) Transmissibility (dimensionless ratio) Shear flow (lb/in.), dynamic pressure (lb/in.) Radius (in.), reaction (lb) Stress ratio (dimensionless), sweep rate (octave/min) Fatigue-cycle ratio (dimensionless) Damping ratio (dimensionless) Frequency ratio (dimensionless) Root mean square Root sum square radius (in.), relative position factor Bending stress (lb/in.2> Endurance limit stress (lb/in.*) Stress (lb/in.*) Surface mounted device Thermal coefficient of expansion (in./in./'C) Kinetic energy Ob-in.), torque (in/. lb) Time (min), temperature, Thickness (in.) Strain energy (lb in.), work (lb in.) Velocity (in./s>, force (lb) Weight (lb)

Dynamic load (lb) Unit load (lb/in.) Displacement along X axis, Xi, E;, Zcoordinate axes First derivative, velocity Second derivative, acceleration Displacement (in.)

CY

s 0 P

P

4 A

R

a"

Angle (degrees, radians), thermal coefficient of expansion (in./in./"C) Displacement (in.) Angular displacement (radians) Poisson's ratio (dimensionless) Density (lb/in.), mass per unit area (lb * in.^) Phase angle (degrees) Difference Angular velocity (rad/s) Natural frequency (rad/s) Subscripts

av b C

d e eq in m ax n 0

out st su t tu tY U

Y ST S T

Average Bending Critical Dynamic, desired Endurance Equivalent Input Maximum Natural Maximum Output or response Static Shear ultimate Tension Tensile ultimate Tensile yield Ultimate Yield Shear tearout Shear Torsion

-

CONTENTS

Preface

xvii

List of Symbols

xix

1. Introduction

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13

Vibration Sources Definitions Vibration Representation Degrees of Freedom Vibration Modes Vibration Nodes Coupled Modes Fasteners Electronic Equipment for Airplanes and Missiles Electronic Equipment for Ships and Submarines Electronic Equipment for Automobiles, Trucks, and Trains Electronics for Oil Drilling Equipment Electronics for Computers, Communication, and Entertainment

2. Vibrations of Simple Electronic Systems

2.1

2.2 2.3 2.4 2.5 2.6

Single Spring-Mass System Without Damping Sample Problem-Natural Frequency of a Cantilever Beam Single-Degree-of-Freedom Torsional Systems Sample Problem-Natural Frequency of a Torsion System Springs in Series and Parallel Sample Problem-Resonant Frequency of a Spring System Relation of Frequency and Acceleration to Displacement Sample Problem-Natural Frequency and Stress in a Beam Forced Vibrations with Viscous Damping Transmissibility as a Function of Frequency Sample Problem-Relating the Resonant Frequency to the Dynamic Displacement

1

1 2 3 3 5 5 6 7 10 13 15 16

16 17

17 19

21 22 23 25 26 27 30 34 34 vii

Viii

CONTENTS

2.7

Multiple Spring-Mass Systems Without Damping Sample Problem-Resonant Frequency of a System

3 Component Lead Wire and Solder Joint Vibration Fatigue Life

3.1 3.2

3.3 3.4 3.5

3.6

Introduction Vibration Problems with Components Mounted High Above the PCB Sample Problem-Vibration Fatigue Life in the Wires of a TO-5 Transistor Vibration Fatigue Life in Solder Joints of a TO-5 Transistor Recommendations to Fix the Wire Vibration Problem Dynamic Forces Developed in Transformer Wires During Vibration Sample Problem-Dynamic Forces and Fatigue Life in Transformer Lead Wires Relative Displacements Between PCB and Component Produce Lead Wire Strain Sample Problem-Effects of PCB Displacement on Hybrid Reliability

4. Beam Structures for Electronic Subassemblies

4.1 4.2

4.3

Natural Frequency of a Uniform Beam Sample Problem-Natural Frequencies of Beams Nonuniform Cross Sections Sample Problem-Natural Frequency of a Box with Nonuniform Sections Composite Beams

5. Component Lead Wires as Bents, Frames, and Arcs 5.1 5.2 5.3 5.4 5.5 5.6 5.7

Electronic Components Mounted on Circuit Boards Bent with a Lateral Load-Hinged Ends Strain Energy-Bent with Hinged Ends Strain Energy-Bent with Fixed Ends Strain Energy-Circular Arc with Hinged Ends Strain Energy-Circular Arc with Fixed Ends Strain Energy-Circular Arcs for Lead Wire Strain Relief Sample Problem-Adding an Offset in a Wire to Increase the Fatigue Life

36 37 39 39 39 40

43 45 46 46 49 50 56

56 60 64 68 69

75 75 77 80 83 90 92 94 97

CONTENTS

6 Printed Circuit Boards and Flat Plates

6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11 6.12 6.13

Various Types of Printed Circuit Boards Changes in Circuit Board Edge Conditions Estimating the Transmissibility of a Printed Circuit Board Natural Frequency Using a Trigonometric Series Natural Frequency Using a Polynomial Series Sample Problem-Resonant Frequency of a PCB Natural Frequency Equations Derived Using the Rayleigh Method Dynamic Stresses in the Circuit Board Sample Problem-Vibration Stresses in a PCB Ribs on Printed Circuit Boards Ribs Fastened to Circuit Boards with Screws Printed Circuit Boards With Ribs in Two Directions Proper Use of Ribs to Stiffen Plates and Circuit Boards Quick Way to Estimate the Required Rib Spacing for Circuit Boards Natural Frequencies for Different PCB Shapes with Different Supports Sample Problem-Natural Frequency of a Triangular PCB with Three Point Supports

7. Octave Rule, Snubbing, and Damping to Increase the PCB Fatigue Life

ix

103 103 106 108 111 116 120 122 127 131 132 137 141 141 142 144 149 150

7.1

Dynamic Coupling Between the PCBs and Their Support Structures 7.2 Effects of Loose Edge Guides on Plug-in Type PCBs 7.3 Description of Dynamic Computer Study for the Octave Rule 7.4 The Forward Octave Rule Always Works 7.5 The Reverse Octave Rule Must Have Lightweight PCBs Sample Problem-Vibration Problems with Relays Mounted on PCBs 7.6 Proposed Corrective Action for Relays 7.7 Using Snubbers to Reduce PCB Displacements and Stresses Sample Problem-Adding Snubbers to Improve PCB Reliability 7.8 Controlling the PCB Transmissibility with Damping 7.9 Properties of Material Damping 7.10 Constrained Layer Damping with Viscoelastic Materials 7.11 Why Stiffening Ribs on PCBs are Often Better than Damping 7.12 Problems with PCB Viscoelastic Dampers

150 154 154 155 155 156 157 159 161 162 162 163 164 164

X

CONTENTS

8. Preventing Sinusoidal Vibration Failures in Electronic Equipment 8.1 8.2

8.3 8.4

8.5 8.6

8.7

8.8

Introduction Estimating the Vibration Fatigue Life Sample Problem-Qualification Test for an Electronic System Electronic Component Lead Wire Strain Relief Designing PCBs for Sinusoidal Vibration Environments Sample Problem-Determining Desired PCB Resonant Frequency How Location and Orientation of Component on PCB Affect Life How Wedge Clamps Affect the PCB Resonant Frequency Sample Problem-Resonant Frequency of PCB with Side Wedge Clamps Effects of Loose PCB Side Edge Guides Sample Problem-Resonant Frequency of PCB with Loose Edge Guides Sine Sweep Through a Resonance Sample Problem-Fatigue Cycles Accumulated During a Sine Sweep

9. Designing Electronics for Random Vibration

9.1 9.2 9.3 9.4 9.5

Introduction Basic Failure Modes in Random Vibration Characteristics of Random Vibration Differences Between Sinusoidal and Random Vibrations Random Vibration Input Curves Sample Problem-Determining the Input RMS Acceleration Level 9.6 Random Vibration Units 9.7 Shaped Random Vibration Input Curves Sample Problem-Input RMS Accelerations for Sloped PSD Curves 9.8 Relation Between Decibels and Slope 9.9 Integration Method for Obtaining the Area Under a PSD Curve 9.10 Finding Points on the PSD Curve Sample Problem-Finding PSD Values 9.11 Using Basic Logarithms to Find Points on the PSD Curve 9.12 Probability Distribution Functions

166

166 167 168 169 171 174 175 177 179 182 185 185 187 188

188 188 189 190 192 193 193 194 195 197 198 200 200 20 1 202

CONTENTS

9.13 Gaussian or Normal Distribution Curve 9.14 Correlating Random Vibration Failures Using the Three-Band Technique 9.15 Rayleigh Distribution Function 9.16 Response of a Single-Degree-of-Freedom System to Random Vibration Sample Problem-Estimating the Random Vibration Fatigue Life 9.17 How PCBs Respond to Random Vibration 9.18 Designing PCBs for Random Vibration Environments Sample Problem-Finding the Desired PCB Resonant Frequency 9.19 Effects of Relative Motion on Component Fatigue Life Sample Problem-Component Fatigue Life 9.20 It’s the Input PSD that Counts, Not the Input RMS Acceleration 9.21 Connector Wear and Surface Fretting Corrosion Sample Problem-Determining Approximate Connector Fatigue Life 9.22 Multiple-Degree-of-Freedom Systems 9.23 Octave Rule for Random Vibration Sample Problem-Response of Chassis and PCB to Random Vibration Sample Problem-Dynamic Analysis of an Electronic Chassis 9.24 Determining the Number of Positive Zero Crossings Sample Problem-Determining the Number of Positive Zero Crossings

10 Acoustic Noise Effects on Electronics 10.1 Introduction Sample Problem-Determining the Sound Pressure Level 10.2 Microphonic Effects in Electronic Equipment 10.3 Methods for Generating Acoustic Noise Tests 10.4 One-Third Octave Bandwidth 10.5 Determining the Sound Pressure Spectral Density 10.6 Sound Pressure Response to Acoustic Noise Excitation Sample Problem-Fatigue Life of a Sheet-Metal Panel Exposed to Acoustic Noise 10.7 Determining the Sound Acceleration Spectral Density Sample Problem-Alternate Method of Acoustic Noise Analysis

xi

202 204 205 206 208 214 215 218 220 221 222 223 224 224 225 226 229 23 1 233

234 234 234 235 236 238 238 239 240 245 246

Xii

CONTENTS

11. Designing Electronics for Shock Environments 11.1 11.2 11.3 11.4

11.5 11.6 11.7 11.8

11.9

11.10 11.11

11.12 11.13 11.14 11.1s 11.16 11.17

11.18

11.19 11.20

11.21

Introduction Specifying the Shock Environment Pulse Shock Half-Sine Shock Pulse for Zero Rebound and Full Rebound Sample Problem-Half-Sine Shock-Pulse Drop Test Response of Electronic Structures to Shock Pulses Response of a Simple System to Various Shock Pulses How PCBs Respond to Shock Pulses Determining the Desired PCB Resonant Frequency for Shock Sample Problem-Response of a PCB to a Half-Sine Shock Pulse Response of PCB to Other Shock Pulses Sample Problem-Shock Response of a Transformer Mounting Bracket Equivalent Shock Pulse Sample Problem-Shipping Crate for an Electronic Box Low Values of the Frequency Ratio R Sample Problem-Shock Amplification for Low Frequency Ratio R Shock Isolators Sample Problem-Heat Developed in an Isolator Information Required for Shock Isolators Sample Problem-Selecting a Set of Shock Isolators Ringing Effects in Systems with Light Damping How Two-Degree-of-Freedom Systems Respond to Shock The Octave Rule for Shock Velocity Shock Sample Problem-Designing a Cabinet for Velocity Shock Nonlinear Velocity Shock Sample Problem-Cushioning Material for a Sensitive Electronic Box Shock Response Spectrum How Chassis and PCBs Respond to Shock Sample Problem-Shock Response Spectrum Analysis for Chassis and PCB How Pyrotechnic Shock Can Affect Electronic Components Sample Problem-Resonant Frequency of a Hybrid Die Bond Wire

248

248 249 25 1 252 253 257 258 260 260 262 264 265 269 269 274 274 275 276 277 278 28 1 282 284 285 285 286 288 288 29 1

292 296 298

CONTENTS

12. Design and Analysis of Electronic Boxes 12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8 12.9 12.10 12.11 12.12 12.13

Introduction Different Types of Mounts Preliminary Dynamic Analysis Bolted Covers Coupled Modes Dynamic Loads in a Chassis Bending Stresses in the Chassis Buckling Stress Ratio for Bending Torsional Stresses in the Chassis Buckling Stress Ratio for Shear Margin of Safety for Buckling Center-of-Gravity Mount Simpler Method for Obtaining Dynamic Forces and Stresses on a Chassis

13. Effects of Manufacturing Methods on the Reliability of Electronics 13.1 13.2

13.3 13.4 13.5 13.6 13.7 13.8 13.9

Introduction Typical Tolerances in Electronic Components and Lead Wires Sample Problem-Effects of PCB Tolerances on Frequency and Fatigue Life Problems Associated with Tolerances on PCB Thickness Effects of Poor Bonding Methods on Structural Stiffness Soldering Small Axial Leaded Components on ThroughHole PCBs Areas Where Poor Manufacturing Methods Have Been Known to Cause Problems Avionic Integrity Program and Automotive Integrity Program (AVIP) The Basic Philosophy for Performing an AVIP Analysis Different Perspectives of Reliability

14. Vibration Fixtures and Vibration Testing

14.1 14.2 14.3 14.4 14.5 14.6

Vibration Simulation Equipment Mounting the Vibration Machine Vibration Test Fixtures Basic Fixture Design Considerations Effective Spring Rates for Bolts Bolt Preload Torque

Xiii

300 300 300 303 305 308 311 316 318 320 324 325 326 328

330 330 331 332 333 334 335 336 338 340 343

346 346 347 347 348 350 352

XiV

CONTENTS

Sample Problem-Determining Desired Bolt Torque Rocking Modes and Overturning Moments Oil-Film Slider Tables Vibration Fixture Counterweights A Summary for Good Fixture Design Suspension Systems Mechanical Fuses Distinguishing Bending Modes from Rocking Modes Push-Bar Couplings Slider Plate Longitudinal Resonance Acceleration Force Capability of Shaker Positioning the Servo-Control Accelerometer More Accurate Method for Estimating the Transmissibility Q in Structures Sample Problem-Transmissibility Expected for a Plug-in PCB

353 353 355 356 357 357 358 359 360 364 365 366

Vibration Testing Case Histories 14.19 Cross-Coupling Effects in Vibration Test Fixtures 14.20 Progressive Vibration Shear Failures in Bolted Structures 14.21 Vibration Push-Bar Couplers with Bolts Loaded in Shear 14.22 Bolting PCB Centers Together to Improve Their Vibration Fatigue Life 14.23 Vibration Failures Caused by Careless Manufacturing Methods 14.24 Alleged Vibration Failure that was Really Caused by Dropping a Large Chassis 14.25 Methods for Increasing the Vibration and Shock Capability on Existing Systems

369 369 370 371

14.7 14.8 14.9 14.10 14.11 14.12 14.13 14.14 14.15 14.16 14.17 14.18

15. Environmental Stress Screening for Electronic Equipment (ESSEE) 15.1 15.2 15.3 15.4 15.5 15.6 15.7 15.8 15.9

Introduction Environmental Stress Screening Philosophy Screening Environments Things an Acceptable Screen Are Expected to D o Things an Acceptable Screen Are Not Expected to D o To Screen or Not to Screen, That is the Problem Preparations Prior to the Start of a Screening Program Combined Thermal Cycling, Random Vibration, and Electrical Operation Separate Thermal Cycling, Random Vibration, and Electrical Operation

367 368

373 375 376 377

379 379 379 381 383 383 384 384 387 389

CONTENTS

15.10 Importance of the Screening Environment Sequence 15.11 How Damage Can Be Developed in a Thermal Cycling Screen 15.12 Estimating the Amount of Fatigue Life Used Up in a Random Vibration Screen Sample Problem-Fatigue Life Used Up in Vibration and Thermal Cycling Screen

XV

389

390 392 395

Bibliography

401

Index

405

-

CHAPTER 1

Introduction

1.I

VIBRATION SOURCES

Electronic equipment can be subjected to many different forms of vibration over wide frequency ranges and acceleration levels. It is probably safe to say that all electronic equipment will be subjected to some type of vibration at some time in its life. If the vibration is not due to an active association with some sort of a machine or a moving vehicle, then it may be due to transporting the equipment from the manufacturer to the customer. Vibrations encountered during transportation and handling can produce many different types of failures in electronic equipment unless the proper consideration are given to the mechanical design of the electronic structure and the shipping containers. Vibration is usually considered to be an undesirable condition, and in most cases it is. However, there are many applications where vibration is deliberately imposed to improve a function. Some sophisticated applications of vibrations are in the use of ultrasonics to clean medical instruments, measure wall thickness, and find flaws in castings. Vibration is also used in sorting rocks of different sizes by passing them over vibrating screens, which have groups of graduated holes. When the first jet airplanes were introduced, standard instruments from airplanes with piston engines and propellers were used. Theres instruments had a tendency to stick when they were used on the jet airplanes. The difference was due to the lower-frequency vibration levels developed in the piston engines. In order to make these same instruments work in the first jet airplanes, small vibrators had to be mounted on the instrument panels. Mechanical vibrations can have many different sources. In household products such as blenders and washing machines, the vibrations are due to the unbalance created by rotating and tumbling masses. In vehicles such as automobiles, trucks, and trains most of the vibration is due to rough surfaces over which these vehicles travel. In ships and submarines the vibration is due to the engines and to buffeting by the water. In airplanes, missiles, and rockets the vibration is due to jet and rocket engines and to aerodynamic buffeting. Most of the vibration in a missile, during subsonic flight, is due to 1

2

INTRODUCTION

+ Displacement ~

Period

-

Displacement

FIGURE 1.1. Simple harmonic motion.

the sound field developed by the rocket engines [1,6].* This is due to the extreme turbulence of the jet exhaust downstream from the rocket engine. 1.2

DEFINITIONS

Vibration, in a broad sense, is taken to mean an oscillating motion where some structure or body moves back and forth. If the motion repeats itself, with all of the individual characteristics, after a certain period of time, it is called periodic motion. This motion can be quite complex, but as long as it repeats itself, it is still periodic. If continuous motion never seems to repeat itself, it is called random motion. Simple harmonic motion is the simplest form of periodic motion, and it is usually represented by a continuous sine wave on a plot of displacement versus time, as shown in Fig. 1.1. The reciprocal of the period is known as the frequency of the vibration and is measured in cycles per second, or hertz (Hz), in honor of the German who first experimented with radiowaves. The maximum displacement is called the amplitude of the vibration. Crede [2,3] defines shock as a transient condition where the equilibrium of a system is disrupted by a sudden applied force or increment of force, or by a sudden change in the direction or magnitude of a velocity vector. Shock is not transmitted easily in the light airframe structures generally used in airplanes and missiles. Impact forces often result in a transient type of vibration, which is influenced by the natural frequencies of the airframe. Only steady-state linear vibrations are considered in this book. Linear vibrations occur with linear elastic media where the displacements are directly proportional to the applied force. If the force is doubled, the displacement is doubled. Only stresses up to the elastic limit of any given material are considered. Higher stresses generally result in permanent deformations, which extend into the plastic deformation region and are not considered here. *Numbers in brackets [ ] refer to bibliography entries in the back of the book.

DEGREES OF FREEDOM

3

FIGURE 1.2. Rotating vector simulating a single-degree-of-freedom system.

1.3 VIBRATION REPRESENTATION

A rotating vector can be used to describe the simple harmonic vibration of a single mass suspended on a coil spring (Fig. 1.2). The vector Yo rotates counterclockwise with a uniform angular velocity of R rad/s. The projection of the vector on the vertical axis represents the instantaneous displacement Y of the mass as it vibrates up and down. This can be written as

When the vector Yo rotates through one revolution, it rotates through an angle of 360°, which is 277- radians, for one complete cycle. The angular velocity is measured in radians per second, and the frequency f is measured in cycles per second. This leads to the relation

1.4

DEGREES OF FREEDOM

A vibrating system requires some coordinates to describe the positions of the elements in the system. If there is only one element in the system that is restricted to move along only one axis, and only one dimension is required to locate the position of the element at any instant of time with respect to some initial starting point, then it is a single-degree-of-freedom system. The same is true for a torsional system. If there is only one element that is restricted to rotate about only one axis so that only one dimension is required to locate the position of the element at any instant of time with respect to some initial starting point, it is a single-degree-of-freedom system. Some samples of systems with a single degree of freedom are shown in Fig. 1.3.

4

INTRODUCTION

Spring and mass

Rod and disk

Pendulum

(a)

(b)

(Ci

FIGURE 1.3. Single-degree-of-freedomsystems.

X+

F

Y

(a)

f6)

(di

(C)

FIGURE 1.4. Two-degree-of-freedomsystems.

A two-degree-of-freedom system requires two coordinates to describe the positions of the elements. Some samples of systems with two degrees of freedom are shown in Fig. 1.4. In rigid-body mechanics, a single body can have six degrees of freedom. These are translation along each of its three mutually perpendicular X , Y , and Z axes, and rotation about each of them, as shown in Fig. 1.5. A typical beam can have an infinite number of degrees of freedom since it can bend in an infinite number of shapes, or modes, as shown in Fig. 1.6.

FIGURE 1.5. A single mass with six degrees

of freedom.

+Z

-Y

VIBRATION NODES

5

_ _ - - -FIGURE 1.6. A simply supported beam showing several degrees of freedom.

k

....

Node point

,/--------

‘.._--

”

First harmonic mode

Second harmonic mode

(a)

(bi

FIGURE 1.7. First and second harmonic modes for a simply supported beam.

1.5 VIBRATION MODES

A standard manner in which a particular system can vibrate is known as a vibration mode. Each vibration mode is associated with a particular natural frequency and represents a degree of freedom. A single-degree-of-freedom system will have only one vibration mode and only one resonant frequency. The six-degree-of-freedom system shown in Fig. 1.5 can have six vibration modes and six resonant frequencies. The simply supported beam shown in Fig. 1.6 can have an infinite number of vibration modes; this is the same as saying this beam can have an infinite number of different shapes for each of its resonances, which are also infinite. The fundamental resonant mode of a vibrating system is usually called the natural frequency or the resonant frequency of the system. Sometimes it is called the first harmonic mode of the system. For example, a simply supported uniform beam vibrating at its fundamental resonant frequency has the . this beam is mode shape of a half sine wave as shown in Fig. 1 . 7 ~When vibrating at its second natural frequency, in its second harmonic mode, it has the mode shape of the full sine wave shown in Fig. 1.7b. The first harmonic mode of a system, with the lowest natural frequency, is the fundamental resonant mode; this often has the greatest displacement amplitudes and usually the greatest stresses. The second harmonic mode, or second resonance, usually has a smaller displacement than the first harmonic mode, so the stresses are usually smaller. The displacements continue to decrease for the higher resonant modes.

1.6 VIBRATION NODES

Vibration nodes are unsupported points on a vibrating body that have zero displacements. Nodes are generally associated with bending or torsion modes. At the first harmonic bending resonant mode in a beam there are no node

6

INTRODUCTION

First mode

Second mode

Third mode

Fourth mode

FIGURE 1.8. First four harmonic modes of a circular membrane.

First

mode

Second mode

Third

mode

FIGURE 1.9. First three harmonic modes of a square plate.

points. At the second harmonic mode there is one node point, the third harmonic mode has two node points, and so on. Figure 1 . 7 ~shows the bending mode of a vibrating beam with no node points, and Fig. 1.7b shows a beam with one node point. Vibrating plates can have straight-line bending nodes and circular nodes. The first four harmonic modes of a circular membrane are shown in Fig. 1.8. The plus ( + ) signs show positive displacements and the minus ( - > signs show negative displacements. The dashed lines represent nodal locations of zero displacement. The first three harmonic bending modes of a square plate with free edges are shown in Fig. 1.9.

1.7

COUPLED MODES

In a system with two or more degrees of freedom, the vibration mode of one degree often influences the vibration mode of the other degree. For example, in Fig. 1 . 4 if~ mass 2 is held rigidly while mass 1 is displaced in the vertical direction, mass 1 will oscillate up and down. Now if mass 2 is released, the motion of mass 1 will act upon mass 2 so mass 2 will begin to oscillate up and down. Because the motion of mass 1 has a direct effect on the motion of mass 2, these two vibration modes are defined as being coupled. In coupled modes, the vibration in one mode cannot occur independently of the vibration in the other mode. Coupled modes can occur in translation, rotation, and combinations of translation and rotation for systems with more than one degree of freedom. For coupled modes in translation and rotation, it is often possible to

FASTENERS

7

Y A

L

3

I I 1

I

> r

L+L+

FIGURE 1.10. A single mass with two springs.

determine whether the system is coupled or uncoupled by making a simple test. Apply a steady load to the body of the system at its center of gravity (CG), in a specific direction. If the body moves in the direction of the applied load without rotation, then the translational mode is not coupled with the rotational mode for motion along the direction of the applied load. Consider, for example, the mass with two springs as shown in Fig. 1.10. If a steady load is applied to the CG along the X axis, the mass will not only translate along the X axis, but it will also tend to rotate at the same time. This test shows that for vibration along the X axis, the translational mode is coupled with the rotational mode. If a steady load is applied to the C G along the Y axis, the mass will translate along the Y axis without rotation. Also, if a moment is applied to the mass about the CG, then rotation will take place without translation. These tests show that for vibration along the Y axis the translational mode is decoupled from the rotational mode.

1.8 FASTENERS

Many different types of fasteners are used in electronic equipment. These include screws, nuts, rivets, and clips. Fasteners are responsible for a very large percentage of field failures and, in most shock tests, they are the largest single source of failure. Although fasteners have been used in very large quantities in electronic equipment for many years, most of the applications are based on static installation considerations. The fastening techniques, which are generally chosen for ease of installation and low cost, are usually not satisfactory for severe shock and vibration environments. A contributing factor is that fasteners are such small items and their use so universal that their application tends to be semiautomatic without regard to their strength. This is particularly the case for machine screws. When large screws are used, it is often because someone has had some association with the automotive or aircraft industry, where very small screws are seldom used.

8

INTRODUCTION

Because fasteners play an important part in the overall reliability of the electronic equipment, extra consideration should be given in their selection as follows [ 5 ] : 1. Select the proper type of fastener (screws, rivets, etc.), considering such trade-offs as environment, strength, maintenance, and cost. 2. Select the correct fastener size and location based on dynamic loads and geometry of the structure. 3. Select the correct locking device for screws and nuts. 4. Select the right installation technique.

Most electronics manufacturers choose screws and rivets on a production basis. If the designer uses screws, the size and locking device are selected according to the tolerances as well as for ease in assembly. In installation, the designer depends on production personnel to use their own judgment in the proper installation of fasteners. The results may be that the wrong fasteners are used, the wrong sizes are used, the wrong lockwashers are used, and the wrong installation torques are used. It has been found, generally speaking, that cold-driven shank-expanding rivets are very satisfactory and should be used more frequently in electronic assemblies. It has been found, too, that when screws are used, they should be of larger sizes than those customarily considered and they should be made of better materials. Many locking devices commonly used are unsatisfactory. Some are often the source of many severe problems. It has also been found that a mechanic’s judgment in tightening a machine screw is usually faulty. Many failures in electronic equipment have resulted because bolts have come loose. Consider what can happen to a sensitive electronic chassis when a large transformer comes loose and rattles around during vibration. Although many investigations have been made with high-speed films and strain gages to study the loosening action of screws and nuts, the mechanism by which this occurs is still not well established. It appears, however, that the bolt stretches slightly under the action of a dynamic load, so that the interface friction forces in the area of the bolt are suddenly and sharply reduced. Since the threads in the screw or nut tend to return to their original shape during the small time increments, the small geometric changes become a driving force, which tends to loosen the screw and nut. Some specific recommendations can be made to improve the quality of the fasteners in electronic equipment: 1. Steel screws should be used in all screw fastenings. The steel should have the minimum properties of SAE 1010. 2. The screws should be tightened by a torque device, which can be preset to the required value.

FASTENERS

9

TABLE 1.1 ~~

Screw Size

Torque

Screw

Torque

(in: lb)

Size

(in: lb)

2-64 4-40 6-32 8-32 10-24 10-32

3-3.5 5-6.5 10-12 20-24 22-27 34-42

12-24 12-28 $20 -28 i-16 f -24

45-56 50-64 65-80

85-100 250-320 330-415

3. The tightening torque should be 6 0 4 0 % of the torque required to twist the head off the screw. These torque values for steel screws are given in Table 1.1 [lll. 4. The screw head should permit a positive nonslipping grip for the driving device and should also withstand the driving forces. A slottedhex-head machine screw seems to be, for general purposes, the most satisfactory. 5 . In all applications involving through-holes, locknuts should be used instead of lockwashers. Most of the standard steel locknuts are satisfactory. 6. Blind-tapped holes should be avoided, if possible. When they are necessary, locking devices such as lockwashers should be used under the screw head in order to prevent the screw from backing out during vibration. The tightening torque should be increased by an amount equal to the resisting frictional torque of the locking device. 7 . The fasteners for a unit should be distributed so that the failure of one fastener will not free the unit or cause it to malfunction. Even for very small components there should be a minimum of two fasteners. Lockwashers and screw-locking inserts should be used with care in electronic systems that will be used in the zero-G environment of outer space. These two devices create a binding friction in the screw by biting into the metal during installation. This action will very often shave metal particles from the screw. If these particles are not removed, they can float around in a zero-G environment and create electrical problems. An actual count was made of the metal particles after inserting 24 screws with external-star-type lockwashers, and a total of about 1000 metal particles were counted. The crossed-recess screwdriver slot (Phillips head) will also tend to be cut by the action of the screwdriver. When the screw is seated and the screwdriver is twisted, the screwdriver will very often twist out of the slot and shave small bits of metal out of the screw head.

10

INTRODUCTION

In order to avoid shaving small bits of metal from the screws with devices that bit into the metal, many electronic firms use liquids such as Loctite and Glyptal, which bind the screws. Some firms use nylon inserts in the screws. Nylon inserts are usually good for about a dozen insertion and removal cycles before the nylon cold flows and reduces the binding torque. NASA will not permit the use of volatile products in the area of displays and optical devices for their spacecraft. These products tend to outgas in a vacuum and then deposit themselves in a thin film; this can coat optical lenses and interfere with the operation of sensitive optical units such as cameras and telescopes. 1.9 ELECTRONIC EQUIPMENT FOR AIRPLANES AND MISSILES

Electronic boxes used in airplanes and missiles often have odd shapes that permit them to make maximum use of the volume available in tight spaces. Since volume and weight are generally quite critical, the electronic boxes usually have a high packaging density. This value normally ranges from about 0.03 to about 0.04 lb/in.3, depending on the severity of the environmental requirements. The weight of a typical electronic box will range from about 10 to about 80 lb. The vibration frequency spectrum for airplanes will vary from about 3 to about 1000 Hz, with acceleration levels that can range from about 1 G to about 5 G peak. The highest accelerations appear to occur in the vertical direction in the frequency range of about 100-400 Hz. The lowest accelerations appear to occur in the longitudinal direction, with maximum levels of about 1 G in the same frequency range. For helicopters, the frequency spectrum will vary from about 3 to about 500 Hz and acceleration will range from about 0.5 to about 4 G. The highest accelerations appear to occur in the vertical direction at frequencies near 500 Hz. The displacement at low frequency are very large, with values of about 0.20-in. double amplitude at about 10 Hz. Missiles have the highest frequency range in this group, with values that generally go up to 5000 Hz [7]. The lower frequency limit is about 3 Hz, and this appears to be due to bending modes in the airframe structure. Acceleration levels range from about 5 to about 30 G peak, with the maximum levels occurring during power-plant ignition at frequencies above 1000 Hz. The vibration environment in supersonic airplanes and missiles is actually more random in nature than it is periodic. However, sinusoidal vibration tests are still being used to evaluate and to qualify electronic equipment that will be used in these vehicles. Because the forcing frequencies in airplanes and missiles are so high, it is virtually impossible to design resonance-free electronic systems for these environments. Of course, it is always possible to completely encapsulate an entire electronic box with some expanding rigid type of foam, which could

ELECTRONIC EQUIPMENT FOR AIRPLANES AND MISSILES

11

drive the resonant frequency well above 1000 Hz (possibly to 2000 Hz) for a small box. This is generally considered to be impractical, however, because it becomes too expensive to maintain, troubleshoot, and repair such a system. The obvious conclusion is that the forcing frequencies present in airplanes and missiles will excite many resonant modes in every electronic box. What becomes equally obvious is that extra care must be taken in the design and analysis of an electronic system, or it can literally shake itself to pieces. The electronic support structure must be dynamically tuned with respect to the electronic components to prevent coincident resonances that can lead to rapid fatigue failures. The first thought that comes to the mind of an experienced mechanical design engineer, when confronted with a severe vibration specification, is to mount the electronic equipment on vibration isolators. There is no doubt that a set of isolators, properly designed, can control shock and vibration. There are four major factors that must be considered when isolation mounts are being discussed. 1. Sway space must be provided all around the electronic equipment to keep it from colliding with other objects. If volume is scarce, it might be more practical to pack more electronics into the same volume by using a larger electronic box with hard mounts. 2. Cold plates are being used more and more in electronic structures to remove the heat dissipated by the electronic equipment. If isolators are used, flexible couplings must be provided between the airframe structure and the electronic box to take care of the large displacements developed by the isolators. Reliable flexible couplings must be used because cooling effectiveness may be sharply reduced if a coupling fails. 3. Electrical wire cables and harnesses must be used to connect the typical electronic box to the main electrical system in the airplane or missile. If isolators are used, these cables and harnesses will be forced through large amplitudes because of the sway space required by the isolators. Special precautions must be taken to prevent fatigue failures in the cables and harnesses. 4. A good vibration isolator is often a poor shock isolator, and a good shock isolator is often a poor vibration isolator. The proper design must be incorporated into the isolator to satisfy both the vibration and the shock requirements.

Cold-plate designs for airborne electronic equipment have become more and more sophisticated with the use of air and liquids. Air heat exchangers make extensive use of multiple fins, wavy fins, split fins, and pin fins in order to improve the heat-transfer characteristics. The multiple-fin heat exchangers are usually dip-brazed aluminum with as many as 22 fins per inch. These fins may be only 0.006 in. thick, but the large number of fins in a typical heat

12

INTRODUCTION

exchanger makes it quite rigid for its weight. Airplanes make extensive use of electronic equipment where a cooling-air heat exchanger is built right into the electronic support structure. The heat exchanger is riveted, brazed, or cemented to the major structural members in the electronic box, so that the heat exchanger itself becomes a major load-carrying member of the system. The cooling air for the cold plate is usually taken from the compressor on the jet engine that powers the airplane. This air must be conditioned before it can be used for cooling, because the air temperature from the first stage is usually greater than 300°F. Liquid-cooled cold plates are usually used to cool electronic equipment on spacecraft or very-high-flying research airplanes. The cooling liquid can be a mixture of ethylene glycol and water, or some other liquids such as FC-75, a fluorocarbon or Coolanol 45, a silicone fluid. The liquid-cooled cold plates are usually made part of the spacecraft airframe structure instead of the electronic box structure. Then, when the electronic box must be removed from the spacecraft, it is not necessary to disconnect fluid lines, which can become quite messy. Since the cold plate stays with the airframe, the heat dissipated by the electronic box is usually transferred to the cold plate through a flat interface on the mounting surface of the electronic box that makes intimate contact with the cold plate. The trend in commercial and military electronics is toward the line replaceable unit (LRU), with which it is possible to replace a defective electronic box right on the flight line in a matter of minutes. This is accomplished by providing all of the required interface connections, both mechanical and electrical, at the back end of the electronic box. The box becomes similar to a printed circuit board (PCB) that can be plugged into its receptacle. If there are several large electrical connectors on the back end of the electronic box, it may be quite difficult to insert the box and engage the electrical connectors properly. Some connectors may require a force of 0.50 lb/pin for proper engagement. When there are 8 connectors, each with 100 pins, there is a total of 800 pins, which will require a 400-lb force to engage them. The plug-in electronic box is usually engaged and locked into position by some mechanism at the front of the box. Since the connectors are at the rear of the box, this means the force required to engage the connectors must pass through the box. When this type of electronic box is subjected to vibration, in many instances the vibration loads must be added to the installation loads to determine the total load acting on the structure. There has been an attempt to standardize electronic equipment used in military and commercial airplanes by establishing certain sizes for modular electronic units. These modular units are then mounted in a standard air transport rack (ATR), which provides rear-located dowel pins and connectors and a quick-release fastener at the front.

ELECTRONIC EQUIPMENT FOR SHIPS AND SUBMARINES

13

1. l o ELECTRONIC EQUIPMENT FOR SHIPS AND SUBMARINES

Ships and submarines will generally make use of console cabinets to support their electronic equipment, since there is usually more room available and weight is not very critical. The electronic components are usually mounted on panels and in sliding drawers. Panels are generally used to support dials, gages, manual controls, and test points. Only small masses are mounted on panels, because they are fastened to a frame or rack in the cabinet and they cannot withstand large dynamic loads. Drawers are often used to support the more massive electronic components such as those normally used in power supplies. The drawers are mounted on telescoping slides to provide access to the equipment. For safety, the drawers usually lock in the open and closed positions. For convenience, the drawers may also tilt to improve access in tight spaces. The vibration frequency spectrum for ships and submarines varies from about 1 to about 50 Hz, but the most common range is from about 12 to about 33 Hz. The maximum acceleration level in this range is about 1 G and appears to be due to vibrations set up by the engines and propellers. In military ships, shock is an important factor, due generally to various explosions, which can do extensive damage to electronic equipment, unless proper consideration is given to the design and installation. For example, it is not desirable to have a very rigid structure supporting the electronics, because a very rigid structure may not deform enough to absorb much stain energy. Theoretically, any structure that does not deform when it is subjected to an impact load will receive an infinite acceleration. A large displacement is desirable, since it can substantially reduce acceleration levels. Either this displacement must be confined to the structure or shock mounts must be used. In either case, provisions must be made in the design and installation to make sure parts will not collide and equipment will not break loose [36]. If shock isolators are used, they should be designed to deflect enough to absorb the shock energy without transmitting excessive loads to the electronic equipment. The shock mounts should have a minimum resonant frequency of about 25 Hz [2,51. Ideally, the resonant frequency of the electronic components should be at least twice that of the shock mounts, but never below 60 Hz. If the electronic components have resonances substantially below 60 Hz, it might bring them into the range of the most common vibration forcing frequencies, which, as previously mentioned, are as high as 33 Hz. If this should happen, the electronic components would be driven continuously near their resonance and could suffer fatigue failures. When the forcing frequency of the ship’s structure is near its higher frequency limit (around 25 Hz), the resonant frequency of the electronic equipment cabinet will be excited, since the shock isolators also have their resonance at 25 Hz. This condition should not impose high stresses on the electronic components mounted in the cabinet if the component resonance is

14

INTRODUCTION

twice that of the isolators. The cabinet support structure will, however, have to withstand the dynamic vibration loads. These loads will be determined by the amplification characteristics of the shock isolators during vibration. Shock isolators are available that provide a vibration amplification of about 3 for the conditions described above. Since the general vibration acceleration input level in these frequency ranges is normally quite low, an amplification factor of 3 for the isolators does not result in high stresses in the equipment cabinet. It is generally not desirable to increase the resonant frequency of the electronic equipment as high as possible. If the equipment is very stiff, it may be good for the vibration condition but poor for the shock condition. A very high spring rate may result in very high shock stresses because of the high acceleration loads. Lee [5] recommends a maximum resonant frequency of about 100 Hz and a maximum acceleration of 200 G on the electronic components. On tall narrow cabinets the load-carrying isolators should be at the base and stabilizing isolators should be at the top. A rigid structure must be used to support the stabilizing isolators at the top. If there is excessive deflection in the top support structure, it can change the characteristics of the entire system because of severe rocking modes. If shock isolators are not used, the shock energy must be absorbed by deflections in the electronic equipment cabinet and in the structure of the ship supporting the cabinet. In this case the natural frequency of the assembly, which consists of the cabinet and the ship’s structure, should be about 60 Hz. The natural frequency of the electronic components mounted on the equipment cabinet should be twice that of the assembly, or about 120 Hz. The ship’s structure must provide a good part of the deflection required to attenuate the shock force, or the dynamic stresses in the equipment cabinet may be high enough to cause structural failures. When the vibration forcing frequency in the ship’s structure is near its normal maximum limit of 33 Hz, the dynamic loads in the equipment cabinet will not be amplified to any great extent since the resonant frequency of the cabinet, at 60 Hz, is almost twice the forcing frequency. Furthermore, the electronic components mounted in the equipment cabinet have a still higher resonant frequency (120 Hz), so their dynamic vibration loads should be relatively small. The materials that are best suited for shock are ductile materials with a high yield point, a high ultimate strength, and a high percentage elongation. In general, metals that are mechanically formed are more desirable than cast metals, which have a relatively low percentage elongation. Since acceleration forces become progressively smaller as they propagate into the interior of the equipment cabinet, the electronic components that can withstand the highest G forces should be mounted near the exterior of the cabinet. Electronic components that have their own rigid structures

ELECTRONIC EQUIPMENT FOR AUTOMOBILES, TRUCKS, AND TRAINS

15

should be used to lend additional strength to the outer structural elements in the cabinet. Electronic components that cannot withstand high G forces should be mounted at the maximum elastic distance from the application points of the shock load. This will usually be at the center of the cabinet. The load path for each mass element in the system should be examined closely to determine the path the load will take as it passes through the structure. For example, a large transformer should be mounted close to a major structural support in order to reduce the length of the load path. This will result in smaller deflections and stresses.

1.1 1 ELECTRONIC EQUIPMENT FOR AUTOMOBILES, TRUCKS, AND TRAINS

Electronic equipment for automobiles and trucks has grown very rapidly in the past few years. Electronics are being used in their antilocking brake systems, fuel-air mixture control, radios, ignition systems, air conditioners, heaters, automatic transmission shifting points, rear view mirrors, door locks, instruments, global positioning systems (GPSs), windows, sun roofs, cruise control, air bags, television, telephones, and many other features, with more being added every year. Anticollision systems are being developed that will automatically engage brakes when vehicles are dangerously close to each other at high speeds. There are plans for computerized car trains that will link several automobiles together going to similar areas. Antitheft systems such as Lojack are available that can be activated by radio to send out a silent alarm to police when an automobile is stolen. Automobile and truck electronics must operate in harsh environments that include wide temperature swings, high under-hood operating temperatures of 140"C, high humidity with condensation, high-velocity splashing water, and subfreezing temperatures with only moderately high vibration and shock levels for the automobiles. Trucks are often required to travel over unpaved country roads for deliveries in outlying areas. Test data on a 2.5-ton truck instrumented with several accelerometers showed acceleration levels between about 15 and 19 G peak at speeds of 10-15 mph, at frequencies between about 15 and 40 Hz. At truck speeds above about 35 mph over rough roads the frequency appears to be random in nature. Diesel electric trains have been making use of electronics on their drive wheels to sense when the wheels are just reaching the slipping point on the tracks. A large engine may have six pairs of driving wheels. The electronics can sense when any one pair of drive wheels will slip. Sand is automatically dispensed to increase friction and the electric motor driving the one set of drive wheels is automatically slowed slightly to prevent slopping on the

16

INTRODUCTION

tracks. A single diesel electric engine with this new feature can go up steeper grades and pull three times as many cars as previous diesel electric trains. 1.12 ELECTRONICS FOR OIL DRILLING EQUIPMENT

Oil drilling equipment probably has the most severe thermal and vibration requirements for electronic systems. Deep drilling at 30,000 feet can encounter temperatures as high as 200°C. The electronics are usually located in a 6-foot long tube section just above the cutters. The coolant is the 200°C mud washing over the electronics tube section. Sometimes the cutters get stuck in a rock section while drilling at 30,000 feet. The turntable at the surface keeps turning while the cutters are stationary. The long drilling pipe winds up and when it breaks loose it generates random vibration acceleration levels as high as 30 G RMS. This means that the electronics section in an oil drilling rig must be very rugged to withstand the high vibration levels and the high temperatures and still provide a high reliability. When an electronic failure occurs, it is a very expensive and time-consuming task to remove 30,000 feet of pipe, replace the electronics section, and drop 30,000 feet of pipe down the hole again. 1. I 3 ELECTRONICS FOR COMPUTERS, COMMUNICATION, AND ENTERTAINMENT

According to many people in the three fields of computers, communication, and entertainment, they are rapidly converging into one giant industry with the internet. Sometimes this is called bandwidth, because fiber optic cables are being added, which can carry a very wide range of frequencies that are required to provide services for television, radio, data processing, stock trading, banking, internet, telephone, security, and more energy efficient homes and offices.

-

CHAPTER 2

Vibrations of Simple Electronic Systems

2.1

SINGLE SPRING-MASS SYSTEM WITHOUT DAMPING

The natural frequency of many single-degree-of-freedom systems can be determined by evaluating the characteristics of the strain energy and the kinetic energy of each system. Considering a single spring-mass system, for example, if there is no energy lost, then the maximum kinetic energy of the mass must be equal to the maximum strain energy in the spring, if the spring mass is negligible. All real systems have some damping. If a mass is suspended on a real coil spring and the spring is stretched and released, the mass will vibrate up and down. This free vibration may continue for a long time, but eventually all free vibrations die out and the mass stops vibrating. This is because damping in the spring dissipates a little energy with each cycle and eventually the mass stops moving. If there is no damping, the mass will theoretically keep on vibrating up and down with the same amplitude and frequency forever. In many systems, the damping is so small that it has very little effect on the natural frequency. Under these circumstances the natural frequency of the damped system can conveniently be approximated by the natural frequency of the undamped system. The maximum kinetic energy of a vibrating spring-mass system with no damping is at the point of maximum velocity. This occurs as the vibrating mass passes through the zero-displacement point, as shown in Fig. 2.1. From elementary physics the maximum kinetic energy T of the vibrating mass is

To = i m V 2 (2.1) The instantaneous tangential velocity I/ can be expressed in terms of rotational velocity as shown in Fig. 1.3:

v=

Yon (2.2) Substituting Eq. 2.2 into Eq. 2.1, the kinetic energy of the system becomes To = ;my,Zn*

(2.3) 17

18

VIBRATIONS OF SIMPLE ELECTRONIC SYSTEMS

T

,

+Displacement

t

< -<

rJ-1

;- - _ - - - - - --

Maximum strain energy

J Time

'Maximum velocity -Displacement

FIGURE 2.1. A vibrating spring-mass system.

The maximum strain energy can be determined from the work done on the spring by the mass as it stretches and compresses the spring during vibration. Since the spring is linear, the deflection is directly proportional to the force as shown in Fig. 2.2. The area under the curve represents the work done on the spring. Since the work is equal to the strain energy, the strain energy becomes

U" = +P,Y"

(2.4)

The spring rate K can be defined in terms of the maximum load P, and the maximum deflection Yo as K = P,/Y,, so

Po = KY,,

(2.5)

Substituting Eq. 2.5 into Eq. 2.4, the maximum strain energy becomes

uo= ;KY; If damping is assumed to be zero, then at resonance the kinetic energy will be equal to the strain energy: i,y2flz 2

=' K y 2

(I

2

0

[E) 1/ 2

Q" =

SINGLE SPRING-MASS

SYSTEM WITHOUT DAMPING

I9

This is the natural frequency in radians per second. Using Eq. 1.2, the natural frequency in cycles per second (Hz) is

The natural frequency equation can be written in a slightly different form by considering that the static deflection a,, of the spring is due to the action of the weight W acting on the spring. The spring rate can then be written as

Expressing the mass in terms of weight W and acceleration of gravity g , we have

Substituting Eqs. 2.8 and 2.9 into Eq. 2.7 gives another relation for the natural frequency: (2.10)

Frequency of a Cantilever Beam

Sample Problem-Natural

Determine the natural frequency of the cantilevered beam with an end mass, as shown in Fig. 2.3, in the vertical direction. Solution. When the weight of the aluminum beam is small compared to the weight of the end mass, it can be ignored without too much error. The cantilever beam can then be analyzed as a concentrated load on a massless beam, which retains its elastic properties. The static deflection at the end of

1

A

y

q

g

O 50

-1rL3 B-r Sect A A

1.4 -100

-

in

---+

FIGURE 2.3. Cantilever beam with an end mass.

20


this beam can be determined with the use of a structural handbook: (2.11) where W = 10.0 Ib (end weight) L = 10.0 in. (length) E = 10.5 x l o 6 lb/in.’ (modulus of elasticity, aluminum) I = bh3/12 = (0.50)(3.0)3/12 = 1.125 in.4 (moment of inertia) (10.0)(10.0)~ =

3(10.5

X

106)(1.125)

=

0.282

in.

X

(2.12)

Substitute Eq. 2.12 into Eq. 2.10 and note that the acceleration of gravity, g , is 386 in./s2. The natural frequency becomes 1 =

187 HZ

(2.13)

The natural frequency of the cantilever beam can also be determined from the spring rate, K , of the beam using Eq. 2.11: (2.14) Using the values previously calculated, the spring rate becomes

K=

3( 10.5 x l o 6 )(1.125)

( 10.0)3

= 3.54 X

l o 4 lb/in.

Also,

w 10 m=-=-= g 386

0.0259 Ib . s’/in.

Substituting into Eq. 2.7, the natural frequency becomes

”’-[-)

1

K

27r m

f,

=

1/2

1

3.54 x l o 4 0.0259

=z(

187 HZ

The results are exactly the same as Eq. 2.13.

1

1/ ?

(2.15)

SINGLE-DEGREE-OF-FREEDOM TORSIONAL SYSTEMS

2.2

21

SINGLE-DEGREE-OF-FREEDOM TORSIONAL SYSTEMS

The energy method is convenient for determining the natural frequency of a torsional system with one degree of freedom, as shown in Fig. 2.4. The torsional system is similar to the spring-mass system shown in Fig. 2.1, except that the spring action is due to the twisting of the rod and the inertia is due to the moment of inertia of the disk about an axis perpendicular to the plane of the disk. Assume the rod mass is small. The maximum kinetic energy of the oscillating disk is

KIN = ;zme2 where I

(2.16)

= mass

moment of inertia of disk 6 = rotational angular velocity

The maximum angular velocity for the oscillating system moving through the angle Bo is

io= eon

(2.17)

Substituting Eq. 2.17 into Eq. 2.16, the maximum kinetic energy becomes (assuming a small rod mass)

(2.18) The maximum strain energy can be determined from the work done on the twisting rod by the disk as it oscillates. Since the spring rate of the rod is linear, the angular deflection will be directly proportional to the torque T applied, as shown in Fig. 2.5. The area under the curve represents the work

22


done on the rod. Since the work is equal to the strain energy, the strain energy becomes

u, = +T,O,

(2.19)

The torsional spring rate K , can be defined in terms of the torque To and the angular displacement On as follows:

K 0 = -TO

(2.20a)

To = K , Bo

(2.20b)

On

so

Substituting Eq. 2.20 into Eq. 2.19, the maximum strain energy becomes

U, = i K , O i

(2.21)

If damping is assumed to be zero, then at resonance the kinetic energy (Eq. 2.18) must equal the strain energy (Eq. 2.21):

(2.22) The natural frequency (in Hz) becomes

j""('= f

Sample Problem-Natural

1/2

2 T I,,

(2.23)

Frequency of a Torsion System

Determine the natural frequency of the torsional system shown in Fig. 2.6.

Aluminum

{-X7" 'i FIGURE 2.6. A single-degree-of-freedom torsional system.

o&

L-

f

R =' 6 0 1" = 0 50 in

23

SPRINGS IN SERIES AND PARALLEL

Solution. The torsional spring rate K , of the rod can be determined using its angular displacement 8 under the action of an external torque To: TOL e= GJ

TO K O =-

and

e

so

where L = 10.0 in. (length of rod) G = 4.0 X l o 6 1b/ine2 (shear modulus of aluminum) J = r d 4 / 3 2 = ~ ( 1 . 0 ) ~ / 3=20.0981 in.4 (polar moment of inertia)

(4.0 X 106)(0.0981) K, =

10.0

= 3.92 X

IO4 in:lb/rad

The mass moment of inertia of the aluminum disk must be taken about the axis perpendicular to the plane of the disk:

I,

’

WR

=-

2g

where W = ~(6.0)~(0.50)/(0.10l b / i r ~ . ~=)5.65 lb (disk weight) R = 6.0-in. disk radius g = 386 in./s2 (gravity)

I,

(5.65) (6.0)2 =

2( 386)

= 0.264

1b.in:s’

The torsional resonant frequency can now be determined from Eq. 2.23:

’=

I

%[

3.92 x 104 0.264

1

1,/2

=

61.5 HZ

(2.24)

2.3 SPRINGS IN SERIES AND PARALLEL If a mass is suspended on two different springs in such a manner that the load path is directly from the mass through one spring, and then through the second spring before it reaches the support, the springs are said to be in series. The series spring, therefore, implies a series load path where the load must first pass through one of the springs before it can pass through the

24


Tension a n d compression

-

Bending

Torsion

FIGURE 2.7. Springs in series.

other spring. Cutting either spring would completely destroy the system. Some samples of springs in series are shown in Fig. 2.7. Springs in series can be combined into one equivalent spring using the relation

1

1 -

Ke,

K,

1

1

K?

K3

+-+-+

(2.25)

If a mass is suspended on two different springs in such a manner that the load path from the mass to the support is split between the springs, then the springs are said to be in parallel. The parallel path then permits the load to pass through one spring without passing through the other. Cutting one spring would still permit the other spring to carry the load. Some samples of springs in parallel are shown in Fig. 2.8. Springs in parallel can be combined into one equivalent spring using the relation

K,,

=K ,

+ K , + K , + ...

(2.26)

Kz = 10001b/in.

-

SPRINGSIN SERIES AND PARALLEL

25

< < K3 = 1000 Ib/in. 6

1 <

5

3

> K4 = 800 Ib/in. >

FIGURE 2.9. A combination of series and parallel springs.

Frequency of a Spring System

Sample Problem-Resonant

Determine the natural frequency of a mass supported by several springs in series and parallel combinations as shown in Fig. 2.9, by obtaining the equivalent spring rate for the system. Solution. Springs K, and K, are in parallel, so they can be combined using Eq. 2.26:

K,

= K,

+ K, = 1000 + 1000

Now there are three springs-K,, combined using Eq. 2.25: 1

--

K,,

_-

1

K,

=

2000 lb/in. series, so they can be

K,, and K,-in

1

1

1

1

K,

K,

4000

2000

+-+-=-+-

+- 1

800

The system now consists of one spring and one mass (Fig. 2.10). The natural frequency can be determined from Eqs. 2.7 and 2.9 as follows:

f,,= 49.5 HZ

3; K,,= 500 >

1

Ib/in.

W = 2 Ib

FIGURE 2.10. A single spring-mass system.

(2.27)

26


2.4 RELATION OF FREQUENCY AND ACCELERATION TO DISPLACEMENT

Many important approximations can be related to the dynamic displacements developed during resonant conditions. For example, if the geometry of a specific structure can be defined, then dynamic bending moments and bending stresses can be calculated from the dynamic displacements. Since displacements are very seldom measured during vibration tests, because optical measurements are often difficult to make, accelerometers are generally used. This permits test data to be taken in terms of frequency and acceleration G forces. The relation of frequency and acceleration to the displacement must therefore be determined to use the test data. Consider a rotating vector that is used to describe the simple harmonic motion of a single mass suspended by a spring. The vertical displacement Y of the mass can be determined by the projection of the vector Yo on the vertical axis as shown in Fig. 2.11. The vertical displacement can be represented by the equation

Y = Yosin nt

(2.28)

The velocity is the first derivative, I/= Y =

dY -=

dt

Szy, cos nr

The acceleration is the second derivative.

A

..

=

Y=

~

d2Y = dt’

-a2yosin f i t

The negative sign indicates that acceleration acts in the direction opposite to displacement.

27

RELATION OF FREQUENCY AND ACCELERATION TO DISPLACEMENT

The maximum acceleration will occur when sin Rt is 1: A,,,

= R2Y0

(2.29)

The acceleration in gravity units (G) can be determined by dividing the maximum acceleration by the acceleration of gravity: g = 32.2 ft/s2

= 386 in./s2

Also, changing radiants to cycles per second, we have fl=2.rrf Substituting into Eq. 2.29,

G = -A,,, g

- 4X2f2YJ 386 (2.30)

Note that the displacement Yo is the single-amplitude displacement. Sample Problem-Natural

Frequency and Stress in a Beam

Determine the resonant frequency, dynamic displacement, and maximum dynamic stress in the bracket with a transformer, as shown in Fig. 2.12, for a 5.0-G peak sinusoidal vibration input. Solution. Consider the transformer as a concentrated mass at the center of a massless beam. Since small screws are used to support the beam, the ends are considered to be simply supported. The natural frequency of a simply supported beam with a concentrated center mass can be determined from the static displacement as shown in Fig. 2.13, using Eq. 2.10. The static displacement can be obtained from a handbook: 6,,

W Transformer Screw

WL3 48 EI

(2.31)

=-

Bracket

L ---

0503 Sect A A FIGURE 2.12. A transformer mounted o n a bracket.

28


W=201b

-- --

FIGURE 2.13. A simply supported beam with a concen-

trated load.

__

~

-8O---.--.

where W = 2.0 Ib (transformer weight) L = 8.0 in. (length) E = 10.5 x l o 6 (aluminum-beam modulus of elasticity) I = bh3/12 = (1.2)(0.50)'/12 = 0.0125 i n 4 (moment of inertia) g = 386 in./s2 (2.0) ( 8.0)3 =

48(10.5

X

106)(0.0125)

=

1.62 X

in.

Substitute into Eq. 2.10 for the natural frequency:

f,

= 246

Hz (simply supported ends)

(2.32)

Equation 2.30 can be used to determine the dynamic deflection at the center of the transformer bracket, using the 246-Hz resonance: 9.8G 9,8Gi,Q yo=-- -

f2

(2.33)

f 2

where G,, = 5.0 G (input acceleration) Q = 30 (transmissibility at resonance) f , = 246 Hz (simply supported beam)

Yo=

9.8(5 .O) (30) =

(246)'

0.0243 in.

(2.34)

The approximate transmissibility for a beam-type structure can be obtained from the resonant frequency. For an electronic subassembly, experience has shown that a value equal to two times the square root of the resonant frequency, or about 30, gives reasonable results.

RELATION OF FREQUENCY AND ACCELERATION TO DISPLACEMENT

29

The dynamic load acting on the bracket can be approximated by letting Eq. 2.34 equal Eq. 2.31 and solving for the dynamic load wd:

w, =

w,=

48 EW, ~

L3 48(10.5 X 106)(0.0125)(0.0243)

(8.0)3

W, = 300 lb

(dynamic load)

(2.35)

If the acceleration loads were assumed to act directly on the transformer, the dynamic load would be ?t$ = WG,,Q

(2.36)

where W = 2.0 lb (transformer weight) Gin= 5.0 G (peak input) Q = 30 (transmissibility at resonance) Thus W, = (2.0) ( 5 .O) (30)

= 300 Ib

(2.37)

This result is exactly the same as that shown in Eq. 2.35, which was obtained from the dynamic deflection of a simply supported beam with a concentrated load. The dynamic bending stress in the simply supported transformer bracket can be determined from the geometry of the structure, using the dynamic loading as shown in Fig. 2.14. The maximum dynamic bending moment

Lr7

Wd = 300 Ib

4.0

Jlransfcyy

bracket

8.0

R = 150

R = 150

_L50_t.-150-~-

Shear diagram

FIGURE 2.14. Shear and bending-moment diagram for a beam

30


occurs at the center of the bracket: =150(4) =600in:lb

(2.38)

The maximum dynamic bending stress can be determined from the standard bending-stress equation:

s, =

MC r

(2.39)

where M = 600 in: lb (bending moment) c = h/2 = 0.50/2 = 0.25 in. (ref. Sect. A.A. Fig. 2.12) I = bh3/12 = (1.2)(0.50)'/12 = 0.0125 ine4 Hence

s, =

(600)(0.25) 0.0125

=

12,000 lb/in.2

(2.40)

A stress concentration factor must be used when several million stress reversals are expected. When only a few thousand stress cycles are expected in a ductile material, then a stress concentration factor is not critical. 2.5

FORCED VIBRATIONS WITH VISCOUS DAMPING

When a harmonic shaking force acts on a spring-mass system with damping, the resulting forced vibration will also be harmonic. The final frequency of the mass will be the same as that of the shaking force, because the initial transient vibrations will eventually be dissipated by the damping. Consider the harmonic shaking force Po cos R t acting on a damped spring-mass system. A free-body force diagram is shown in Fig. 2.15. The differential equation of motion with an external force exciting the system is mY+

C Y + Ky=

P" cos at

(2.41)

The solution of this equation consists of a complementary function plus a particular function. The complementary solution is the free vibrations. These will die out because of the damping. The particular solution can be taken in the form Y=Y,cos(Rt-

e)

(2.42)


t

1

Po cos Ltt

31

Po cos nt

FIGURE 2.15. Forced vibration forces acting on a simple system.

The maximum displacement, Yo,can be expressed in terms of the maximum impressed force, Po, as follows:

Yo=

PO

[ ( K - mQ2)' + c 2 R 2 ]'''

(2.43)

Dividing the numerator and denominator of the above equation by K and substituting Eq. 2.6 leads to the following:

Po -

--

K

(2.44:)

Let Y,,= P o / K , where Y,, is the deflection of the system due to the maximum dynamic input load acting as a static load. For additional simplification, let

R R,=-

a"

C

and

R , = -cc

(2.45)

This leads to the general amplification (not transmissibility) equation A = -YO =

1

(2.46)

A plot of the dynamic amplification ratio A against the frequency ratio R , is shown in Fig. 2.16. This is not a force-transmissibility curve. A force-transmissibility curve is shown in Fig. 2.17.

32


10 8

6 4

2

10 08

06 04

02 01

0

10

0.5

15

R"

=

25

20

Forcing frequency

30

f

--

f,

Natural frequency

FIGURE 2.16. A dynamic amplification curve for a simple system.

10

a 6 4

02 01

0

05

10

R"

15

25

20

30

Forcing frequency - f Natural frequency f,

_-

=

FIGURE 2.17. A transmissibility curve for a simple system.

33


The instantaneous magnitude of the force experienced by the support is the vector sum of the spring force and the damper force. These two forces have a 90” phase angle between them, so the force becomes

Fo = Yo(K 2

+~~fl’)’~’

Substituting Eq. 2.43 into the above expression,

Fo =

+C2fl2)li2 [ ( K - mn’)2 + C W ] 1/ 2 Po( K 2

The transmissibility Q, which represents the ratio of the maximum output force F,, to the maximum input force Po, can be determined by dividing the numerator and denominator by K and substituting Eq. 2.6 into the above equation:

By substituting Eq. 2.45 into Eq. 2.47, the transmissibility Q becomes

1 + (2R,Rc)2

(2.48)

(1 - R $ ) ’ + ( 2 R , R c ) 2

A plot of the force transmissibility Q against the frequency ratio R , is shown in Fig. 2.17. This curve is very similar to the amplification curve shown in Fig. 2.16. In the resonant area, where R , = 1.0, the two curves are almost identical for systems with a low damping ratio. For lightly damped systems, where R: is small enough to be assumed zero, the transmissibility expression reduces to

1 e=--1-Rk

(2.49)

34


A very convenient relation can be obtained from Eq. 2.48 by considering the transmissibility at resonance, where R , = 1.0: 1

+ (2RC)' W C l 2

For lightly damped systems, the ratio R: is very small compared to the value of 1 in the numerator; thus the transmissibility equation can be reduced to Q=-

1

(2.50)

2Rc 2.6

TRANSMISSIBILITY AS A FUNCTION OF FREQUENCY

The transmissibility of a system can be related to the resonant frequency and damping as follows: c=

/E Q2- 1

(2.51)

For a lightly damped system, where Q is greater than about 10, the relation can be solved approximately for the transmissibility:

Q=

(Km)"l

(2.52)

When the spring rate K is increased, the resonant frequency is increased. This decreases the dynamic displacement and the stresses, so the damping c is decreased, which increases the transmissibility Q. When the mass m is increased, the resonant frequency is decreased. This increases the dynamic displacement and the stresses, so the damping is increased, which decreases the transmissibility. Sample Problem-Relating the Resonant Frequency to the Dynamic Displacement

The natural frequency of a printed circuit board (PCB) is 100 Hz, with a transmissibility of 10 at the center of the circuit board. It is desirable to keep the maximum dynamic deflection at the center of the circuit board to 0.020 in. (single amplitude), in order to keep from overstressing the electrical lead wires on the resistors, capacitors, and diodes mounted on the circuit board. What should the natural frequency of the circuit board be for a 5-G peak sinusoidal vibration input?

TRANSMISSIBILITY AS A FUNCTION OF FREQUENCY

35

Solution. The PCB can be approximated as a single-degree-of-freedom system when it is vibrating at its fundamental resonant mode. Therefore Eq. 2.33 can be used to estimate the single-amplitude displacement for the 100-Hz resonant frequency:

Yo=

9.8Gi, Q

f2

where Gin= 5.0-G input (peak) Q = 10 (transmissibility at resonance) f, = 100 Hz (resonant frequency)

Yo=

9.8( 5 .O) (10) (

= 0.0490

in.

(single amplitude) (2.53)

This displacement is much too high. If the resonant frequency is increased, the displacement will decrease very rapidly as shown by Eq. 2.33. However, Eq. 2.52 shows that the transmissibility will also increase as the resonant frequency increases. Extensive test data on PCBs, with various edge restraints, indicate that many epoxy fiberglass circuit boards, with closely spaced electronic component parts, have a transmissibility that can be approximated by the relation

Q=fl

(2.54)

Substitute into Eq. 2.33 and solve for the resonant frequency:

( 2 3)

To obtain a single-amplitude displacement of 0.020 in. with a 5-G peak vibration input, the circuit board resonant frequency must be increased to the following: (2.56)

36 2.7


MULTIPLE SPRING-MASS

SYSTEMS WITHOUT DAMPING

The fundamental resonant frequency of a multiple spring-mass system, without damping, can be determined by considering the strain energy and the kinetic energy of the system. If there is no energy lost, then the total kinetic energy of the masses must be equal to the total strain energy of the springs. Consider the multiple spring-mass system as shown in Fig. 2.18. The total kinetic energy of the system is the sum of the individual kinetic energies of each mass as shown by Eq. 2.3:

T,,,,,

=

$m,Y:02 + i2 m 2 Y’O’ 2

+ $m,Y;’02

This can be written as

(2.57)

The total strain energy of the system is the sum of the individual strain energies of each spring, as shown by Eq. 2.4:


( 2S 8 )

FIGURE 2.18. A multiple spring-mass system without damping.

rF1 yj

MULTIPLE SPRING-MASS

SYSTEMS WITHOUT DAMPING

37

Since the total kinetic energy must equal the total strain energy, set Eq. 2.57 equal to Eq. 2.58 and solve for the natural frequency:

(2.59)

Note that the deflections in the above equation are the total deflections. For example, in Fig. 2.18 the total deflection of mass 3 must include the deflections of masses 1 and 2, and the total deflection of mass 2 must include the deflection of mass 1. Also, the effective weight of mass 1 on spring 1 must include all three weights. Sample Problem-Resonant Frequency of a System

Calculate the fundamental resonant frequency of the two spring-mass system shown in Fig. 2.19. Solution. The static deflection of weight 1 will be influenced by weight 2, since weight 2 is attached to weight 1; thus both weights will act on spring 3 . We have

Yl =

wl+ w,=--2 + 4 Kl

1000

- 0.006 in

(2.60)

The static deflection of weight 2 must include the static deflection of weight 1, since weight 2 is attached to weight 1:

Yz

1

1

w2

Yl + - = 0.006 K,

4

+ -- 0.00733 in. 3000

(2.61)

2.0 Ib

. 5

3 K2 = 3000 Ib/in. 4.0 Ib

FIGURE 2.19. A two-degree-of-freedom system without damping.

38


FIGURE 2.20. A free-body force diagram for a two-mass system.

Substitute into Eq. 2.59:

[(2)(0.006)

1

f f,

=-[

+ (4)(0.00733)](386)

(2)(0.006)2 + (4)(0.00733)*

2~

= 37.3

(2.62)

HZ

The static deflections of weights 1 and 2 in Fig. 2.19 could also have been determined by the simultaneous solution of the deflection equations for each mass. A free-body force diagram of the system is shown in Fig. 2.20. Consider the forces in the vertical direction: For weight 1,

- K , Y , + W , + K 2 ( Y 2 - Y Y =i )O -(

K , + K 2 ) Y ,+ K 2 Y 2+ W , = O

(2.63)

For weight 2,

- K , ( Y 2 - Y,)+ w,= 0 -KzYz

+ K,Y, + Wz = 0

(2.64)

Substituting values for K , , K 2 , W,, and W, into the above equations and solving,

+2 = 0

+

- 4000Y1 3000Y2 3000Y, - 3000Y,+ - IOOOY,

4= 0

+6=0

-

h

Y , = -= 0.006 in 1000

Yz =

3000(0.006) 3000

+4

(2.65) = 0.00733

in.

(2.66)

These results are exactly the same as those shown by Eqs. 2.60 and 2.61.

Component Lead Wire and Solder Joint Vibration Fatigue Life

3.1

INTRODUCTION

Electronic components are available in a large variety of types, sizes, and materials for leaded and leadless mounting and for through-hole or surface mounted applications. Most of these components will end up on printed circuit boards (PCBs) for use in everything electronic from cameras and washing machines to telephones and space shuttles. Some types of electronic equipment will be used in homes and offices that are air conditioned with very quiet environments. Other types of equipment will be used in military programs with high vibration and shock levels. It does not matter what the electronic equipment is doing, or what functions it performs, or how the equipment is used-in all cases the buyers or the consumers want a low cost, reliable, easy to use piece of equipment. Sometimes they get it and sometimes they don’t. Luck often seems to play a large part in the quality of electronic products. With good luck they last for years. With bad luck they last for weeks. A satisfied customer is a repeat customer. A quality product will usually attract repeat customers, even when the price of the product is slightly higher than the price of the competition. Most electronic failures are mechanical in nature. Many of these mechanical failures occur in the component lead wires and solder joints. Extensive military testing experience over a period of many years has shown that about 80% of the electromechanical failures are due to some type of thermal condition and about 20% of the failures are due to some form of vibration and shock. This chapter is devoted to the examination and evaluation of sinusoidal vibration, and its fatigue effects on the electrical lead wires and solder joints on different types of components and PCBs [25]. VIBRATION PROBLEMS WITH COMPONENTS MOUNTED HIGH ABOVE THE PCB

3.2

Electronic components are usually attached to PCBs with the use of solder. Different attachment methods can be used including dip soldering, wave 39

40

COMPONENT LEAD WIRE AND SOLDER JOINT VIBRATION FATIGUE LIFE

soldering, and infrared, vapor phase, oven, and even hand soldering. Eutectic solder is a mixture of 63% tin and 37% lead. It has a melting temperature of about 183°C. A wave soldering machine for through-hole devices uses a solder temperature of about 230°C to get the proper wicking action up the plated through-holes with the lead wires. Most semiconductor electronic components are rated below a temperature of about 200°C. Some types of capacitors are only rated at 85°C ambient. When these components are wave soldered to a PCB, the heat from the solder can be conducted up the metal lead wires directly into the silicon semiconductor chips, causing them to malfunction. The cure for this problem is often to raise the components high above the PCB. Longer lead wires increase the length of the heat flow path to the temperature-sensitive chips. This increases the thermal resistance so the temperature rise to the chip is reduced. The thermal problem is solved but it creates a vibration problem as shown below. Sample Problem-Vibration TO-5 Transistor

Fatigue Life in the Wires of a

A TO-5 transistor is mounted on the top of a 0.25-in. high plastic tube, which is sitting on the top surface of a PCB. The purpose of the high mounting is to reduce the excessive heat that is conducted up the three wires during the wave soldering operation, causing electrical malfunctions in the transistor. The longer wires reduce the temperature rise, which solves the thermal problem. Will the tall transistor wires survive a qualification dwell test requirement of 30 min with a 4.0-G peak sinusoidal vibration input level up to a frequency of 2000 Hz?

Solution Find the Response of the Transistor and the Wire Fatigue Life. The approximate natural frequency of the tall transistor can be obtained by considering it to be similar to a cantilevered beam with a concentrated end mass, using the static displacement natural frequency relationship:

2%-

(see Eq. 2.10)

where W = 1.0 g = 0.0022 lb (transistor weight) L = 0.25-in. (length of three lead wires) E = 20 x lo6 lb/in.* (kovar modulus of elasticity)

I

=

%-dJ/64= %-(0.016)'/64 = 3.22 X (one wire moment of inertia)

in.4 (3.1)

VIBRATION PROBLEMS WITH COMPONENTS MOUNTED HIGH ABOVE THE PCB

g

= 386

in./s2 (acceleration of gravity)

WL3

Sst =

S,,

41

(0.0022) (0.25)

E + (3)(20 X 106)(3 X 3.22 X l o p 9 )

= 5.93 X

f,,= % .2

lo-’ in.

(see Eq. 2.11)

(displacement for three wires)

/-

5 . 9 3 x lo-’

= 406

Hz (natural frequency)

(3.2) (3.3)

The dynamic force Pd acting on the transistor can be obtained from the acceleration and the mass of the transistor. The acceleration depends on the transmissibility expected at the natural frequency of the transistor. A good approximation for a beam-type structure is about two times the square root of the natural frequency. However, since the transistor is sitting on (and not cemented to) the top of a plastic tube, the interface slapping and impacting of the transistor on the tube during vibration will tend to reduce the expected transmissibility by about 50%. A transmissibility equal to the square root of the natural frequency will therefore be used here:

Pd = WGinQ (see Eq. 2.36) where W = 0.0022 lb (transistor weight) Gin= 4.0 peak (acceleration input level) = 20 (approximate transmissibility expected) Q=

Pd= (0.0022) (4.0) (20)

= 0.176

Ib

(dynamic force on transistor) (3.4)

The dynamic bending stress in the three lead wires can be obtained from the standard bending stress equation shown below. Stress concentrations due to cuts, scrapes, and scratches in the wires must be considered when many thousand stress reversal cycles are expected. The stress concentration can be used in the bending stress equation below, or it can be used to define the slope of the S-N fatigue curve for the kovar lead wire material. The stress concentration factor is normally used only once in either location. In this problem the stress concentration factor will not be used in the bending stress equation below. The stress concentration will be used in the kovar S-N fatigue curve, which is shown in Fig. 3.1. S,

where M

MC = -

I

(see Eq. 2.39)

= PdL = (0.176)(0.25) = 0.044

lb *in.(dynamic bending moment (3.5) c = d / 2 = 0.016/2 = 0.008 in. (wire radius)

42


Kovar wire

I

1

N , cycles to fail

FIGURE 3.1. S-N log-log fatigue curve for kovar wire, with a stress concentration factor of 2.

I

=

(3X3.22 x = 9.66 x wires) (see Eq. 3.1)

s, =

in.4 (moment of inertia, three

(0.044) (0.008)

9.66 x 10-9

= 36,439

lb/in.'

(3.6)

The fatigue life expected for the transistor kovar lead wires can be obtained from the S-N fatigue curve shown in Fig. 3.1. This is a log-log plot of the stress S and the number of cycles N to fail. Fatigue tests are run to failure on many similar closely machined samples. The results show there is a lot of scatter in the failure data. The best average straight line is drawn through the scattered test points to represent the typical fatigue properties of the material. The slope of the fatigue curve is adjusted to include a stress concentration factor of 2.0. This results in a fatigue exponent slope of b = 6.4. Every point on the straight sloped line represents a failure. The equation for every point on the sloped line is shown below: N,SP

=

N2Sf

(3.7)

Then

where N2 = 1000 cycles to fail at 84,000 lb/in.' (kovar wire reference point) S, = 84,000 lb/in.* (ultimate tensile strength of kovar lead wire)

VIBRATION FATIGUE LIFE IN SOLDER JOINTS OF A TO-5 TRANSISTOR

43

S , = 36,439 lb/in.’ (wire bending stress) (see Eq. 3.6) b = 6.4 (slope of kovar fatigue curve with stress concentration of 2)

N,

=

Wire life

=

84,000 6 . 4 (1000) 36,439

]

(

= 2.096 X

2.096 x l o 5 cycles to fail (406 cycles/s) (60 s/min)

l o 5 wire cycles to fail (3.8)

= 8.6

minutes to fail

(3.9)

3.3 VIBRATION FATIGUE LIFE IN SOLDER JOINTS OF A TO-5 TRANSISTOR Vibration forces generate overturning moments in the three lead wires. These moments result in shear tearout stresses in the solder joints, which can be obtained from the equation shown below. Microscopic examinations of failed solder joints have been made. This shows the solder appears to fail someplace between the outer diameter of the wire and the inner diameter of the plated through-hole. Therefore, an average of the wire diameter and the plated through-hole diameter is used to find the average of the solder joint [121.

S,,

=

A4 (solder shear tearout stress) hA

(3.10)

where A4 = 0.044 lb in. (see Eq. 3.5) h = 0.062 in. (solder height assumed equal to PCB thickness) A = (3>.rrd2/4 = ( 3 ) ~ ( 0 . 0 2 2 ) ~ /= 4 0.00114 i n 2 (average area of three solder joints)

”‘

=

0.044 (0.062) (0.00114)

=

622.5 Ib/in.*

(3.10a)

The vibration fatigue life for solder can be obtained from the log-log fatigue curve shown in Fig. 3.2, and with the use of Eq. 3.7. Very rapid fatigue cycles are considered here since the vibration frequency is so fast that the solder will not have a chance to creep and relax the stresses. Solder creep is common in thermal cycling environments, where the stress cycles are often very slow. This allows the solder to creep at high temperatures and relax a large percentage of any high stresses that are developed. Note that if forced

44


63/37 Tinilead solder

I

h’,cycles to f a i l FIGURE 3.2. S-N log-log vibration and thermal cycling fatigue curve for 63/37 tin/lead solder.

cooling is used immediately after the soldering operation it will increase the solder fatigue life.

1000 cycles to fail at 6500 Ib/in.2 (solder vibration life reference) S , = 6500 lb/in.l (solder ultimate shear reference) S , = 622.5 Ib/in.* (see Eq. (3.10a) b = 4.0 (slope of solder fatigue curve for vibration rapid cycles)

where N ,

=

N,

=

Solder life

=

6500 (1000) 622.5

[

‘

=

1.19 X lo7 solder cycles to fail (3.11)

1.19 x IO’ cycles to fail (406 cycles/s) (3600 s/h)

=

8.14 hours to fail

(3.12)

Comparing the solder fatigue life in Eq. 3.12 with the wire fatigue life of 8.6 minutes in Eq. 3.9 shows the wires are far more critical than the solder joints in vibration. Sinusoidal vibration tests were run using the same 4.0-G peak input level to establish the approximate fatigue life of the lead wires. The test results

RECOMMENDATIONS TO FIX THE WIRE VIBRATION PROBLEM

45

showed the wires failed after several minutes of vibration. The solder joints showed no visible cracks using a 35 power microscope. The vibration analysis and the vibration tests showed the 0.25-in. high transistor wires are critical. They will not be able to survive a 30-minute sinusoidal vibration qualification test using an input acceleration level of 4.0 G peak. Several changes were recommended to improve the fatigue life of the lead wires.

3.4 RECOMMENDATIONS TO FIX THE WIRE VIBRATION PROBLEM 1. Remove the 0.25-in. tall tube spacer and bring the transistor down to a height of about 0.060 in. above the top surface of the PCB. Attach heat sink clips to each of the transistor lead wires just before the wave soldering operation. This will thermally short-circuit the heat flow into the clips and reduce the high temperatures produced in the transistor. Remove the heat sink clips after the wave soldering operation. Experience with this approach has been very good in reducing hot spot temperatures. The reduced lead wire length will drive the transistor natural frequency above the 2000-Hz range, so there will be no vibration problems. This fix is labor intensive because it requires hand labor to add and to remove the heat sink clips. It is only cost effective for a production run of several hundred units. This fix would not be acceptable for production runs that involve several million units, as in the communication and automobile business. 2. Cement the 0.25-in. tall tube spacer to the PCB and to the transistor body to increase the stiffness of the assembly. This should drive the transistor natural frequency above the highest forcing frequency range of 2000 Hz, so there will be no vibration problems. This fix can be automated for high production runs. It may still not be acceptable for the super high communication and automobile production runs because their costs are calculated down to 0.001 penny. 3. Automate an operation that dimples the wires to automatically position the transistor 0.025 in. above the PCB for the wave solder operation. After soldering, automate another operation that adds a measured amount of viscous material, such as RTV, that encloses the three wires and fills most of the 0.025-in. space under the transistor. It is not necessary to fill the entire area under the transistor to substantially increase the damping and easily reduce the transmissibility more than 60%. This will reduce the wire bending stresses more than 60% and increase the wire fatigue life to about 50 hours. A fatigue life of only 30 minutes is required, plus some additional time to account for the usual false starts in most qualification test programs. This fix will solve the vibration problem with a slight increase in the production costs.

46


3.5 DYNAMIC FORCES DEVELOPED IN TRANSFORMER WIRES DURING VIBRATION

Transformers are made with copper and iron so they are often quite heavy. When these components are mounted on PCBs that are exposed to vibration, high dynamic forces and stresses can be produced in the lead wires and solder joints. Heavy components must be carefully mounted, properly restrained, and adequately supported to keep the lead wires and solder joints from failing in severe vibration and shock environments. Transformers are often in the shape of cylinders or cubes. Surface mounted transformers have the lead wires extending from the sides so they can be bent down 90" to meet the PCB. For through-hole mounting, straight wires are typically used, where they extend from the bottom surface of the component directly into the PCB. When transformers are mounted on through-hole PCBs that are exposed to vibration, high acceleration forces can be generated in the lead wires and solder joints when the natural frequency of the PCB is excited. The arching of the PCB under these conditions produces axial forces and bending forces in the lead wires. When the lead wires are not spaced very far apart, there will be very little bending in the wires, so almost all of the inertia force produced in the transformer wires will be carried in direct tension. Copper wires are typically used for the transformer core winding and for the external leads. The fatigue properties of the copper wire must be known to find the approximate fatigue life expected in vibration conditions.

Sample Problem-Dynamic Transformer Lead Wires

Forces and Fatigue Life in

A 0.80-lb transformer, with four 0.028-in. diameter copper wires, is mounted near the center of a 0.062-in. thick plug-in PCB that has a natural frequency of 225 Hz. The system is required to operate in a 3.0-G peak sinusoidal vibration environment. Find the following: (a) Dynamic force in the wires and solder joints. (b) Dynamic stresses in the lead wires and solder joints. (c) Approximate fatigue life expected in the lead wires and solder joints. Solution Use the Square Root of the Natural Frequency to Find the PCB Transmissibility. (a>The dynamic force Pd can be obtained from the following equation: Pd = WG,,Q

(see Eq. 2.36)

47

DYNAMIC FORCES DEVELOPED IN TRANSFORMER WIRES DURING VIBRATION

where W = 0.80 lb (transformer weight) Gin= 3.0 (peak sine input) Q= = \/225 = 15 (estimated PCB transmissibility)

a

Pd

=

(0.80)(3.0)(15)

=

36.0 lb

( 3.13)

(b) The dynamic tensile stresses in the wire can be obtained from a handbook, as shown below:

s,=

Pd

-

where Pd = 36.0 lb (shown above) d = 0.028 in. (copper wire diameter) A , = n d 2 / 4 = [ ~ ( 0 . 0 2 8 > ~ /X4 4] wires copper wires) 36.0

sw=---= 0.00246

(3.14)

A,

= 0.00246

i n 2 (area of four

14,634 lb/in.* (wire tensile stress) (3.15)

The dynamic shear stress in the solder joint can be obtained from Eq. 3.14 with a slight change as shown below. Test data show that solder joints appear to fail at some point between the wire and the plated through-hole. Therefore, the average solder joint diameter is obtained by using the average of the lead wire diameter plus the plated through-hole diameter. The thickness of the PCB,without any solder fillets, is normally used for the solder shear area. This is a conservative estimate since visual inspections show solder fillets of about 0.010 in. on the top and bottom surfaces of the PCB for through-hole mounted components. Pd

ss= A,

(3.16)

where Pd= 36.0 lb (dynamic load) (see Eq. 3.13) d = (0.028 + 0.040)/2 = 0.034 in. (average solder joint diameter) A s = (4)(n)(0.034)(0.062) = 0.0265 ine2(area of four solder joints) 36.0

s,=-0.0265

- 1358 lb/in.2

(solder shear stress) (3.17)

(c) The approximate fatigue life of the lead wire can be obtained from the log-log S-N fatigue curve for the copper lead wire shown in Fig. 3.3. This curve includes a stress concentration factor of 2.0, to account for cuts,

48


Copper wire

108

103 iV, cycles to f a i l

FIGURE 3.3. S-N log-log fatigue curve for copper wire, with a stress concentration factor of 2.

scrapes, and scratches normally found in component lead wires. The fatigue damage relation is used to find the number of stress cycles required to produce a fatigue failure, as well as the expected fatigue life.

N,S,b = N,S,b (see Eq. 3.7) where N , = 1000 stress cycles to fail at 45,000 lb/in., (copper reference point) S , = 45,000 lb/in., (stress reference point) b = 6.4 (reference slope of fatigue curve for copper lead wire with k = 2) S , = 14,634 lb/in.’ (wire stress) (see Eq. 3.15) Nl

=

Wire life

=

(1000)

[

~

45,000 14,634

1

6.4

=

1.33 x l o 6 wire cycles to fail (3.18)

1.33 x lo6 cycles to fail

(225 cycles/s) (3600 s/h)

=

1.64 hours

(3.19)

The approximate fatigue life of the solder can be obtained from the log-log fatigue curve for solder shown in Fig. 3.2. Solder creep is not considered here because the stress cycles are very fast, so the solder does not have a chance to relax any internal stresses. The fatigue damage equation (Eq. 3.7) is used to find the number of stress cycles to fail, as well as the expected fatigue life.

DISPLACEMENTS BETWEEN PCB AND COMPONENT PRODUCE LEAD WIRE STRAIN

49

where N2 = 1000 stress cycles to fail at 6500 Ib/in2 (solder reference point) S , = 6500 lb/in.* (stress reference point) SI = 1358 Ib/in.* (solder shear) (see Eq. 3.17) b = 4.0 (reference slope of fatigue curve for solder vibration)

[

6500

1

N,= (1000) Solder life =

1358

= 5.25 X

5.25 x 10' cycles to fail (225 cycles/s) (3600 s/h)

IO5 solder cycles to fail (3.20) = 0.65

hour

(3.21)

Comparing Eq. 3.19 (for the wire life) and Eq. 3.21 (for the solder life) shows the solder is more critical than the lead wire in this case. 3.6 RELATIVE DISPLACEMENTS BETWEEN PCB AND COMPONENT PRODUCE LEAD WIRE STRAIN

Plug-in PCBs can produce large dynamic displacements in high vibration and shock environments, when their natural frequencies are excited. This often results in large relative displacements between the PCB and any large electronic components mounted on the PCB. The body of a large component may be thicker and stiffer than the PCB that supports the component. When the component is attached to the PCB with many lead wires in a through-hole or surface mounted arrangement, the relative displacement between the PCB and the component can generate a significant amount of strain in the lead wires, as shown in Fig. 3.4. The lead wires that are soldered to the PCB will

50


stay perpendicular to the PCB when it bends away from the component body. The lead wires will also stay perpendicular to the component body. This motion will generate an axial force and a bending moment in the lead wires. When the length of the component body is small compared to the length of the PCB, the axial force in the lead wire will be much greater than the bending moment in the lead wire. The bending moment in the wire can then be ignored without introducing a large error. Finite element methods (FEMs) of analysis show that a long component body will bend much more than the lead wires will stretch. Also, when the component body is stiff and when it has many relatively stiff wires, such as a dual inline package (DIP), the PCB will be stiffened in the local area of the component. This will tend to reduce the relative displacement between the component and the PCB. In the following analysis only the component body bending and the lead wire stretching will be considered. The change in the PCB curvature is too complex to consider here for hand calculations. Approximations mentioned above will significantly reduce the amount of calculation time-and still give reasonable results that are conservative-and can be used to estimate the fatigue life. A trigonometric expression can be used to describe the PCB displacement at any point. The relative displacement ( 6 ) between the component and the PCB will be the sum of the component body bending deflection and the lead wires stretching deflection. The component considered in the analysis is a ceramic hybrid with kovar lead wires spaced on 0.050-in. centers. It becomes too complex to examine the forces in every wire so only four end wires are assumed to carry the dynamic loads produced by the relative displacements. Chapter 8 has a section that relates to finding the minimum desired PCB natural frequency to achieve a 10 million cycle fatigue life with sinusoidal vibration, based on the component type, the PCB geometry, and the expected environment, This information was based on testing many different types of PCBs with different types of components until they failed. The data were used with FEMs to cross check the results analytically. The following sample problem starts with the minimum desired PCB natural frequency equation to show how the approximate fatigue life in the lead wires and solder joints can be established using approximate hand calculation methods. Sample Problem-Effects

of PCB Displacement on Hybrid Reliability

A ceramic hybrid 1.5 in. long, with 0.025-in. diameter kovar lead wires, is through-hole mounted 0.050 in. above the top surface and at the center of a 0.062-in. thick epoxy fiberglass plug-in PCB, parallel to the 9.0-in. edge, as shown in Fig. 3.4. The PCB is required to operate in a 5.0-G peak sinusoidal vibration environment. Simplify the analysis by assuming only four wires at each end of the component carry the dynamic load. Find the following:

(a> Minimum desired PCB natural frequency for a 10 million cycle fatigue life.


51

(b) Relative dynamic displacement expected between the PCB and the component. (c) Dynamic forces and stresses in the lead wires and solder joints. (d) Approximate fatigue life expected in the lead wires and solder joints. (e) Bending stress in the ceramic component body.

Solution. (a) The minimum desired PCB natural frequency equation is shown below. -

[

9.8GinChr&

)2’3

0.00022B

(see Eq. 8.13)

where Gin= 5.0 (peak sine vibration) C = 1.26 (component type for hybrid with pins extending from bottom surface) h = 0.062 in. (thickness of PCB) r = 1.0 (relative position factor for component at center of PCB) L = 1.5 in. (length of component) B = 9.0 in. (length of PCB parallel to length of component)

fd =

[

(9.8) (5 .O) ( 1.26) (0.062) ( 1.O) (0.00022)(9.0)

=

177 Hz (3.22)

(b) The relative dynamic displacement difference ( 6 ) between the component and the PCB can be obtained from the dynamic displacement curve, which is trigonometric, as shown below. 7TX

2,= Znsin-

and

U

6= Z ,

(3.23)

- 2,

Then 6 = Z , 1 -sini

)

(3.24)

7aT x

where Gin= 5.0 (peak sine vibration) f , = 177 Hz (to achieve 10 million cycle fatigue life) = 13.3 (approximate PCB transmissibility expected) Q= =

fi

2,= 9.8Gi, Q/(fnI2 = (9.8)(5.0)(13.3)/(177)2

= 0.02080 in.

(3.25)

x = 3.75 in. (to edge of component) u

=

9.0 in. (length of PCB)

1

.rr(3*75) 9.0

= 0.00071

in.

(3.26)

52


This represents the deflection between the PCB and the edge of the component. This also represents the bending deflection of the component and the stretching deflection of the four lead wires assumed to carry the dynamic force. (c) There is not enough information at this point to find the forces and stresses in the lead wires and solder joints. Additional information must be developed. There are two unknowns-the deflection of the component and the deflection of the lead wires-so two equations must be generated to solve the problem. The first equation says that the force in the component must be the same as the force in the lead wires. The second equation says that the sum of both deflections must equal 0.00071 in. Start with the deflection of the component body (Z,) as a cantilevered beam with an end load generated by the four lead wires. PC L3, z,= 3% 1,

where E , I, L,

so

Pc

=

3ECICZC L3,

(3.27)

= 40 X

lo6 1b/in2 (ceramic component modulus of elasticity) = bh3/12 = (0.562)(0.170)'/12 = 2.301 X l o p 4 = 0.75 in. (half the full length of the component) (3)(40 x 106)(2.301X 10-4)Zc P- =

'L

( 0.75)

= h 545 .- X

10JZc (3.28)

Find the axial deflection of the four lead wires (Z,) assuming pure tension only. 2,

where d, A, E, L,.

=

PWL, ( 4 wires) AWE,

so

P,

4A,EwZ, =

(3.29)

L,

in. (wire diameter) = 7r(d,)2/4 = ~(0.025)'/4 = 0.000491 in.2 (area of one wire) = 20 x 10 Ib/ine2 (kovar wire modulus of elasticity) = 0.050 2(0.025) 2(0.025) = 0.150 in. (includes two wire diameters into the component and two wire diameters into PCB for equivalent length) = 0.025

+

P,

+

(4)( 0.000491)( 20 x 1 0 6 ) Z , =

0.150

= 2.618

x 105ZW (3.30)

For equilibrium conditions the bending force in the component must equal the axial force in the four wires so Eq. 3.28 must equal Eq. 3.30. 6.545 x 104Z,

= 2.618

x 105Zz, then

Z,

=

4.002,

(3.31)

DISPLACEMENTS B E W E E N PCB AND COMPONENT PRODUCE LEAD WIRE STRAIN

53

The relative displacement (6) between the component and the PCB will be equal to the bending displacement of the component and the axial displacement of the four wires. 6=Z,+Z, where S Z,

= 0.00071 = 4.002,

(3.32)

in. (see Eq. 3.26) (see above)

Then 0.00071 = 4.002,

+Z,

= 5.00Z,

so Z,

=

1.420 X

in.

(expected wire displacement)

(3.33)

The resulting dynamic force in one lead wire can be obtained from Eq. 3.29 considering one wire. This will be the same force acting on the solder joint of the lead wire.

P,

A,E,Z, =

L

-

,

(0.000491)(20 X 106)(1.420X 0.15

= 9.296

lb (3.34)

(d) The dynamic tensile stress expected in the lead wire can be obtained as follows:

P,

sW -- - A=,

9.296 0.000491

=

18,933 lb/in.,

(3.35)

The approximate fatigue life expected for the kovar lead wire can be obtained from Eq. 3.7 and the kovar fatigue curve shown in Fig. 3.1, using the information shown below.

where

N,= 1000 stress cycles (reference point at 84,000 lb/in.*) = 84,000 lb/in.2 (reference point) S , = 18,933 lb/in.* (tensile stress shown above) b = 6.4 (slope of fatigue curve with a stress concentration of 2.0 to account for cuts, scrapes, and scratches in the lead wires)

S,

Nl

=

84,000 6.4 (1000) 18,933)

[

=

13.83 X 106 wire cycles to fail (3.36)

54


This is close to the 10 million cycle fatigue life typically expected for components in a sinusoidal vibration environment when the PCB natural frequency is obtained using Eq. 8.13. The approximate wire fatigue life for a sinusoidal vibration resonant dwell test condition can be obtained as shown below. Life

13.83 X l o 6 cycles to fail =

(177 cycles/s) (3600 s/h)

= 21.7

hours to fail

(3.37)

The dynamic shear stress expected in the solder joint can be obtained using the following information.

& = - ps -

ps --

rrdh

where P, d h A,

A,

= 9.296

lb (same as Eq. 3.34) = (0.025 + 0.040)/2 = 0.0325 in. (average solder joint diameter) = 0.062 + 2(0.015) = 0.092 in. (solder joint height including fillets) = n-dh = n-(0.0325)(0.092) = 0.00939 in.* (solder joint shear area)

Ps 9.296 S S -- - = A, 0.00939 ~

=

990 lb/in.2

(3.38)

The approximate fatigue life for the solder joint can be obtained using Eq. 3.7 and the vibration portion of the solder fatigue curve shown in Fig. 3.2, with the following data:

where N , = 1000 cycles to fail at 6500 1b/im2 (reference point) S , = 6500 1 b / h 2 (stress reference point) S , = 990 1 b / h 2 (shear stress in solder) b = 4.0 (slope of solder fatigue curve for vibration) N,

=

[ 7;: il

( 1000) -

=

1.86 x l o 6 solder cycle to fail (3.39)

The approximate solder joint time to fail in a sinusoidal vibration condition will be as follows: Life

1.86 x l o 6 cycles to fail =

(177 cycles/s) (3600 s/h)

= 2.92

hours to fail

(3.40)


55

The solder joint with this geometry and application is more critical than the lead wire. (e) The bending stress in the ceramic hybrid body can be obtained from the standard bending stress equation with the following information:

where P = 9.296 lb (in one wire, but four wires carry the load) (see Eq. 3.34) L = 0.75 in. (half of the component length) M = PL = (4)(9.296)(0.75) = 27.88 lb . in. c = 0.17/2 = 0.085 in. (half the component thickness) in.4 (moment of inertia) (see Eq. 3.28) I = 2.301 X s b =

(27.88)(0.085) 2.301 X

=

10,300 lb/in.’

(3.41)

Ceramics have a compressive strength of about 250,000 1b/ins2 but the tensile strength is only about 25,000-30,000 lb/in.2, so the ceramic component body appears to have a good safety factor.

-

CHAPTER 4

Beam Structures for Electronic Subassemblies

4.1 NATURAL FREQUENCY OF A UNIFORM BEAM The natural frequency of a uniform beam can usually be determined by considering the strain energy and the kinetic energy of the vibrating beam. There are several different ways of expressing these quantities. Some methods are approximate, while others are exact. The exact methods usually require more work, and the approximate methods are often satisfactory for engineering solutions, so the latter will be given more consideration. Boundary conditions are very important in exact methods as well as in approximate methods. For approximate methods, if the geometric boundary conditions are satisfied, the resulting natural frequency equations will usually be quite accurate for the fundamental resonant frequency. The geometric boundary conditions consist only of the slope and the deflection. However, these boundary conditions must be met by the type of deflection curve that is used to describe the vibrating beam. The technique of assuming a deflection curve for a vibrating system, from which the natural frequency equation is determined, is called the Rayleigh method, after Lord Rayleigh [28], who first proposed it. If the deflection curve happens to be the exact curve for the boundary conditions, then the frequency equation will also be exact. A trigonometric function can be used to describe the deflection curve of the simply supported beam shown in Fig. 4.1. Consider the deflection equation

TX

Y = Yosin L This curve satisfies the geometric boundary conditions of slope and deflection as follows: Deflections:

x = 0, x =L , X=L/2,

56

Y=O Y=O Y=Y,

NATURAL FREQUENCY OF A UNIFORM BEAM

57

I yo

FIGURE 4.1. Deflection curve for a simply supported beam.

Slopes:

If then

X=O,

8=-Y0 L

X=L,

8=--Y LO

x = -2 ’

8=0

77

The strain energy can be obtained from the general bending moment equation d2Y E I y = M dX

as

From Eq. 4.1,

dY 7~ “iX _- -Yo cosdX L L d2Y “i2 - - -Yo dX L2

--

($1

2

T4 =

“iX sin L

“iX

F Y i sin’-- L

(4.4)

58

BEAM STRUCTURES FOR ELECTRONIC SUBASSEMBLIES

Substituting Eq. 4.4 into Eq. 4.3 for the strain energy of the beam, we find

U=

(4.5)

The kinetic energy of the vibrating beam will be as follows:

At resonance, the kinetic energy is equal to the strain energy, so Eq. 4.6 is equal to Eq. 4.5:

EIli Y i 4 L3

-

w O2Y;L

4g li 'EIg

=-

="i"i wL4

f

1/ 2

(4.7)

21i w L 4

This results in L e exact frequency equation because L e exact deflection curve was used in Eq. 4.1. A polynomial can also be used to describe the deflection curve for a simply supported beam, shown in Fig. 4.2. Consider the deflection equation

w (lb/in )

[La-* I

E -x

FIGURE 4.2. A simply supported beam with the Y axis at the center.


59

This curve satisfies the geometric boundary conditions of slope and deflection as follows: Deflections:

x =0 ,

Y=Yo

X=a, X=-a,

Y=O Y=O

e = -d=Y- -

2Y0X

Slopes:

dX

so

x =0 ,

a*

o=o

From Eq. 4.8,

--

dY

2Y0X

dX

a*

d2Y dX

2Y0 a*

--

(4.9) For the strain energy, substituting Eq. 4.9 into Eq. 4.3,

2 EIY: u= 2 EIY,, ( a ) = ___ ~

a'

a3

(4.10)

The kinetic energy can be obtained by using Eq. 4.8 as follows:

(4.11)

60


At resonance, the kinetic energy must equal the strain energy, so Eq. 4.11 must equal Eq. 4.10:

2EIY;

--

U3

Since u

-

4wR2Y,'u 15g

=L/2,

(4.12) Comparing these results with the exact frequency equation (Eq. 4.7) shows the polynomial results in a frequency that is about 11.1% too high. In fact, the Rayleigh method always results in a frequency value that is greater than the true value. unless the exact deflection curve is used. Sample Problem-Natural

Frequencies of Beams

Determine the approximate resonant frequency of the beam in Fig. 4.3 when it is made of aluminum and when it is made of steel, and vibration is in the vertical direction. Solution. If the beam is made of aluminum, = 10.5 x 106 1 b / h 2 (modulus of elasticity) b = 0.50 in. (beam width) h = 1.0 in. (height of beam) I = b h 3 / 1 2 = (0..50)(1.0)'/12 = 0.0417 in.4 (moment of inertia) g = 386 in./s2 (gravity) L = 10.0 in. (length) p = 0.10 1 b / h 3 (density of aluminum) W = b h L p = (0.50)(1.0)(10.0)(0.10)= 0.50 lb (weight of beam) w = W / L = 0.50/10.0 = 0.05 lb/in.

E

Section A A

FIGURE 4.3. A uniform beam with simply supported ends.


61

Substituting into Eq. 4.12,

fn= fn =

i

10.96 (10.5 X 106)(0.0417)(386)

(0.05) ( 10.0)'

1013 Hz

(4.13)

Now assume the beam is made of steel: 29 x lo6 lb/in.* (modulus of elasticity) p = 0.283 lb/in.3 (density) W = (0.50)(1.0)(10.0>(0.283)= 1.415 lb (weight) w = W / L = 1.415/10.0 = 0.1415 lb/in.

E

=


fn= fn =

i

10.96 (29 x 106)(0.0417)(386) (0.141) ( 10.0)'

1005 HZ

(4.14)

Comparing Eqs. 4.13 and 4.14 shows the natural frequency of the steel beam is almost the same as that of the aluminum beam, even though the steel beam is almost three times heavier. A closer examination of the natural frequency equation shows that it depends on the ratio of the modulus of elasticity to the density. These two factors can be examined for the beam shown in Fig. 4.3, considering several different materials:

E

6.5 x lo6

100 x l o 6 in.

Magnesium:

-P

Aluminum:

-P

E

10.5 X l o 6 = 105 x l o 6 in. 0.10

Steel :

-=

E

29.0

Beryllium :

0.065

=

P

lo6 0.283

E

42.0 x lo6

P

0.068

_ --

X

=

102 x IO6 in.

= 619

x l o 6 in.

(4.15)

The only two items that will change in the natural frequency equation are the modulus of elasticity and the density. An examination of the ratios shown above indicates that the natural frequency of the beam will be about the

62


same for magnesium, aluminum, and steel. A beryllium beam, however, would have a much higher natural frequency. The Rayleigh method can be used to determine the fundamental resonant frequency for beams with various end conditions. This method can be used with a trigonometric or a polynomial expression as long as the geometric boundary conditions of slope and deflection are satisfied. Some typical deflections for different boundary conditions are shown, along with the resulting natural frequency equations, in Fig. 4.4. The resonant frequencies of beams with different edge conditions are shown in Fig. 4.5. CASE

1

Cantilever Beam- Uniform Load

Trigonometric

Polynomial

Y

Y - L -

-x

(

Y = Y,, l-cos-

CASE

2

2L

-

-X

Y

=

Y,,

(;I2

Fixed-Fixed Beam- Uniform Load

Trigonometric

Y

Polynomial

Y

FIGURE 4.4. Displacement curves and natural frequency equations.

fn =

C

J@$

Where C = Modal constant

BEAM TYPE

Cantilever

t-

$JL

k C = 56

Simply supported ends or hinged-hinged

MODE 3

MODE 2

MODE 1

,644

C = 351 500

r====7 4-7

&

C = 982 ___ ___~

Fixed ends

=

C = 6.28

m C

F ree-free

1.57

=

224

+ I-+C = 1 4 1

I

19.2

=

500

~

C

=

_

252

_

Fixed-hinged

=

C

=

c =192

9.82

132

277

723

868

I , , i

356

C

=

b (' =

C = 394

F & C = 166

245

('=

795

446

1<-

=

853

I,

7 95

zlY\8 C = 166

,

409 511, 774 910 0602i7, -

C = 318

982

616

H inged-free

_

500

/

C

* 400 800

359 .641

356

776

C

333 667

w * L C

MODE 5

MODE 4

C

=

284

JJLI 235

(' =

A

-

('=475 429 810

v=---=2 3 8 1 6/91

<- =

43 3

381 763

707

28 4

FIGURE 4.5. Resonant frequencies of uniform beams [51].

c

=

43 3

64 4.2


NONUNIFORM CROSS SECTIONS

Electronic equipment in close quarters must make use of every cubic inch of volume. When there is a little extra space, it is usually located in an area that is difficult to reach. Often structural modifications must be made to accommodate equipment that seems to grow larger with every redesign. This usually requires notching, reinforcing, or moving major load-carrying members, The net result is a chassis or a structure that supports all of the electronics at the expense of a cross section with notches and cutouts along its length. In order to determine the resonant frequency of this type of structure, it is necessary to consider the effects of nonuniform cross sections. If there are many different cross sections that must be considered, most methods of analysis can become very long and time consuming. However, there is a trade-off that can be made, which reduces the amount of work at the expense of some accuracy. This trade-off involves the calculation of an equivalent moment of inertia for a uniform structure that will have approximately the same stiffness as the nonuniform structure. The loss of accuracy is only about 5-1096 in the natural frequency for many cases, which is well worth the amount of work that can be saved. Consider a cantilever beam with two different cross sections that form two steps (Fig. 4.6). Castigliano’s strain energy theorem can be used to determine the deflection under the load at the free end of the beam. The general deflection equation is

The deflection for the step beam shown in Fig. 4.6 is (4.16)

FIGURE 4.6. A cantilever beam with two different cross sections.


65

The bending moment at point 1 is

M,

= PX,

The bending moment at point 2 is

M,

=P("

+X,)


P a 6=--/X EI, o

3EI,

"2 + 2 ax,+ x2') dX, ;d X , + -Ib( o

P

(4.17)

EI,

Let

L "=b=2

(4.18)

PL3 6s-+24

[

E;*

I

(4.19)

The deflection of a uniform cantilever beam with a concentrated load at the end can be obtained from any structural handbook: (4.20) An average moment of inertia I,, is used in the above expression so that the average moment of inertia can be determined by forcing the deflections in Eqs. 4.19 and 4.20 to be equal:

PL3

--

3EIa,

-+&I

PL3 24

EI,

Solving for the average moment of inertia, we have (4.21)

66


The average moment of inertia for the two-step beam in Fig. 4.6 can be determined for two different moments of inertia. Let

I , = 2 in.4 and

I 2 = 4 in.'

(4.22)

Substituting Eq. 4.22 into Eq. 4.21, (4.23) Now let us see if there is some other way that an average moment of inertia might be determined, so as to give results similar to Eq. 4.23 without going through so much work. Consider a method for averaging the moment of inertia as a function of the length of each section for the stepped beam shown in Fig. 4.6: aI, + b12 I' = (4.24) a' a+b Substituting Eqs. 4.18 and 4.22 into Eq. 4.24, (4.25) Comparing Eq. 4.25 with Eq. 4.23 indicates the approximate averaging method shown in Eq. 4.24 is only about 15.5% lower than the true value. Since the natural frequency varies as the square root of the moment of inertia, the error in the natural frequency will be only about 8.1% too low. Some other proportions can be examined. For example, consider the proportion a=2b so a = $ L and b = S L (4.26) Substituting Eqs. 4.26 and 4.17, (4.27) To get the same deflection as a uniform cantilever beam with a concentrated load, force Eq. 4.27 to equal Eq. 4.20:

PL'

PL'

3EIa,

S1E

-=-

(

(

191, + 8 1 2

I,I,

I , I2 = 27 191, + 81,)

I


67

Substituting Eq. 4.22 into the above,

I,,

= 27

8 (j8+32/

- = 3.08 in.4

(4.28)

This is the correct value for the average moment of inertia. Compute the approximate average value with the method shown by Eq. 4.24. This can be done with Eqs. 4.26 and 4.22 as follows: (4.29) Comparing Eq. 4.29 with Eq. 4.28 indicates the approximate averaging method shown by Eq. 4.24 is only about 13.6% lower than the true value. There is a second method of averaging the moment of inertia as a function of the length of each section for the stepped beam shown in Fig. 4.6. Consider the following: (4.30)

Substituting Eqs. 4.18 and 4.22 into Eq. 4.30,

-L+ - L I"

=

L/2

-+2

L/2

= 2.66

in.4

(4.31)

4

Comparing this with the correct value of 3.55 in.4 shown by Eq. 4.23 indicates the approximate averaging method shown by Eq. 4.30 is about 25% lower than the true value when a = b. The resonant frequency would be 13.4% lower. When a = 2b (as shown by Eq. 4.26 along with Eq. 4.22), Eq. 4.30 becomes (4.32)

- 2+ -

4

Comparing this with the correct value of 3.08 i n 4 shown by Eq. 4.28 indicates the approximate averaging method shown by Eq. 4.30 is about 22% lower than the true value when a = 26. The resonant frequency would be about 11.7% lower.

68


There are, of course, many other methods that can be used to determine an average moment of inertia for a stepped beam. It is generally desirable to choose one that is conservative, so that it will produce a moment of inertia that is slightly lower than the true value. This will then result in a calculated natural frequency that is slightly lower than the true value. This is preferred because most electronic structures have many bolted and riveted joints, which tend to reduce the stiffness and therefore the natural frequency. Sample Problem-Natural Frequency of a Box with Nonuniform Sections

An electronic box has mounting flanges at each end and a cross section that has four different moments of inertia along its length (Fig. 4.7). It is required to determine the approximate resonant frequency during vibration in the vertical direction (along the Y axis) if

I,

= 3.5

in.4, Ib = 2.5 in.',

I,

= 4.2

in.4 Id = 5.5 in.4

Solution. The approximate average moment of inertia for an equivalent uniform beam can be determined with an equation similar to Eqs. 4.24 and 4.30. Using Eq. 4.24 first, we have a[,

+ bIb + CI, + dId

I'a\

=

I'a b

=

I:,

= 3.96

a+b+c+d (4.5)(3.5) + ( 6 . 5 ) ( 2 . 5 ) + (3.0)(4.2) 4.5 + 6.5

(4.33)

+ (7.0)(5.5)

+ 3.0 + 7.0

in.'

Approximating the chassis as a simply supported uniform beam, the natural frequency is (see Eq. 4.7) (4.34)

Mounting flange -,

FIGURE 4.7. A n electronic chassis with four different cross sections.

COMPOSITE BEAMS

69

where E = 10.5 X lo6 lb/in.’ (aluminum) I = I:” = 3.96 in.4 g = 386 in./s2 (gravity) W = wL = 28.5 Ib (weight) L = 21.0 in. (length)

so we have T

fn =

f,

i

= 387

(10.5 X 106)(3.96)(3.86 x l o 2 ) (28.5) (21 .O)

Hz

(4.35)

If there is some question concerning the integrity of several structural members that are part of a riveted assembly and there is a fear of some relative motion in this assembly, there could be a reduction in the stiffness of the electronic box, which would reduce the natural frequency. A conservative approximation of the natural frequency could then be made by using an approximate average moment of inertia that is known to be relatively low. Under these circumstances it might be desirable to use an expression similar to Eq. 4.30:

1’’ a =v

a+b+c+d a b c d

-+-+-+Io

I”

=

av

I:”

Ib

IC

(4.36)

Id

+

4.5 + 6.5 3.0 + 7.0 4.5 6.5 3.0 7.0 -+ - + - + 3.5 2.5 4.2 5.5

= 3.57

in.4

Substituting this value into Eq. 4.34, fn = 370 Hz

(4.37)

For this particular electronic box, there is not much difference in the resonant frequency for the two different methods used in determining the approximate average moment of inertia.

4.3 COMPOSITE BEAMS Composite laminations are often used in electronic boxes because of electrical, thermal, and vibration requirements. Sometimes a printed circuit board (PCB) is mounted so close to a metal bulkhead that it is possible for bare

70


metal lead wires on the circuit board to contact the bulkhead during vibration. Short circuits can result, and they may damage the electronic equipment. In order to prevent this, a thin strip of epoxy fiberglass, 0.005 in. thick, may be cemented to the bulkhead. Epoxy fiberglass is hard and tough and can resist the pounding of many sharp points during a resonant condition. The bulkhead, of course, then becomes a composite laimination. PCBs often use metal-strip laminations of aluminum or copper to conduct away the heat. In some cases, thin copper strips are bonded to a circuit board; electronic component parts such as resistors, diodes, flatpacks, and transistors are cemented to these copper strips; and their electrical lead wires are soldered to the PCB. Aluminum plates have been laminated to PCBs in order to conduct away the heat. In some cases, the components are mounted directly on the aluminum, which has clearance holes for the electrical lead wires to permit soldering to the circuit board on the back side of the aluminum plate. In other cases, the components may be mounted on a circuit board only 0.005 in. thick. The heat will then flow right through the epoxy fiberglass circuit board, with a moderate temperature rise, and be conducted away by the aluminum plate [54]. Sometimes it is necessary to thermally isolate an electronic component, such as a gyro, which must be maintained at a constant temperature to provide the required accuracy. A closely controlled heating system may be used to maintain a constant gyro temperature over a wide external temperature range. Since most structural members are made of metal, and are good heat conductors, the thermal isolation system will usually consist of a metal laminated to a plastic or a ceramic. Two very common materials that are often used in electronic box structures are aluminum and epoxy fiberglass. The natural frequency of a composite beam of these materials can be determined by considering a combination of the physical properties of both materials. Consider the case of a simply supported laminated beam that has a uniform load distribution along its length (Fig. 4.8). If the lamination of aluminum and epoxy fiberglass is side by side, as shown in the cross-section view of Fig. 4.9, then an equivalent beam of one material can be used to find the natural frequency.

FIGURE 4.8. A simply supported laminated beam.

COMPOSITE BEAMS

71

Epoxy fiberglass

h= 10

Section AA

I

FIGURE 4.9. Cross section of a lami-

I

nated beam.

The width of the epoxy fiberglass lamination can be reduced to make it equivalent in its stiffness to an aluminum beam, using the EI stiffness factor. The subscripts a and e refer to aluminum and epoxy, respectively:

E , I, E,b,h:

=E, I, -

12

E,b,hg

12

Since the height of the aluminum section, h a , is the same as the height of the epoxy section, h e , they cancel. The aluminum equivalent of the epoxy section is (4.38) where b, = width of aluminum equivalent to epoxy be = 0.50-in. (width of epoxy) E , = 2.0 X l o 6 lb/in.2 (epoxy fiberglass) E , = 10.5 X l o 6 lb/in.2 (aluminum) Thus b,

=

(0.50)

(

2.0 x l o 6 106

= 0.0952

in.

(4.39)

The equivalent width of a solid aluminum beam can now be determined by adding the increment shown above to the width of the aluminum section: be, = 0.75

+ 0.0952 = 0.8452 in.

(aluminum)

The natural frequency of this beam in the vertical direction can be determined from the equation of a uniform beam as follows: 1/2

(see Eq. 4.7)

72


10.5 x IO6 1b/in2 (aluminum) = b,,h3/12 = (0.8452)(l.0)3/12 = 0.0704 in.4 g = 386 in./s2 (gravity) L = 15.0 in. (length) W, = (15)(1.0)(0.75)(0.10 lb/in.3) = 1.125 lb (aluminum) We= (15~(l.0>(0.50>(0.065 lb/in.3) = 0.487 lb (epoxy) W = total beam weight = 1.612 lb

where E I

=

Thus

f"

7~

(10.5 X 106)(0.0704)(386)

2

(l.612)( 15.0)3

=-[

f , = 360 HZ

(4.40)

If the lamination of aluminum and epoxy is stacked in the vertical direction as shown in the cross-section view of Fig. 4.10, then a composite moment of inertia can be computed for the two materials. The same cross-sectional area for the aluminum and the epoxy fiberglass is used to keep the same weights. The following tabulations can be set up to simplify the calculation of the composite moment of inertia for the beam stiffness factor EI: Section

Area

Eo

AE

Y

AEY

EPOXY Aluminum

0.50 0.75

2 x lo6 10.5 X IO6

1.00 x l o 6 7.87 X IO6

0.80 0.30

0.80 x l o 6 2.36 X 106

Total

8.87 X IO6 3.16 x l o 6

CAEY

-

y=

~

CAE

-

8.87 X IO6

=

0.356 in.

3.16 X l o 6 (centroid)

Y

FIGURE 4.10. Cross section of a laminated beam.

10

Aluminum

COMPOSITE BEAMS

Section EPOXY Aluminum

C

C2

0.444 0.056

0.196 0.0031

O I 0.0066 0.0225

AEC 0.196 x lo6 0.024 X lo6 0.220 x lo6

Total

73

EO Io 0.0132 x lo6 0.2360 X lo6 0.2492 x lo6

The composite beam stiffness factor is

C A E c 2+ C E o I , ,= 0.220 X lo6 + 0.249 X lo6

EI

=

EI

= 0.469 X

lo6 in.in.2

(4.41)

Substituting into Eq. 4.7 for the natural frequency,

f

=-[ 7~

2

(0.469 X 106)(386) (1.612)(15.0)3

I”

= 286

HZ

(4.42)

Equation 4.38 can also be used with the composite cross section shown in Fig. 4.10. The result will then be a T section with only one material involved. The equivalent width of an aluminum section is

b,

=

2.0 x l o 6

I.,,(

10.5 x lo6

)

= 0.238

in.

This results in the aluminum T section shown in Fig. 4.11. A table can be set up to simplify the calculation for the moment of inertia of the cross section: Item 1 2

Area 0.095 0.750

Total

0.845

Y 0.80 0.30

--1.254

=

C

C2

0.444 0.056

0.196 0.0031

0.301

- CAY =-=-Y= EA I

AY 0.076 0.225

0.301

- 0.356 in

0.845

cAc2 + cIo

= 0.0209

Ac 0.0186 0.0023

IO 0.00127 0.02250

0.0209

0.02377

. (centroid)

+ 0.02377 = 0.04467 in.4

FIGURE 4.11. A T section used to simulate a laminated beam.

74


Y

\--

125--

s 10

FIGURE 4.12. Cross section of a beam with three

laminations.

I 1

,

7

Aluminum

EPOXY

0 30

Aluminum -

b

The beam stiffness factor becomes

EI

=

10.5 x 106(0.04467)

=

0.469 x l o 6 Ib.in.*

(4.43)

Comparing Eq. 4.43 with Eq. 4.41 shows the beam stiffness factors are exactly the same for both methods. If the composite lamination of aluminum and epoxy fiberglass is stacked in three layers like a sandwich, as shown in the cross-section view of Fig. 4.12, then the beam stiffness factor EZ can be calculated more easily. The same cross-sectional area for the aluminum and the epoxy fiberglass is used to keep the same weights. For the aluminurn section, (0.40)3] = 1.025 x 106 1 b . h . ' For the epoxy section,

The composite beam stiffness factor is

EI

= E,Za

EI

=

+ E,Z,

=

1.025 X l o 6

1.0383 x l o 6 Ib'in.'

+ 0.0133 X lo6 (4.44)

Use Eq. 4.7 to determine the natural frequency of the sandwich cross-section composite beam:

The composite beams shown in this section must be cemented together in order to eliminate relative motion at the interface between the aluminum and the epoxy. If bolted joints are used, relative motion will usually occur, which will reduce the stiffness and the natural frequency. It takes a large number of large bolts to prevent relative motion between two members at high frequencies and high G forces.

-

CHAPTER 5

Component Lead Wires as Bents, Frames, and Arcs

5.1

ELECTRONIC COMPONENTS MOUNTED ON CIRCUIT BOARDS

Electronic boxes are being required to occupy less space while providing more functions, so emphasis has been put on reducing the size of electronic component parts. The development of solid-state electronic parts, such as integrated circuits, sharply reduced the physical size of the parts and permitted more functions to be included in a smaller volume. Even small electronic component parts, however, must be mounted to provide the proper heat removal, accessibility, and structural integrity, depending on the environment. Because space is very limited in most aircraft, spacecraft, submarines, and even automobiles, electronic equipment must be supported by many different types of structures that can be adapted to the geometry of the system. Also, the physical size and shape of many electronic component parts themselves may permit them to be analyzed as structural members. Consider, for example, some typical electronic component parts such as integrated circuits, resistors, capacitors, and diodes mounted on printed circuit boards (PCBs) by their electrical lead wires. This is a common practice in the electronics industry because it permits low-cost production and easy maintenance. These component parts are generally soldered to the PCB by infrared, vapor phase, dip, and wave soldering, and even hand soldering. The PCBs are usually of the plug-in type, which are guided along the edges to permit easy connector engagement. Under these conditions, the edges of the PCB can usually be considered as simply supported. A typical installation might be like that shown in Fig. 5.1. During vibration in an axis perpendicular to the plane of the PCB, the acceleration forces will produce deflections in the circuit board. As the circuit board bends back and forth, bending stresses are developed in the electrical lead wires that fasten the electronic component parts to the circuit board (Fig. 5.2). Electronic component parts can be mounted on PCBs in many different ways. The ability of these components to survive a severe vibration environment will depend on many different factors such as component size, resonant 75

76

COMPONENT LEAD WIRES AS BENTS, FRAMES, AND ARCS Resistor

edge-guide

Printed-circuit board

FIGURE 5.1. Electronic component parts mounted on a printed circuit board.

-

coniporient

Electrical lead

Simply supported edge Circuit board bending

w

FIGURE 5.2. Bending in the component lead wires on a vibrating circuit board.

frequency of the circuit board, acceleration G forces, method of mounting components, type of strain relief in the electrical lead wires, location of the component, and duration of the vibration environment. Most component failures in a severe vibration environment will be due to cracked solder joints, cracked seals, or broken electrical lead wires. These failures are usually due to dynamic stresses that develop because of relative motion between the electronic component body, the electrical lead wires, and the PCB. This relative motion is generally most severe during resonant conditions that can develop in the electronic component part or in the PCB. Resonances may develop in the component part when the body of the component acts as the mass and the electrical lead wires act as the springs. These resonances are usually not too severe if the body of the component is in contact with the PCB, since this contact will sharply reduce the relative motion of the component. If resonances of this type do develop, it is an easy task to tie or cement the component part to the circuit board. If resonances develop in the circuit board, large displacements can force the electrical lead wires to bend back and forth as the circuit board vibrates up and down (Fig. 5.2). If the stress levels are high enough and if the number of fatigue cycles is great enough, then fatigue failures can be expected in the solder joints and the electrical lead wires.

BENT WITH A LATERAL LOAD-HINGED

ENDS

77

The most severe stress condition will be for a component part mounted at the center of a PCB. For a rectangular board, the most severe condition will occur when the body of the component part is parallel to the short side of the circuit board. This is due to the more rapid change of curvature for the short side of the board than for the long side, for the same displacement. Tying the electronic component part to the circuit board with lacing cord, at the center of the component body, will generally have very little effect on the relative motion between the component body and the circuit board when the circuit board is in resonance. Cementing the component body to the circuit board or tying both ends of the component part to the circuit board will greatly improve the fatigue life in the solder joints and in the electrical lead wires. Conformal coatings are quite often used on PCBs to protect the circuitry from moisture. Although most conformal coatings are quite thin, from about 0.0005 in. to about 0.010 in. thick, they will still act as a cement, so that they will also improve the fatigue life of the electronic component parts. PCB resonances can lead to high bending stresses in the lead wires on electronic component parts. Every time the circuit board bends, it forces the electrical lead wires to bend, since the wires are normally soldered to the circuit board. The greater the circuit board deflection, the greater the bending stresses in the lead wires. If the component body is cemented to the circuit board, the relative motion between the board and the component will be substantially reduced. Stresses in the component lead wires can be determined by examining the geometry of the electronic component part, the natural frequency of the circuit board, and the acceleration G forces. For example, considering the geometry, components such as resistors, capacitors, diodes, and flatpacks have small electrical lead wires compared to a large component body. Almost all of the relative deflection between the circuit board and the component will therefore occur in the electrical lead wires. If the body of the component is ignored and only the electrical lead wire deflections are examined, the resulting errors will be very small. The relative slope of the electrical lead wires, where they are soldered to the circuit board, will not change as the board vibrates up and down. The wires will always remain perpendicular to the board. Also, each electrical lead wire will always remain perpendicular to the component body where the lead joins the body, because the body is so much stiffer than the lead. Therefore, as the circuit board bends back and forth during vibration, the electrical lead wires will bend as shown in Fig. 5.2. If only the electrical lead wires are considered, they will approximate the shapes of the various frames and bents shown in the following sections. 5.2


ENDS

Consider a bent with hinged ends and a concentrated load acting in the lateral direction. The deflection can be broken up into two parts. The first

78

COMPONENT LEAD WIRES AS BENTS, FRAMES, AND ARCS

I"

Deflection due to bending of members AB and CD only

Deflection due to bending of member BC only

FIGURE 5.3. Bent with a lateral load and hinged ends.

FIGURE 5.4. Free-body force diagram of the vertical leg.

v

part considers bending of the vertical legs A B and CD only, with no bending in the horizontal leg BC. The second part considers bending of the horizontal leg BC only, with no bending in the vertical legs. When the horizontal leg bends, the angular displacement causes an additional displacement of the vertical leg through pure rotation (Fig. 5.3). The vertical leg A B is shown in Fig. 5.4. Considering forces in the horizontal direction,

Considering moments about points B ,


79

ENDS

FIGURE 5.5. Free-body force diagram of the horizontal leg.

Considering leg AB as a cantilevered beam with a concentrated end load,

a,=--H h 3

Ph3

--

3EI,

6EI,

(5.3)

The horizontal leg BC is shown in Fig. 5.5. The angular displacements are 0,

MBL 3E I ,

=-

and

MCL 02= 6E I ,

(5.4)

Substitute Eq. 5.4 into Eq. 5.5 and note that MB = M , due to symmetry:

Substituting Eq. 5.2 into Eq. 5.6,

The vertical leg will rotate through the angle 0, due to bending of the horizontal leg shown in Figs. 5.3 and 5.5:

The total deflection of the bent will then be the sum of 6, and 6,:

f3T =

6,

Ph3 Ph2L + 6,= + 6E12 12EI,

80


Let K

= hIl/L12.Then

Ph (5.9)

5.3 STRAIN ENERGY-BENT WITH HINGED ENDS Castigliano's strain energy theorem can be used to determine the loads, moments, and deflections in rectangular bents. This theorem states that the partial derivative of the strain energy, with respect to an applied force, will give the deflection produced by that force in the direction of the force. The total strain energy in a system can usually be accounted for by considering tension and compression, bending, torsion, and shear. Considering each one individually, they are as follows [13]:

Tension and compression:

Ut =

[g

(5.10)

Bending:

(5.11)

Torsion:

(5.12)

Shear:

q"" 2 GA

(5.13)

For example, the lateral displacement of a rectangular bent with hinged ends and a concentrated lateral load can be determined by considering only the strain energy of bending. Consider the rectangular bent shown in Fig. 5.6. The bending deflections are so much greater than all of the other deflections, such as tension and compression, torsion and shear, that only the bending need be considered unless the beams are very short (length less than about three times the depth). If stresses are within the elastic limit and deflections are small, there is very little error involved in ignoring the other deflection sources. The strain energy of bending must be considered for two vertical legs and one horizontal leg. Since the bent is symmetrical, only one-half of the system need be considered (Fig. 5.7).

STRAIN ENERGY-BENT

WITH HINGED ENDS

81

Y

Erin( I I

V

V

FIGURE 5.6. Bent with a lateral load and hinged ends.

1

FIGURE 5.7. Half of a bent with a lateral load and hinged ends.

V

The strain energy of bending was shown to be

lJb =

/te

2EI

Using Castigliano's theorem to find the deflection, we have

b2M(dM/dP)dX

-=

8P

[

2EI

1 =-

EI

["7 b

dM

dX

With the X and Y axes as shown, the bending moment at point vertical leg is

M ,= H Y

(5.14)

0 on the (5.15)

82


Considering all the forces in the X direction, P H=2

(5.16)

Substituting into Eq. 5.15 PY

M1 =

(5.17)

2

(5.18) With the X and Y axes as shown, the bending moment at point horizontal leg is

M 2 = HI.2 - I/x

@

on the

(5.19)

The force V can be determined by taking moments about point A in Fig. 5.6: Ph

=

VL

Ph v=-

L

(5.20)

Substituting Eqs. 5.16 and 5.20 into Eq. 5.19, (5.21) (5.22) Considering the strain energy in two vertical legs and two halves of the horizontal leg as shown in Fig. 5.7, the deflection equation 5.14 can be written as follows (note that E , = E? = E ) :

STRAIN ENERGY-BENT

WITH FIXED ENDS

83

Substituting Eqs. 5.17, 5.18, 5.21, and 5.22 into the above,

a=-+-2Ph3

12E12

2 P (h’u EI, 4

h2a2 I h2a3 2L

)

3L2

Since a =L/2

we have Ph3

Ph2L

a = - 6E12 +-1 2 E I , Let K = h I , / L 1 2 , and solve for 8: (5.23)

An examination of Eq. 5.23 shows it is exactly the same as Eq. 5.9, which was obtained by using superposition.

5.4

STRAIN ENERGY-BENT

WITH FIXED ENDS

If a rectangular bent has fixed ends, as in Fig. 5.8, it becomes statically indeterminate. Castigliano’s theorem can then be utilized by considering various physical characteristics of the structure. For example, it is obvious from Fig. 5.8 that the angular rotation at points A and D will be zero, because these ends are fixed. Also, since the structure is symmetrical, only one-half of the structure need be examined, as shown in Fig. 5.9. What may not be obvious from Fig. 5.9 is that the center of the horizontal leg BC becomes a point of inflection, at point E , so the moment ME must be zero. Also, point E will move only in the horizontal direction and not in the

84


FIGURE 5.8. Bent with a lateral load and fixed ends.

H

Y

h I

V

FIGURE 5.9. Half of a bent with a lateral load and fixed ends.

vertical direction. If the X and Y axes are taken as shown in Fig. 5.9. Castigliano’s theorem can be used by knowing that the vertical deflection at point E , due to the force V , must be equal to zero. This can be written as follows: (5.24) Since only bending deflections will be considered here and since there are and @ in Fig. 5.9, the bending moments to be considered at points deflection equation can be established using Eqs. 5.11 and 5.14:

0

The bending moment at point

0on the horizontal leg is M , = vx

(5.26)

STRAIN ENERGY-BENT

WITH FIXED ENDS

85

so JMI

-=x

(5.27)

av

@


on the vertical leg will be (5.28)

The horizontal force H can be determined by considering all of the forces in the X direction. This results in

P H=2

(5.29)


VL

M

, - 2

dM,

PY

(5.30)

2

_ -L

--

av

(5.31)

2

Substituting Eqs. 5.26, 5.27, 5.30, and 5.31 into Eq. 5.25,

1

VL3

EI,( 24

VL2h

VL2 611

-+-=-

pLhzI

)+g(T--

VLh I,

8

=O

Ph2 21,

Let K = hZ,/LZ, and solve for I/:

V=

3 PhK L(1+6K)

(5.32)

86


The moment M D can be determined from Fig. 5.9 by taking moments about point D :

CMD=O

(:i

M D + V - -Hh=O

(5.33)

Substituting Eqs. 5.32 and 5.29 into the above,

3 PhK MD'L(1+6K)(i)

Ph -0 2

With a little algebra this can be written as

(5.34) The deflection can be determined using the same method, making sure to include both vertical legs and both halves of the top horizontal leg:

The moment at point

0is &I1 = m =

3 PhK

L( 1

+ 6 K )X

(5.36)

so

aM,

3hK

aP

L(1+6K)

-=

The moment at point

@

(5.37)

X

is

VL

PY

3 PhK

PY

(5.38)

so

(5.39)

STRAIN ENERGY-BENT

WITH FIXED ENDS

87

Substituting Eqs. 5.36-5.39 into Eq. 5.35,

I(

3 PhKX

L(1+ 6 K )

EI,

+

E I l L 2 ( 1 6K)’

[ ’+

L(1+ 6 K )

0

s-

Y2

6hK

9h2K2Y

2E12 (1 + 6 K ) 2 - ( 1 + 6 K )

6=

IdX

[:IL’,

18Ph2K2

6=

3hKX

Y3

h

(T)+i], + 6 K ) + (1 + 6 K ) ’ 3( 1 + 6 K ) ,

27K2 - 9 K ( l

3 Ph2K ,L 4E11(1 + 6K)’

2EI,

Ph3 j 3 6 K 2 + 3 0 K + 4 / 24E12 3 6 K 2

+ 12K + 1


6=

-(

Ph

24E12 Ph

1+

+ i

( 6 K3+( 61K) ( 6 K + 1) 3

(5.40)

The previous bent with fixed ends could also have been analyzed by knowing that, for small angular displacements, the rotation angle on the vertical leg shown by 6,, will be the same as the rotation angle on the horizontal leg shown by e,, and indicated in the free-body force diagram of Fig. 5.10. A close inspection of the figure shows that the slope is positive at O B 2 , but negative at e,,; thus 6 7 2 = - OB1

(5.41)

Castigliano’s theorem can be applied here, since the partial derivative of the strain energy with respect to an applied moment will give the angular rotation produced by that moment in the direction of the moment. Since the strain energy was due only to bending, Eq. 5.11 is used again:

88


L

X

4

Y

FIGURE 5.10. Free-body force diagram of a bent with a lateral load.

Using Castigliano's theorem to find the angular rotation due to an applied moment,

In Fig. 5.10 the angular rotation due to moment M B acting at the top of the vertical leg will be approximately the same as the angular rotation due to moment M B acting on the horizontal leg. Then using Eqs. 5.41 and 5.42, this can be written as follows:

If the X and Y axes are taken as shown in Fig. 5.10, the bending moment at on the vertical leg is point

0

M 1 -- M

B

-fl=M

B

PY 2

--

(5.44)

so

(5.45)

STRAIN ENERGY-BENT


@

WITH FIXED ENDS

89

on the horizontal leg is

M2= -M,+Vx

(5.46)

The vertical force can be expressed in terms of M B by considering the sum of the moments about the left end, at point B , on the horizontal leg (note that from symmetry M , = M E ) : 2ME I/= -

(5.47)

L Substituting Eq. 5.47 into Eq. 5.46,

(5.48) (5.49) Substituting Eqs. 5.44, 5.45, 5.48, and 5.49 into Eq. 5.43,

MBh M E L -+-=I, 61,

Let K

= hI,/L12

Ph2

41,

and solve for M E : Ph - 4( 1 1 / 6 K )

M E

+


"(

M -_ 4

6Kl+I)

Substituting Eq. 5.47 into the above,

"(

I/=-

2L

1--

6K+1

(5.50)

90



V=

3 PhK L(6K

(5.51)

+ 1)

An examination of Eq. 5.51 shows it is exactly the same as Eq. 5.32.

5.5

STRAIN ENERGY-CIRCULAR

ARC WITH HINGED ENDS

Castigliano’s theorem is very convenient for determining forces, moments, and deflections in circular arcs. For example, consider the semicircular arc with hinged ends and a concentrated load (Fig. 5.11). Since the figure is symmetrical, it is possible to simplify the problem by considering only half of the system. The vertical deflection produced by the force P acting on the full system will then be the same as the vertical deflection produced by the force V acting on half of the system. The vertical force V can be determined by considering the sum of all the forces acting in the vertical direction. This leads to P v=-

(5.52)

2

The horizontal force H is not quite as easy to determine. An examination of the structure shows that the horizontal deflection at point B , due to force H , will be zero. Using Castigliano’s theorem, this can be written as follows:

a,,

IP

dU

-

JH

(5.53)

0

6 I

VI

FIGURE 5.11. Circular arc with hinged ends and a vertical load.


91

ARC WITH HINGED ENDS

When the angle 0 is between 0" and 90", the bending moment at point will be M,=VR(l-cosO)

-HRsine

0

(5.54)

so (5.55) The horizontal deflection at point B can be determined from Eqs. 5.11 and 5.53: (5.56) Substituting Eqs. 5.54 and 5.55 into the equation above, 1

S""=E/O [ VR( 1 - cos e ) - HR sin 0 ] ( -R sin 6 ) R d 0 = 0 Integrating the above equation and collecting terms,

Only the factor in parentheses can be zero, so this leads to 2v H=IT

(5.57)

The vertical deflection due to force V can be determined from the following equation: (5.58) From Eq. 5.54, dM1 dV

-= R ( 1 -cos 0 ) Substituting Eqs. 5.54 and 5.59 into Eq. 5.58, 1 6= EI

0

[ VR( 1 - cos e ) - HR sin e ] [ R( 1 - cos e ) ]R d e

(5.59)

92


Substituting Eq. 5.57 into the above equation and integrating,

(5.60) Substituting Eq. 5.52 into Eq. 5.60, 6=-

5.6

PR

'

(5.61)

52.6EI


ARC WITH FIXED ENDS

If the circular arc has fixed ends, it becomes statically indeterminate and requires more work to solve for the redundant forces. A semicircular arc with fixed ends and a concentrated load is shown in Fig. 5.12. Since the figure is symmetrical, the problem can be simplified by considering only half of the system. There are two unknowns, H and M,, so that two equations must be developed to find these forces. The force I/ is known, since an examination of all the forces in the vertical direction will show the following:

P

I/= -

(5.62)

2

The two physical properties that can be used to find the unknown forces are that the horizontal deflection at point C due to force H is zero, and that the relative slope at point C due to the moment M , is zero. The bending moment at point @ is

M , = H R ( l - c o s O ) +M,-VRsinO

forOsOsT/2

FIGURE 5.12. Circular arc with fixed ends and a vertical load.

(5.63)


ARC WITH FIXED ENDS

93

Thus (5.64) The horizontal deflection at point C can be written as

( 5 -65) Substitute Eqs. 5.63 and 5.64 into Eq. 5.65:

a,,

1 z/o

v/2

=

[ HR( 1 - cos e ) + M ,

- VR

sin

e ]( R ) ( 1 - COS e ) R d e = 0

Integrating the above equation and collecting terms,

i

- 0.35618 + 0.5707-MC R EI3

R

- OSOOV

(5.66)

This is the first equation that contains the two unknowns, H and M,. The second equation can be determined from the relative slope at point C, which can be written as follows: (5.67) From Eq. 5.63, (5.68) Substituting Eqs. 5.63 and 5.68 into Eq. 5.67, 1 O = ~ j g a i i [ H R ( 1 - c o s 8 ) +MC-VRsin8](1)RdB=O

Integrating the above equation and collecting terms, 0.5707H +

(5.69)

Only the factors in parentheses in Eqs. 5.66 and 5.69 can be zero, so these two equations can be used to solve for H and Mc. These two equations are

94


repeated below: Mc 0.3561H + 0.5707R 0.5707H

+ 1.5707-MC R

= 0.5OOV

=

V

Solving these two equations simultaneously results in the following:

H Mc

= 0.458P =

0.152 PR

(5.70) (5.71)

The vertical deflection due to force I/ can now be determined from the following equation:

=

r2

aM, Ml-Rd9

(5.72)

-Rsin 9

(5.73)

av

From Eq. 5.63,

aMl dV

-- -


1

6= EI

I

[ HR( 1 - cos 0 ) + Mc - VR sin 81 ( - R sin 8 ) R d 8

Integrating the above equation and collecting terms,

Substitute Eqs. 5.62, 5.70, and 5.71 into the above equation: PR3 PR' 6 = 0.0115 -= EI 87EI STRAIN ENERGY-CIRCULAR STRAIN RELIEF

5.7

(5.74)

ARCS FOR LEAD WIRE

Electrical lead wires are often subjected to high axial dynamic loads during vibration when the PCB natural frequency is excited. Many large surface mounted and through-hole mounted electronic components have several


ARCS FOR LEAD WIRE STRAIN RELIEF

95

straight leads that extend from the bottom surface of the component body down to the PCB. When the PCB bends during its resonant condition, the lead wires will be forced to deform as shown in Fig. 5.2. This type of installation often leads to very high axial forces, high stresses, and a reduced fatigue life in the solder joints and the lead wires at the ends of a long component, where the relative displacements are the greatest. These failures can usually be avoided by adding a strain relief in the wire in the form of an offset. When the offset is properly designed, the axial load in the wire is changed to a bending load in the wire. Axial loaded wires usually have very high spring rates and early fatigue failures. Wires that are loaded in bending will typically have much lower spring rates, so they produce lower forces, resulting in an increased fatigue life. In a linear system the force P is related to the spring rate K and the deflection Y as shown below:

P=KY

(5.75)

The displacement Y is often a constant for given sets of conditions in vibration, shock, and thermal cycling environments. An examination of the above equation shows that when the displacement Y is constant, the best way to reduce the force P in the system is to reduce the spring rate K . This can usually be accomplished by adding an offset in the straight lead wire. Offsets in the form of circular arcs as shown in Fig. 5.13 work very well. The displacement and the resulting spring rate of the offset due to an axial load can be obtained with the use of Castigliano’s strain-energy methods as shown below.

FIGURE 5.13. Half of the geometry proposed for a component lead wire strain relief.

96


The bending moment at point M,

0is = PR( 1 - COS

e,)

(5.76) (5.77)


0 is

M 2 = PR( 1 + sin

el)

(5.78)

aM2 -=R(l+sin@,) aP

(5.79)

The deflection Y can be obtained from the strain energy of the bending in the system. ( 53 0 )

Substitute Eqs. 5.76-5.79 into Eq. 5.81: (1-cos8,)’de1+/

T I 2R 3 ( 1 + s i n e , ) * d e 2

n y=

PR

[ivi2(l -2

PR3

EI

~ 0 8, s

8, - 2sin 8,

1

+ cos2 0 , ) de, + /“12(1 + 2sin el + sin2 8,) d e , n

1

91 1 ++ -sin28, 2 4

PR3

-(O)-(-2)]=

4

4

4.71PR3 EI

The above displacement value is for only one-half of the circular arc offset. The other half of the offset is symmetrical, and it is in series, so the final displacement will be two times greater:

Y=

9.42 PR3

EI

(5.82)


Sample Problem-Adding the Fatigue Life


97

an Offset in a Wire to Increase

A 1.2-in. long application-specific integrated circuit (ASIC) is through-hole mounted at the center of a plug-in PCB 8.0 in. long and 0.080 in. thick, as shown in Fig. 5.14. The component has several kovar wires, 0.012 in. in diameter, that extend from the bottom surface down to the PCB. The PCB has a natural frequency of 190 Hz. It must pass a 30-minute resonant dwell safety of flight test using a sinusoidal vibration input level of 3.4 G peak applied perpendicular to the plane of the PCB. Find the dynamic forces, stresses, and fatigue life in the end lead wires and solder joints. Add an offset strain relief in the wire as shown in Fig. 5.15 and recalculate the dynamic forces, stresses, and fatigue life in the end lead wires and solder joints.

kL+

0.6-

0.080 l 1 I I l l l I I l I l

t L = 8.0

1

t i c

FIGURE 5.14. Relative motion between the vibrating circuit board and the component produces strain in the lead wires.

R

i

FIGURE 5.15. Full geometry proposed for a component lead wire strain relief.

98


Solution Use Eqs. 2.33 and 4.1 to Find Relative Displacement Difference 6 Between the P C B and the Component as Shown in Fig. 5.14. The displacement shape of the PCB at its natural frequency will be a half sine wave. The maximum displacement Z , will be at the center. The displacement Z , at any point on the curve will be as shown below: T X

2,=&sin-

so

L

S = Z , 1 -sin-i

(5.83)

L

where G = 3.4 (peak input acceleration level) Q = V‘i!% = 13.78 (approximate transmissibility expected for PCB) f , = 190 Hz (PCB natural frequency) Z,

=

9.8Gi,Q/(f,,)’

=

(9.8)(3.4)(13.78)/(190)’

=

0.0127 in. (5.84)

x = 3.4 in. (location of edge of component) L

= 8.0

in. (length of PCB parallel to component)

(

6 = (0.0127) 1 - sin-

T(3.4) 8.0

i

= 3.51 X

in. (5.85)

The above expression represents the relative displacement between the component and the PCB, which represents the stretch in the lead wire. This conservatively assumes that the PCB deflects in a perfect sine wave, there is no bending in the component body, and the lead wire is loaded in perfect tension only. These conditions are only approximations, so they will result in values that are slightly greater than the true values. A stress concentration is not used in the calculation of the wire axial stress, because a stress concen-

tration of 2.0 will be used in the fatigue exponent b when the fatigue life of the wire is calculated in Eq. 5.90. The spring rate K of a straight wire that is loaded in pure tension can be obtained from a handbook, as shown below:

( 52 6 ) where d, A E L,

= 0.012

in. (wire diameter) in.’ (wire area) = .rr(0.012)’/4 = 1.131 X = 20 x 106 lb/in.’ (kovar wire modulus of elasticity) = 0.068 4(d,) = 0.068 4(0.012) = 0.116 in. (effective length of wire for axial load goes into component 2 diameters and into PCB 2 diameters) =

n-(d,,)’/4

+

+

K,

=

(1.131 x 10-4)(20 x 106) 0.116

QQ



=

1.950 x lo4 lb/in.

(spring rate) (5.87)

The dynamic force in the straight wire P , can be obtained from the wire spring rate K , and the relative displacement S from Eq. 5.85 as shown below:

P,=K,S=(1.950x104)(3.51X10-4)

=6.84lb

(5 .88)

The dynamic axial stress in the wire can be obtained from the standard stress equation, the dynamic load, and the wire area shown in Eq. 5.86: 6.84

p, w- A

s

--=

1.131 x

=

60,477 lb/in

.*

(wire axial stress) (5.89)

The approximate fatigue life of the kovar wire can be obtained from Fig. 3.1 with the general fatigue equation as shown below. Note that the fatigue exponent 6.4 has a stress concentration of 2.0 included to account for any possible deep cuts or scratches due to careless handling or manufacturing problems. N,

= N2[;]’

=

(1000)

84,000

6.4

=

8189 cycles to fail

(5.90)

The approximate time for the wire to fail can be obtained from the 190-Hz PCB natural frequency as shown below: Wire fatigue life

8189 cycles to fail =

(190 cycles/s) (60 s/min)

=

0.718 minute to fail (5.91)

The qualification test requires a 30-minute dwell. The rapid wire fatigue failure shows the proposed design is not acceptable. The dynamic shear stress in the PCB wire solder joint can be obtained from the force in the wire and the average shear area of the solder joint as shown below:

ss= where P , d, d,,

p, A,

(5.92)

lb (see Eq. 5.88) in. (wire diameter) = 0.024 in. (expected diameter of the plated through-hole in the PCB) = 6.84

= 0.012

100


d,, t A,

= (0.012

+ 0.024)/2

0.018 in. (average solder joint diameter) = 0.080 in. (PCB thickness) = rd,,,t = r(0.018)(0.080) = 0.00452 i n 2 (average solder shear area) (5.93)

&=--

6.84

=

-

1513 lb/in.*

(solder shear stress) (5.94)

0.00452

The approximate vibration fatigue life of the solder joint can be obtained from Fig. 3.2 with the general fatigue equation as shown below:

The approximate time for the solder joint to fail can be obtained from the 190-Hz PCB natural frequency as shown below: Solder fatigue life

3.41 X 10' cycles to fail =


= 29.9

minutes to fail (5.96)

The solder joint fatigue life is better than the lead wire fatigue life shown in Eq. 5.91, but the solder joint will not meet the 30-minute resonant dwell requirement with a safety factor. Now add an offset strain relief in the lead wire as shown in Fig. 5.15 and recalculate the dynamic forces, stresses, and fatigue life in the lead wires and solder joints. The new spring rate of the wire with the strain relief can be obtained from Eq. 5.82 as shown below: (5.97) where E = 20 x 106 lb/in.* (kovar wire modulus of elasticity) d , = 0.012 in. (wire diameter) I , = n-(d,I4/64 = .rr(O.O12)'/64 = 1.018 X i n 4 (wire moment of inertia) Y = 0.017 in. (radius of the wire offset strain relief) K=

(20 X 106)(1.018 X (9.42) (0.017)

= 440

lb/in.

(5.98)

The dynamic force in the wire can be obtained from the relative displacement shown in Eq. 5.85 and the above spring rate. P,,=K13=(440)(3.51xlO-~) =0.154lb

(5.99)

101



Although there is an axial load in the wire, the strain relief offset will produce a bending moment and a bending stress in the wire. The offset distance will be equal to 2 radii.

M = 2rP,

=

(2)(0.017)(0.154)

= 0.00523

1b.in.

(5.100)

The bending stress equation in the wire can be obtained from a structural handbook as shown below. A stress concentration factor is not used here because it will be used in the fatigue exponent b when the fatigue life of the wire is calculated in Eq. 5.103. (5.101) where C = d,/2 = 0.012/2 = 0.006 in. (wire radius) M = 0.00523 lb-in. (see Eq. 5.100) I , = dd,I4/64 = ~ ( 0 . 0 1 2 ) ~ / 6=4 1.018 X inertia) (0.00523)(0.006) s b =

1.018 X

= 30,825

lb/in.2

in.4 (wire moment of

(wire bending stress) (5.102)

The approximate fatigue life of the kovar wire can be obtained using Fig. 3.1 with the general fatigue equation shown below. A stress concentration factor of 2.0 was included in the 6.4 fatigue exponent to account for any deep cuts or scratches that may be caused by the wire-forming dies.

Nl = N 2 [ $ I b = (lOOO)(w] 84,000

=6.115 x 10’cycles to fail (5.103)

The approximate time for the wire to fail can be obtained from the 190-Hz PCB natural frequency as shown below: Wire fatigue life

6.115 x 10’ cycles to fail =


= 53.6

minutes to fail (5.104)

The system requires a qualification fatigue life of 30 minutes. The lead wire has a safety factor of about 1.8, which should be adequate since the analysis method was conservative. The shear stress in the solder joint can be obtained using Eq. 5.92, the average solder shear area from Eq. 5.93, and the wire force from Eq. 5.99, as shown below:

s

P, As

--=-

0.154 0.00452

= 34

lb/in

.’

(solder shear stress)

(5.105)

102


The approximate vibration fatigue life of the solder joint can be obtained from Fig. 3.2 and the general fatigue equation as shown in Eq. 5.95. 1012 cycles to fail (5.106) The approximate time for the solder joint to fail can be obtained from the 190-Hz PCB natural frequency as shown below: Solder fatigue life

1.33 x lo1' cycles to fail =

(190 cycles/s) (3600 s/h)

=

1.94 x 106 hours to fail (5.107)

The solder joint in this case is very safe. It is much less critical than the lead wire so the lead wire is the weaker element in this system.

-

CHAPTER 6

Printed Circuit Boards and Flat Plates

6.1

VARIOUS TYPES OF PRINTED CIRCUIT BOARDS

Electronic systems make extensive use of plug-in printed circuit boards (PCBs) because they are very easy to service. Defective circuit boards can be removed and replaced quickly and easily without bothering with wires and a soldering iron. After a defective circuit board has been removed, troubleshooting is easy and repairs can be made by a skilled technician. Many different types of PCBs are manufactured by the electronics industry. Epoxy fiberglass is the most common material used, with laminated layers of copper on one or both sides of the board to form the electrical conductors. The overall PCB thickness can vary from about 0.006 to 0.125 in. Board sizes can vary from about 2 to 16 in. Many different shapes can be found, ranging from small squares to large circular plates and triangles, depending on the shape of the electronic box used to support the circuit boards. Since electronic equipment is being packed into every available inch of space in most airplanes, missiles, and even television sets, the shape of the circuit board is often dictated by the geometry of the available space. The rectangular PCB is the most common shape used by the electronics industry, since this shape is easily adapted to the popular modular plug-in type of assembly, which utilizes an electrical connector along the bottom edge of the circuit board. PCBs with many high-power-dissipating components will run very hot unless the heat is removed. Therefore aluminum and copper, which have high thermal conductivities, are often bonded to the epoxy fiberglass circuit boards to act as heat sinks. Ribs are often added to PCBs that must operate in a severe vibration and shock environment. Ribs increase the stiffness of the circuit board, which in turn will increase the resonant frequency. This will reduce the board deflections during resonant conditions, thus reducing the stresses developed in the electronic component parts mounted on the circuit board. Ribs can be fabricated of steel, aluminum, or epoxy fiberglass. If metal ribs are used, caution must be exercised to prevent short circuits across exposed electrical 103

104

PRINTED CIRCUIT BOARDS AND FLAT PLATES

printed circuit strips. Ribs can be bolted, riveted, soldered, cemented, welded, or cast integral with heat-sink plates. PCBs may be supported by the electronic box in many different ways, depending on such factors as the environment, weight, maintainability, accessibility, and cost. For example, in a vacuum environment where there is no air to provide convection heat transfer, heat will often be conducted from the circuit board to a heat exchanger. High-pressure thermal interfaces must then be provided, using materials that have a high thermal conductivity, in order to prevent excessive temperatures from developing in the electronic components mounted on the circuit boards. Screws are ideal for providing high-pressure interfaces, but they are generally time-consuming to install, since they take many turns to insert and remove. Maintenance people prefer simply to unplug one module and plug in another. This reduces maintenance time substantially, since electronic systems can easily contain over a hundred plug-in PCBs. Sometimes a fast-acting cam-and-wedge type of clamp is used to provide a high-pressure interface, if there is enough room available in the electronic box. The manner in which the PCBs are supported in the electronic box can be an important factor in determining just how the boards will respond to vibration and shock. If the electronic box is fabricated from light sheet metal and manufacturing tolerances are loose, a loose fit is desirable for the plug-in circuit boards. This will permit easy connector engagement, with a fixed connector position, when there is a substantial mismatch in the mating box connector. This type of installation is not desirable, however, for a system that will be subjected to severe vibration and shock. A loose circuit board will often develop high acceleration loads, which will lead to high deflections and stresses in the electronic component parts mounted on the circuit boards. The edges of the PCBs should be supported if they will be subjected to severe vibration or shock environments. Board edge guides are available that will support a wide variety of PCBs. These guides are usually fabricated of beryllium copper, but they are also available in a wide range of metals and molded plastics. A board edge guide that grips the edge of a PCB firmly is very desirable. This firm grip can reduce deflections due to the edge rotation and translation, which will increase the natural frequency of the circuit board. In addition, a firm grip will tend to dissipate more energy during vibration because of friction and relative motion between the edges of the board and the edge guide. This, in turn, will reduce the transmissibility experienced by the printed circuit board during resonance. A tight board edge guide may require tight manufacturing tolerances, plus a floating chassis connector, to permit accurate connector alignment during the circuit board installation, to prevent connector damage. The transmissibility developed by a PCB during resonance will depend on many factors, such as the board material, number and type of laminations in a multilayer board, natural frequency, type of mounting, type of electronic component parts mounted on the circuit board, acceleration G levels, type of

VARIOUS TYPES OF PRINTED CIRCUIT BOARDS

105

Side guide

Circuit board laminated to

4.0 Electronic component parts

I I I l l

1 1 I I I I ~ I 1 I1 1 1 1 1 I

\Connector

FIGURE 6.1. Laminated circuit board with a stiffening frame.

conformal coating, type of connector, and shape of the board. Slight modifications in the installation geometry of the circuit board can have a sharp effect on the transmissibilities and the mode shapes developed at resonance. This can be demonstrated by changing the edge restraints of a PCB from approximately simply supported to approximately fixed. Vibration test data on one group of boards, which used Birtcher guides on two opposite sides of the board to simulate a simply supported condition, showed natural frequencies of about 260 Hz. The transmissibility was about 16 for a 5-G peak sinusoidal vibration input. This is about equal to the square root of the natural frequency. The opposite sides of the circuit boards were then clamped to simulate a fixed-edge condition. This increased the typical natural frequency to about 380 Hz and increased the typical transmissibility to about 25. This is equal to 1.28 times the square root of the typical natural frequency. The group of PCBs used in these tests had the dimensions and construction shown in Fig. 6.1. These tests were run in a rigid vibration test fixture that included a mating connector for each circuit board. Accelerometers were mounted at several points on the vibration fixture and each circuit board to monitor the input and output acceleration G forces for the system (Fig. 6.2). These same PCBs were then installed in an electronic box, and the 5-G peak sinusoidal vibration tests were repeated. The circuit boards with Birtcher guides showed a typical natural frequency of about 225 Hz with a typical transmissibility of about 14. This is equal to 0.93 times the square root of the natural frequency. The circuit boards then had their opposite sides clamped to the chassis structure to simulate a fixed-edge condition. The natural frequency increased to about 340 Hz, and the transmissibility, from the chassis to the center of the circuit board, increased to about 20. This is equal to 1.08 times the square root of the natural frequency.

106


accelerometers

Birtcher guide for a supported edge

Vibration input direction Rigid vibration test fixture

I

Vibration shaker head

FIGURE 6.2. PCB mounted in a rigid vibration fixture.

6.2

CHANGES IN CIRCUIT BOARD EDGE CONDITIONS

In order to calculate the natural frequency of a PCB, it is necessary to estimate the edge conditions of the board. Much insight can be obtained into edge conditions by using a strobe light to watch board motions during vibration. This shows very quickly whether the edges are translating or rotating. Sometimes edge conditions can change when the acceleration forces change. An edge that appears to be simply supported with a 2-G peak input may appear to be free with a 5-G peak input. The natural frequency with a 2-G input will then be different from the natural frequency with a 5-G input. Sinusoidal vibration tests were run on a group of PCBs that had the opposite sides supported by a preloaded beryllium-copper wavy spring, which acted as a board edge guide, in a channel section. The bottom edge of each board had a 100-pin connector that plugged into the sockets of a mating connector mounted in a rigid vibration fixture. The top edge of each board was restrained by three foam-rubber strips 1 in. by 0.125 in. thick and compressed about 60% when the board was installed. These rubber strips were used to restrain the unsupported top edge of the circuit board, to simulate the action of a cover using the same type of rubber strips. Strips of this type are convenient to use because they provide some support and damping without the high fabrication costs associated with close tolerances on machined parts.A sketch of this type of installation is shown in Fig. 6.3. Vibration tests were run with peak inputs of 2, 5 , 10, and 15 G using a strobe light to observe the action of the circuit board edges during resonant conditions. The 2-G tests showed that all four edges of the circuit board appeared to be simply supported. The edges were observed to be rotating without translation. The 5-G tests showed that the sides of the circuit boards acted as though they were supported, but the top edge at the rubber pads and the bottom edge at the connector showed some translation in addition to the rotation. The 10-G tests still showed that the sides of the circuit boards acted as though they were simply supported, but the top edges at the rubber strips and the bottom edges at the connectors all experienced much more

CHANGES IN CIRCUIT BOARD EDGE CONDITIONS

107

/ Side channel guide

Wavy spring

Section AA

/ l I ' l l I l ~ l ~ / ~ l l ~ l l l ~ I ~ ~ 100-pin connector

FIGURE 6.3. Typical installation of a PCB.

-*

8 0 +----

-F-

-c

0 3 8 rib ai u in i n urn

1

i 0 12 rib

NO 2-56

70 I I

! I 1

"1

I

1.1 I 1

I I 1 I I I 1 I I 1 1 I I I I 1

150-pin connector

k

fiberglass board 0 062 in

- .

FIGURE 6.4. PCB with ribs fastened by screws.

translation than rotation. The 15-G tests showed some translation at the sides in addition to the rotation, while the top and bottom edges appeared to be free because there was very little rotation, only large translation amplitudes. Resonant dwell tests at a 10-G peak resulted in many broken connector pins. Stiffening ribs were added to some of the circuit boards to increase their resonant frequencies. Some of these ribs were screwed to the faces of the circuit boards (Fig. 6.4).

108


TABLE 6.1 Input G Level

Resonant Frequency

Transmissibility

(Hz)

Q

2

215

15.0

5 10

182

11.2 8.2

161

Vibration tests were run on these boards using the same type of foamrubber strips at the top of each circuit board and the same type of wavy-spring channel guides at the sides of each board. When the input G levels were changed, there were substantial changes in the natural frequencies and transmissibilities (Table 6.1).

6.3 ESTIMATING THE TRANSMISSIBILITY OF A PRINTED CIRCUIT BOARD

Printed circuit boards can have a wide variety of sizes and shapes along with many different mounting arrangements. In the early stages of a design, these circuit boards must be analyzed to make sure they will function properly in the required environment. One critical part of the analysis relates to the dynamic loads developed in these circuit boards at their fundamental resonant frequencies. These loads are closely associated with the transmissibilities, developed by the circuit boards, but transmissibilities are very difficult to estimate without previous test data on similar boards. Obviously, test data are the best sources for information on the transmissibility characteristics for various types of circuit boards. However. there are times when test data are not available because of a new design or a geometry that has not been used before. Under these conditions it becomes necessary to estimate the response characteristics of the circuit board, at least until a model circuit board can be fabricated and tested. There are a number of factors that should be considered when the transmissibility of a PCB must be estimated without the use of test data. These factors are all related to the damping characteristics of the board, which determine the amount of energy lost during the vibrating condition. When more energy is lost, or transformed to heat, there is less energy remaining and the transmissibility is lower; hence the dynamic loads and stresses are also lower. The greatest energy losses are probably due to hysteresis and friction. Hysteresis losses are generally due to internal strains that are developed during bending deflections in the circuit board. Friction losses are generally due to relative motion between high-pressure interfaces such as mounting

ESTIMATING THE TRANSMISSIBILITY OF A PRINTED CIRCUIT BOARD

109

surfaces, stiffening ribs, and edge guides. These energy losses are greatest when the deflections are greatest and smallest when the deflections are smallest. Since higher frequencies have smaller deflections, they will also have less damping. This means higher frequencies will usually have higher transmissibilities at resonant conditions. Terms such as “low” resonant frequency and “high” resonant frequency are only relative. In general, however, the term “low” applies to resonant frequencies below about 100 Hz. The term “high” applies to resonant frequencies above about 400 Hz. Most circuit boards appear to have their fundamental resonant frequencies between about 200 and 300 Hz when used in military electronic systems. Test data show the transmissibility of a PCB can generally be related to the square root of the natural frequency of the board. For rectangular boards, the transmissibility will normally range from about 0.50 to 2.0 times the square root of the natural frequency, depending on many factors. A small circuit board with small electronic component parts and no stiffening ribs, with a resonant frequency of about 400 Hz, can have a transmissibility as high as 2.0 times the square root of the natural frequency for a low input G force below a 2-G peak. The transmissibility for this case will then be about 40. A large circuit board with large electronic component parts and several stiffening ribs, with a resonant frequency of about 100 Hz, can have a transmissibility as low as 0.50 times the square root of the natural frequency for a high input G force above a 10-G peak. The transmissibility for this case will then be about 5. Over the middle frequency ranges (200-300 Hz), the transmissibility will often be about equal to the square root of the natural frequency for a 5-G peak sinusoidal vibration input. Again, there are many other factors that will influence the transmissibility of a PCB. If there are no test data available on the particular type of circuit board being analyzed, then the approximations just outlined are suggested as a good starting point. Some of the factors that should be considered when the transmissibility of a circuit board is being estimated are as follows: 1. The Natural Frequency of the Circuit Board. A high natural frequency means low displacements and low strains, so the transmissibilities are usually higher. Conversely, a low natural frequency means high displacements and high strains, so the transmissibilities are usually lower. 2. The Input G Force for Sinusoidal Vibration. A lower input G force means low displacements and low strains, so the transmissibilities are usually higher. A high input G force means high displacements and high strains, so the transmissibilities are usually lower. 3. Ribs. Riveted and bolted ribs will generally permit some relative motion to occur at the high-pressure interface between the ribs and the circuit board, which will dissipate energy and reduce the transmissibility at resonance.

110


Welded, cast, and cemented ribs are usually stiffer than riveted and bolted ribs, so that they raise the resonant frequency more, but they do not provide as much damping. 4. Circuit Board Supports. Circuit board edge guides that grip the edges of the board firmly provide a high-pressure interface that will dissipate energy and lower the transmissibility. If mounting bolts are used to fasten the boards, high-pressure interfaces in the bolt areas will tend to dissipate energy. More mounting bolts will dissipate more energy and increase the stiffness at the same time. 5. Circuit Board Connectors. Circuit board connectors, such as the edge type or the pin-and-socket type, will provide some damping that will tend to reduce the transmissibility. A longer connector will provide more support to the circuit board edge while providing additional damping. 6. Type of Electronic Component Part. Large electronic component parts that are in intimate contact with the circuit board will tend to dissipate more energy than small electronic component parts. This is because larger components cover a bigger span on the circuit board, resulting in larger relative deflections at the circuit board interface over a larger area, so the damping is slightly greater. 7. Heat-Sink Strips. Heat-sink strips are often laminated to the circuit board to conduct the heat away from the electronic component parts. These strips will act as laminations that will dissipate energy through relative interface shear deflections. At high frequencies, where the displacements are small, the damping is small, so transmissibilities are not reduced very much. 8. Multilayer Circuit Boards. Many PCBs must use extra layers to provide all of the electrical interconnections required by the circuits. These extra layers will increase the damping slightly and thus tend to reduce the transmissibility. At high frequencies, where deflections are small, transmissibilities are not reduced very much. 9. Conformal Coatings. Conformal coatings are generally used to protect electronic components from dust accumulation. Many manufacturers of electronic equipment also use conformal coatings to protect sensitive circuits from water vapor and condensation. A thin polyurethane or epoxy film about 0.005 in. thick is usually sprayed or dipped on the circuit board. Unless the adhesion is good and unless all of the sharp points are coated, water vapor can wick in between the coating and the circuit board and condense. Once this happens, the moisture will usually stay where it is unless it is baked out at high temperatures. Small amounts of moisture will not have much effect on the electrical operating characteristics of the circuits unless the circuits have a high impedance. Conformal coatings also act as a cement, which provides good adhesion for electronic component parts to the circuit board. During vibration, relative motion between the components and the circuit board will produce strains in the conformal coating and will tend to provide some damping.

NATURAL FREQUENCY USING A TRIGONOMETRIC SERIES

11 1

z FIGURE 6.5. A flat rectangular plate supported on four sides.

6.4 NATURAL FREQUENCY USING A TRIGONOMETRIC SERIES Most PCBs can be approximated as flat rectangular plates with different edge conditions and different loading conditions. General plate equations can then be used to determine the strain energy and the kinetic energy of the vibrating plate, which leads to the natural frequency equation. One very convenient method for analyzing plates is the Rayleigh method [28]. A deflection curve is assumed that satisfies the geometric boundary conditions, which are the deflection and the slope for a particular plate. Once these boundary conditions are satisfied, the assumed deflection curve is used to obtain the strain energy and the kinetic energy of the particular plate. If there is no energy dissipated, the strain energy will be equal to the kinetic energy and the approximate natural frequency can be determined [32]. The Rayleigh method results in a natural frequency that is slightly higher than the true natural frequency for a given set of conditions, unless the exact deflection curve is used. Consider a flat, rectangular plate, with four simply supported edges and a uniformly distributed load, being vibrated in a direction perpendicular to the plane of the plate (Fig. 6.5). The deflection curve for the simply supported plate can be represented by a double trigonometric series [341: I Z mn-X nrY Z= A,,,,,sin-sina b m= n= 1.3:5. 1,3:5,

Extensive vibration test data on PCBs show that most of the damage occurs at the fundamental resonant mode where the displacements and the stresses are the greatest. The above equation can then be simplified to the

112


following expression:

Z

=Z,

ITX %-Y sin -sin a b

The assumed deflection curve must be checked to make sure it meets the geometric deflection boundary conditions. This means the edges of the plate must have a zero deflection and the center of the plate must have the maximum deflection. Then from Eq. 6.1, at X = O atX=O atX=a atX=a a at X = 2

and y = O , andY=b, andY=O, andY=b, b andY=-, 2

Z=O Z=O Z=O Z=O Z=Z,

The deflection boundary conditions are satisfied: there are no deflections at the edges and the maximum deflection is at the center. Next check the geometric boundary conditions for the slope of the plate at different points. There must be a finite slope at all four edges, but the center of the plate must have a zero slope. Considering first the slope along the X axis, the required equation can be determined from the partial derivative of Z with respect to X. Then from Eq. 6.1, az 7i 7ix 7iY 8 - - = Z,-cos-sin-ax a a b so

at X = a

b and Y = -, 2

7T

O x = -Z,-

a

The slope boundary conditions are satisfied at the edges and at the center of the plate along the X axis. The same check can be made for the slope along the Y axis. The total strain energy I/ of the vibrating plate can be represented in the following form [41:

NATURAL FREQUENCY USING A TRIGONOMETRIC

SERIES

113

where D = Eh3/12(l - p’) (plate stiffness factor) E = modulus of elasticity (lb/in.2) h = plate thickness (in.) p = Poisson’s ratio (dimensionless) The total kinetic energy T of the vibrating plate can be represented in the following form [41:

T = c / u / b Z 2dXdY 2 0 0 where p = W/abg = u h / g (mass per unit area) W = total weight of plate (lb) u = material density (lb/in.3) a = length of plate (in.) b =width of plate (in.) h = plate thickness (in.) g = acceleration of gravity (386 in./s2) a=circular frequency (rad/s) Performing the operations on Eq. 6.1 required by Eq. 6.2,

aZ

%-x

n

- = 2,-cos-sina a

7TY

ax

a22

--

n2

-

ax2

a2z 2

b 7 T x

7TY

a

b

-2 -sin-sin-

‘a2

nY

(z =ziTsin’-sin2] nX n4

b

a

az

n

nx

nY

3Y

b

a

b

- = 2,-sin-cos-

a2z

-- -

dY2

n2

-Z,,sin-sinb

nX a

~

nY b

114


From Eqs. 6.5 and 6.8,

(6.10)

From Eq. 6.4 or 6.7,

a22

T2

ax dY

= z,-cos-cos-

nx

77-Y

a

b

ab

[SY)n-x 2

7i-4

=Z~a"COS--COS

7

a

,n-Y b

(6.11)

Since these equations must be integrated, the following relations are used:

(6.12)

Following Eq. 6.2 and integrating Eq. 6.6 using Eq. 6.12,

(6.13)


(6.14)


NATURAL FREQUENCY USING A TRIGONOMETRIC SERIES

115


(6.16)

Substituting Eqs. 6.13-6.16 into Eq. 6.2 for the strain energy of the vibrating plate,

V=

2

(6.17)

8

The kinetic energy of the vibrating plate can be determined from the deflection (Eq. 6.1) along with the kinetic energy (Eq. 6.3): (6.18) Substituting Eq. 6.12 into Eq. 6.18 for the integration, we have

( i b Z 2 dXdY

ab

=Zi-

4

(6.19)

Substituting Eq. 6.19 into Eq. 6.3 for the kinetic energy of the vibrating plate, we have (6.20) Since the strain energy of the vibrating plate must equal the kinetic energy at resonance, if there is no energy dissipated, then Eq. 6.17 must be equal to Eq. 6.20:

P

116


Solving for the natural frequency of the rectangular plate, we have

(6.21)

In this particular case the natural frequency equation shown above is exact using the Rayleigh method, because the deflection curve used for the vibrating plate is exact for the boundary conditions. 6.5

NATURAL FREQUENCY USING A POLYNOMIAL SERIES

The natural frequency equation for a uniform, flat, rectangular plate simply supported on four edges can also be derived with the use of a polynomial series. Consider the coordinate system for the rectangular plate when the X and Y axes are taken at the center of the plate (Fig. 6.6). The deflection curve for the simply supported plate can be represented by a double polynomial series as follows [30]: -x

-x

Z = h C m=O n=O

[

1 - 7

) ( &' ) [ ;) ( ;) I--

'l*

A,,,, cos R t

Since most of the damage on a PCB will occur in the fundamental resonant mode, where the displacements and the stresses are the greatest, the above

I

Supported

I

z FIGURE 6.6. A simply supported plate with the axes at the center.


117

equation can be simplified to the following: (6.22) The assumed deflection curve must be checked to make sure it meets the geometric deflection boundary conditions. This means the edges of the plate must have a zero deflection and the center of the plate must have the maximum deflection. Then from Eq. 6.22, at X = O a n d Y = O ,

Z=Z,

at X = O a n d Y = d ,

Z=O

at X = c a n d Y = d ,

Z=O

at X = c a n d Y = O ,

Z=O

The deflection boundary conditions are satisfied, since there are no deflections at the edges and the maximum deflection is at the center. Next check the geometric boundary conditions for the slope of the plate at different points. There must be a finite slope at all our edges, but the center of the plate must have a zero slope. Considering first the slope along the X axis, the required equation can be determined from the partial derivative of Z with respect to X. Then from Eq. 6.22,

and at X=O

andY=O,

8,=0

at X=O

andY=d,

O,=O

The slope boundary conditions are satisfied at the edges and at the center of the plate along the X axis. The same check can be made for the slope along the Y axis. The strain energy equation (Eq. 6.2) and the kinetic energy equation (Eq. 6.3) can also be used with the polynomial equation (Eq. 6.22). Perform-

118


ing the operations on Eq. 6.22 as required by Eq. 6.2 for the strain energy,

( 6.23)

(6.24)

(6.25)

(6.26)

(6.27)

(6.28) From Eqs. 6.24 and 6.27,

From Eq. 6.23 or 6.26, d2Z

2x 2Y

ax aY =z,---;c 2 d' (6.30) When the integration is performed according to Eq. 6.2 for the strain energy, observe that the X and Y reference axes for this geometry are not at the edges of the plate; they are at the center of the plate. Therefore, to consider the full plate, we use Z ) dXdY = 4

jc jdf( Z ) dXdY 0 0

(6.31)


119

From Eqs. 6.2, 6.25, and 6.31, 2

4l[[$]

16Zicd d X d Y c= r

From Eqs. 6.2, 6.28, and 6.31, 2

46/i$) From Eqs. 6.2, 6.29, and 6.31,

From Eqs. 6.2, 6.30, and 6.31,

J2Z

2

-(-I(

64Zicd 1 4 / i / d ( ~ y ) dXdY= c2d2 3 0

0

'j

(6.35)

3

The strain energy of the plate can be obtained by substituting Eqs. 6.32-6.35 into Eq. 6.2:

1 V = 64DZicdjjj-p

1

1

1

+9c2d2 15d4 +

(6.36)

The kinetic energy of the plate can be determined from Eqs. 6.3 and 6.22 as follows:

+-+,+, xc 4

y4

d

x4y4]

c4d4

(6.37)

Since the X and Y reference axes are through the center of the plate and integration must take place over the entire plate area. Equation 6.31 must be

120


used with the kinetic energy (Eq. 6.3). Then from Eq. 6.37,

Substituting Eq. 6.38 into Eq. 6.3 for the kinetic energy of the vibrating plate,

T = 0.566ZicdpR2

(6.39)

Since the strain energy must equal the kinetic energy at resonance, Eq. 6.36 must be equal to Eq. 6.39:

64DZ;cd

( 3+ 9 c 2 d 2 1

1

1

-

= 0.566Z;cdpR2

+

Solving for the natural frequency of the plate,

f

( 6.40)

=-=-

Sample Problem-Resonant

Frequency of a PCB

Determine the resonant frequency of a rectangular PCB that will be subjected to a 2-G peak sinusoidal vibration test. With a low input G level, the four edges of the circuit board will act as though they were simply supported. The board has a uniformly distributed weight of 1 lb, with dimensions as shown in Fig. 6.7.

Y

. 1

a = 83

. , 7

Supported U

?

g a

Typical electronic component part

a f n

I

Supported

-jkh=0062

%

S

E

5

I

7' O

I

FIGURE 6.7. Dimensions of a simply supported rectangular plate.

121


Solution. Using the natural frequency equation derived with the use of a trigonometric expression,

f

-(-I

n- D

=

n

2

1 (2 +

P

$1 1

(see Eq. 6.21)

where a = 8.0 in. (board length) b = 7.0 in. (board width) W = 1.0 lb (board weight) g = 386 in./s2 (gravity) h = 0.062 in. (board thickness) E = 2.0 X l o 6 lb/in.2 (epoxy fiberglass) p = 0.12, Poisson’s ratio (dimensionless) we have

D=

Eh 12(1 - P 2 )

area

2 x 106(0.062)3 =

12[1 - (0.12)*] 1.o

W

mass

’=

-

-

gab = (386)(8.0)(7.0)

40.1 Ib.in.

= 0.463 X

lb.~’/in.~

Substituting into Eq. 6.21, 40.1

1/ 2

1

1

(61+9]

0.463~10-~]

f,, = 52.6 HZ

(6.41)

The natural frequency of the circuit board can also be determined from the equation derived with the use of a polynomial expression:

1

where D

= 40.1

lb . in.

lb . s 2 / i n 3 p = 0.463 x c = a / 2 = 8.0/2 = 4.0 in. d = b/2

= 7.0/2 = 3.5

in.

(see Eq. 6.40)

122


Substituting into Eq. 6.40, 1

+

9(4.0)2(3.5)2

15(3.5)4

f, = 56.1 HZ

(6.42)

6.6 NATURAL FREQUENCY EQUATIONS DERIVED USING THE RAYLEIGH METHOD

The Rayleigh method is very convenient for determining approximate resonant frequencies of rectangular plates with various edge conditions. All that is required is to assume a deflection curve that satisfies the geometric boundary conditions of deflection and slope for the particular plate. Consider the case of a rectangular plate that is fixed on all four edges. The deflection will be zero at all four edges and maximum at the center. The slope will also be zero at the four edges and zero at the center. If a trigonometric expression is used to describe this plate, the coordinate axes can be placed at the edges of the plate as shown in Fig. 6.8. One trigonometric expression that will satisfy these boundary conditions is

i

z=z,1-cos-

a

This deflection curve can be examined to make sure it meets the geometric

Fixed 0

2

t i

Fixed

FIGURE 6.8. A rectangular plate with four sides fixed.

1

z

I

b

NATURAL FREQUENCY EQUATIONS USING THE RAYLEIGH METHOD

123

deflection requirements: at X = O a n d Y = O ,

Z=O

at X = O a n d Y = b ,

Z=O

at X = a a n d Y = b ,

Z=O

a b at X = - and Y = -, Z 2 2

= 42,

Now examine this curve to make sure it meets the geometric slope requirements. Consider first the slope along the X axis, which can be determined by taking the partial derivative of Z with respect to X :

Then b atX=OandY=-, 2

O,=O

b at X = a and Y = -, 2

O,=O

a b at X = - and Y = -, O , = O 2 2

The same check can be made for the slope along the Y axis, which shows the curve will satisfy the geometric slope requirements. Then using Eqs. 6.2 and 6.3 for the strain energy and the kinetic energy, the fundamental natural frequency becomes

A polynomial expression can also be used to describe a rectangular plate fixed on all four sides. The coordinate axes for this plate can be shifted to the center (Fig. 6.9). One polynomial expression that will satisfy these boundary conditions is

124


Y

t--7

Fixed

u ! I

u.

>X

Fixed

--

& i Q-x

_-

c

FIGURE 6.9. Axes at the center of a plate with four fixed edges.

z

This polynomial will result in the following natural frequency:

The natural frequency of a rectangular plate with three supported edges and one free edge can be derived using a trigonometric function or a polynomial function. A sketch of the coordinate axes for each plate, along with the deflection expression and the resulting natural frequency equations, for a trigonometric function and for a polynomial function, are shown in Fig. 6.10. A comparison can be made of the deflection equation and the resulting natural frequency equation using a trigonometric function or a polynomial function for plates with combinations of free edges, supported edges, and fixed edges, as shown in Figs. 6.11 and 6.12. Trigonometric functions can also be combined with polynomial functions in the same deflection equation to describe the deflection of a uniform rectangular plate. For example, considering a plate that is simply supported on two opposite edges and fixed on two opposite edges, the coordinate axes and the deflection equation that meets the geometric boundary conditions will be as shown in Fig. 6.13:

z=z, ( l-cos-

2"x)/I c1

-

$1

NATURAL FREQUENCY EQUATIONS USING THE RAYLEIGH METHOD Trigonometric

125

Polynomial

Y

Y

a m a

a m a

X

U

Supported

X

(61

fa)

i?xsin irY z = Z,,sin 2a h

z = z , , x1-d'

y2)

FIGURE 6.10. T h r e e sides supported and o n e side free: ( a ) trigonometric and ( b ) polynomial.

Trigonometric

a a m

Polynomial

n Free

a

a

m

X

i?X

Z = Z , ,sin (I

Free

( 1 -):7

z = z,,

FIGURE 6.11. Two opposite edges free and two opposite edges supported: ( a ) trigonometric and ( b ) polynomial.

126

PRINTED CIRCUIT BOARDS AND FLAT PLATES Trigonometric

Y

I Supported

Supported

I 1 ----

I

X

(

z=z,1 - - :*2)2(

-

p)

FIGURE 6.12. Two opposite edges fixed and two opposite edges supported: ( a > trigonometric and ( b ) polynomial.

Y

Supported

FIGURE 6.13. Combining a trigonometric function and a polynomial function.

1

-x

Supported

There has been an extensive amount of work on the derivation of natural frequency equations for uniform, flat, rectangular plates with various edge conditions. Little [30], Warburton [31], and Laura and Saffel [33] made extensive use of trigonometric and polynomial series to determine the various resonant modes of different plates. If only the fundamental resonant mode of a uniform, flat, rectangular plate is desired, then the simplified types of trigonometric and polynomial expressions, just outlined, can be combined to determine the fundamentalresonant-frequency equations for many different types of plates. Many of these equations for plates with various edge conditions are shown in Figs. 6.14-6.16.

DYNAMIC STRESSES IN THE CIRCUIT BOARD

127

xxxxxxxxxxxxxxxxx

Free edge

Supported edge

Fixed edge

Equation

Plate a

X X

b I

XL

xxxxxxxxxxxxxxxxxxxxxx

j b " !


X



ZTbji X

X



\ " I


FIGURE 6.14. Natural frequency equations for uniform plates.

6.7


Dynamic bending stresses in a uniformly loaded PCB can be determined for the resonant condition by considering a sinusoidal load variation for the dynamic load over the board surface when the edges of the board are simply supported. This follows from the relative dynamic deflections, which will be maximum at the center of the board and zero at the edges (Fig. 6.17).

128

PRINTED CIRCUIT BOARDS AND FLAT PLATES xxxxxxxxxxxxxxxxx

Free edge

Supported edge

Fixed edge Plate

Equation l

0

I

j, =

+)3 55 (i-

D p

h

I "

$ " l


a

031'

b


The load intensity at any point can be expressed in a trigonometric form that satisfies the boundary conditions: q

?Tx

= q,, sin-sin-

a

7TY

b

The differential equation for the deflection of the circuit board then becomes [291

a4z

a4z

a4z qo

7Tx

7TY

- -sin -sin axj+2aX2aY'+iiYc-D a b

(6.43)

DYNAMIC STRESSES IN THE CIRCUITBOARD

Plate

Equation

f n

D 2 [p

= -x

D 2.45

Nib'

2.68 2 45 f n = -2 - -+-(,4 c r 2 b b "h + J x

Lzizi3 xxxxxxxxxxxxxxxxxxxxx

-+-+F)] 0.127 0.707 2.44

( u4

[, (

129

il I"'

xxxxxxxxxxxxxxxxxxxxxxx X

a

I '' \


:v

xxxxxxxxxxxxxxxxxxxxxxx

:m

xxxxxxx

x x

x xx


FIGURE 6.17. Dynamic deflection mode of a circuit board.

130

PRINTED CIRCUIT BOARDS

A N D FLAT PLATES

The deflection of the circuit board at any point can also be represented by a trigonometric expression similar to the one shown in Eq. 6.1:

%-x

%-Y

Z = A sin ---sina

(6.44)

b

Performing the operations on Eq. 6.44 as required by Eq. 6.43,

a4z ax4 a4z --

-- -

a4z

%-Y b

a

a4

%-x

ITTT1

-

dY4

%-x

IT4

-Asin-sin-

%-Y

,Asin-sinb a

b

%-x

7i4

A sin -sin ax2ay2 a2b2 a =-

TY b

-

(6.45) (6.46) (6.47)

The deflection form factor A can be determined by substituting Eqs. 6.45-6.47 into Eq. 6.43: A=

40 1 2 n4D T + F [a a2b-

1

(6.48)

+g)

Substitute Eq. 6.48 into Eq. 6.44, and the deflection at any point on the circuit board is

%-x

q0 sin -sin Z=

ITY -

b

a

2

i l

The maximum deflection, Z,, will occur at the center of the board where X = a / 2 and Y = b / 2 . Substituting into the above equation, 40

z0= r4D

1

2

1

?+- a2b2' 2 )

(6.49)

[ a

Assuming that the circuit board acts like a single-degree-of-freedom system at its fundamental resonant mode, the maximum dynamic displacement can be approximated by Eq. 2.30 as follows:


131

Substitute this expression into Eq. 6.49 and solve for the maximum dynamic pressure intensity qo as follows: 2 4n

=

(6.50)

f,'

The plate stiffness factor D can be determined from the natural frequency equation for a uniform, simply supported rectangular plate, using Eq. 6.21: 4f,2 P 2

D=

1

(6.51)

Substitute Eq. 6.51 into Eq. 6.50, and the maximum dynamic load intensity at the center of the plate simplifies to the following expression: (6.52) The maximum dynamic bending moment in the rectangular circuit board will occur in the center. The bending moment must be greater for the section along the shorter plate dimension, b, where it has the value M y because the bending is along the Y axis. The maximum bending moment then becomes [291

My=

Sample Problem-Vibration

(6.53)

Stresses in a PCB

Determine the dynamic bending stresses in a 1.0-lb epoxy fiberglass PCB 7.0 X 8.0 X 0.062 in. thick when it is exposed to a 2-G peak sinusoidal vibration input and a transmissibility Q of 8.0 (see Fig. 6.7). Solution. Substitute into Eq. 6.53 for the dynamic bending moment, where

a = 8.0 in. (board length) b = 7.0 in. (board width) Gout= G,,Q = (2)(8.0) = 16 at center W = 1.0 lb (board weight) p = 0.12 (Poisson's ratio for G-10 epoxy fiberglass) 40 = WG,,,/ab = (1.0)(16)/(8.0)(7.0) = 0.286 Ib/in.2

132


Substituting into Eq. 6.53 for the bending moment, we have

M y=

= 0.496

in.lb/in.

(6.54)

The dynamic bending stress at the center of the plate can then be determined from the standard plate equation. Since PCBs normally have holes in them for the component lead wires, a stress-concentration factor should be used: (6.55) where M y = 0.496 in: lb/in. (see Eq. 6.54) h = 0.062 in. (circuit board thickness) K , = 3.0 (theoretical stress-concentration factor for a small hole in the circuit board) Substituting into the above equation,

s, =

6(3.0)(0.496) =

(0.062)

2320 lb/in.’

(6.56)

An examination of the fatigue S-N curve for G-10 epoxy fiberglass shows that the circuit board will never fail with a stress so low. 6.8

RIBS ON PRINTED CIRCUIT BOARDS

If the stiffness of a PCB is increased without a significant weight increase, the natural frequency will also increase and the deflection at the center of the board will decrease rapidly. A decrease in the circuit board deflection means a decrease in the electrical lead wire stresses and the circuit board stresses during vibration. One simple method for increasing the stiffness of a circuit board is to add ribs. If the ribs are made of thin steel, copper, or brass, they can be cemented or soldered to the copper cladding on the circuit board. This forms a very stiff section because these materials have a high modulus of elasticity. These ribs can be undercut between supports to allow the printed circuits to run under the ribs without causing electrical shorts. Consider the PCB shown in Fig. 6.7. If two steel ribs are soldered to the board in a symmetrical pattern along the length, the circuit board will appear as shown in Fig. 6.18.

133


Board supported on 4 sides

Y h=O062 epoxy

/

"if

n

a

and cemented to board

b=70

-

t

= 0040 rib (steel)

1

b/2 = : 5 0 = d -L

U, -

b/4 = 175

1

X

'

Connector

FIGURE 6.18. Stiffening ribs soldered to a circuit board.

If a 2-G peak sinusoidal vibration input is used to excite the circuit board, the four edges of the board will act as though they were simply supported. If a substantially higher G force is used, such as 10 G, the connector edge may act more like a free edge. Considering four simply supported edges, the natural frequency for the circuit board with ribs can be determined by making a slight modification in Eq. 6.21 to take account of the different stiffnesses along the X and Y axes. The natural frequency equation then takes the following form [291:

+3 ] l ' *

(6.57)

Here D, is the bending stiffness of the composite board along the X axis, and D , is the bending stiffness along the Y axis. Since the circuit board ribs are parallel to the X axis, they will increase the bending stiffness along the X axis but not along the Y axis. The torsional stiffness of the plate-and-rib combination is represented by DxY, If the ribs are in a symmetrical pattern on one side of the board, then the moment of inertia of one T section can be used to compute the bending stiffness along the X axis. A table (see Tables 6.2 and 6.3) is used to make it easier to compute the moment of inertia of the composite T section shown in Fig. 6.19.

z f

0062J

L-----35

-_I

Epoxy board

@

FIGURE 6.19. Dimensions for a rib section soldered to a circuit board.

134


TABLE 6.2

Item

Area

E X 10'

Z

A E x 10'

A E Z X 10'

I,, x 10-3 3.5(0.062)'

1

0.2170

2.0

0.031

0.434

0.0134

2

0.0152

29.0

0.252

0.441

0.1110

0.875

0.1244

= 0.070

12 0.040(0.380)3

Total

= 0.183

12

TABLE 6.3

2

EI,, x 103

C

2

0.140 5.310

0.111 0.110

Total

5.450

Item

AEC'

x 10'

~~~~~~~~~

1

0.0123

5.34 5.34

0.0121

10.68

The centroid of the T section is at

z=XCAAEEZ

0.1244 -

X

30'

0.875 X l o 6

= 0.142

in.

The bending stiffness of the T section is

C E I = EI, +AEc'

= 5.450

x 10'

+ 10.68 x l o 3 (6.58)

C E I = 16.13 x l o 3 lb.in.2

The bending stiffness of the circuit board along the X axis, with the ribs, is

D,=T where EI d

EI

(6 3 9 )

16.13 x l o 3 1 b . i n 2 (see Eq. 6.58) = b/2 = 3.5 in. (see Fig. 6.18) =

Substituting into the above equation, D,

16.13 X 10' =

3.5

= 4610

1b.h.

(6.60)


135

The bending stiffness along the Y axis will be approximately the same as the epoxy board:

Eh

D,= where E h

(6.61)

12(1 - p2)

=2X

l o 6 1b/in2 (G-10 epoxy fiberglass) = 0.062 in. (circuit board thickness) p = 0.12 (Poisson’s ratio)

Substituting into the above equation, (2 X 106)(0.062)3

D,=

12[ 1 - (0.12)?]

= 40.2

1b.in.

(6.62)

The torsional stiffness can be determined by considering a unit board width along with one rib (Fig. 6.19). The subscripts e and r refer to the epoxy board and to the rib, respectively:

D,,

= G, J ,

Gr +2d Jr

(6.63)

where G, = 0.90 X l o 6 1b/ine2 (shear modulus, epoxy fiberglass) J, = $h3 (unit torsional stiffness, epoxy fiberglass) in3 J, = 3(0.062>3= 79.3 X GI = 12 X l o 6 lb/in.2 (shear modulus, steel rib) J , = i L t 3 (torsional stiffness, steel rib) JI = $(0.38>(0.040>3= 8.10 x l o p 6 in.4 d = b/2 = 3.5 in. (rib spacing; see Fig. 6.18) Substituting into the above equation,

D,,=

(0.90 X 106)(79.3 X

D,,=

85.3 1b.in.

+ (12

X

106)(8.10 X 2( 3 S ) (6.64)

With the addition of the two steel ribs, the circuit board weight will increase to about 1.06 lb. The mass per unit area will then be mass P = --

area

p = 0.490 X

W

-

1.06 (386)(8.0)(7.0)

lb

in.^

gab

(6.65)

136


The natural frequency of the circuit board with the two ribs can be determined by substituting Eqs. 6.60, 6.62, 6.64, and 6.65 into Eq. 6.57: 4610

1

f,

=

1/2

4( 85.3)

251 HZ

(6.66)

It would appear from Eq. 6.66 that the natural frequency of the circuit board with two ribs will be about 251 Hz. However, since the center section of the circuit board, between the ribs, is only 0.062 in. thick, as shown in Fig. 6.18, it may be possible for this section to develop a resonance below 251 Hz. This can be checked by considering the center section of the circuit board, between the ribs, as a simply supported rectangular plate. Equation 6.21 can then be used to determine the natural frequency as follows:

-[-I 7T

f

"

=

2

D P

1

1

(2+ 2 )

(seeEq.6.21)

where a = 8.0 in. (board length) b = 3.5 in. (board width) D = 40.2 lb .in. p = 0.463 x lb . s 2 / i d Substituting into the above equation,

'= f,

7T

=

40.1 0.463 X l o p 4

142 HZ

1

1

)1'2[m m] +

(6.67)

Since the natural frequency of the unstiffened center section is only 142 Hz, a third rib may have to be added to raise the circuit board fundamental resonant frequency to 251 Hz. When the fundamental resonant frequency of the circuit board is increased to 251 Hz, the transmissibility of the circuit board will also increase. Considering the geometry and the mounting of the circuit board, if a low-input vibration-acceleration force with a 2-G peak is used, the transmissibility at resonance will probably be slightly greater than the square root of the natural frequency, as explained in Section 6.3. Vibration test data on many circuit boards indicate that the transmissibility is related to the natural

RIBS FASTENED TO CIRCUIT BOARDS WITH SCREWS

137

frequency as follows: Q = 1.2(fn)”’ Q = 1.2(251)”’

=

19.0

The dynamic deflection at the center of the circuit board with the ribs can be determined from Eq. 2.30 by approximating the circuit board as a single-degree-of-freedom system in its fundamental resonant mode:

6,=

9.8GinQ

f,’

where Gin= 2.0 (peak input acceleration) Q = 19.0 (transmissibility at resonance) f, = 251 Hz (natural frequency of board with ribs) Thus 6, =

(9.8) (2.0) (19.0) (251)*

= 0.00591

in.

(6.68)

Electronic component parts mounted on the circuit board will experience lower stresses in their electrical lead wires when the circuit board deflections are reduced. These stresses are directly proportional to the circuit board deflections at the fundamental resonant mode. This means that the deflections can be used in a direct ratio to determine the new stresses developed in the circuit board components when the ribs are added. 6.9 RIBS FASTENED TO CIRCUIT BOARDS WITH SCREWS

Extensive vibration tests on bolted assemblies show that bolted joints will experience a substantial amount of relative motion during major structural resonances. This relative motion may add damping to the system, which reduces the transmissibility at resonance. It also reduces the stiffness of the structure, and this tends to reduce the natural frequency. The ability of two bolted interfaces to remain rigid is a function of the stiffness of the joint, the number of screws, the screw size, the screw spacing, the screw torque, the interface conditions, the vibration force, and the vibration frequency. In order to correlate test data with a theoretical analysis, a bolted efficiency factor is very convenient. This efficiency factor will range from 0%, for a joint that has no physical connection, to loo%, for a joint that is welded. Evaluating the types of bolted joints normally found in electronic

138


Z 7 k 0040 original thickness

' 2

-~+0010

equivalent thickness

Bolted steel'

__

._._ ~.

I

b

y

r Bolted ~ interface

Y

4

L = 0 38 -T

h=0062

subassemblies such as brackets, covers, ribs, and stiffeners, which must transfer an alternating load, the typical efficiency of a bolted joint used for airborne electronic structures appears to be about 25%. It may be as high as 50% for small sheet-metal panels with large, closely spaced screws, and as low as 10% for large sheet-metal panels held down with quarter-turn quickdisconnect fasteners. The natural frequency of a PCB, with stiffening ribs bolted to the board, can be determined by using a bolted efficiency factor. For example, if the ribs on the circuit board shown in Fig. 6.19 were bolted to the board, instead of soldered, the resulting T section through one rib would be similar to the section shown in Fig. 6.19, except for the efficiency of the bolted interface. The 25% efficiency factor for this bolted joint will be analyzed by reducing the effective thickness of the rib to 25% of its true thickness (Fig. 6.20). The moment of inertia of the composite T section, with the bolted rib, will be determined with the use of Tables 6.4 and 6.5. Note that the equivalent thickness of the rib has been reduced from 0.040 to 0.010 in. to account for the 25% bolted efficiency factor. The centroid of the T section is at '=

CAEZ 0.O411X1O6 CAE 0.544 X 106

=

0.0756 in

TABLE 6.4 ~~

Item

Area

E X lo6

Z

1

0.2170

2.0

0.031

0.434

0.0134

2

0.0038

29.0

0.252

0.110

0.0277

0.544

0.0411

Total

A E X l o 6 A E Z X lo6

I" x lo-' 3.5(0.062)3 12

(0.010)(0.380)3 12

= 0.070

= 0.046

RIBS FASTENED TO CIRCUIT BOARDS WITH SCREWS

139

TABLE 6.5

EI, x 103

Item

C

C2

A E C x~ io3

0.0446 0.1764

0.00199 0.0311

0.864 3.420

~~

1 2

0.140 1.335

Total

1.475

4.284

The bending stiffness of the T section is

LEI

= El,

XEI

= 5.76

+ A E c 2 = 1.475 X l o 3

+ 4.28 X l o 3

x lo3 lb.in.2

(6.69)

The bending stiffness of the circuit board along the X axis, with the bolted rib, is

where EI d

= 5.76 X

lo3 1 b . h 2 (see Eq. 6.69) in. (see Fig. 6.18)

= b/2 = 3.5

Substituting into the above equation,

D,

5.76 x 103 =

3.5

=

1645 1b.in.

(6.70)

The bending stiffness along the Y axis will be approximately the same as that of the epoxy board, which is shown by Eq. 6.62 to be DY=40.2lb.in.

(6.71)

The torsional stiffness of the bolted T section can be determined by considering a unit board width, along with one rib, as shown in Fig. 6.20. The subscripts e and r refer to the epoxy board and to the rib, respectively: Gr Jr D,yy=G,J,+ - (see Eq. 6.63) 2d where G, = 0.90 X l o 6 lb/in.’ (shear modulus, epoxy fiberglass) J, = +A3 = +(0.06213= 79.3 X in3 G, = 12 X lo6 1b/in2 (shear modulus, steel rib) Jr = iLt3 3 = f(0.38)(0.010)3 = 0.126 X in.4 d = b/2 = 3.5 in. (rib spacing; Fig. 6.18)

140


Substituting into the above equation,

D,,

=

+

(0.90 X 106)(79.3 X

(12 X 106)(0.126 X

2( 3 S)

DX,=71.6lb.in.

(6.72)

If the circuit board weight does not change, the mass per unit area will be the same as that shown in Eq. 6.65. The natural frequency of the circuit board with the bolted ribs can then be determined by substituting Eqs. 6.70-6.72 and 6.65 into Eq. 6.57:

f=Z[ ”

f,

2 0.490 =

[-+1645

1 X

lo-‘

(8.0)4

“il

1/2

4( 71.6) (8.0)2(7.0)2

160 HZ

+

(7.0)‘

(6.73)

Since the center section of the circuit board is not stiffened between the ribs, as shown in Fig. 6.18, it may be possible for this section to develop a resonance below 160 Hz. The natural frequency of this section was previously checked in the approximation of a simply supported rectangular plate. These results are shown in Eq. 6.67, which shows a natural frequency of 142 Hz. If a natural frequency of 160 Hz is required, then a third rib may have to be bolted to the circuit board. The deflection at the center of the circuit board, with the bolted ribs, can again be determined from Eq. 2.30 by approximating the board as a singledegree-of-freedom system at its resonance. The transmissibility developed with the bolted ribs should be slightly less than with the soldered ribs because of the additional damping at the rib interfaces and the bolt interfaces. Vibration test data on several groups of circuit boards using a 2-G peak sinusoidal vibration input indicate the transmissibility was approximately related to the natural frequency as follows:

Q

=

( 160)1’2 = 12.7

Substituting into Eq. 2.30. the single-amplitude displacement with the bolted ribs becomes 6, = 6,

9.8G,,Q - 9.8(2.0)(12.7) f ;1 (160)‘

= 0.00972

in.

(6.74)

Circuit board displacements are important because they can be used to estimate the fatigue life, or the change in the fatigue life, of the electronic components mounted on the circuit board.

PROPER USE OF RIBS TO STIFFEN PLATES AND CIRCUIT BOARDS

141

Y

t

I

Circuit

board

h-rrmrmrrrml FIGURE 6.21. A circuit board with ribs in two directions.

6.10 PRINTED CIRCUIT BOARDS WITH RIBS IN TWO DIRECTIONS

If two sets of ribs are fastened to one side of a circuit board in a symmetrical pattern, a T section will have to be analyzed to determine the bending stiffness along the Y axis as well as the X axis (Fig. 6.21). Again, be sure to check each plate section between the ribs to make sure each section has a natural frequency greater than the full-plate natural frequency including the ribs.

6.11 PROPER USE OF RIBS TO STIFFEN PLATES AND CIRCUIT BOARDS

Ribs will not increase the stiffness of a plate or a circuit board if they are not properly used. In order for a rib to be effective, it must carry a load. Since all loads must eventually be transferred to the supports, the rib is most effective when it carries a load directly to the supports. For example, consider a flat square plate simply supported on two opposite edges and free on two opposite edges. If ribs are added to the plate so that they join the two free edges, the ribs will not carry the load directly to the supports and these ribs will not be very effective in stiffening the plate. If the ribs are added to the plate so that they join the two supported edges (Fig. 6.22), then these ribs will carry the load directly into the supports and they will be very effective in stiffening the plate. If ribs cannot be carried directly into the supports, then a secondary member should be provided to carry the load to the supports to make the

142

PRINTED CIRCUIT BOARDS AND FLAT PLATES Poor

Good Free

r-----l

Free

Free

FIGURE 6.22. Adding ribs to stiffen a flat plate effectively.

Poor

Good

Free

Free

FIGURE 6.23. Adding an additional rib to stiffen a flat plate.

ribs more effective. For example, consider a flat square plate simply supported on three sides and free on a fourth side. If ribs must be added to the plate so they end at the free edge, then an additional rib should be placed across the free edge to act as a secondary member, which will carry the load to the supports (Fig. 6.23).

6.12 QUICK WAY TO ESTIMATE THE REQUIRED RIB SPACING FOR CIRCUIT BOARDS

A quick method for estimating the required spacing for stiffening ribs on flat plates was shown by Bruhn [13]. H e uses stiffening rib spacing criteria based on the thickness of the plate. The stiffness (or modulus of elasticity) of the rib material should be at least equal to or greater than the stiffness of the plate material. The height and thickness of the ribs and the attachment to the plate must add enough local stiffness to the assembly to substantially reduce the displacements when the plate is subjected to a transverse load. The effective plate width B that the rib can reinforce is shown in Fig. 6.24 as 30t, where t is the plate thickness. The required rib height for a metal rib will vary from about 0.25 in. for a small PCB about 4 in. square, to a metal rib height of about 0.50 in. for a large PCB about 10 in. square.

143

QUICK WAY TO ESTIMATE THE REQUIRED RIB SPACING FOR CIRCUIT BOARDS

Rib,

\n

,Bonded

I

I

if

I

I-

B----l

B = 30t

t

FIGURE 6.24. Effective width of a flat uniform plate with a stiffening rib.

An examination of the sample problem rib pattern in Figs. 6.18 and 6.19 shows the rib is centered on a PCB section that is 3.5 in. wide. When the Bruhn criteria are used for the spacing, the allowable PCB plate width should be limited to the value shown below:

B

= 30t =

(30)(0.062) = 1.86 in.

(6.75)

The obvious conclusion from this quick approximation is that the rib spacing in Figs. 6.18 and 6.19 is inadequate. More stiffening ribs will be required to reach the 251-Hz natural frequency shown in Eq. 6.66. The same conclusion was reached after a great deal of work in the paragraph following Eq. 6.67, and in the paragraph following Eq. 6.73. If a third rib is added to the PCB as shown in Fig. 6.25, the effective plate width associated with each rib will be 2.33 in. This is still greater than the maximum allowable plate width of 1.86 in. shown in Eq. 6.75. This means that even with three ribs on the PCB, the natural frequency will still be

T 7.0 I

1 -

FIGURE 6.25. Suggested stiffening rib placement to achieve effective circuit board frequency.

144


slightly under the desired 251-Hz level. If the thickness of the PCB is increased to about 0.0776 in., the effective allowable plate width will be as shown below:

B

= 30t =

( 3 0 ) ( 0 . 0 7 7 6 ) = 2.33 in.

(6.76)

The desired plate width of 2.33 in. now matches the plate width shown in Fig. 6.25 for three ribs.

6.13 NATURAL FREQUENCIES FOR DIFFERENT PCB SHAPES WITH DIFFERENT SUPPORTS

Any time the natural frequency of a PCB is excited, problems can develop. Electrical lead wires, solder joints, and connector pins can crack, closely spaced PCBs can impact against each other and cause short circuits, and screws can become loose. Most of the PCBs made today are rectangular in shape. Some of these PCBs are plug-in types and other are bolted to some type of support. Natural frequencies for uniform rectangular plates and PCBs with various supports are readily found in technical literature. However, there has been a large increase in the use of electronics in unusual new applications, where volume and size are limited, so new shapes with new support methods are often required. There is very little data available for finding natural frequencies for odd-shaped plates and PCBs with different edge and point supports. Computers are available with various FEM software programs that can calculate the natural frequencies and mode shapes for plates with different shapes and supports. When access to a computer is available, when the operator is familiar with the FEM software, and when there is time available, odd-shaped plates and PCBs are easy to evaluate. When time is short, or when there are no computers available, the analysis methods shown here can be very convenient for obtaining the natural frequencies of some odd-shaped PCBs. The analysis methods shown here can be used for a sanity check of FEM analyses of odd-shaped plates. When the computer analyst is tired, especially right after lunch or late in the afternoon, it is very easy to make data entry errors. One bad data entry error on a computer can ruin weeks and months of hard work. In addition, it is often very difficult to locate and correct the bad data entry point. Natural frequencies for a variety of PCB shapes with different supports were calculated using a computer. The results for the first three natural frequencies are shown in Figs. 6.26-6.29 for different shapes. The first number is for the fundamental or the lowest natural frequency. The next two numbers are for the higher harmonics. Most of the vibration and shock damage will usually occur at the lowest natural frequency where the dynamic displacements are the greatest. The higher harmonics usually have smaller

NATURAL FREQUENCIES FOR DIFFERENT PCB SHAPES

145

Rectangles

[XI

Case 1

(1) 450.60 (2) 647.31 (3) 887.04

Hz

Case 5

Case 2

Case 3

243.10 Hz 452.94 731.52

72.21 Hz 123.16 208.36

1

Case 4

1

111.27 Hz 136.32 252.29

I

1

Case 6

iU (1) 72.12 (2) 200.80 (3) 223.21

71.78 174.72 210.68

Case 9

Case 10

174.72 210.68 243.48

217.88 270.59 278.93 I

a3 (1) 270.59 (2) 278.93 (3) 281.00

42.8 1 164.27 275.81

Case 13

Case 14

, , 107.72 225.38 343.32

Caai5

(1) 217.30 (2) 255.83 (3) 423.79

142.20 212.87 305.91

160.49 166.69 176.11

164.74 217.30 342.75

I

[ a i 7

160.49 176.11 178.21

FIGURE 6.26. Natural frequencies for flat uniform rectangular plates with different types of supports.

displacements so they produce less damage. These frequencies were tabulated for easy reference. The frequencies are for 0.10-in. thick epoxy fiberglass PCBs with a 24-in.’ area and a density of 0.208 Ib/ine3. This resulted in slightly different dimensions for the different shapes. The density was based on the total weight of a fully assembled PCB. This included all of the electronic component parts as well as the weight of the bare epoxy circuit board. The density was based on the total weight divided by the volume of

146

PRINTED CIRCUIT

BOARDS AND FLAT PLATES Hexagons

Case 1

( 1 ) 379.62 Hz ( 2 ) 718.78 (3) 8 0 6 . 3 1

Case 5

00 G Case 2

Case 3

Case 4

212.42 Hz 513.24 534.05

72.52 H z 89.17 235.97

117.56 Hz 134.47 272.42

Case 6

Case 7

Case 8

00 (1) 61.36 (2) 166.94 ( 3 ) 184.45

97.55 142.27 233.46

142.27 169.85 36 1.43

147.86 184.45 216.28

Case 9

Case 10

Case 11

Case 12

170.27 210.54 350.92

135.45 340.55 36 1.43

@ ( 1 ) 184.45 (2) 216.45 (3) 260.60

0 55.11

208.86 229.82

Case 1 3

( 1 ) 340.55 (2)361.43 ( 3 )3 9 3 . 3 1

189.24 265.18 265.98

@ 184.45 194.95 214.54

184.45 214.54 233.23

FIGURE 6.27. Natural frequencies for flat uniform hexagonal plates with different types of supports.

the epoxy board by itself. The PCB shapes are rectangles, hexagons, triangles, and circles with 16 different support arrangements for each shape shown in Figs. 6.26-6.29. The natural frequencies of a PCB that matches the dimensions, material, and support method of one of the tabulated PCBs can be obtained directly from the tabulated natural frequencies shown for each geometry. The approximate natural frequencies for PCBs with similar shapes and similar supports, but with different sizes and different materials, can be obtained with some simple conversion calculations using the equation shown below.


147

Triangles

I

Case 3

I

Case 1

Case 2

Case 4

(1) 443.23 Hz (2) 858.95 (3) 971.22

270.02 Hz 657.43 759.54

44.01 Hz 87.93 99.66

75.25 Hz 139.12 150.74

Case 5

Case 6

Case 7

Case 8

(1) 88.10 (2) 89.97 (3) 132.27

88.70 118.79 241.04

88.74 120.78 259.15

88.13 89.97 134.44

10

Case 11

Case 12

( 1 ) 39.31 ( 2 ) 98.49 (3) 126.02

77.21 290.66 335.03

263.63 306.07 456.64

226.94 237.77 261.01

Case 13

Case 14

Case 1 5

Case 16

(1) 95.48 (2)127.65 13) 227.91

227.54 245.94 326.47

AAAA Case 9

Case

A AIA1A I

I

115.57 122.67 199.19

122.50 141.72 210.29

--1 1

1

~~

Free edge

Supported edge

Fixed edge

Point support ~

FIGURE 6.28. Natural frequencies for flat uniform triangular plates with different types of supports.

148


Circles Case 1

Case 2

@ (1) 384.96Hz (2)775.09 (3)775.40 Case

5

(1) 59.55

(2)183.34 ( 3 ) 196.05 Case

9

4

Case

@ 177.02Hz 505.87 506.06 Case

6

119.73 166.89 249.89 Case

10

81.15Hz 94.16 258.04 Case

124.97Hz 148.03 306.04

7

166.89 182.12 441.24 Case

8

Case

156.94 196.05 255.55

11

Case

12

~

(1) 196.05 (2) 255.56 (3)255.91 Case

67.04 219.59 234.61

13

172.81 441.23 442.49

162.43 234.61 342.68 Case

15

Case

@olf '

(1) 441.23 (2)442.49 (3)458.35

Free edge

210.37 290.44 335.79

16

!$-

196.05 216.08 252.96

196.05 252.96 253.17

Fixed edge

Point s u p p o r t

............//////////// -I

Supported edge

FIGURE 6.29. Natural frequencies for flat uniform circular plates with different types of supports.


149

The subscript 1 refers to the tabulated values and geometry shown for the four different PCB shapes. The subscript 2 refers to the properties of a new specimen plate or PCB.

(6.77)

where f = natural frequency (Hz) a = width (in.) b = height (in.) E = modulus of elasticity (lb/in.,> h = thickness (in.) y = density (lb/in.3) p = Poisson’s ratio (dimensionless) Sample Problem-Natural Three Point Supports

Frequency of a Triangular PCB with

Find the natural frequency of a polyimide glass triangular PCB that is 8.5 in. wide, 7.5 in. high, and 0.090 in. thick, with a total weight of 0.70 Ib. It is fastened to a support with three small screws located at the center of the three sides, near the edges, similar to Fig. 6.28 Case 5. The following information is required to obtain a solution: a, = 7.44 in. wide

a, = 8.5 in.

b , = 6.45 in. high E , = 2.0 X l o 6 lb/in.’ modulus h , = 0.10 in. thick p1 = 0.12 dimensionless

in. E , = 3.0 X l o 6 lb/in.’ h , = 0.090 in. p2= 0.12 dimensionless

y , = 0.208 1b/ins3

f,= 88.10 Hz

b,

= 7.5

0.70 lb

yz

(8.5)(7.5)(0.090)(0.50) in.

0.244 1 b / h 3

f, = ? (3.0 X 106)(0.090)2(0.208) (2.0 X 106)(0.10)2(0.244)

= 67.5

Hz (6.78)

I CHAPTER 7

Octave Rule, Snubbing, and Damping to Increase the PCB Fatigue Life

7.1 DYNAMIC COUPLING BETWEEN THE PCBs AND THEIR SUPPORT STRUCTURES

Vibration and shock failures in PCBs can often be avoided by understanding how energy is transferred directly to the PCBs from their support structures. The support structures are used to provide easy access to the PCBs for troubleshooting, repairs, or replacement. Since the PCBs are attached to the support structure, the dynamic response of the support structure becomes the input to the PCBs. In any type of dynamic vibration or shock environment, the natural frequencies of the support structure and the PCBs can become excited. If the natural frequencies of the PCBs are close to the natural frequency of the support structure, their transmissibility Q values can couple, which means that their Q values will multiply. This will force the PCB response acceleration G levels to be magnified substantially, resulting in rapid PCB failures. For example, when the chassis has an input acceleration level of 10 G and a transmissibility Q of 10, the chassis will experience 10 X 10 or 100 G. If the PCB also has a transmissibility Q of 10 at the same frequency, the PCB will respond at a level of 100 X 10 or 1000 G. Acceleration levels this high can result in very rapid fatigue failures. The natural frequencies of the chassis and the various PCBs must be well separated to prevent severe coupling. An octave apart represents a good separation. Octave means to double. The natural frequencies of the chassis and the PCBs should be separated by a ratio of more than 2 : 1. The properties, locations, and arrangements of the various PCBs and the properties of the support structure will affect the responses and the fatigue life of the PCBs. There are also feedback effects, where the dynamic feedback from the PCBs to the support structure can produce acceleration G levels high enough to damage the support structure, in poorly designed electronic systems. Electronic systems with various PCBs enclosed within some type of chassis, form a complex dynamic structure. This usually requires the use of a 150

DYNAMIC COUPLING BETWEEN PCBs AND SUPPORT STRUCTURES

-,

151

PCB

Chassis mass 1

FIGURE 7.1. Chassis and PCB modeled as a two-degree-of-freedom lumped mass dynamic

system.

high-speed computer with the proper software to perform a dynamic frequency response analysis and to obtain the coupling properties between the chassis and the PCBs. This can be a very expensive and time-consuming analysis. This study has already been done using a computer to perform a parametric analysis of many two-degree-of-freedom systems with springs, masses, and dampers. The chassis is represented as mass 1, since it receives the energy first, so it is the first degree of freedom. The PCB is represented as mass 2, since it receives its energy from the chassis, so it is the second degree of freedom. This is shown in Fig. 7.1. The spring rates were changed to change the natural frequencies, the masses were changed to change the weight ratios between the chassis and the PCBs, and the dampers were changed to change transmissibility Q values of the chassis and the PCBs. The energy transferred from the chassis to the PCB for different dynamic combinations was used to find the acceleration G levels acting on the PCB. The information was arranged in the form of curves with several different PCB and chassis weight ratios, shown in Fig. 7.2-7.5, to make it easier to find the dynamic acceleration G response of different types of PCBs mounted in different types of chassis. Vibration test data show that most of the damage in electronic assemblies occurs at the fundamental natural frequency of the PCBs. These data also show that the response of the chassis is important since this represents the input to the PCB. The manner in which the chassis influences the PCB response determines the coupled acceleration G level acting on the PCB. The coupled response of the PCB is shown as Q,. The uncoupled response of the chassis acting by itself is shown as ql. The uncoupled response of the PCB acting by itself is shown as q2. The ratio R is used to show the Q2/q1 values that were obtained from the computer parametric studies. The fundamental natural frequency for the chassis is shown as f l and the fundamental natural frequency of the PCB is shown as f i . An examination of Fig. 7.2 shows that when the natural frequency of the PCB is about the same as the natural frequency of the chassis, the frequency ratio f2/fi is about 1.0. The

152

OCTAVE RULE TO INCREASE THE PCB FATIGUE LIFE

-

Dangerous area

-

T

E=4.0

/k

\\

\

7 %=0.05

l.-==u.3u

i

1

0 0

0.5

2

1

3

Ratio = f2/fl

FIGURE 7.2. Dynamic coupling effects between the chassis and the P C B for various ratios of transmissibility when the weight of the chassis is 20 times the P C B weight.

0

1

2

3

Ratio = f2/fi

FIGURE 7.3. Dynamic coupling effects between the chassis and the P C B for various ratios of transmissibility when the weight of the chassis is four times the P C B weight.

DYNAMIC COUPLING BETWEEN PCBs AND SUPPORT STRUCTURES

153

FIGURE 7.4. Dynamic coupling effects between the chassis and the PCB for various ratios of transmissibility when the weight of the chassis is two times the PCB weight.

FIGURE 7.5. Dynamic coupling effects between the chassis and the PCB for various ratios of transmissibility when the weight of the PCB is two times the chassis weight.

154


weight ratio of the PCB to the chassis is shown as W,/W,. When the weight of one of the PCBs is less than one-tenth the weight of the chassis, the weight ratio will be less than 0.10. Under these conditions the dynamic coupling between the PCB and the chassis, shown as R = Q 2 / q 1 ,can reach a value that is close to about 5.0. This means that the acceleration G levels acting on the PCB will be about 5 times greater than the expected acceleration G levels acting on the chassis. These conditions can produce high forces and stresses on the PCB, which can result in very rapid PCB fatigue failures. This condition can be avoided by following the octave rule. 7.2

EFFECTS OF LOOSE EDGE GUIDES ON PLUG-IN TYPE PCBS

The computer parametric studies for the octave rule were based on plug-in types PCBs that have three or four restrained edges that are simply supported (hinged) or clamped (fixed). Vibration test data show that a typical plug-in type of electrical connector on one edge of the PCB will act like a support. If there are any two unsupported PCB edges that are free to translate instead of being restrained in some way, the increased displacement amplitude will increase the period of vibration. This condition can occur, for example, when loose PCB edge guides are used. This tends to decrease the effective natural frequency of the PCB. This condition often brings the PCB natural frequency closer to the chassis natural frequency, which increases the dynamic coupling between the chassis and the PCB. This is a dangerous condition that can sharply increase the acceleration G levels produced in the PCB, resulting in more rapid PCB fatigue failures. The purpose of the octave rule is to reduce the dynamic coupling effects between the chassis and the PCBs, to improve the fatigue life of the PCBs in severe vibration and shock environments. Loose PCB edge guides will often result in increased coupling between the PCB and the chassis, which is not desirable. Loose PCB edge guides should be avoided to improve the reliability of the PCBs in dynamic conditions. DESCRIPTION OF DYNAMIC COMPUTER STUDY FOR THE OCTAVE RULE 7.3

Dynamic response data generated by the computer for the chassis and the PCB were translated into a graphic form with curves, to cover a wide range of parameters in order to simplify analysis methods that are normally very complex. The purpose of this technique was to permit the use of hand calculations for finding quick. but approximate, dynamic acceleration G levels acting on the PCB. These analysis methods have also been used as a sanity check for finite element modeling (FEM) calculations performed on the computer. When there are large differences in the results of the computer analysis and the hand calculations, something is usually wrong, so

THE REVERSE OCTAVE RULE MUST HAVE LIGHTWEIGHT PCBs

155

further investigations should be made. It is very easy to make data entry errors on a large computer program. Data input errors are very easy to make, but very hard to find in a large program. One bad data entry can ruin months of hard work. This is called GIGO, for garbage in garbage out, by computer users. 7.4

THE FORWARD OCTAVE RULE ALWAYS WORKS

Natural frequencies for the chassis and the PCBs should be planned in advance to take advantage of the octave rule. Once the design has been completed and orders have been placed, it is usually too late to try to utilize the octave rule. In the analysis phase the natural frequency for the chassis, with the weights of the PCBs included, must be obtained. Then the natural frequency of each PCB, assumed to be acting by itself outside the chassis, must be obtained. The resulting natural frequencies of the chassis and the various PCBs should be separated by more than one octave to avoid severe dynamic coupling. For example, when the chassis considered as mass 1 has a natural frequency of 100 Hz, then each PCB considered as mass 2 should have a natural frequency greater than 2 X 100 or greater than 200 Hz. This is called the forward octave rule. The forward term is used to describe the direction of the load path. The chassis, mass 1, gets the dynamic load first and transfers the load in the forward direction to the PCBs, mass 2. An examination of Figs. 7.2-7.5 shows that the dynamic coupling ratio R between the PCBs and the chassis is always sharply reduced when the forward octave rule is used, where the frequency ratio f2/fi is greater than 2.0. This means that the acceleration G levels transferred from the chassis to the PCBs are also sharply reduced. This reduction increases the fatigue life of the PCBs in severe dynamic environments.

7.5 THE REVERSE OCTAVE RULE MUST HAVE LIGHTWEIGHT PCBs

The reverse octave rule is used to describe the load path in the opposite direction from the PCB (mass 2) back to the chassis (mass 1). For the reverse octave rule to work the chassis should have a natural frequency that is more than two times greater than the natural frequency of the PCB. When the chassis has a natural frequency of 200 Hz, then the PCB should have a natural frequency less than 100 Hz. An examination of Fig. 7.2 reveals that a frequency ratio f2/fl of 0.5 or less represents the reverse octave rule. This shows the ratio R is approximately 1.0. This means that the acceleration level acting on the PCB will be about the same as the acceleration level acting on the chassis. This is good for vibration and shock, because the acceleration G levels transferred to the PCB from the chassis are low. Remember, these conditions are for a very small PCB weight, where the weight ratio W,/W, is

156


typically less than about 0.10. This covers the vast majority of the electronic equipment produced. Care must be used when the PCB weight ratio is increased to a value of 0.25, as shown in Fig. 7.3. Then the reverse octave rule, with a frequency ratio of f 2 / f l of 0.5, shows an R value of about 4.5. This means that the acceleration level acting on the PCB will be 4.5 times greater than the acceleration level acting on the chassis. The reverse octave rule in this case is a very dangerous condition because it can lead to high PCB acceleration G levels with a reduced PCB fatigue life. Some companies in the automobile industry tend to use a single PCB clamped to a sheet metal tray, where the PCB weight may be half the weight of the tray, This results in a weight ratio W,/W, of 0.50. An examination of Fig. 7.4 shows that if the reverse octave rule is used with the frequency ratio of 0.50, the coupling ratio R is about 4.0. The PCB acceleration G level under these conditions can be about 4 times greater than the G levels developed in the supporting tray. This condition may result in a reduced PCB fatigue life. An examination of Fig. 7.4 for the forward octave rule, with a frequency ratio of 2 or more, shows a coupling R value of about 1.2. This means that acceleration G levels acting on the PCB will be about 1.2 times greater than the acceleration G levels acting on the chassis. This shows that the forward octave rule will always work well, even for conditions that do not work well for the reverse octave rule. Sample Problem-Vibration

Problems with Relays Mounted on PCBs

Sinusoidal vibration tests were run on a small electronic chassis that contained several relays, with an acceleration rating of 75 G peak, mounted at the center of a plug-in PCB. During a 5 G peak qualification test, using a sinusoidal vibration sweep from 20 to 2000 Hz, the relays chattered, resulting in a test failure. An examination of the PCB by itself showed it had a weight of 2.5 lb, a natural frequency of 81 Hz, and a transmissibility Q of 9.0. An examination of the chassis by itself showed it had a weight of 10.0 lb, a natural frequency of 120 Hz, and a transmissibility Q of 3.0. The questions here are why did the relays chatter, and what can be done to the system so it will pass the test? Solution The Curves in Fig. 7.3 Can Be Used to Solve This Problem. Several different relations must be obtained to solve this problem as shown below.

fi/fl W,/W,

=

=

81/120 = 0.675 (uncoupled natural frequencies of PCB to chassis)

(7.1)

2.5/10

(7.2)

=

0.25 (weight ratio of PCB to chassis)

q 2 / q 1= 9/3 = 3.0 (uncoupled transmissibility ratio of PCB to chassis)

(7.3)

PROPOSED CORRECTIVE ACTION FOR RELAYS

G,

= Ginql

G 2 = G,Q, R

= Q2/q1

157

(uncoupled acceleration response of chassis)

(7.4)

(coupled acceleration response of PCB)

(7.5)

(ratio of the coupled transmissibility of PCB Q2 to uncopuled transmissibility of chassis q11

(7.6)

(coupled transmissiblility of the PCB) (rewriting Eq. 7.6) (7.7)

Q2 = Rq,

Substitute Eqs. 7.4 and 7.7 into Eq. 7.5 to obtain the coupled response of the PCB: G,

=

(Gi,q,)( Rq,)

= RGin(q , ) ,

(coupled response of the PCB) (7.8)

An examination of Fig. 7.3 shows there is no curve with a ratio q2/ql = 3, so a curve with this ratio must be generated by following the profiles between uncoupled transmissibility ratios 2 and 4. When this new curve is added with a dashed line, using the frequency ratio of 0.675 from Eq. 7.1, it results in the coupled transmissibility ratio R from the chassis to the PCB as shown below:

Substitute Eq. 7.9 into Eq. 7.8 along with an input Gin level of 5.0 and a chassis uncoupled transmissibility q , of 3.0 to obtain the coupled acceleration G, response of the PCB as shown below:

G,

=

(3.6)(5.0)(3.0),

=

162-G peak response of PCB

(7.10)

The 162-G peak response of the PCB far exceeds the 75-G peak rating for the relays, so the relays are not compatible with the PCB design and the environment. Some type of corrective action must be taken to allow the system to pass the qualification test. 7.6 PROPOSED CORRECTIVE ACTION FOR RELAYS

1. Mount the relays away from the center of the PCB, where acceleration levels are the highest. Mount the relays at the edges of the PCB where the acceleration levels are the lowest. This will require new PCB layouts with new PCB fabrication and assembly costs and delayed delivery schedules. 2. Make a search of the various relay manufacturers to see if they may have another relay off the shelf, with the same form, fit, and function, that can meet the requirements. 3. Examine the PCB layout to see if it is possible to add stiffening ribs to increase the natural frequency to a value that is two times the chassis natural

158


frequency. This would result in a new PCB natural frequency of 2 X 120 = 240 Hz, which would then follow the forward octave rule. Increasing the PCB natural frequency will also increase the PCB Q value. 4. The reverse octave rule with a PCB natural frequency of 120/2 = 60 Hz will not work here because the coupling ratio R for the PCB would be about 3.3. An examination of Eq. 7.10 shows the PCB acceleration response would be about 148 G peak, which is still well above the 75-G peak relay rating. 5. Adding snubbers to the PCB will reduce the PCB dynamic displacement and stress, but that is not the problem here. The problem is high acceleration levels. Snubbers can cause impacting shocks when the snubbers strike each other, which can produce high shock acceleration pulses. This can make the relays chatter and fail. 6. Damping is often recommended. Increased damping will decrease the transmissibility Q value during vibration. The best damping is constrained layer damping. This consists of alternate layers of thin high-energy dissipating viscoelastic strips separated by thin strips of aluminum foil. Damping works well on structures with low natural frequencies, below about 50 Hz. The dynamic displacements are then large enough to produce large relative shear displacements that convert kinetic energy into heat. When the natural frequencies are above about 100 Hz, the relative displacements are not large enough to provide significant damping. The amount of room required by the constrained layer dampers can usually be better utilized by employing stiffening ribs to increase the PCB natural frequency. The best option is corrective action item 3 above, adding stiffening ribs to the PCB to increase the natural frequency to 240 Hz. This may be difficult to do on a PCB that is fully populated so it may require a new design. This change will follow the forward octave rule. The new information required is shown below. =

f2/f,

g,

ql/q,

R

=

=

=

240/120

=

2.0 (frequency ratio of PCB/chassis)

(7.11)

= 15.5 (approximate uncoupled transmissibility of PCB)

(7.12)

15.5/3 = 5.16 (uncoupled transmissibility ratio of PCB/chassis)

(7.13)

1.4 (value of Q2/g1 estimated from Fig. 7.3)

(7.14)

Substitute Eq. 7.14 into Eq. 7.8 to find the coupled acceleration response of the PCB: G,

=

( 1.4) (5 .O) ( 3 ) 2 = 63 .O-G peak PCB response

(7.15)

USING SNUBBERS TO REDUCE PCB DISPLACEMENTS AND STRESSES

159

The peak response acceleration level of 63.0 G for the modified PCB is now well below the relay rating of 75 G peak, so the new PCB design should be acceptable. 7.7 USING SNUBBERS TO REDUCE PCB DISPLACEMENTS AND STRESSES

Snubbers can often be used to make PCBs work smarter in severe vibration and shock environments. Sometimes electronic equipment designers pay no attention to the octave rule so their PCBs have a high failure rate when they are exposed to vibration. Once the design is in full production and equipment is failing in the field, it can be very difficult to find a cost-effective fix. This is where snubbers can often save the day [431. High procurement costs for military electronic equipment have forced governments to go to the best available commercial electronic equipment manufacturers for many of the new military programs. Sometimes the commercial equipment lacks the ruggedness necessary for military programs, so the ruggedness must be increased, but at a reasonable cost. Again snubbers can often save the day. Snubbers are very effective for reducing problems in PCBs associated with large dynamic displacements. This condition can cause high lead wire and solder joint stresses, cracking of components, short circuits due to impacting between adjacent PCBs, and broken pins on electrical connectors. Snubbers, sometimes called bumpers, are small devices that can be attached to adjacent PCBs so they strike each other in dynamic conditions. This reduces the bending displacements in severe vibration and shock conditions. Reducing the deflection reduces the dynamic forces and stresses. This increases the fatigue life of the assembly. The best locations for snubbers and bumpers are close to the centers of the various PCBs, where the maximum displacements are expected. Soft rubber snubbers work best on PCBs with natural frequencies below about 50 Hz. Half-sphere shaped snubbers with diameters of about 0.20 in. can be used on small PCBs. Large PCBs work well using similar half-sphere shapes with diameters up to about 0.50 in. Soft rubber snubbers should be installed so they interfere slightly with other snubbers on adjacent PCBs for improved performance, as shown in Fig. 7.6. PCBs with natural frequencies above about 100 Hz must use more rigid materials for snubbers to reduce the PCB relative motion. Higher natural frequencies result in smaller dynamic displacements, so soft rubbery materials will not work. High-level vibration and shock qualification test programs using plug-in PCBs with epoxy fiberglass snubbers have been very successful. The snubbers in these programs were short posts with a cross-section diameter of 0.25 in., epoxy bonded to the center of each PCB. Hard snubber materials require a small clearance between adjacent snubbers to allow the PCBs to be inserted and removed with no interference. Smaller clearances

160


FIGURE 7.6. Soft rubber snubbers recommended for low natural frequency circuit boards.

work better. The best snubbers are ones that just touch each other with no space between them. This acts like a center support, which substantially increases the natural frequency. Small gaps are very desirable for reducing displacements, but they can be expensive because they require very close manufacturing tolerance control. The clearance between adjacent snubbers should be less than half of the dynamic PCB displacement expected in some particular dynamic environment, as shown in Fig. 7.7. Most snubber applications will be related to modifying existing hardware. The best location is at the center of the PCB. It will be very difficult to find any space at the center of any existing hardware. Therefore, a search must be made of the various PCBs to find free spaces for attaching the snubbers. One method that works well is to make transparent copies of all the PCB layouts. These are then arranged in their proper assembled order. Looking through the transparent assembled layouts will often reveal desirable places for locating adjacent snubbers.

Small gap

--I&

Rigid

FIGURE 7.7. Rigid epoxy glass rod snubbers with a small clearance gap, recommended for high natural frequency circuit boards.

USING SNUBBERS TO REDUCE PCB DISPLACEMENTS AND STRESSES

Sample Problem-Adding

161

Snubbers to Improve PCB Reliability

Several different military and commercial electronic systems have been experiencing component failures on plug-in type PCBs during sinusoidal vibration tests. The PCBs that have been experiencing problems are listed below. 1. PCBs with a natural frequency of 125 Hz during a 5-G peak sine vibration test. 2. PCBs with a natural frequency of 160 Hz during a 5-G peak sine vibration test. 3. PCBs with a natural frequency of 260 Hz during a 6-G peak sine vibration test.

An examination of the problem PCBs shows there is room available near the center for mounting small snubbers. The proposed plan is to use 0.25-in. diameter epoxy glass dowel rods as snubbers, bonded to the PCBs, leaving a nominal clearance gap of 0.012 in. between adjacent snubbers. Is this an acceptable course of action?

Solution Find the Expected Dynamic Displacements and Compare with 0.012-in. Gap. PCB dynamic displacements can be obtained from the general deflection equation: 2, =

9.8GinQ

(see Eq. 2.33)

(fn)'

Table 7.1 lists the results when values are inserted in above equation. The recommended gap between the snubbers is 0.012 in. for all three problem PCBs. The first PCB needs a gap of 0.017 in. or less so a smaller gap of 0.012 in. is good. The second PCB needs a gap of 0.012 in. or less so it just meets the requirements. The third PCB needs a gap of 0.007 in. or less, so a gap of 0.012 in. does not meet the requirements. Such a small gap will be very difficult and very expensive to achieve due to the large number of tolerances involved in the materials, the manufacturing processes, and the

TABLE 7.1 PCB

Gin

fn

1 2 3

5.0

125 Hz 160 Hz 260 Hz

5.0 6.0

Q

dm = 11.2 dm 12.6 =

= 16.1

ZO

Gap = Z0/2

Acceptable

0.035 in. 0.024 in. 0.014 in.

0.017 in. 0.012 in. 0.007 in.

Yes Yes no

162

OCTAVE RULE T O INCREASE T H E PCB FATIGUE LIFE

assembly. However, it can be done with the use of matched assemblies. This involves matching each individual PCB with its own set of snubbers in its own chassis. There can be no switching of similar PCBs to another chassis because the accumulation of tolerances might result in a much large gap that can reduce the PCB fatigue life.

7.8 CONTROLLING THE PCB TRANSMISSIBILITY WITH DAMPING

Damping is usually defined as the conversion of kinetic energy into heat. Material damping relates to the energy lost due to internal friction or hysteresis in the molecular structure of the material. Structural damping relates to the energy lost due to friction in rubbing, scraping, slapping, and impacting at various interfaces and joints. The total system damping is the sum of the material damping and the structural damping. All real systems produce some damping when they are vibrated. When a system with light damping is disturbed, it will continue to oscillate for a long time after the disturbing force has been removed. A system with heavy damping may oscillate once or twice after the disturbing force has been removed, similar to a good automobile shock absorber. The damping in all systems will always bring the system back to a state of rest some time after the disturbing force has been removed. Damping removes some of the energy within the system so there is less energy available to distort and damage the system. This means there is less energy available to deform structures such as PCBs in vibration and shock environments. A reduction in the kinetic energy reduces the forces and stresses in PCBs, which increases the fatigue life, so damping increases the PCB reliability. Increasing the damping in a PCB increases the amount of energy that is lost, which decreases the transmissibility, and further increases the fatigue life and the reliability of the PCB. 7.9

PROPERTIES OF MATERIAL DAMPING

Internal energy is lost, or converted into heat, any time a structure is deformed. Applying a tensile load to a bar does not instantly deform the bar. There is a small time lag as the bar elongates to its new stabilized length. This entire process is reversed when the applied load is removed. However, the bar never returns to its exact original length since some of the strain energy has been converted into heat, so less energy is available to restore the original bar length. Applying an equal compressive load on the bar reverses the direction of the events just described. The positive and negative strains can be plotted on a graph to show the typical hysteresis loop formed by materials that are deformed. The area in the enclosed hysteresis loop becomes a measure of the energy lost with each stress cycle. A larger area

CONSTRAINED LAYER DAMPING WITH VISCOELASTIC MATERIALS

163

enclosed within the loop shows there is more damping. Again, more damping reduces the PCB transmissibility, which increases the fatigue life and the reliability. 7.10 CONSTRAINED LAYER DAMPING WITH VISCOELASTIC MATERIALS

Most of the damage produced in a vibrating PCB occurs at the natural frequency, where the transmissibility is the greatest, so the displacements, forces, and stresses are also the greatest. When the damping is increased, the PCB reliability is increased. More damping in a PCB is desirable, if it does not result in an increase in the size, weight, and cost. One good method for increasing the damping is with the use of viscoelastic materials with constrained layers to increase the damping. Viscoelastic materials are very similar to rubber materials. They are quite elastic so they can deform, compress, and stretch through large displacements without failing. These materials can also dissipate large amounts of energy when they are deformed. Even more energy can be dissipated with the use of materials such as asbestos mixed in with the viscoelastic materials. Pure natural rubber has similar problems-it does not have much strength or much damping. The addition of carbon black improves these properties. Low temperatures and very high frequencies make viscoelastic materials act like hard materials. Their ability to perform well under these conditions is reduced. High temperatures are also a problem, because these materials lose much of their damping properties under these conditions. Their modulus of elasticity is difficult to measure because it changes with the speed of the applied load. The modulus for a rapidly applied load can be three times greater than for a slowly applied load. Viscoelastic materials can be effective in reducing the dynamic displacements, forces, and stresses in vibrating beam and plate structures that are not required to operate over a wide range of high to low temperatures. These materials are available in various forms of paints and tapes that can be applied in single and multiple layers. Multiple layers of damping materials are much more effective than single-layer applications. More damping layers are more effective but they also add more weight and may make repairs more difficult. One of the most effective methods for increased damping is constrained layer damping. This is the application of many layers of damping material that are applied in the form of a high narrow strip. Alternate layers of viscoelastic materials are separated by layers of aluminum foil about 0.005 in. thick. The typical height is about 0.50 in. This can often be added across the center of a plug-in type PCB, in the form of a strip similar to a stiffening rib, as shown in Fig. 7.8. The center of the PCB is the best place for a damping strip because the displacements are the greatest at that position. This results in large relative shear deflections in the alternate damping layers, which improves the damping.

164

OCTAVE RULE TO INCREASE THE PCB FATIGUE LIFE Constrained layer viscoelastic

Adhesive bonding

PCB

Alumi

fo ‘Connector

Section BB enlarged

FIGURE 7.8. Constrained layer viscoelastic damper for possible use on low-frequency

boards.

7.1 1 WHY STIFFENING RIBS ON PCBs ARE OFTEN BETTER THAN DAMPING

Vibration testing experience with damping techniques has shown that damping can be quite effective for PCBs with low natural frequencies, below about 50 Hz. However, the tests show that stiffening ribs usually work even better. When the natural frequencies are well above about 100 Hz damping is not a very effective means for reducing the transmissibility. High operating temperatures also tend to reduce the damping properties and effectiveness of viscoelastic materials. Constrained layer damping strips take away valuable space that could be used for mounting additional electronics. When this space is already lost, experience has shown that it is often much better to use the space for bonding stiffening ribs to the PCB, instead of constrained layer damping strips. Stiffening ribs will increase the natural frequency, which will more rapidly reduce the dynamic displacements, because displacements are inversely related to the square of the natural frequency.

7.12

PROBLEMS WITH PCB VISCOELASTIC DAMPERS

A very large supplier of sophisticated electronic systems received a large contract for a military electronic system that had to operate with a high reliability in severe thermal, vibration, and shock environments. There was not enough sway space available for an isolation system, so the mechanical design engineers selected to use viscoelastic damping techniques to reduce the plug-in PCB response accelerations and displacements. The design ap-

PROBLEMSWITH PCB VISCOELASTIC DAMPERS

165

proach used two, one-sided surface mounted PCBs assembled back to back, with a constrained layer viscoelastic material bonding both halves of the PCBs together into one double PCB plug-in module. Vibration and shock tests were run on prototype models using extensive instrumentation. The test data looked very good, so the design was released to the production group. The first few production models were shipped to the customer for a preliminary inspection and evaluation. Production unit number 8 was selected for the qualification test program. Many military qualification test programs require running electrically operating vibration tests at low temperature, room temperature, and high temperature. The first two vibration tests were satisfactory. However, the high-temperature vibration test resulted in many PCB failures. Component lead wires were breaking, solder joints were cracking, and components were flying off the PCBs. The high-temperature vibration tests were a complete disaster. The mechanical engineers were stunned. They could not understand what went wrong. The previous vibration tests on prototype models had been very successful. An investigating failure analysis team was formed and additional vibration tests were run on the viscoelastic damped PCBs. The tests were instrumented and run at room temperature. The resulting test data looked good. The vibration tests were then run again at high temperatures. The test results were dramatically different, with substantially lower PCB natural frequencies producing sharp increases in the dynamic displacements, with components flying off the PCBs. This was caused by the dramatic reduction in the stiffness and the damping properties of the viscoelastic materials at the high temperatures. The system had to be completely redesigned, using stiffening ribs on the PCBs, to obtain a high natural frequency to keep the dynamic displacements and stresses low to achieve the required fatigue life. The production group had to redesign their tools for the new production program. Needless to say that a lot of money was lost on this program, and product deliveries were substantially delayed, because the mechanical engineering group overlooked the adverse effects of high temperatures on the damping properties of the viscoelastic materials.

DCHAPTER 8

Preventing Sinusoidal Vibration Failures in Electronic Equipment

8.1

INTRODUCTION

Sinusoidal vibrations are often found in machines and devices that rotate or oscillate, such as electric motors. wheels, engines, turbines, gears, washing machines, blenders, oil drilling equipment, springs, and even bridges. Sinusoidal vibrations are also extremely useful as a diagnostic tool for evaluating the dynamic characteristics of various types of structures, and for examining resonant conditions with the use of a strobe light. Vibration failures are often difficult to trace, since there are so many different factors to be considered. Not only must the environment be considered, but also the mechanical and electrical design, manufacturing tolerances, resonant frequency of the outer housing structure, resonant frequency of internal elements, possible dynamic coupling effects, sensitivity of electronic components, cables and harnesses, effects of conformal coatings or lack of coatings, humidity effects, and dozens of other factors. Sometimes there does not appear to be any connection between parts that are failing and exposure to vibration, as shown in the following case history. A large auto manufacturer was having problems with dead batteries on a particular automobile model, when the auto was shipped across the country by rail. The batteries were charged when they left the factory and were discharged when they arrived at their destination. The most obvious sources for the problem were the batteries themselves and the electrical connections related to the battery. When these suspected sources were cleared of blame, the investigation was expanded to other areas, but always within the engine compartment where the battery was located. It appeared logical that the battery failures must be related to the battery location. After many hours of investigation and many meetings, one clever individual observed that the horn ring on this particular car model had a very soft touch. The horn would sound when a small force was applied to the horn ring. H e had a hard time convincing the company officials to send an observer along with the next rail shipment of cars across the country. His theory was correct. Along one long stretch of track through a desolate area, the speed of the train and the 166

ESTIMATING THE VIBRATION FATIGUE LIFE

167

condition of the tracks set up a disturbing force that excited the resonant frequency of the horn ring on its springs. Over an extended period of many hours, the constant sounding of the horn drained the battery. The springs on the car horn ring were made stiffer, which raised the resonant frequency of the horn ring above the disturbing force, and the problem was solved. ESTIMATING THE VIBRATION FATIGUE LIFE

8.2

The approximate fatigue life of a vibrating system can often be estimated from the fatigue characteristics of the various members that carry the major structural loads. These fatigue properties are usually obtained from controlled stress cycle tests performed on many parts that have been manufactured from the same material, with close dimensional tolerances. These parts are stress-tested to failure; then the data points are plotted on log-log paper, with stress on the vertical axis and the number of cycles to failure on the horizontal axis. A straight line that represents the best average fatigue properties is drawn through the scattered data points as shown in Fig. 8.1. The equation for the sloped portion of the curve can be expressed as follows [l]:

N,S,b = N,S,b where N = number of stress cycles to produce a fatigue failure S = stress level at which these failures occurred b = fatigue exponent related to the slope of the line

(8.1)

The exponent b is related to the fatigue life of the structure, and it is useful in predicting the expected fatigue life of other structures fabricated of the same material and exposed to similar environments. In order to reflect the real structural properties of these materials, a stress-concentration factor K must be included in the evaluation of the exponent b. A typical stress-concentration factor for an electronic box structure or for a PCB within the box is about 2.0. Since the stress concentration has little importance with a small number of fatigue cycles, and more importance with more fatigue cycles, the slope of the S-N curve increases with the stress concentration as shown by the dashed line in Fig. 8.1.

I Cycles to failure.

I

FIGURE 8.1. Typical plot of the data obtained

A’

from fatigue tests of metal structures.

168

PREVENTING SINUSOIDAL VIBRATION FAILURES

The value of the exponent b can be determined for aluminum by using typical physical properties for aluminum alloys, where the endurance stress is typically one-third of the ultimate tensile stress. With a stress concentration of 2, the endurance is one-sixth of the ultimate tensile stress. Rewrite Eq. 8.1 as follows:

where Nl = lo8 cycles N2 = l o 3 cycles S , = S, = +St, (one-sixth the ultimate tensile stress) S, = S,, = ultimate tensile stress Substitute into Eq. 8.2 and simplify:

Take the logarithm of both sides and solve for the exponent b:

b=

log,, 105

--

log,, 6

5

0.778

=

6.4

This fatigue exponent b can be used to determine the approximate fatigue life for a number of different conditions. For linear systems, the number of fatigue cycles will be directly related to the time, so the time T can be used to replace the number of cycles ( N ) .Also, the stress S will be directly related to the acceleration G and to the displacement Z . The dynamic load can also be included if it is desired. These relations can be written as follows:

T,G:

=

N,ZF

= N2Z!

(8.4b)

N,G!

=N , G ~

(8.4~)

Sample Problem-Qualification

T,G~

(8.4a)

Test for an Electronic System

An electronic control system is to be used on subway trains that are expected to operate 6 hours per day, 4 days per week, for 52 weeks per year, with a desired life of 20 years. Field test data show that the subway track system produces harmonic vibrations over a frequency range from 10 to 300 Hz, with typical accelerations of 1.1 G peak. A sinusoidal vibration qualification test is

ELECTRONIC COMPONENT LEAD WIRE STRAIN RELlEF

169

desired to prove that the electronic system will be capable of operating in this environment for the 20-year period. The proposed qualification test plan specifies a sinusoidal test where the forcing frequency sweeps back and forth over the observed bandwidth using a 4.5-G peak acceleration input level. Determine the length of time that should be used for these sweep tests to qualify the electronic system. Solution. Equation 8.4a can be used to determine the period of time for the qualification test. The following information is required:

T2 = (6 h)(4 day)(52 wk)(20 yr) = 24,960 h (life desired) G, = 1.1 G (peak field test data) GI = 4.5 G (peak proposed qualification test level) b = 6.4 (vibration fatigue exponent for electronic structures) Substitute into Eq. 8.4a and solve for the desired vibration qualification test time period:

T , = (24,960)

(

:::r4

-

= 3.03

hours

(8.5)

The 3.03 hours represents the testing time with no safety factor. Since there are many variables involved in the determination of the field test data and in the design, manufacture, and assembly of this electronic equipment, some safety factor must be included. Therefore, recommendations should be made to vibrate the electronic equipment for 3.03 hours along each of three mutually perpendicular axes for a total of 9.09 hours of vibration for the qualification test. The above recommendation is equivalent to a safety factor of 3.0, which will require a slightly heavier structural assembly to meet these levels. The weight penalty is not very large here, because the fatigue life changes very rapidly with a slight reduction of the stress level. For example, when the stress level changes by a factor of 2, the fatigue life changes by a factor of 2 raised to the 6.4 power, which is 84.4.

8.3 ELECTRONIC COMPONENT LEAD WIRE STRAIN RELIEF Electronic systems that are used in any vibration or shock environments will expose the electronic components to some level of vibration and shock. These levels may be amplified or attenuated, depending on the characteristics of the dynamic system. When PCBs are used in these electronic systems, the resonant frequencies of the PCBs are often excited, which forces the PCBs to bend back and forth. This motion forces the electrical lead wires on the components to also bend back and forth, as shown in Fig. 8.2.

170


Strained lead wires

+L

\

Relative

PCB Bending’

1

B -_____i_/

FIGURE 8.2. Relative motion produced between component and circuit board when the board resonance is excited.

Different lead wire geometries will result in different stress levels in the lead wires and solder joints of these components. Even when the dynamic displacements of a group of PCBs are the same, the stresses developed in the lead wires and solder joints of similar components can still be sharply different, depending on the geometry of the wire. When the dynamic displacement of a particular PCB has been determined for a particular environment and a particular PCB resonant frequency, the displacement amplitude can be treated as a fixed condition. Under these circumstances, it can be shown that the best way that the forces and stresses can be reduced in the lead wires is to reduce the spring rate of the wires. This appears to be contrary to the normal approach generally taken when a structure fails in a vibration or shock environment. Stronger is normally considered to be better, so most of the time the structure will be made stiffer (stronger) when a failure occurs. However, in the case of lead wires, a stiffer wire will fail faster. This can be demonstrated by examining the force P developed in a structure as a function of the spring rate K and the displacement Y as follows:

P=KY

(8.6)

When the displacement is held constant, the best way that the force can be reduced is to reduce the spring rate. This is the same condition that exists in the component lead wires for a specific dynamic displacement generated by the resonant frequency and acceleration level. In the typical application, where the electrical lead wires are forced to bend back and forth during a resonant condition, the lead wire stiffness is related to the following parameters:

K wire .

EI = -

L3

(wire flexing stiffness parameters)

The above relation shows that increasing the length L of the wire will reduce the stiffness K very rapidly, since it is a cubic function. For example, when the wire length is doubled, the stiffness, the force, and the stress in the wire are reduced by a factor of two cubed, or eight. The above relation also shows

DESIGNING PCBs FOR SINUSOIDAL VIBRATION ENVIRONMENTS

171

that decreasing the moment of inertia I of the electrical lead wire will rapidly reduce the wire stresses, since the moment of inertia varies as the cube of the wire height. When the wire height, or thickness, is reduced by a factor of two, the spring rate, stiffness, force, and stress in the wire are decreased by a factor of two cubed, or eight. This can be accomplished by coining the lead wire, that is, squeezing it into a flat thin metal strip. The cross-sectional area of the wire stays the same, but its thickness is reduced, so the spring rate, stiffness, force, and stress in the wire are also reduced. The above relation also applies to surface mounted electronic components as well as the poke-through type of mounting, where electrical lead wires are forced to bend during resonant conditions. It also applies to leadless ceramic chip carrier (LCCC) devices, which are surface mounted with or without electrical lead wires. In applications that do not use lead wires, the above relations then apply to the shear stiffness of the solder joints that bond the component to the PCB. Here again the stiffness of the solder joints must be reduced to reduce the forces and stresses in them. An examination of the shear stiffness will show which parameters will affect the solder stresses: AG Kshear =

(solder shear stiffness) L

(8.8)

where A = cross-sectional area of the solder joints (he2) L = height of solder joint (in.) G = shear modulus of the solder (lb/in.’> The shear stiffness can be decreased by reducing the cross-sectional area of the solder, reducing the shear modulus of the solder, or increasing the height of the solder joint. Some experiments are being carried out to reduce the solder shear modulus, but so far this has been unsuccessful. Some success has been achieved by reducing the solder cross-sectional area. The greatest success in improving the fatigue life of the solder has been in increasing the height of the solder joint. The typical solder joint height for LCCC devices is about 0.003-0.005 in. This has successfully been raised to 0.010 in. with the use of small solder mask pads, 0.010 in. thick, under the LCCC devices. 8.4 DESIGNING PCBs FOR SINUSOIDAL VIBRATION ENVlRONMENTS

In typical electronic systems, the electronic components are mounted on plug-in PCBs. This makes it easy to repair an electronic system if an electrical failure occurs and the failure can be traced to a faulty PCB. The faulty PCB is simply replaced with a good one.

172


f

PC pcB

2nd

degree

Chassis

I

1st degree Rigid fixture

3

1 Vibrat

3n

lnpu!

I

1

V i brat ion

input

FIGURE 8.3. A simple electronic chassis assembly with PCBs, simulated as a

lumped-mass system. When the plug-in PCBs are mounted within a chassis housing, and the system is required to operate in a sinusoidal vibration environment, then the chassis will act as the first degree of freedom, because the vibration energy will excite the chassis structure first. Since the PCBs are normally attached to the chassis structure (side walls or base of the chassis), the PCBs receive their dynamic excitation from the chassis. This makes the PCBs act like the second degree of freedom. A typical system is shown in Fig. 8.3. Resonant conditions can occur in the chassis and also in the PCBs during excitation by sinusoidal vibration. Since the chassis and the PCBs are coupled together, any resonant condition in any one of the masses will force some response in the other masses. When the resonant frequency of the chassis is close to the resonant frequency of any of the PCBs, and when the chassis and the PCBs have high transmissibilities Q, then the chassis response can augment the PCB response. This will produce very high acceleration levels in the PCBs, which can result in rapid fatigue failures. In order to prevent this from occurring, the chassis resonance and the PCB resonances must be well separated. This can be accomplished by using the octave rule. An octave means a factor of two in frequency. When the resonances of the chassis and the PCBs are separated by one octave or more, then the coupling effects between those members are sharply reduced. It makes very little difference whether the chassis or the PCBs have the higher resonant frequency, as long as their resonant frequencies are well separated. Vibration testing experience and associated finite element studies have shown that the fatigue life of many different types of electronic components can be related to the dynamic displacements developed by the PCBs during vibration [38]. These studies have shown that these components can achieve a fatigue life of about 10 million stress reversals in a sinusoidal vibration environment, when the peak single-amplitude displacement of the PCB is limited to the value shown below and in Fig. 8.4 [371:

Z=

0.00022 B Chra

(maximum desired PCB displacement)

(8.9)

DESIGNING PCBs FOR SINUSOIDAL VIBRATION ENVIRONMENTS Bending lead wires

I

FIGURE 8.4. Axial-leaded component wires will bend when the PCB resonance is excited.

-2 displacement where B

173

of PCB edge parallel to component (in.) L = length of electronic component (in.) h = height or thickness of PCB (in.) C = constant for different types of components (see Eq. 9.50) r = relative position factor for components (see Eq. 9.50) = length

The peak single-amplitude displacement expected at the center of a PCB can be estimated by assuming the PCB acts like a single-degree-of-freedom system at its resonant condition, as shown in Eq. 2.30: 9.8G 9.8Gi,Q z=-

f'

f,'

The displacement value obtained from the above relation will be a little less than the correct value obtained from the actual dimensions of the PCB plate geometry. When the PCB is approximated as a single degree of freedom, and a single mass is used to represent the PCB, the single mass will show the displacement of the centroid of the PCB and not the actual displacement of the PCB itself. Vibration test data have shown that the transmissibility Q of a PCB can typically be related to its resonant frequency. Although the test data show that high input acceleration levels (above 8 G) will reduce the Q values, and low input acceleration levels (below 2 G ) will increase the Q values, a good approximation of the Q value can still be obtained with the following relation (also see Eq. 14.21):

Q=&-

(8.10)

The desired PCB resonant frequency that will provide a fatigue life of about 10 million stress cycles for the component lead wires and their solder joints can be determined by combining Eqs. 2.30, 8.9, and 8.10:

( fd =

9.8Gi,Chra 0.00022 B

)2'3

(minimum desired PCB frequency) (8.11)

174


Sample Problem-Determining

Desired PCB Resonant Frequency

A 40-pin DIP with standard lead wires will be mounted at the center of a 6.0 x 8.0 x 0.10-in. thick plug-in PCB. The DIP will be mounted parallel to the 8-in. edge. The system will be subjected to a sinusoidal vibration qualification test where the maximum acceleration input level is expected to be 7.0 G peak. Determine the minimum desired PCB resonant frequency, and the approximate fatigue life considering a resonant dwell condition.

Solution. The octave rule must be used to make sure the resonant frequency of the PCB is well separated from the resonant frequency of the chassis, so coupling effects between the chassis and the PCB are reduced. Under this condition it is convenient to assume that the acceleration input to the PCB is the same as to the chassis. This is not really true, but it is convenient for a quick approximate answer. Then

B = 8.0 in. (length of PCB parallel to component) h = 0.10 in. (PCB height or thickness) L = 2.0 in. (length of 40-pin DIP) C = 1.0 (constant for standard DIP geometry) P = 1.0(relative-position factor for component at center) Substitute into Eq. 8.11 for the desired PCB resonant frequency:

fd =

[

0.666

i

9.8( 7.0) ( l . O ) ( O . l O ) ( 1. O ) m 0.00022(8.0)

= 310

HZ

(8.12)

The fatigue life of the DIP can be estimated from the expected 10million-cycle fatigue life:

Life

10 x l o 6 cycles to fail =

(310 cycles/s) (3600 s/h)

= 8.96

hours to fail

(8.13)

A resonant frequency of 310 Hz will be difficult to achieve; a stiffening rib will probably be required. A stiffening rib will use up some of the valuable surface area of the PCB, which may reduce the number of electronic components that can be mounted on a tightly packaged PCB. When several PCBs require stiffening ribs, the amount of surface area may be reduced, so an additional PCB may have to be added to the system. This may result in a larger, heavier, and more expensive assembly, which will make many company executives very unhappy. These factors must be considered with respect to a possible qualification test failure, or to failures in the field.

HOW LOCATION AND ORIENTATION OF COMPONENT ON PCB AFFECT LIFE

175

The probable fatigue life of 8.96 hours to fail, as shown by Eq. 8.13, is not really a long period of time. An examination of MIL-STD-810D shows it requires sinusoidal sweep tests of 1 hour per axis for three axes, plus resonant dwell periods of 30 minutes at each of four major resonances, in each of three axes, for a total resonant dwell period of 6 hours, and a total testing time of 9 hours for one qualification test. Past experience in qualification test programs has shown that many different types of problems and failures can occur, which will require the test to be repeated. Sometimes these tests may have to be repeated several times before the whole qualification test can be completed. A safety factor is therefore necessary to ensure the successful completion of the tests. A recommended practice is to design the electronic system with enough fatigue life to permit the system to pass five qualification tests. This does not represent a safety factor of 5, as typically computed using stress levels as criteria. Instead it represents a safety factor of 1.286, since the fatigue life is related to the stress level raised to the 6.4 power as shown by Eq. 8.3: Fatigue-life change

=

( ~ 2 8 6 )=~5.0 '~

(8.14)

8.5 HOW LOCATION AND ORIENTATION OF COMPONENT ON PCB AFFECT LIFE

Electronic components can be mounted in any location and in any orientation on a PCB. Rectangular components are usually mounted with their sides parallel to the sides of rectangular PCBs. Smaller components, less than about 1 in., very seldom cause any vibration problems unless the component happens to be a very tall and heavy transformer. This type of component is not considered here. Larger components, greater than about 1 in. are where vibration problems typically start, depending on the type of PCB and the vibration-level requirements. Vibration problems become very severe when the component size reaches 2 in. or more. When plug-in PCBs are being evaluated, the most critical location for large components will be at the center of the PCB. This is where the most rapid change of curvature occurs during the fundamental resonant frequency of a PCB with simply supported edges. As the component is moved away from the center, the curvature is reduced, so the relative motion between the PCB and the component body is reduced, as shown in Fig. 8.5. This reduces the forces and stresses in the lead wires and solder joints, so the fatigue life is increased. The most dangerous orientation for a long component is parallel to the shorter side on a rectangular plug-in PCB. This is due to the more rapid change of curvature of the shorter side than of the longer side of the PCB. This results in more relative motion between the PCB and the component, which increases the loads and stresses in the lead wires and solder joints as

176

PREVENTING SINUSOIDAL VIBRATION FAILURES Large relatlve rnat10n

__

,

Small relative no!icl J

1'-

FIGURE 8.5. Relative motion between the component and the P C B is reduced when the component is mounted near the P C B edge.

it--L---L-

Sma I rela!lve notlo'-

Large relative rnotlon

S a n e d i s p l a c e m e i t a t center

FIGURE 8.6. Relative motion between the component and the P C B is reduced when the component is mounted parallel to the long edge of the PCB.

shown in Fig. 8.6. Increased stresses are undesirable because they reduce the fatigue life of the system. The forces and stresses in the lead wires and solder joints of electronic components mounted on PCBs can be related to the relative curvature of the PCB during its resonant condition. The relative curvature can be related to the displacement of the PCB, and the displacement can be related to the location of the component on the PCB. When the edges of the PCB are simply supported (or hinged), then the displacement at any PCB location can be determined from the following relation:

Z

=Z,

7Tx

sin-sina

irY b

(see Eq. 6.1)

HOW WEDGE CLAMPS AFFECT THE PCB RESONANT FREQUENCY

177

When the component is located at the center of the PCB, X is a / 2 and Y is b/2. The PCB displacement at the center simply becomes 7 T 7 T

Z

=Z,

sin-sin2 2

=Z,

(8.15)

When the component is located off center, at the position where X is a / 2 and Y is b/4, then the PCB displacement is 7

Z

T

X

= 2, sin -sin - = 0.707Z,

2

4

(8.16)

When the component is located at the quarter mounting points, X is a / 4 and Y is b/4. The PCB displacement at this point is

Z

=Z,

X

T

sin -sin - = o.50ZO 4 4

(8.17)

These relative displacements at different positions on the PCB will determine the forces and stresses in the component lead wires and solder joints, which will have a direct effect on the fatigue life of these components. These relative displacements can therefore be used to find the approximate fatigue life of different types of components, in different environments, based on the component location on the PCB. 8.6 HOW WEDGE CLAMPS AFFECT THE PCB RESONANT FREQUENCY

Wedge-clamp-type edge guides are often used to grip the side edges of the PCB to reduce the thermal resistance for lower temperatures and to increase the PCB resonant frequency. A typical wedge clamp is shown in Fig. 8.7. Vibration tests with wedge-clamp edge guides have shown that their effectiveness is related to the stiffness of the PCB, and therefore to the resonant frequency of the PCB. A PCB with a low resonant frequency, less than 100 Hz, is not very stiff. The wedge clamp is then capable of holding the PCB edge guide rigidly to prevent any rotation during the resonant condition, when the input acceleration level is less than about 8 G peak.

FIGURE 8.7. Wedge-clamp edge guides grip the PCB edges firmly, which increases the PCB stiffness Wedge clamD

and resonant frequency.

178


? -

z

P,

L

FIGURE 8.8. Curve for estimating the per-

centage fixity for a wedge-clamp edge guide.

;r

i -i*-

I?C 366

693

1,

ireaJercy

-2

A PCB with a high resonant frequency, greater than about 600 Hz, is very stiff. The wedge clamp is not capable of holding the PCB edge guide rigidly to prevent rotation, so some rotation will occur at this interface. The higher the frequency, the greater the amount of relative motion. Thus the frequency can be used to estimate the percentage of fixity for the stiffness characteristics of a wedge clamp, as shown in Fig. 8.8. When wedge clamps are used as side edge guides for plug-in PCBs, the accuracy of the resonant-frequency calculation can be improved by using the following equation:

f n =fs

+Pf(ff - f s >

(8.18)

where f , , = expected natural frequency of PCB f , = natural frequency of PCB with simply supported sides f t = natural frequency of PCB with fixed (clamped) sides Pf = percentage fixity of PCB side edges from Fig. 8.8 The fixity cannot be determined until the resonant frequency is known, and the resonant frequency is not known yet. However, a closed-form solution can be obtained by using similar triangles in Fig. 8.8 as follows:

f,,- 100 600 - 100 (fn

-

-

1.00 - Pf 1.00 - 0.50

100)(0.50) =(500)(1.00-Pf) Pt = 1.1 - O.0Olfn

(8.18a)

Substitute Eq. 8.18a into Eq. 8.18 and solve for f,:

f n = f s + (1.1 -0.00lfn>(ff-fs) f s + l W f *-f,> fn =

1

+ 0.001( ff - f , )

(8.19)

The use of this equation can be demonstrated with a sample problem.


Sample Problem-Resonant Clamps

179

Frequency of PCB with Side Wedge

Determine the resonant frequency of the 0.5-lb plug-in PCB with side wedge clamps shown in Fig. 8.9. Rubber strips under the top cover of the chassis are expected to grip the top edge of the PCB, so the top of the PCB will act as a simply supported (hinged) edge. The electrical connector across the bottom edge of the PCB is also expected to support this edge, so it acts like a hinge. For a preliminary analysis, assume a uniformly distributed weight across the face of the PCB. Solution. When the resonant frequency of the PCB is less than about 100 Hz, the side wedge clamps will prevent rotation of the side edges. The PCB resonant frequency can then be determined by assuming the side edges are clamped. When the PCB resonant frequency is greater than about 600 Hz, the side wedge clamps are not capable of preventing rotation, so the side edges cannot be evaluated for a clamped condition. Under these circumstances, the sides will act somewhere between a fixed edge and a hinged edge. A good approximation of the resonant frequency for the PCB with side wedge clamps can be obtained from Eq. 8.19. In order to use this expression, it is necessary to determine the resonant frequency of the PCB with two different side edge conditions. The first condition is with hinged side edges, and the second condition is with fixed side edges. First Condition: PCB with Hinged Side Edges For the first condition, all four edges of the PCB will act as hinges. The resonant frequency can be determined from Eq. 6.21: 1 \ m i 1 p ( a- 2+ - b 2 )

f "= -21:'

Top rubber st' ps

6 0-

FIGURE 8.9. Plug-in PCB with supported (hinged) top and bottom edges, and wedge clamps at the sides (dimensions in inches).

180


where E h

= 3.0

x lo6 I b / h 2 (modulus of epoxy glass with copper planes)

= 0.10

in. (thickness of epoxy fiberglass PCB) (Poisson's ratio for epoxy glass with copper planes) W = 0.50 Ib (weight of PCB) g = 386 in./s2 (acceleration of gravity) a = 6.0 in. (length of PCB) b = 4.0 in. (width of PCB) D = Eh3/12(l - p ' ) = (3 X 106)(0.100)'/(12)(1 - 0.0324) = 258.4 Ib . in. p = W/gab = 0.50/(386)(6.0)(4.0) = 0.000054 Ib * s 2 / h 3 p = 0.18

Then f n = ~ 0.000054 / ~ ( i +36 i / 16 = 3 ~ o H z

(8.20a)

Second Condition: PCB with Fixed Side Edges The resonant frequency of the PCB with opposite edges clamped and opposite edges supported can be determined from Fig. 6.12. Note that clamped edges as defined here mean that the side edges of the PCB do not undergo any rotation during the resonant condition. We have 7~

where D

8

10116

31

258.4 Ib in. p = 0.000054 Ib s'/inS3 =

n = 6.0

b

=

in.

4.0 in.

Thus ir fn =

4

3.46 45:::.0

(&

8

++ 576

-1

3 256

=387Hz

(8.20b)

P CB with Side Wedge Clamps, Supported on Top and Bottom Edges The resonant frequency of the PCB with side wedge clamps can be determined using two different methods. In the first method a trial-and-error solution is used with Fig. 8.8 to find the percentage fixity. In the second method a direct solution for the PCB resonant frequency is obtained in closed form, using Eq. 8.19.


181

Trial-and-Error Method .for Resonant Frequency. Assume a resonant frequency of 366 Hz. From Fig. 8.8 find the percentage fixity using similar triangles:

X

-

100 - 50

366 - 100 600 - 100

or X = 2 6 . 6 %

This value is measured from the top down, so that one has 100 - 26.6 = 73.4% fixed edges at the wedge clamps. Substitute into Eq. 8.18 to obtain the calculated PCB resonant frequency,

f, = 310 + (0.734)(387 - 310)

= 366.5

HZ

(8.21a)

Since the assumed PCB resonant frequency of 366 Hz is approximately the same as the calculated value of 366.5 Hz, the correct PCB resonant frequency with the side wedge clamps is about 366 Hz. Closed-Fom Solution for Resonant Frequency. The resonant frequency of the PCB with side wedge clamps can also be determined directly with the use of Eq. 8.19, where

f,= 310 Hz for simply supported (hinged) sides (see Eq. 8.20a), and

ff = 387 Hz

for clamped sides

(see Eq. 8.20b). Thus 310 fn =

+ (1.10)(387 - 310)

1 + (0.001)(387 - 310)

= 366.4

HZ

(8.21b)

Wedge-clamp side edge guides will increase the resonant frequency of the

PCB,which will reduce the dynamic displacements and improve the fatigue life of large components mounted on the PCB.The fatigue-life improvement is not quite as good as might be expected from the reduction of the dynamic displacement alone. This is due t o the differences in the mode shapes for the simply supported side edge guides and the clamped side edge guides as shown in Fig. 8.10.

Supported sides

ClamFed s des

Small -3 -Large

P C 6 Gendtng

"

FIGURE 8.10. Differences in relative motion between the component and the PCB for supported sides and clamped sides.

182


An examination of the relative displacements between the component body and the PCB for large components shows that the clamped-edge condition will result in a slightly larger relative displacement. This means that the fatigue life for a large component mounted at the center of a PCB with a clamped-edge condition will not be quite as good as the fatigue life of the same component mounted on the same PCB with a simply supported edge condition, when the dynamic displacements of both PCBs are the same. 8.7

EFFECTS OF LOOSE PCB SIDE EDGE GUIDES

Plug-in PCBs must have some type of guide at the sides to control the position and maintain alignment accuracy for the blind-mating electrical connector at the bottom edge of the PCB. Many different types of edge guides are available, from spring clips to wedge clamps. Sometimes slots or grooves are cast or machined in the chassis side walls to act as edge guides. These slots or grooves are usually made extra wide to allow for manufacturing and assembly tolerances on the chassis and the PCBs. The end result is a loose side edge guide, which will not adequately support the PCB in a vibration or shock environment. The idea behind this looseness is to reduce manufacturing costs and still provide a rugged electronic chassis assembly. At first glance it might appear that a loose PCB side edge guide might even act like an isolation system, which would reduce the acceleration levels transferred to the PCBs by the chassis. However, vibration test data show that just the opposite is true. A loose PCB side edge guide can actually increase the acceleration transferred to the PCBs in a vibration and shock environment, with light damping [17]. Investment castings are often used for electronic chassis housings, and it is tempting to cast slots in the chassis side walls to be used as PCB edge guides. These slots are almost free, since you only pay for the slots in the initial tooling cost; so machining costs are reduced. However, these cast-in slots should not be used, because they can sharply increase the acceleration levels and stresses in the PCBs. This can reduce the fatigue life of the system. Vibration test data on chassis assemblies with loose PCB side edge guides showed that the PCB resonant frequency was always coupled to the chassis resonant frequency, even when the chassis resonant frequency was increased to try to separate the two resonances. When the PCB has loose side edge guides, the period of the vibration depends on its amplitude. The clearance of the loose PCB edge guide as well as the spring rate of the PCB and the acceleration level will determine the resonant frequency of the PCB. For a given set of conditions, a higher input acceleration level will result in a higher PCB resonant frequency. A smaller clearance will also result in a higher PCB resonant frequency. The effects of a loose PCB side edge guide can be evaluated by considering the PCB to be a mass vibrating between two springs when there is a clearance distance a between the mass and the spring for some initial condition, as shown in Fig. 8.11.

EFFECTS OF LOOSE PCB SIDE EDGE GUIDES

4 a L-

y\

183

4Fa

Clearance

Plug-in connector acts as a hinge

FIGURE 8.11. A loose edge guide permits the PCB to slap back and forth through the clearance, resulting in a reduced PCB resonant frequency.

The resonant frequency of a PCB with loose side edge guides can be determined by computing the PCB period of vibration in three steps:

Step 1: First compute the resonant period of the PCB with a zero side edge clearance. The period is simply the reciprocal of the resonant frequency of the PCB with simply supported (or hinged) edges. Step 2: Next compute the period from the velocity of the PCB as it swings through the loose side edge clearance. The velocity will be proportional to the input acceleration and the transmissibility. Step 3: Simply add the periods from steps 1 and 2, and invert, to obtain the true PCB resonant frequency with the clearance. The velocity of the PCB can be determined from the edge clearance a, considering harmonic motion and starting with the displacement: Y = a sin R t

(8.22)

The first derivative of displacement is the velocity:

dY

I/= - = Ru cos Rt

(8.23)

dt

The maximum velocity will occur when cos R t

=

1.0:

v= f l u Let R

=

2 r f , so

v= 2 T f U

(8.24)

184


The single-amplitude displacement due to vibration was shown in Eq. 2.30. Substitute this value into Eq. 8.24 for the velocity: V=

19.6rrGinQ

fn

(8.25)

The total period of the PCB can be obtained from the motion through one complete cycle. When the PCB is in contact with the springs, it is hard against the stops with no more side edge clearance. For step 1, the period is just the reciprocal of the PCB resonant frequency:

P,= 2 s g

(period of PCB)

(8.26)

For step 2, the PCB period depends on the clearance a and the velocity of the PCB as it passes through the clearance. Since the total clearance during one full cycle is 4a, and velocity x time is distance, the period of the PCB passing through the clearance becomes 4a

7

(8.27)

4afn 19.6rrGinQ

(8.28)

P2

=

Substitute Eq. 8.25 into Eq. 8.27,

P,

=

The total period of the PCB is the sum of the two periods:

P = P , + P2 = 27T +

4afn 19.6rrGinQ

(8.29)

The resonant frequency of the PCB is the reciprocal of the period P : f

1

= -

"

P

(8.30)

If a periodic disturbing force has a frequency lower than the PCB resonant frequency (with no side edge clearance), it will always be possible to set up a condition where the PCB resonant frequency (with side edge clearance) will become equal to the disturbing-force frequency and produce severe coupling effects. In other words, when loose PCB edge guides are used in a chassis and the chassis has a lower resonant frequency than the PCB (with no side edge clearance), it will always be possible to set up a condition where the PCB resonant frequency (with side edge clearance) will become equal to the chassis resonant frequency and produce severe coupling effects on the PCB. This increases the stress levels in the PCB, which reduces the fatigue life [171.

SINE SWEEP THROUGH A RESONANCE

Sample Problem-Resonant Loose Edge Guides

185

Frequency of PCB with

A plug-in PCB has a resonant frequency of 187 Hz when the PCB is simply supported on all four sides. The PCB is to be used in a chassis that has clearance slots in the side walls instead of side edge guides. The single-amplitude clearance expected between the PCB and the chassis side wall slots will be 0.008 in. minimum and 0.012 in. maximum. Determine the expected PCB resonant frequency range for a sinusoidal vibration input level of 6.0 G peak. Solution 0.008-in. Clearance Between PCB and Chassis Side Wall Slot Test data show that loose side edge guides result in higher PCB transmissibility Q values, because of more severe dynamic coupling between the chassis and the PCB. Therefore a slightly higher than normal Q must be used in this solution. The following information is required:

f,, = 187 Hz (PCB resonant frequency, no side edge clearance) PI = 1/187 = 0.005347 s (PCB period, no side clearance) a = 0.008 in. (minimum side edge clearance) Gin= 6.0 G (peak input acceleration level) Q = 1 . 2 m = 16.4 (expected PCB transmissibility) Substitute into Eq. 8.29 for the resonant frequency of the PCB with side edge clearance:

P = 0.005347

(4)(0.008)( 187) + (:19.6)( T )(6)( 16.4)

= 0.00633

second

1L

fn=

o.00633 = 158 HZ

(for PCB)

(8.31)

0.072-in. Clearance Between PCB and Chassis Side Wall Slot

P = 0.005347

(4) (0.012) (187) + ( 19.6) ( (6) (16.4)

=

0.00683 second

T )

1 fn=--

8.8

0.00683

-

146 Hz

(for PCB)

(8.32)

SINE SWEEP THROUGH A RESONANCE

Several military specifications (MIL-E-5400, MIL-T-5422, and MIL-STD-810) recommend using sinusoidal sweeps through resonant points as part of a qualification test on many different types of electronic systems. Such a sine

186


I

--J 1' - /n (c'

--

I

-L_*

rreaiie"

I

I

F-

FIGURE 8.12. Half-power points for a transmissibility curve.

sweep test is also convenient for evaluating electronic subassemblies, such as PCBs, to determine their resonant frequencies and transmissibilities. The test will usually start at a low frequency, around 10 Hz, with a sweep up to 2000 Hz, then back to 10 Hz once again. A logarithmic sweep is usually used, so that the time to sweep from 10 to 20 Hz is the same as the time to sweep from 1000 to 2000 Hz. The sweep rate is usually specified in terms of octaves per minute. an octave being a factor of 2 in frequency. This rate is thus related to the time it will take to sweep from 10 to 20 Hz, from 20 to 40 Hz, from 40 to 80 Hz, and so on. Most of the damage accumulated during a sweep through the resonant points of an electronic structural assembly will occur near the peak response points. A convenient reference is the half-power points, used extensively by electrical engineers to characterize resonant peaks in electronic circuits. These are the points where the power that can be absorbed by damping is proportional to the square of the amplitude at a given frequency. For a lightly damped system, where the transmissibility Q is greater than about 10, the curve in the region of the resonance is approximately symmetrical. The half-power points are often taken to define the bandwidth of the system as shown in Fig. 8.12. The time it takes to sweep through the half-power points can be determined from the following expression:

t=

R log, 2

where t = time (minutes) R = sweep rate (octaves/min) Q = transmissibility at resonance (dimensionless)

(8.33)

SINE

Sample Problem-Fatigue

SWEEP THROUGH A RESONANCE

187

Cycles Accumulated During a Sine Sweep

A qualification test program requires several sine vibration sweeps through the 320-Hz resonant frequency of a PCB with a transmissibility Q of 20. Previous vibration tests were run on a similar PCB using the same input acceleration level with a resonant dwell condition, which demonstrated a component fatigue life of approximately 10 min. (a) How many fatigue cycles are accumulated during one sweep through the resonance using a sweep rate of 0.5 octaves/min? (b) How many single sweeps through the resonant point are required to produce a failure? Solution. (a) The time it takes to sweep through the PCB resonant point can be determined from Eq. 8.33, where t = time (minutes) R = 0.5 octaves/min (sweep rate) Q = 20 (transmissibility of PCB)

The number of fatigue cycles accumulated during one sweep through the resonant frequency of 320 Hz will be

n = (320 cycles/s) (8.66 s)

= 2771

cycles per sweep

(8.35)

(b) The number of single sweeps through the resonant point to produce a fatigue failure can be determined as follows:

10 min to fail NS =

0.144 min/sweep

= 69.4

sweeps to fail

(8.36)

-

CHAPTER 9

Designing Electronics for Random Vi bration

9.1

INTRODUCTION

Random vibration is being specified for acceptance tests, screening tests, and qualification tests by commercial, industrial, and military manufacturers of electronic equipment, because it has been shown that random vibration more closely represents the true environments in which the electronic equipment must operate. This includes airplanes, missiles, automobiles, trucks, trains, and tanks as well as chemical processing plants, steel rolling mills, foundries, petroleum drilling machines, and numerically controlled milling machines. Random vibration has also proved to be a very powerful tool for improving the manufacturing integrity of electronic equipment by screening out defective components and defective assembly methods, which results in a sharp improvement in the overall reliability of the system. Electronic packaging designers and engineers must understand the fundamental nature of random vibration and fatigue, in order to design, develop, and produce cost-effective and lightweight structures that are capable of operating in the desired environments with a high degree of reliability. They must examine the path that the dynamic load takes as it passes through the structure, to make sure there are no weak links that can cause catastrophic failures. They must also examine the load-carrying capability of the structural elements, to make sure they will not buckle under the expected dynamic loads. They must constantly be on the alert for major structural resonances that can magnify dynamic loads and stresses. If two major structural resonances occur close to one another, they may produce severe dynamic coupling effects that can produce rapid fatigue failures.

9.2

BASIC FAILURE MODES IN RANDOM VIBRATION

There are four basic failure modes that must be considered and controlled in order to produce a reliable electronic system. These failures are the results of 188

CHARACTERISTICS OF RANDOM VIBRATION

189

the following conditions: 1. 2. 3. 4.

High acceleration levels. High stress levels. Large displacement amplitudes. Electrical signals out of tolerance.

Considering high acceleration levels, there are a number of electronic components, such as relays and crystal oscillators, that will malfunction electrically when their internal resonances are excited. These components will not suffer catastrophic failures during the vibration; they will just not operate properly. For example, if the relays chatter, the electrical signal may be impaired and an electrical failure can occur. When the resonance is excited in a crystal oscillator, the electronic signal may be distorted enough to produce an electrical failure. Therefore, it is important to use electronic components that will be capable of proper operation in the expected environment. It is also important to locate these sensitive electronic components in areas that will not respond to severe resonant conditions. For example, mount the components at the edges of a plug-in PCB and not at the center. Considering high stress levels, if they occur in major structural elements, then catastrophic failures can occur. This can usually be avoided by increasing the stiffness of the structure to raise the resonant frequency. This reduces the dynamic displacements and stresses, so the fatigue life is improved. Considering large displacement amplitudes, this often results in collision between adjacent PCBs when they are too close together, or when insufficient attention has been given to the placement tolerances. Collision between components on adjacent PCBs can result in broken and cracked components, circuit traces, and solder joints. PCB resonant frequencies must be high enough to limit the dynamic displacements, and positioning tolerances must be controlled to properly locate the PCBs. Considering electrical signals out of tolerance, this can be caused by relative motion in cables and harnesses, high temperatures, relative motion within some capacitors, resonances within crystal oscillators, a slipping potentiometer, relative motion in a transformer core, and excessive motion in a tube filament.

9.3 CHARACTERISTICS OF RANDOM VIBRATION

The most obvious characteristic of random vibration is that it is nonperiodic. A knowledge of the past history of random motion is adequate to predict the probability of occurrence of various acceleration and displacement magnitudes, but it is not sufficient to predict the precise magnitude at a specific instant.

190

DESIGNING ELECTRONICS FOR RANDOM VIBRATION

1

D sp'ace-nent-trace

FIGURE 9.1. Typical plot of acceleration o r displacement with respect to time.

FIGURE 9.2. R a n d o m motion is made up of many different overlapping sinusoidal curves.

A typical time curve for random vibration would appear as shown in Fig. 9.1. Analysis of Fig. 9.1 reveals that the random motion can be broken up into a series of overlapping sinusoidal curves, with each curve cycling at its own frequency and amplitude, as shown in Fig. 9.2.

9.4 DIFFERENCES BETWEEN SINUSOIDAL AND RANDOM VIBRATIONS

Random vibration is unique in that all of the frequencies within a given bandwidth are present all of the time, and at any instant of time. This means that when an electronic system is subjected to a random vibration environment over a frequency bandwidth from 20 to 2000 Hz, all of the structural resonances of the electronic system within the same bandwidth will be excited at the same time. In other words, the fundamental resonant frequency of the chassis will be excited along with many of the higher harmonics. The same thing is true for all of the circuit boards in the system.

DIFFERENCES BETWEEN SINUSOIDAL AND RANDOM VIBRATIONS ,--Mass

191

2

V bration

direction

I

b Oil film

/

FIGURE 9.3. Two spring-mass systems with two different resonant frequencies on an

oil-film slider plate.

When the same electronic system is subjected to a sinusoidal sweep vibration input from 20 to 2000 Hz, each structural resonance in the given frequency band for the chassis and the various circuit boards will be excited individually. Since electronic systems respond differently in random vibration than in sinusoidal vibration, this means that fatigue failures due to random vibration can be different than fatigue failures due to sinusoidal vibration. This can be demonstrated by considering two cantilevered structures with two different resonant frequencies as shown in Fig. 9.3. Mass 1 is shown as having a low resonant frequency, and mass 2 a high resonant frequency. During a sinusoidal sweep, mass 1 with a lower resonant frequency will develop its resonant peak first. Mass 2, with a higher resonant frequency, will develop its resonant peak later, as shown in Fig. 9.4. While the resonant frequency of mass 1 is being excited, mass 2 is quiet, so mass 1 will not strike mass 2 and there will be no damage to the system. When the resonant frequency of mass 2 is excited, mass 1 will be isolated and so will be quiet. Therefore, mass 1 will not strike mass 2, and there will be no damage to the system.

Resonance of mass 1

81

fl

fz

fl

fz

FIGURE 9.4. Two masses with different resonant frequencies.

192


In the random vibration environment all of the exciting frequencies within the given bandwidth are present at the same time. Therefore, the forcing frequencies that correspond to the resonant frequencies of masses 1 and 2 are present at the same time, so masses 1 and 2 will be excited at their individual resonant frequencies at the same time. Both masses can then produce large displacement amplitudes at the same time, so they can strike each other and produce impact failures. This shows that random vibration environments can produce failures that cannot be duplicated in a sinusoidal vibration environment. 9.5

RANDOM VIBRATION INPUT CURVES

There are many different types of curves that can be used to show the random vibration input requirements. The most common curve, which is also the simplest, is the white-noise curve shown in Fig. 9.5. Random vibration input and response curves are typically plotted on log-log paper, with the power spectral density, expressed in squared acceleration units per hertz (G2/Hz), plotted along the vertical axis, and the frequency (Hz) plotted along the horizontal axis. The power spectral density P is often referred to as the mean squared acceleration density, and it is defined by G2 P = lim Af-0 Af

In the above equation, G is the root mean square (RMS) of the acceleration expressed in gravity units, and Af is the bandwidth of the frequency range expressed in hertz. Root mean square acceleration levels are related to the area under the random vibration curve. The input RMS acceleration levels can be obtained by integrating under the input random vibration curve, and the output (or response) RMS acceleration levels can be obtained by integrating under the

FIGURE 9.5. Typical white-noise curve with a constant input power spectral density (PSD).

fi

/1

iog (frequency)

.Y

RANDOM VIBRATION UNITS

10

193

2000 Frequency, Hz

FIGURE 9.6. White-noise input curve with a constant input PSD of 0.20 G2/Hz.

output (or response) random vibration curve. The square root of the area then determines the RMS acceleration level, with units

Sample Problem-Determining the Input RMS Acceleration Level

Determine the input RMS acceleration level resulting from the white-noise (flat-top) random vibration input specification shown in Fig. 9.6.

Solution. GRMs

=

GRMS =

9.6

JET= d0.20(2000 - 10) 19.95 (RMS input acceleration level)

(9.3)

RANDOM VIBRATION UNITS

Random vibration environments in the electronics industry normally deal in terms of the power spectral density P (or mean squared acceleration density), which is measured in gravity units so that it is dimensionless. That is, the acceleration is divided by the acceleration of gravity: a

acceleration

g

gravity

G=-=

(dimensionless)

An acceleration level of 10 G means that the acceleration has a magnitude that is 10 times greater than the acceleration of gravity. The most common method used for evaluating random vibration is in terms of the power spectral density. However, random vibration can also be expressed in terms of velocity spectral density, and in terms of displacement

194


\

Negative

(-

1 slope

spectral density as shown below: Quantity

Dimensions

Unit

(acceleration)'

Power SD

Velocity SD

(in./s'):

frequency

Hz

(velocity)' frequency

(in/s)' Hz

(displacement)'

Displacement SD

in.' -

Hz

frequency

Random vibration input curves are usually plotted on log-log paper, using straight sloping lines. A straight line sloping up to the right is considered to have a positive ( + ) slope, and a straight line sloping down to the right is considered to have a negative ( - ) slope, as shown in Fig. 9.7. Accelerations, velocities, and displacements for random vibration are typically expressed in terms of root mean square (RMS) values. These should not be confused with root sum square (RSS) values. The difference between the RMS and the RSS values for a set of numbers can be shown by selecting the set 2,6,10,14, and determining the RMS and the RSS values: RMS =

4

2'-+ 6'

+

RSS = \/22 + 6* 10' 9.7

+ 14'

=

9.165

+ 142 = 18.330

SHAPED RANDOM VIBRATION INPUT CURVES

Input random vibration curves can come in a wide variety of shapes, depending on the type of environment or condition the curve is trying to simulate. Two of the more common types of curves are shown in Fig. 9.8. The square root of the area under the curve still represents the input RMS acceleration level. Since these curves are plotted on log-log paper, special equations must be used to determine the areas under the sloped sections of the curves, Only straight vertical lines must be used to break up

SHAPED RANDOM VIBRATION INPUT CURVES

Frequency

195

Frequency

FIGURE 9.8. Two typical random vibration input PSD curves.

the area segments under the sloped sections of the curves. Horizontal lines must not be used. The subscript 1 below refers to the left side of the area segment being analyzed, and the subscript 2 refers to the right side. Areas under the positive- or negative-slope sections can be determined with the following equations, when the slope is not -3:

or (9.4a) When the sloped section of the PSD curve has a value of -3, then the following equations can be used to find the area under the curve:

A

= -f 2

fl

P, log, f 2

(9.5)

or A

f 2

= flP, loge-

(9 S a )

fl

The area under the flat-top section, where the slope is zero, can be determined from the following relation:

Sample Problem-Input

RMS Accelerations for Sloped PSD Curves

Determine the input RMS acceleration level for the random vibration curve when it is drawn on log-log paper, as shown in Fig. 9.9.

196


~;:l~-y Slope

Slooe

~

6 dB octave

FIGURE 9.9. Breaking u p an input PSD curve using straight vertical lines to produce three area sections.

5

75

200 2000

Frequency i i z

Solution. Vertical lines are used to break up the area into three sections. The area of each section is determined individually; then they are added together to obtain the total area under the curve. Area 7

Using Eq. 9.4, where fl

=5Hz

f2 =

75 HZ

P? = 0.20 G ~ / H ~

we have S

=3


A, =

Area 2

3(0.20) ~

3+3

dB/octave

1 [ 7t)i3 75 - -

( 5 ) =7.47G2

Using Eq. 9.6, A , = (0.20)(200 - 75)

Area 3

(slope)

=

25.0 G 2

Using Eq. 9.4, where p 2 = 0.002 G ~ / H Z

f,= 200 Hz fi = 2000

Hz

(9.7)

RELATION BETWEEN DECIBELS AND SLOPE

S = - 6 dB/octave

3(0.002)

A,

=

3-6

[

(slope)

[ ,'"o"o

2000-

)-6'3

(200)

-

]

= 36.0

G2

197

(9.9)

The total area under the curve leads to the RMS acceleration: A,=A, +A, +A,

7.47

=

GRMS= 468.47 = 8.27 G

+ 25.0 + 36.0 = 68.47 G2

(RMS input acceleration)

(9.10)

The areas under the PSD curves in the previous sample problem can also be determined by using integration methods. The slope in dB per octave must first be changed to a pure slope number. 9.8 RELATION BETWEEN DECIBELS AND SLOPE

The term decibel, as it is used in random vibration, is used to measure PSD ratios P,/P, as follows: Number of decibels (dB)

=

p2 10 log,, -

(9.11)

PI

When a PSD plotted on log-log paper gives a straight line, its slope can conveniently be expressed in dB/octave. For example, when the PSD at point 2 is two times greater than the PSD at point 1, then in terms of decibels this ratio is expressed as follows:

1010g1,(2)

=

lO(0.301)

= 3.01 dB

(9.12)

Another way of saying the same thing is that to increase by 3 dB means (nearly) to double; to decrease by 3 dB means (nearly) to cut in half. Figure 9.10 shows several frequently used positive and negative slopes expressed in dB/octave, using straight lines on a log-log plot. Note that a slope of 3 dB (which is the same as 3 dB/octave) is represented by a line at a 45" angle, which represents a pure-number slope value of one. A slope of 6 dB/octave then represents a pure-number slope value of two, which is represented by an angle of 63.4'. Another relation that is often convenient for plotting straight lines on log-log paper is as follows: 3 dB/octave

=

10 dB/decade

(9.13)

198


30 dB

1003

- 9 dB’octave

9 dB octave N

I

100

>. 5 d B ortave L

-6 dB’octave a ‘A a

10

3 dB octave

- 3 dB octave 1

‘1

2

10

100

1000

Frequency, Hz

FIGURE 9.10. Representing the sloped lines on a PSD plot in terms of decibels.

9.9 INTEGRATION METHOD FOR OBTAINING THE AREA UNDER A PSD CURVE

Areas under PSD curves can also be determined from the basic area relation using logarithm equations, Consider the following point-slope equation from analytical geometry:

Y=SX+b

(9.14)

where Y = log P (where P = PSD in G’/HZ> S = slope (pure number) X = log f (frequency) b = log b intercept on the Y (or PSD) axis Then log P

= S log f

+ log b

(9.15)

Rewriting the above expression in exponent form, P=bfs

(9.16)

The area under the curve can be determined from the standard integration equations as follows: A

= /’YdX 1

= /’Pdf 1

= 12bf 1

df

(9.17)

INTEGRATION METHOD FOR OBTAINING THE AREA UNDER A PSD CURVE

199

Starting with area 1 in Fig. 9.9, the b intercept on the P axis must be determined where the frequency f equals 1 Hz. This can be obtained from point 1 or point 2, since they are both on the same line with the same slope:

P,

= 0.0133

G2/Hz

f,=5Hz or

P,

= 0.20

G*/HZ

f 2 = 75 HZ

The slope S = 1.0 (equivalent to 3 dB/octave). Substitute into Eq. 9.16 for the intercept b: b=-

0.20 or -- 0.00266 G2/Hz (5)’ (75)’

0.0133

(9.18)

Substitute into Eq. 9.17 and integrate between the limits of 5 and 75 Hz for area 1 as shown in Fig. 9.9:

[;J

A , = ~ ~ 0 . 0 0 2 6 6 dff ’= 0.00266 -

0.00266 A, = 7 [ ( 7 5 ) , - ( 5 ) 2 ]= 7.46 G 2

(9.19)

i-

Comparing the results with Eq. 9.7 shows that the two different methods of analysis agree very well. The integration method for finding area 2 will be exactly the same as Eq. 9.8. The integration method for finding area 3 will be the same as the method shown for area 1, except that a new intercept b must be obtained using Eq. 9.16. The 200-Hz point or the 2000-Hz point can be used in Fig. 9.9, since they are both on the same line:

P,

= 0.2

f l = 200

G2/Hz Hz

or

p2 = 0.002 G ~ / H ~

f 2 = 2000 Hz

200


The slope S = - 2 (equivalent to - 6 dB/octave). Substitute into Eq. 9.16 for the intercept b: b=

0.20 (200) - 2

0.002 Or

(2000) -*

= 8000

G2/Hz

(9.20)

Substitute into Eq. 9.17 and integrate between the limits of 200 and 2000 Hz for area 3, as shown in Fig. 9.9: 2000

A,

=~

A,

=

~ ~ 0 8 0 0 0df f -=28000

8000 (200)-’] ~ [ ( 2 0 0 0 )- ~

= 36.0

G2/Hz

(9.21)

Comparing the results with Eq. 9.9 shows the two areas and the two different methods of analysis agree very well. 9.10 FINDING POINTS ON THE PSD CURVE

Random vibration input PSD curves are often specified in terms of the frequency break points and the slope in dB, with only one G2/Hz point defined as shown in Fig. 9.11. When it is necessary to find the PSD level at these break points, then it is convenient to use the relation

(9.22)

Sample Problem-Finding

PSD Values

Determine the PSD values at break points 1 and 2 as shown in Fig. 9.11.

Slope

FIGURE 9.11. Locating break points on a PSD curve.

Sloae

Frequency Hz

USING BASIC LOGARITHMS TO FIND POINTS ON THE PSD CURVE

201

Solution. At break point 1, f1=5Hz 75 HZ

f,

=

P,

= 0.20

S

=3

G~/HZ

dB/octave

Substitute into Eq. 9.22: 5

P, = 0.20(

3/3

E)

= 0.0133

G2/Hz

(9.23)

Use Eq. 9.22 to find the PSD level at break point 2:

Pz =

0.20 (200/2000)

= 0.002

G~/HZ

(9.24)

-6’3

USING B SIC LOGARITHMS TO FIND POINTS ON THE PSD CURVE

9.1

The two-point slope equation from analytical geometry can also be used to obtain PSD values at break points 1 and 2 in Fig. 9.11:

y , -Y,

= S ( x 2 -1 .>

(9.25)

Since the PSD relation is plotted on log-log paper, Fig. 9.25 has the following meaning: log Y2- log Y,= S(l0g x,- log X , )

(9.26)

At point 1 substitute the known values from Fig. 9.11 into Eq. 9.26 and note that a slope of 3 dB/octave is the same as a pure-number slope of 1.0 at point 1: log 0.2 - log Yl = ( 1.O) (log 75 - log 5) log Yl = - 1.875

Yl = 0.0133 G2/Hz

(9.27)

At point 2 substitute the known values from Fig. 9.11 into Eq. 9.26 and note

202


that a slope of - 6 dB/octave is the same as a pure-number slope of -2: log Y2- log 0.2 = ( - 2) (log 2000 - log 200) log Y, = - 2.7

y2= 0.002 G?/HZ

(9.28)

Comparing Eq. 9.23 with 9.27 and Eq. 9.24 with 9.28 shows they agree very well. 9.12

PROBABILITY DISTRIBUTION FUNCTIONS

In order to predict the probable acceleration levels the electronic equipment will see in a random vibration environment, it is necessary to understand probability distribution functions. The distribution most often encountered, and the one that lends itself most readily to analysis, is the Gaussian (or normal) distribution, which is defined by

e -x Y=

/2u

(9.29)

a+%

The right side of the above equation represents the probability density function, or the probability, per unit of X , for the ratio of the instantaneous acceleration ( X I to the RMS acceleration ( a1. 9.1 3

GAUSSIAN OR NORMAL DISTRIBUTION CURVE

The Gaussian distribution curve shown in Fig. 9.12 represents the probability for the value of the instantaneous acceleration levels at any time. The abscissa is the ratio of the instantaneous acceleration to the RMS acceleration, and the ordinate is the probability density, sometimes called the

-4

-3

-2

-1

0

1

2

3

4

-

'1

FIGURE 9.12. Gaussian distribution curve. T h e probability of a range of X is given by the area under the curve in that range.

GAUSSIAN OR NORMAL DISTRIBUTION CURVE

203

c x

0.1

46 (95.4%) 20

n W

-

001

R

fl

0

0027 (99 73%,) 3n

0001

Q

0 0001

0

1

2

3

n/n

FIGURE 9.13. Another method for showing the Gaussian distribution.

probability of occurrences. The total area under the curve is unity. The area under the curve between any two points then directly represents the probability that the accelerations will be between these two points. For example, the shaded area under the curve in Fig. 9.12 shows that the instantaneous accelerations will be between l a and - l a about 68.3% of the time. Figure 9.12 can be presented in another way, as shown in Fig. 9.13. This plot shows the probability that a given acceleration level will be exceeded. Figures 9.12 and 9.13 show how the Gaussian distribution relates to the magnitude of the acceleration levels expected for random vibration. The instantaneous acceleration will be between the + 1a and the - l a values 68.3% of the time. It will be between the +2a and the - 2 a values 95.4% of the time. It will be between the + 3 a and the - 3 a values 99.73% of the time. Another way of expressing the Gaussian distribution is shown in Fig. 9.13 as follows. The instantaneous acceleration will exceed the l a value, which is the RMS value, 31.7% of the time. It will exceed the 2a value, which is two times the RMS value, 4.6% of the time. It will exceed the 3 a value, which is three times the RMS value, 0.27% of the time. It is important to remember that in a random vibration environment, all of the frequencies in the bandwidth are present instantaneously and simultaneously. Likewise, the l a (or RMS), the 2a, and the 3 a acceleration levels are all present at the same time in the proportions shown above. Remember also that the square root of the area under the PSD-versus-frequency curve represents the RMS accelerations in gravity units (GI. The square root of the area under the input PSD curve represents the input RMS acceleration level, and the square root of the area under the response (or output) PSD curve represents the response RMS acceleration level. The maximum acceleration levels considered for random vibrations are the 30 levels, because the instantaneous accelerations are between the + 3 a and the - 3 a levels 99.73% of the time, which is very close to 100% of the time. Higher acceleration levels of 4a and 5a can occur in the real world, but they are usually ignored because virtually all of the test equipment for random vibration have 3 a clippers built into the electronic control systems.

+

204


FIGURE 9.14. A time history of a typical random vibration acceleration trace.

These clippers limit the input acceleration levels to values that are 3 times greater than the RMS input levels. A real plot of the random vibration acceleration or displacement with respect to time will appear as shown in Fig. 9.14. This same practice is typically applied to the response (or output) acceleration levels. For the purposes of analysis, the maximum acceleration levels considered are the 3a levels. This means that when a 10-G RMS random vibration environment is being evaluated (it does not matter if it is an input or a response), then 2a accelerations of 2 X 10 or 20 G can be expected some of the time, and 3a accelerations of 3 X 10 or 30 G can be expected some of the time. Most of the damage will be generated by the 3a levels, since they are the maximum levels expected in this environment. Displacements, forces, and stresses will occur in exactly the same proportions as the accelerations described above, for linear systems. In other words, the maximum displacements, forces, and stresses expected in a random vibration environment will be 3 times greater than the RMS displacements, forces, and stresses. Random vibration qualification tests performed in a laboratory will typically be run using much higher acceleration levels than the values found in the actual environments. The laboratory may use input test levels of 8.0 G RMS for a period of 2 hours, to try to accumulate the same amount of damage that is expected in a 1.5-G RMS environment over a 15-year period. Random vibration qualification tests are often considered to be accelerated life tests.

9.14 CORRELATING RANDOM VIBRATION FAILURES USING THE THREE-BAND TECHNIQUE

Random vibration is a complicated phenomenon that often produces rapid and unexpected fatigue failures. Many large companies have extensive computer databases that contain a vast amount of stress and fatigue data from testing programs on specific types of structures. The computers are used to sort the fatigue data to evaluate the approximate fatigue life of other similar structural members in similar environments. Many small companies do not have the time or money to run large testing projects with sophisticated

RAYLEIGH DISTRIBUTION FUNCTION

205

programs on high-speed digital computers. A simplified approach to the evaluation of random vibration fatigue, without the use of a large computer, would therefore be a valuable tool. The three-band technique offers such a tool. This technique is based on a large volume of test data that was arranged and rearranged to see if a simplified method could be developed to analyze random vibration fatigue failures with reasonable accuracy. The basis for the three-band technique is the Gaussian distribution. The l a and - l a are assumed to act at instantaneous accelerations between the l a level 68.3% of the time. The instantaneous accelerations between the + 2a and the - 2 a are assumed to act at the 2a level 95.4 - 68.3, or 27.170, of the time. The instantaneous accelerations between +3a and -3a are assumed to act at the 3a level 99.73 - 95.4, or 4.3370,of the time. These values are shown below for the three bands:

+

l a values occur 68.3% of the time 2a values occur 27.1% of the time 30. values occur 4.33% of the time

(9.30)

RAYLEIGH DISTRIBUTION FUNCTION

9.15

The Rayleigh distribution function shows the probability for the distribution of the peak accelerations in a random vibration environment. The curve is shown in Fig. 9.15. The total area under the curve is equal to unity. The area under the curve between any two points then represents the probability that a peak amplitude will be between these two points. Another way in which the Rayleigh distribution can be presented is shown in Fig. 9.16. This plot shows what the probability is for exceeding some peak acceleration level. The Rayleigh distribution shows the following relations: Peak accelerations will exceed the l a level 60.7% of the time. Peak accelerations will exceed the 2a level 13.5% of the time.

0

1

2 no

3

4

FIGURE 9.15. The curve.

Rayleigh

probability

distribution

206


0 0!2

0 01

(98 B o o )

0 001 0 0003 (999 7 % ) 0 0031

0

1

2

3

4

0,'"

FIGURE 9.16. Another method for showing the Rayleigh distribution.

Peak accelerations will exceed the 3a level 1.2% of the time. Peak accelerations will exceed the 4a level 0.03% of the time. Extensive random vibration testing and analysis have shown that fatigue failures can be correlated with much greater accuracy using the Gaussian distribution rather than the Rayleigh distribution. Therefore, only the Gaussian distribution will be used for fatigue failure analysis with random vibration in this book.

9.16 RESPONSE OF A SINGLE-DEGREE-OF-FREEDOM SYSTEM TO RANDOM VIBRATION

When a single spring-mass system is excited by random vibration, the system responds by vibrating at its resonant frequency, with a varying displacement amplitude, as shown in Fig. 9.17. When the base of the single-degree-of-freedom system is excited by a continuous random vibration whose spectrum is nearly flat in the area of the resonance, the mean square acceleration response of the mass can be obtained by considering the area under the PSD-versus-frequency curve as defined in Eq. 9.2. The area can be obtained by integrating across the

Randon

!

Input

FIGURE 9.17. A displacement history for the response of a spring-mass system to a

random vibration input.

207

RESPONSE OF A SINGLE-DEGREE-OF-FREEDOMSYSTEM

frequency band from f i and

f2

Area

using the output PSD (Pout)as follows: = G& = /"Pout df

(9.31)

fl

The response (or output) Poutis needed to obtain the response of the mass to the random vibration input. Since only the input PSD P is usually known, it is desirable to use P in the above equation. This can be accomplished by using the following expression [l]:

Pout= Q2P

(9.32)

The transmissibility Q for the single-degree-of-freedom system is obtained from Eq. 2.48. For a lightly damped system, where the Q is greater than about 10, the transmissibility will be as shown in Eq. 2.46. Substitute Eqs. 2.46 and 9.32 into Eq. 9.31:

The above equation can be integrated to obtain the mean square acceleration response of the mass to the random vibration input: (9.34)

In a lightly damped system the damping ratio can be related to the transmissibility as follows: (9.35) Substitute into Eq. 9.34 to obtain the RMS response of the mass to the broadband random vibration input: (9.36) where P = input PSD in units of G2/Hz at the resonant frequency f, = resonant frequency (Hz) Q = transmissibility at the resonant frequency

208


--Random

--_,Response

v bration input curve

for a simple system

Freauency. Hz

FIGURE 9.18. The random vibration input PSD curve should be flat in the area of

the resonance.

The above equation is valid when the random vibration PSD input is flat in the area of the resonance, as shown in Fig. 9.18. The amount of error is small even when the slope of the random vibration input curve is 6 dB/octave in the area of the resonant frequency. Therefore, Eq. 9.36 can be used to obtain good values under most conditions. Sample Problem-Estimating

the Random Vibration Fatigue Life

A 6061-T6 cantilevered aluminum beam 6.0 in. long by 0.50 in. wide by 0.30 in. high has a transformer mounted at the end of the beam as shown in Fig. 9.19. The total weight of the assembly is 0.50 Ib, and it is restricted to move only in the vertical direction. The assembly must be capable of operating in a white-noise random vibration environment with an input PSD level of 0.30 G*/Hz, from 20 to 2000 Hz, for a period of 4.0 hours. Determine the approximate dynamic stress and the expected fatigue life of the assembly. Solution No. 7 Use the Three-Band Technique and Miner's Cumulative Damage Ratio. The natural frequency is the heart of the system. The analysis can be simplified by finding the static displacement of the beam with an end mass and consider it to be a single-degree-of-freedom system. This will save time and the resulting error is small.

l+jJ

-

6.0 Sect. AA

FIGURE 9.19. Cantilever beam with an end mass.


209

A. Static Displacement of the Cantilever Beam with an End Mass

y

WL3 =-

3EI

St

(see Eq. 2.11)

where W = 0.50 Ib (total weight) L = 6.0 in. (beam length) E = 10.5 X lo6 Ib/im2 (aluminum modulus of elasticity) I = (0.50)(0.30)3/ 12 = 0.00112 in.4 (moment of inertia)

(0.50) ( 6.0)3

E’=

(3)(10.5 X 106)(0.00112)

st

= 0.00306

in. (static displacement)

The natural frequency can be obtained using Eq. 2.10 when gravity is 386 in./s2:

6. Natural Frequencj Based on the Static Displacement and Gravit]

C. Response of Beam to Random Vibration The response of the beam to random vibration can be obtained from Eq. 9.36 using the following information:

P = 0.30 G2/Hz (PSD input) f,, = 56.5 Hz (beam natural frequency) Q =2fi =2 m = 15 (approximate beam transmissibility) G (RMS)

=

/:(0.30)(56.5)(15)

= 20.0

G

(RMS acceleration response)

(9.38)

D. Dynamic la RMS Bending Stress in the Beam Acting 68.3% of the Time Alternating stresses must always include stress concentrations when thousands of stress cycles are expected in a structure. The stress concentration K can be used in the stress equation or in defining the slope b of the S-N fatigue curve for alternating stresses. The stress concentration should be used only once in either place. For this sample problem a stress-concentration factor K = 2 will be used in the S-N fatigue curve as shown in Fig. 9.22, where slope b = 6.4. MC

S, = - (see Eq. 2.39) I

210


Frequency,

Hz

FIGURE 9.20. Transmissibility Q response plot of the cantilever beam with a n e n d mass.

-67.5

56.5 Frequency, Hz

FIGURE 9.21. PSD response plot of the cantilever beam with a n e n d mass.

WG (RMS) L = (0.50)(20.0)(6.0) = 60.0 lb .in. (RMS bending moment) c = 0.30/2 = 0.15 in. (distance to neutral axis) I = 0.00112 in.4 (moment of inertia)

where M

=

A plot of the transmissibility curve versus frequency will appear as shown in Fig. 9.20. A plot of the PSD response curve can be obtained with the use

RESPONSE OF A SINGLE-DEGREE-OF-FREEDOM SYSTEM

I

21 1

6061-T6 Aluminum

I

103

108

N,cycles to fail FIGURE 9.22. S-N fatigue curve for 6061-T6 aluminum beam with a stress concen-

tration of 2. of Eq. 9.32. The resulting curve is shown in Fig. 9.21. la&=

(60.0)( 0.15) 0.00112

= 8036 lb/in.2

(RMS bending stress) (9.39)

E. Number of Stress Cycles Needed to Produce a Fatigue Failure Using the Three-Band Method The approximate number of stress cycles Nl required to produce a fatigue failure in the beam for the l a , 2 a , and 3a stresses can be obtained from Fig. 9.22 and the following equation: b

(seeEq.3.7) where

N,= 1000 cycles to fail at 45,000 lb/in.* (reference point) S,

= 45,000

lb/in.* (stress to fail reference point) S, = 8036 1 b / h 2 ( l a RMS stress) b = 6.4 (slope of fatigue line with stress concentration K careful to use the stress concentration K only once)

RMS l a N ,

(

45,000

]

be

6.4 = 6.14 X

lo7 cycles to fail

2 0 N2= (1000)

= 7.27 X

l o 5 cycles to fail

3a N3 = (1000)

= 5.43

(1000) 8036

= 2;

x i o 4 cycles to fail

I

(9.40)

21 2


F. Actual Number of Stress Cycles Accumulated During a 4-Hour Vibration Test The actual number of fatigue cycles n accumulated during 4 hours of vibration testing can be obtained from the percent of time exposure for the l a , 2 a , and 3 u values in the three-band method of analysis shown in Eq. 9.30. 1a n , = (56.5 cycles/s)(3600 s/h)(4 h)(0.683)

= 5.56 X

10’ cycles

2a n 2 = (56.5 cycles/s)(3600 s/h)(4 h)(0.271)

=

2.20

10‘ cycles

3 u n3 = (56.5 cycles/s)(3600 s/h)(4 h)(0.0433)

X

= 3.52 X

104 cycles

I

(9.41) G. Use Miner’s Cumulative Fatigue Damage Ratio to Estimate Fatigue Life Miner’s cumulative fatigue damage ratio is based on the idea that every stress cycle uses up part of the fatigue life in a structure. It does not matter if the stress cycle is due to sinusoidal vibration, random vibration, thermal cycling, shock, or acoustic noise. A fatigue cycle ratio of n / N is used to show the percentage of life used up. The actual number of stress cycles generated in a specific environment is shown as M . The number of cycles required to produce a fatigue failure in a specific environment is shown as N . When the ratios are all added together, a sum of 1.0 or greater means that all of the life has been used up so the structure should fail. The cumulative fatigue damage ratio equation is shown below:

(9.42) Substitute Eqs. 9.40 and 9.41 into Eq. 9.42 to obtain Miner’s fatigue damage cycle ratio: 2.20 X l o 5 3.52 X l o 4 7.27 x 10’ 5.43 x l o 4 R,, = 0.009 +- 0.303 + 0.648 = 0.960 5.56

X

loi

R1’= 6.14 x 10’

+

+

(9.43)

An examination of the above fatigue cycle ratio shows that the 1a RMS level does very little damage even though it acts about 68.3% of the time. Most of the damage is generated by the 3a level even though it acts only about 4.33% of the time. The 3a level generates more than two times as much damage as the 2 a level, which acts about 27.1% of the time. The above fatigue cycle ratio shows that about 96% of the life is used up by the 4-hour vibration test. This means that 4% of the life is left. The expected life of the structure can be obtained as shown below. Expected life

= 4.0

h

+ (4.0)(1.0 - 0.960) = 4.0 + 0.16 = 4.16 hours

(9.44)


21 3

Since fatigue has a large amount of scatter, the proposed beam does not have a sufficient safety factor to ensure its fatigue life for the environment. The beam design should be changed to provide a fatigue life of about 8 hours. This is equivalent to a safety factor of 2 on the fatigue life. The beam S-N fatigue curve already included a safety factor of 2 based on the structural stress level. This resulted in a fatigue exponent slope b of 6.4. Therefore, the real fatigue life is expected to be greater than about 4.16 hours. Because fatigue is very difficult to predict accurately, it is better to play it safe by adding another safety factor of 2 on the life. This is only equivalent to a structural safety factor of 1.11, as shown below. Structural safety factor

=

(2)1'6.4 = 1.11

Upper management executives may grumble a little bit when there is a slight increase in the size, weight, and cost, as long as the product passes the qualification tests and shows a high reliability with low maintenance costs in the field. However, if the product fails its required qualification tests, or if it has a very poor reliability record in the field, the people associated with the poor product will probably be looking for new jobs. Solution NO. 2 Quick Method for Finding Approximate Random Fatigue Life. The three-band technique for estimating the random vibration fatigue life was based on test data failures accumulated over a period of several years, to permit the use of hand calculations before computers were available. The previous sample problem required a substantial amount of work to reach a final conclusion. A simplified method for finding the random vibration fatigue life was developed, based on the total damage accumulated in the three-band method of analysis. This new quick method of analysis is based on the percentage of time exposure at the l a , 2 a , and 3 a levels as shown below. Damage D

=

C N S " = N,S,b + N2S,b+ N,S,b

Substitute the Gaussian probability distribution function shown in Eq. 9.30 for the percent of time spent at each level for the three bands, using the fatigue exponent b of 6.4:

+

+

D = (0.683)(1.0)6'4 (0.271)(2.0)6.4 (0.0433)(3.0)6'4= 72.55 (9.45) Now find one random v acceleration level that will generate the same total damage as the three-band technique for the full Gaussian distribution shown above, assuming that it acts 100% of the time.

G6,4= 72.55

SO

G = 1.95

(9.46)

214


This means that an acceleration response level of 1 . 9 5 ~can ~ be assumed to act 100% of the time to quickly estimate the dynamic stress level and the number of fatigue cycles required to produce a fatigue failure in a structure. The approximate fatigue life can then be obtained from the natural frequency of the structure. The previous sample problem can now be reexamined using this quick method for calculating the approximate fatigue life of the cantilevered beam structure. The previous l o RMS stress level of 8036 lb/in.2 was shown in Eq. 9.39. The number of fatigue cycles needed to produce a failure can be obtained by using the 1 . 9 5 ~level in Eq. 9.40 as shown below: 1 . 9 5N ~

=

[

(1000) ( 1 . ~ ~ ~ )6'~4 = ~ 8.55 ~ 3 X610') cycles to fail (9.47)

The approximate fatigue life of the beam can be obtained from the natural frequency of 56.5 Hz shown in Eq. 9.37: Life

=

8.55 x 10' cycles to fail (56.5 cycles/s)(3600 s/h)

= 4.20

hours to fail

(9.48)

Comparing the fatigue life of 4.16 hours using the three-band technique in Eq. 9.44 with the fatigue life of 4.20 hours using the quick analysis method shows good correlation. One note of caution is advised here. The quick method of analysis shown above only applies to the determination of the approximate fatigue life. This method of quick analysis must not be used for finding the maximum e.xpected displacements or maximum acceleration levels. The real maximum displacements expected will still be three times the RMS displacement levels and the maximum expected acceleration levels will still be three times the RMS acceleration levels. Some people like to round off the 1.95 factor to an even 2.0 factor because it is easier to remember. There is nothing wrong with this idea since it provides a slightly greater safety factor. This can be demonstrated by using Eq. 9.40, which shows a 2 o fatigue life of about 7.27 X l o 5 cycles. The approximate fatigue life can be obtained as shown below: Life 9.17

7.27 x 10' cycles to fail =

(56.5 cycles/s) (3600 s/h)

= 3.57

hours to fail

(9.49)

HOW PCBs RESPOND TO RANDOM VIBRATION

Extensive testing and analysis of PCBs show that they can be evaluated as single-degree-of-freedom systems if they are supported or restrained around the perimeter. In a random vibration environment, Eq. 9.36 can therefore be used to predict the response characteristics of typical PCBs (circular, rectangular. etc.). This applies to plug-in PCBs and to PCBs that are bolted around the perimeter. As long as the fundamental resonant frequency of the PCB

DESIGNING PCBs FOR RANDOM VIBRATION ENVIRONMENTS

/Plug-in

215

Low-level higher harmonics

PC.B

osci~ioscope

FIGURE 9.23. Oscilloscope pattern shows a PCB acts like a single-degree-of-freedom system when it is exposed to random vibration.

can be determined, the approximate response can be obtained for a random vibration environment. When an accelerometer is mounted at the center of a plug-in PCB that is excited by random vibration, and when the accelerometer response signal is picked up and displayed on an oscilloscope, it shows that the fundamental resonant frequency comes through with a large amplitude, as shown in Fig. 9.23. Since the input and the response are both random, the amplitude and the frequency can be observed to be constantly changing with small variations. The higher harmonic resonant frequencies come through as very low, constantly changing amplitudes. 9.18


Electronic equipment must be designed with an understanding of the operating environment, the capability of the electronic components, and the characteristics of the support structure in order to achieve a high degree of reliability for systems exposed to random vibration. In most electronic assemblies the PCBs are fastened to a chassis, and the chassis is mounted to a structure that is excited by random vibration. Since the chassis receives the dynamic energy first, the chassis becomes the first degree of freedom. Since the PCBs must receive their dynamic energy from the chassis, the PCBs become the second degree of freedom, as shown in Fig. 9.24. Chassis

PCB /

Leaf sorine \

2nd degree of freedon PCB mass

FIGURE 9.24. A typical electronic chassis with PCBs, modeled as a spring-mass system.

216


FIGURE 9.25. PCB resonances can produce stresses in the electrical lead wires due to relative motion between the component body and the PCB.

Care must be used in the design of the chassis and the PCBs to avoid severe coupling of the chassis resonances and the PCB resonances. This is usually accomplished by following the octave rule: the resonant frequencies of the chassis and the PCBs are separated by an octave (a factor of 2). When the resonant frequency of the chassis is at least two times the resonant frequency of the PCB, then the PCBs will not be severely amplified by the chassis resonance. The same is also true when the resonant frequency of the PCB is at least two times the resonant frequency of the chassis. The PCBs are usually considered to be the heart of the electronic system, because almost all of the electronic components are mounted on PCBs. This makes it convenient and easy to service and repair electronic equipment by simply removing and replacing defective PCBs. The typical PCB consists of several flat, thin plastic plates, with printed electrical traces, all laminated together. This type of construction results in an oil-canning of the PCB in any vibration or shock environment. Since the electronic components are mounted on these flexible assemblies, and since some of the electronic components are large and heavy, a significant amount of relative motion can occur between the PCB and the electronic component, as shown in Fig. 9.25. Extensive finite element analysis and vibration testing of electronic systems show that the fatigue life of many different types of electronic components can be related to the dynamic displacements experienced by the PCBs that support these components. When the PCB resonant frequency is excited, the plate structure is forced to bend back and forth. When the displacement amplitudes are high, the relative motion between the components and the PCB can be high, which often results in cracked solder joints and broken electrical lead wires. The fatigue life of these components can often be increased by reducing the dynamic displacements of the PCB. Since the displacements can be controlled through the resonant frequency, it follows that the fatigue life of many different types of components can also be controlled by controlling the PCB resonant frequency. When the dynamic single-amplitude displacement at the center of a perimeter-supported PCB is limited to the value shown below, the component can be expected to achieve a fatigue life of about 20 million stress

217


reversals in a random vibration environment, based on the 3a PCB displacements [38]: Z=

where B L h

0.00022 B

Chrc

(maximum desired PCB displacement)

(9.50)

length of PCB edge parallel to component (in.) = length of electronic component (in.) = height or thickness of PCB (in.) C = constant for different types of electronic components =

'1.0 1.26 1.26

1.o -

2.25 1.o 1.75 0.75

t r

=

for a standard dual inline package (DIP) for a DIP with side-brazed lead wires for a pin grid array (PGA) with two parallel rows of wires extending from the bottom surface of the PGA for a PGA with wires around the perimeter extending from the bottom surface of the PGA for a leadless ceramic chip carrier (LCCC) for leaded chip carriers where the lead length is about the same as for a standard DIP for a ball grid array (BGA) far axial-leaded component resistors, capacitors, and fine pitch semiconductors

relative position factor for component on PCB (see Eqs. 6.1 and 8.15 and Fig. 6.5) 1.0 0.707

when component is at center of PCB (i point X and Y ) when component is at point X and

f point Y on a PCB supported on four sides 0.50

when component is at

point X and

f point Y on a PCB supported on four sides The actual dynamic single-amplitude displacement expected for the PCB is shown by Eq. 2.30. This relation can also be used for random vibration when the RMS displacements are utilized in the evaluation. Since this analysis is based on a single degree of freedom, the true resonant frequency

21 8


can be used in place of the number of positive zero crossings associated with random vibration. For a single-degree-of-freedom system, the number of positive zero crossings is the same as the resonant frequency. Thus (9.51) The approximate transmissibility Q of the PCB can be related to the resonant frequency as follows (also see Eq. 14.21): (9.52)

Q=&

The desired PCB resonant frequency that will provide a fatigue life of approximately 20 million stress reversals can now be determined by combining Eqs. 9.50, 9.51, 9.52, and 9.36. Equation 9.36 must be modified slightly by multiplying by 3, because we are using 3 a acceleration levels. The desired PCB resonant frequency is then 2 9 . 4 C h r d m 0.00022 B Sample Problem-Finding

(minimum desired)

(9.53)

the Desired PCB Resonant Frequency

A 40-pin dual inline package (DIP) with side-brazed lead wires is to be flow-soldered to a typical 7.0 X 9.0 X 0.090-in. thick plug-in PCB. The DIP is to be mounted parallel to the 7.0-in. edge. The input PSD in the area of the PCB resonant frequency is expected to be flat at 0.080 G2/Hz. Determine the minimum desired PCB resonant frequency and the approximate fatigue life for the DIP wires on the simply supported PCB for the following mounting positions:

(i

(a) Component mounted at the center of the PCB point X and Y > . (b) Component mounted at the quarter points ( f point X and point Y ) on the PCB.

Solution

When the octave rule is followed, the PCB resonant frequency will be well separated from the chassis resonant frequency, so coupling between the chassis and the PCB will be reduced. Under these conditions it is convenient to assume that the PSD input to the PCB is the same as the input to the chassis. This is not really true, but in most cases the error is small, and the method provides quick approximate answers. This method is conservative (so the PCB will have a resonant frequency that is a little too high) when the PCB resonant frequency is two or more

Part (a)


21 9

times greater than the chassis resonant frequency, for the forward octave rule. This method is slightly unconservative (so the PCB will have a resonant frequency that is slightly too low) when the chassis resonant frequency is two or more times greater than the PCB resonant frequency, for the reverse octave rule. When these conditions are followed, the fatigue life of the electronic component lead wires and solder joints will be approximately 20 million stress cycles. We have

B

= 7.0

in. (length of PCB parallel to DIP) h = 0.090 in. (PCB thickness) L = 2.0 in. (length of 40-pin DIP) C = 1.26 (constant for side-brazed DIP lead wires) P = 0.080 G2/Hz (input PSD to PCB) r = 1.0 (for component at center of PCB) Substitute into Eq. 9.53 to obtain the desired PCB frequency:

f d =

[

(29.4)( 1.26)(0.090)( 1.0)4( v/2)(0.08)(2.0)

f d = 268

(0.00022) (7 .O)

Hz

(9.54)

(minimum desired)

The fatigue life of the component can be estimated as follows: Life

20 x l o 6 cycles to fail =

(268 cycles/s) (3600 s/h)

= 20.73

(9.55)

hours to fall

A resonant frequency of 268 Hz is difficult to achieve with the PCB described above. Some type of stiffening will be required to raise the resonant frequency to this level. This may require removing some electronic components from the PCB to make room for the ribs, which will make many electrical engineers and program managers very unhappy. Reducing the number of electronic components that can be mounted on each PCB means that more PCBs will be required in the system, which may increase the size, weight, and cost. These factors must be carefully considered against the possibility of qualification test failures, or failures in the field. Part (b) When the component is mounted at the quarter points (+ point X and point Y),the relative position factor Y is reduced to 0.5 because the

relative motion between the component and the PCB is less. Substitute a

220


value of r = 0.5 into Eq. 9.53 and solve: fd =

154 Hz

(minumum desired)

(9.56)

Substitute into Eq. 9.55 to obtain the approximate fatigue life when the PCB has a resonant frequency of 154 Hz: Life

= 36.1

hours to fail

(9.57)

9.19 EFFECTS OF RELATIVE MOTION ON COMPONENT FATIGUE LIFE

The component fatigue life was based on the relative motion that occurs between the component body and the PCB, because this relative motion produces stresses in the lead wires and solder joints. When the stress levels are high, the fatigue life is low, and vice versa. Therefore, the stress levels in the lead wires must be reduced. This can be done in several different ways: 1. Increase the damping, which will reduce the displacements. This works well for PCBs that have a low resonant frequency, below about 50 Hz, because then dynamic displacements are large, so there is a good transfer of kinetic energy into heat. However, when PCB resonant frequencies are high, above about 200 Hz, the displacements are small, so damping is not very effective. 2. Increase the PCB resonant frequency. This is a very effective method for improving the fatigue life, because the PCB displacements are reduced very quickly. However, it is not always easy to increase the stiffness. Ribs may have to be added, which reduces the surface area available for electronic components. If the thickness is increased to increase the stiffness. electrical lead wires from components such as DIPs may not extend through the PCB, so flow-soldering reliability is reduced. 3. Reduce the spring rate of the electrical lead wires. This can be done by increasing the length of the wires, perhaps by looping them. The spring rate can also be reduced by coining the wires. In this operation the wire cross section is squeezed to make it thin and flat, to reduce the moment of inertia of the cross section. These operations will, of course, increase the manufacturing costs. 4. Add local stiffeners under the problem electronic components. Stiff metal shims can be epoxy-cemented to the PCB, under components such as large DIPs, to reduce the relative motion between the PCB and the DIP body. This will, of course, increase the cost of the assembly. 5. Reduce the acceleration input level to the system. Sometimes it is possible to obtain a change in the environmental requirements by just asking

EFFECTS OF RELATIVE MOTION ON COMPONENT FATIGUE LIFE

221

the customer for a reduction. This is always easier than modifying the design. Many companies increase the acceleration requirements to provide an additional safety factor. Sometimes it is possible to show them that their high safety factor will increase their weight and cost, so a less severe requirement is desirable. 6. Mount the large components away from the center of the PCB. Since the maximum curvature change on the PCB usually occurs at the center, it is desirable to mount large components near the sides. Of course, it is not always possible to mount electronic components where they are best suited for mechanical reasons. After all, the purpose of an electronic system is to work electrically. However, if the mechanical requirements are ignored, then it is possible for many fatigue failures to occur. Extensive failures may not only be very expensive to repair, they may ruin the reputation and the business of a company.

Sample Problem-Component

Fatigue Life

Determine the fatigue life of the PCB component lead wires in the previous sample problem, when the PSD input level is increased from 0.080 to 0.120 G2/Hz, and the PCB resonant frequency remains at 268 Hz, as shown in part (a).

Solution. Use the fatigue-damage method as shown in Eq. 8.1, and rewrite the fatigue expression to utilize the time ( T ) to fail and the acceleration ( G ) values as follows:

T,Gf

=

T,G,b

(see Eq. 8.4a)

where b = 6.4 (fatigue exponent) P , = 0.120 G2/Hz (PSD level at 268 Hz) P, = 0.080 G2/Hz (PSD level at 268 Hz) T2= 20.73 hours to fail (see Eq. 9.55) = 16.37 Q , = Q, = Substitute into Eq. 9.36 to obtain the RMS G level: G,

=

4(1.57)(0.12)(268)(16.37)

= 28.75

G RMS

G , = ~(1.57)(0.08)(268)(16.37)= 23.47 G RMS

222


Then for the expected life we have

1

23.47 6 , 4 T , = 20.73 - = 5.66 hours to fail 28.75

(

(9.58)

The approximate fatigue life can also be determined by considering a ratio of the displacements or the stresses. Since this is a linear system, these quantities are all directly proportional to each other. IT’S THE INPUT PSD THAT COUNTS, NOT THE INPUT RMS ACCELERATION

9.20

References are often made to the magnitude of the input RMS acceleration levels to establish the severity of some random vibration test or environment. Although the magnitude of the acceleration level can be of some importance, the magnitudes of the input PSD levels in the regions of the structure’s resonant frequencies are far more important. The input (and also the output or response) acceleration level was shown by Eq. 9.2 to be directly related to the square root of the area under the PSD curve. However, there can be an infinite number of input PSD curves that have the same area, and therefore the same input RMS acceleration level. Several typical curves with the same area, but with different shapes and different PSD levels, are shown in Fig. 9.26. When the input PSD and the resonant frequency are known, then Eq. 9.36 shows that the response acceleration is related to the value of the input PSD level at the resonant frequency. A low input PSD level will result in a low RMS acceleration response value. A high input PSD level will result in a high response value. Therefore, the RMS response levels can be different for different-shaped PSD curves, even when the RMS acceleration input levels are exactly the same. A large area under the high-frequency segment of the input curve may not produce a high acceleration response level, unless the structure has its resonant frequency and a high Q i n the region of the curve where the PSD is also high. Similarly, a low input acceleration level does not mean that the system must have a low acceleration response level.

FIGURE 9.26. Various random vibration input curves with the same areas, but with different PSD values at the same frequencies.

CONNECTOR WEAR AND SURFACE FRETTING CORROSION

9.21

223

CONNECTOR WEAR AND SURFACE FRETTING CORROSION

Extensive testing experience with electronic equipment in random vibration environments has shown that fretting corrosion can occur at electrical contacts when high interface pressures and high relative velocities are present for long periods of time. What appears to happen is that wear occurs through the outer layer of gold (approximately 20-50 millionths of an inch thick) at the contact area first. With continuing random vibration, wear then occurs through the nickel undercoating (approximately 100-200 millionths of an inch thick) so that now the electrical contact interface is between a brass pin and a brass socket. The brass-on-brass interface with a high relative velocity produces a high local temperature, which produces an oxide of brass that is an electrical insulator. Although there is mechanical contact between the pin and the socket at the electrical interface, there is no electrical continuity for approximately one microsecond of time. In an analog electrical system, this is no problem. However, in a high-speed digital system where multiplexing is being used, the loss of information for that time period can result in electrical failures. These failures are very difficult to find, because the problem appears to be very similar to a broken wire or a cracked solder joint. Also, when the vibration is stopped the problem always disappears. A great deal of time is then lost in looking for broken cables, wires, solder joints, and components [521. Inspecting electrical contacts with a microscope for fretting corrosion can be very difficult. Polished brass looks very much like gold on the contacts, so the wearthrough cannot be observed easily and is almost always overlooked. However, when a 5% solution of sodium sulfide in water is brushed on the connector surface in the contact area, a small spot will turn black in a couple of seconds when wear through the gold and nickel has occurred and there is any copper base alloy present. This is a nondestructive test; the solution can be washed off with water with no harm to the contacts. This problem with contact fretting corrosion and wear appears to occur only in severe random vibration environments and is most severe on electrical connectors that have only two points of contact. Contacts with a flat blade pin and a tuning-fork socket (and only two points of electrical contact) have shown very high failure rates in severe random vibration tests for long time periods. This problem has never been observed to occur in any sinusoidal vibration environment. or in random vibration tests on connectors with more than three points of electrical contact. This does not mean that fretting does not occur. This means that all of the electrical points of contact do not open (lose continuity) at the same time under these conditions [52]. Test data show that this condition may first occur after extensive highacceleration random vibration exposure, and then not reappear even after many hours of additional vibration testing. The failure mode appears to follow the previously defined exponential fatigue laws (Eq. S.l), except that the fatigue exponent b appears to have a value of about 4.0. When high-level

224


random vibration tests are run on individual PCBs in a rigid vibration fixture, and when blade-and-tuning-fork electrical contacts are used, the first fretting corrosion failures have been observed to occur after about 15 min [521 using a white-noise input level of 40 G RMS, with a PSD of 0.80 G2/Hz. Sample Problem-Determining

Approximate Connector Fatigue Life

A blade-and-tuning-fork electrical connector, with two contact points, is being proposed for plug-in PCBs that must be capable of passing a random vibration qualification test program with an input acceleration level of 10 G RMS for 1 hour along each of three axes, for a total test time of 3 hours. Will the proposed contacts be capable of reliable operation in this environment?

Solution. In any type of qualification test program, there is such a high probability of malfunctions occurring due to improper instrumentation, calibration, and data accumulation and to recording-device failure, accidents, and human errors, that these tests often have to be repeated several times in order to complete only one qualification test. Therefore, electronic systems must be designed with some margin of safety. One good approximate rule is to design the system so it is capable of withstanding five qualification tests. This safety factor adds very little weight, and it provides that extra safety margin that is often necessary to complete a difficult qualification test schedule. Therefore, let

b = 4.0 (fatigue exponent for electrical contact failure) T2 = 15 min (baseline time to fail, from test data [52]) G2 = 40 G (RMS acceleration, baseline for failure) Substitute into Eq. 8.4a for the approximate time to fail:

[

T , = ( 15 min) 40 RMS 10 G RMS

14”

= 3840

minutes

= 64.0

hours

(9.59)

Since the maximum qualification testing time expected is 3 X 5 = 15 hours, and the system is capable of withstanding about 64.0 hours of random vibration testing, the proposed design will be satisfactory. The fatigue life of these electrical contacts will be affected by the manufacturing tolerances, spring rate of the tuning-fork socket, surface finish, gold thickness, nickel thickness, and any lubrication used. 9.22

MULTIPLE-DEGREE-OF-FREEDOM SYSTEMS

When a multiple-degree-of-freedom system is subjected to a random vibration input, the motion of each mass in the system will influence the motion of every other mass in the system.

OCTAVE RULE FOR RANDOM VIBRATION

225

-.

I .._ 3. L

d

A '

a

r"?, 1

1

1

I

/

FIGURE 9.27. Integrating under the transmissibility Q response and the input PSD levels to obtain the RMS acceleration re-

sponse.

The response of the system can be determined by using an integration technique. The l a RMS acceleration response can be obtained for any or all masses in the system by integrating under the input PSD curve and under the frequency response curve for any of the masses at the same time, as shown in Fig. 9.27 and in Eq. 9.60:

Go,, =

P,A f , Qf

(9.60)

This is just another form of Eq. 9.36, as can be demonstrated by substituting the expression for the bandwidth Af of the resonance into the above relation:

f" Q

Af= -

9.23 OCTAVE RULE FOR RANDOM VIBRATION In Chapter 7 it was shown that severe coupling can occur between the chassis and the PCBs within it when the octave rule is not followed in a sinusoidal vibration environment. Similar problems can occur in random vibration environments when the octave rule is not followed. When the resonant frequencies of the chassis and the PCBs are not separated by at least one octave, then the chassis resonance can magnify the PCB resonance. This will increase the PCB acceleration and displacement levels, which will reduce the PCB fatigue life. When the chassis and PCB resonances are separated by one octave or more, then the acceleration and displacement levels are reduced, so the fatigue life of the PCB is increased. It makes no difference whether the forward octave rule or the reverse octave rule is followed, as long as the chassis and the PCB resonances are well separated, and the relative PCB mass is small. The approximate responses of the chassis and PCBs in a random-vibration environment can be determined by assuming that these elements will act like lumped spring-mass systems. When the resonances are well separated, the chassis mass and each PCB mass can be treated as a single-degree-offreedom system.

226


FIGURE 9.28. Chassis and PCB modeled as a spring-mass system.

Vlb

A

c

10

Irlplt

2x0

: 400 Hz

~

FIGURE 9.29. Response of the chassis, mass 1, to sinusoidal vibration.

Sample Problem-Response

1L1

i '

10,

I

I 1 102

1

103

104

Frequency H r

of Chassis and PCB to Random Vibration

An electronic chassis with a total weight of 30 lb and a resonant frequency of 400 Hz holds several plug-in PCBs, each with a weight of 1 Ib and an approximate resonant frequency of 200 Hz. Figure 9.28 shows the chassis as mass 1 and one of the PCBs as mass 2. The system must be capable of operating in a white-noise random vibration environment with a PSD input of 0.100 G2/Hz. Determine the RMS acceleration response levels and the RMS displacement levels for the chassis and PCB, masses 1 and 2. Solution Part 1-Chassis

RMS Response The RMS response of the chassis can be determined with the use of Eq. 9.36. Since the chassis and PCB resonances are well separated, and since the PCB mass is small compared to the chassis mass, there will be very little effect of the PCB resonance on the chassis resonance. The chassis will therefore have the typical single-degree-of-freedom response as shown in Fig. 9.29, where the chassis resonance is strong at 400 Hz and the 200-Hz PCB resonance shows up as a small spike on the chassis response curve. Then

P = 0.10 G'/Hz (PSD random vibration input to chassis) Q = 0 . 5 m = 10 (estimated chassis transmissibility) f , = 400 Hz (chassis resonant frequency)


227

Substitute into Eq. 9.36 to obtain the RMS acceleration level:

GRMS = (5(0.10)(400)(10)

= 25.06

(9.61)

The dynamic single-amplitude displacement expected for the chassis can be obtained from Eq. 2.30. This represents the relative motion of the chassis, mass 1, with respect to the support: G R b f S = 25.06

f,

= 400

(RMS acceleration response) (chassis resonant frequency)

Hz

(9.8) (25.06) ‘RMS

=

(400)’

= 0.00153

in.

(9.62)

Part 2-PCB RMS Response The PCB has its resonant frequency at 200 Hz, and the chassis has its resonant frequency at 400 Hz. When the PCB is excited at its resonant frequency, the uncou )ed transmissibility normally expected for the PCB would be about 200 = 14.14. The PCB can also receive additional energy from the chassis due to the coupling with the chassis at 200 Hz. The uncoupled transmissibility expected for the chassis at a frequency of 200 Hz can be obtained with Eq. 2.49 and the following information:

9

ff = 200 Hz (forcing frequency) f , = 400 Hz (chassis resonant frequency) R = ff/f, = 200/400 = 0.5 (frequency ratio) Thus 1 e=-= 1- R 2

1 1 - (0.5)’

=

1.33 (chassis at 200 Hz)

This shows that the PCB resonance will couple with the chassis resonance, and that the PCB transmissibility will be 1.33 times greater than previously calculated at a frequency of 200 Hz. The coupled transmissibility for the PCB at 200 Hz then becomes Q p = (1.33)(14.14)

=

18.8 (PCB at 200 Hz)

(9 -63)

A second resonant peak can be expected for the PCB at a frequency of 400 Hz, because the chassis has its resonance at 400 Hz. The chassis uncoupled transmissibility at 400 Hz was shown to be about 10 in part 1 above. The PCB uncoupled transmissibility at 400 Hz can be obtained from

228


Eq. 2.49 with the following information: ff = 400 Hz (forcing frequency)

f,,

= 200

R

= 400/200 = 2.0

Hz (PCB resonant frequency) (frequency ratio)

Thus

Q=

1 1 - (2)2

= - 0.333

(PCB at 400 Hz)

The negative sign means that the 200-Hz response for the PCB is out of phase with the 400-Hz response. This is not important here, so the sign can be ignored. The coupled transmissibility for the PCB at 400 Hz becomes Q p = (10)(0.333)

=

3.33

(PCB at 400 Hz)

(9.64)

The frequency response curve for the PCB with its two resonant peaks will now appear as shown in Fig. 9.30. The relative RMS acceleration response of the PCB with respect to the chassis can be determined with the use of Eq. 9.36 or Eq. 9.60. When Eq. 9.36 is used, the two PCB resonant peaks at 200 and 400 Hz can be treated as two single-degree-of-freedom peaks, to obtain the RMS response at the center of the PCB using Eqs. 9.63 and 9.64: 7T

GRMs=

- (0.10) (200) (18.8) 2

G R M S = 28.3

7T

+ -2 (0.10) (400) (3.33)

(response at center of PCB)

(9.65)

Equation 9.36 is very accurate when the transmissibility of each resonant peak is greater than 10. Some accuracy is lost when the transmissibility is

I

200 Hz

I 10

FIGURE 9.30. Response of the PCB, mass 2, to sinusoidal vibration.

'1 10'

1 1

102

" loa

Frequency Hz

104


229

significantly less than 10. When this condition exists, better accuracy will be obtained with Eq. 9.60. The relative dynamic single-amplitude displacement of the PCB, mass 2, with respect to the chassis, mass 1, can be determined with the use of Eq. 2.30 and the following information:

f, = 200 Hz

(PCB resonant frequency)

Thus (9.8) (28.3) =

'RMS

(200) *

Sample Problem-Dynamic

= 0.00693

in. (PCB)

(9.66)

Analysis of an Electronic Chassis

A 30-lb chassis with four mounting flanges contains some 1.0-lb plug-in PCBs with relays as shown in Fig. 9.31. The relays have a maximum rating of 40 G peak up to a frequency of 2000 Hz. A lumped spring-mass system was used to model the assembly with two degrees of freedom as shown in Fig. 9.28. The transmissibility curves expected for the PCB and the chassis were shown in Figs. 9.30 and 9.29, respectively. The operating environment will have a PSD input level of 0.10 G'/Hz. Determine the following:

(a) Will the relays be capable of operating in the environment without chattering? (b) Will the chassis mounting lugs be capable of operating in the environment without failing?

Solution. (a> The RMS response of the PCB to the random vibration was shown by Eq. 9.65 to be 28.3 G. Since the relays are rated at 40 G peak, it

-

. . ~

! I

Relays on PCB

/

4 '

Vibration

fi. i ~ p u t

FIGURE 9.31. PCBs with relays, mounted within an electronic chassis (dimensions in

inches).

230


would appear at first glance that the relays are acceptable for the random vibration environment. However, the 3 a peaks for the random vibration must be considered even though they only occur about 4.33% of the time. Therefore, the maximum levels the relays are expected to see would be 3 x 28.3 or 84.9 G . This is well above the 40-G peak rating for the relay, so these relays are unacceptable for the application when they are mounted at the center of the PCB. If the relays are mounted at the edges of the PCB, they will see the 25.06-G RMS levels developed by the chassis resonance at 400 Hz, as shown by Eq. 9.61. The 317 acceleration levels will be 75 G. so again the 40-G peak relay rating will be exceeded. Therefore, these relays are unacceptable for this application. (b) The dynamic stresses in the mounting feet can be determined from the spring rate of the chassis using Eq. 2.7 with the following information: f, = 400 Hz (chassis resonant frequency) W = 30 lb (chassis total weight) g = 386 in./s2 (acceleration of gravity)

Thus

K=

47r (400) '( 30) 386

=

490,940 lb/in.

(chassis spring rate)

The dynamic load acting on the chassis can be determined from the chassis dynamic displacement shown in Eq. 9.62:

Pd= KZ

=

(490,940)(0.00153)

= 751.1

lb RMS on four lugs

The dynamic bending moment acting on the lugs can be obtained from the mounting-lug geometry shown in Fig. 9.31:

M=

(751.1) ( 1.4) 4 lugs

=

263 1b.in. RMS

Substituting into Eq. 2.39 for the dynamic bending stresses the quantities

I = bh'/12 = (0.75)(0.50)'/12 C = 0.50/2 = 0.25 in. K = 1.5 (stress concentration)

= 0.00781

in.'

we obtain

s, =

( 1.5) (263) (0.25) 0.00781

=

12.628 lb/in.' RMS (bending)

(9.67)

DETERMINING THE NUMBER OF POSITIVE ZERO CROSSINGS

231

The 3 a bending stress in the mounting lug is S,(3a)

=

(3)(12,628)

= 37,884

Ib/in.’

(bending)

(9.68)

Since this exceeds the 12,000-lb/in.2 fatigue bending endurance limit for the aluminum-alloy mounting lug, the design is unacceptable. In order to improve the structural integrity of the design, gussets can be added to the mounting lug as shown in Fig. 9.32. Gussets will increase the stiffness and strength of the mounting feet, which will increase the resonant frequency of the chassis. The previous analysis must now be revised to account for the new chassis stiffness. In the preliminary design phase of a new electronic chassis design, this process is repeated until a satisfactory design has been achieved for the environment.

9.24 DETERMINING THE NUMBER OF POSITIVE ZERO CROSSINGS The number of fatigue cycles accumulated during a resonant dwell condition in a sinusoidal vibration test is easy to determine. It is simply the product of the resonant frequency (cycles/second) and the time (seconds). The number of fatigue cycles accumulated during a random vibration test is also related to the duration of the test. However, the actual number of stress reversals is more difficult to evaluate. Since all of the exciting frequencies within the bandwidth are present all of the time, the number of fatigue cycles must relate to the frequencies that will do the most damage. These are the frequencies with the highest transmissibilities, where the damping is small. These frequencies will influence the number of times, on the average, where the displacement trace crosses the zero axis with a positive slope. This is called the number of positive zero crossings and is denoted by the symbol

No+. The number of crossings, with both positive and negative slopes, was determined by Rice [41] to be (see Fig. 9.1)

(9.69)

232


In order to obtain position, to show zero crossings, it is necessary to divide by

R4 and simplify as follows:

1

N’

= -

2T

O

For lightly damped systems with several resonant peaks, the above equation can be simplified by considering each peak to act like a single degree of freedom. Then using Eq. 9.36 and simplifying,

There are a number of other random vibration relations that are convenient for determining displacements, velocities, and accelerations in random vibration environments for single-degree-of-freedom systems. For example, when the input PSD is known and the RMS response displacement is desired, the following relation is convenient:

z,,, = 1

2 2 F

(9.72)

where Q = transmissibility at resonance (dimensionless) P = PSD at resonance f, (G*/Hz) f, = resonant frequency of system (Hz) The displacement, velocity, or acceleration response of a single-degreeof-freedom system can be determined form Eq. 9.36, or from the same equation written in a slightly different form, as follows: Output

where R , f,

=

[0.7”i( 1 +R4R: , ](f,,)(input)]

0.5

1/2Q = damping ratio (dimensionless) = resonant frequency (Hz) = C/C, =

(9.73)

DETERMINING THE NUMBER OF POSITIVE ZERO CROSSINGS

233

The units are as follows: Quantity

Input

Displacement Velocity Acceleration

Output (RMS)

in.’/Hz (in./s>’/Hz G~/HZ

in. in./s G

Sample Problem-Determining the Number of Positive Zero Crossings

Sinusoidal vibration tests were run on an electronic system that had an accelerometer mounted at the center of a plug-in type PCB. The fundamental natural frequency and the next two higher harmonics with the following transmissibility Q values were obtained: Frequency (Hz)

Transmissibility Q

345 604 1010

18.7 8.4 4.8

Determine the number of positive zero crossings, considering a white-noise random vibration excitation with a 0.10-G2/Hz PSD input level.

1 No - 2

[ 2 7 (345)]’

i-

[2 a (604)12

+

+ - _

( a/2) (0.1)(345)( 18.7)

2.155X lo-‘ 4.587x l o - ”

+ 5.531 X

+

( a/2) (0.1)(604) (8.4)

+

+ 1.889 X lo-’

+ 3.840 X lo-’* + 4.692 X

=

[ 2 a (lolo)]* ( a/2) (0.1) ( 1010) (43)

383 Hz

(9.74)

-

CHAPTER 10

Acoustic Noise Effects on Electronics

10.1

INTRODUCTION

Acoustic noise (pressure waves) can be generated by almost any source, from a loud radio or a rocket engine to a siren or a ship’s propeller. This noise can be radiated through the air or conducted through liquids and solids. High acoustic energy levels can lead to rapid fatigue failures in large, thin panels and can also affect the operation of certain sensitive electronic components. Acoustic noise is measured in decibels (dB), implying a ratio of the mean square sound pressure to a reference mean square pressure. The reference selected is the threshold of hearing. Thus

Number of decibels (dB)

P

= 20 log,, -

(10.1)

Pref

where P

= sound

pressure level (lb/in.’)

Pref= sound pressure reference = 2.90

x

lb/in.*

= 0.0002

dyn/cm*

The above expression should not be confused with Eq. 9.11, where the coefficient is 10 rather than 20 because a power ratio rather than an amplitude ratio is used. The sound pressure P is of interest because it represents the level of the pressure wave that can cause acoustic damage.

Sample Problem-Determining the Sound Pressure Level

Acoustic noise tests yielded sound pressure-level measurements of 150 dB. Determine the equivalent sound pressure level in terms of a pressure in pounds per square inch RMS. 234

MICROPHONIC EFFECTS IN ELECTRONIC EQUIPMENT

235

Solution. Substitute into Eq. 10.1: P

150 = 20 log,,

2.9 x 10-9

150

P - log,, 2.9 x 10-9 20 lOg,,P=7.5+(-8.537) = -1.037 - - - log,,

P = 0.092 lb/in.2 RMS

(10.2)

The threshold of acoustic damage is normally considered to be at about 150 dB. Values below this level generally do not result in any serious structural damage in electronic equipment. Microphonic effects can, however, occur at lower values. 10.2

MICROPHONIC EFFECTS IN ELECTRONIC EQUIPMENT

When an electronic component is sensitive to acoustic noise, it is called microphonic. Sound pressure waves can physically alter the shape of the component slightly, and therefore its electrical characteristics will fluctuate. Certain types of ceramic and paper capacitors are affected by acoustic noise, as well as certain types of crystals, which are often used in oscillators and piezoelectric accelerometers. However, most of the so-called microphonic effects are not really caused by microphonic components. Instead, the signal distortions, modulations, and noise are most often due to loose wires, loose ground screws, poor wire routing, improper use of shields, poor solder joints, and many other electromechanical defects. Acoustic tests on some types of electronic components have shown that they can produce malfunctions in the electronic equipment. The equipment items most susceptible to malfunctions are components whose function involves motion of flexible parts (e.g., relays and pressure switches) and components that contain relatively flexible structural parts but are otherwise rigid (e.g., vacuum tubes and piezoelectric devices). Capacitors, resistors, and inductors also have a history of malfunctions in severe acoustic environments. Film capacitors from some manufacturers have shown problems at acoustic levels as low as 132 dB. Similar components from other sources showed no problems at 166 dB. Ceramic capacitors showed problems at levels of 159 dB. Electrolytic capacitors showed problems at acoustic levels of 125 dB [451. In general, structural members that have large, flat, thin, lightweight sections are most sensitive to high-level acoustic environments, so members with these characteristics should be stiffened or damped to reduce the acceleration response levels. When large flat thin panels are subjected to acoustic noise, the acoustic fluctuating pressures force the panels to vibrate. The panels will have a

236

ACOUSTIC NOISE EFFECTS ON ELECTRONICS

dynamic response related to the geometry, method of support, and construction. The response influences the stress patterns and the stress levels, which determine the fatigue life.

10.3 METHODS FOR GENERATING ACOUSTIC NOISE TESTS Acoustic noise tests are usually generated by air jets, sirens, and loudspeakers. Air jets and loudspeakers generate a broader band of frequencies; sirens generate several narrow bands. A siren can generate discrete frequencies close to the fundamental resonant frequency of the panel, which will increase the response of the panel. Air jets and loudspeakers generate a broader frequency spectrum, so the panel has a lower response. Therefore, the stress levels in a panel excited by a siren will be greater than the stress levels developed by an air jet or loudspeaker, for the same sound pressure level. The time to failure for a 0.032-in. thick flat 2024-T3 aluminum panel subjected to acoustic noise from a siren and from an air jet is shown in Fig. 10.1.These tests show that the siren produces more rapid fatigue failures at the same sound pressure level. This is probably due to the greater amount of energy developed by the siren at the fundamental resonant frequency of the panel [461. The fatigue life of the panel can be increased by adding a bonding in addition to the rivets holding the panel to the supports. A further increase in the fatigue life can be achieved by forming a radius in the panel to increase the panel stiffness. This increases the resonant frequency, which reduces the deflections and stresses. These structural modifications are shown in Fig. 10.2 [46].

3

Time to f a i l u r e hours

FIGURE 10.1. Test data showing failures of 0.032-in. thick 2024-T3 flat aluminum panels subjected to air jet and siren acoustic noise (dimensions in inches).

METHODS FOR GENERATING ACOUSTIC NOISE TESTS Fixture opening

237

0 032 in

Radius = 8 ft

Differential

pressure = 6 l b

in

1

* 0'---r%-++0 Relative time to failure

FIGURE 10.2. Effects of different mounting methods and curvature on the panel fatigue life (panel gage and size constant). 5

1 2 000,

0 01

-

01

Air

jet

Sireq

10

Time to failure, hours

FIGURE 10.3. Effects on the RMS stress levels and failures induced in 0.032-in. thick 2024-T3 aluminum panels subjected to air jet and siren acoustic noise.

Strain gages on the flat panel show that, for a given RMS stress level, the air jet produces more rapid fatigue failures than the siren. This may be because the peak stress responses for the air jet are several times greater for a given RMS value than they are for the constant-level siren tests. The test results are shown in Fig. 10.3. No significant differences were noted in the nature of the failures experienced for the two types of loading. Similar acoustic noise tests were run on a similar aluminum panel 0.064 in. thick for a comparison. It was found that doubling the panel thickness increased the panel fatigue life by a factor of about 20. These results were confirmed for both the air jet and the siren test. Air jet and loudspeaker acoustic noise tests are very similar to random vibration tests, since they cover a broad frequency spectrum. Random vibration analysis methods can therefore be used to determine the fatigue life of

238


thin flat panels exposed to these acoustic noise environments. The method of analysis is to determine the pressure spectral density in units of pressure squared per hertz [(lb/in.2)2/Hz] in place of the power spectral density in units of acceleration squared per hertz (G2/Hz) used for random vibration, as shown in Eq. 9.36.

10.4

ONE-THIRD OCTAVE BANDWIDTH

Sound pressure levels are usually measured with filters set up for one-third octave bands. Each octave is then divided into three bands. The ratio of the upper band frequency to the lower band frequency for the one-third band increments can be shown to be +log,,

f2

=

+log, 2 = f(0.693)

= 0.231

Take the antilogarithm of both sides: f 2

-=

1.2598 or

f2 =

1.2598f,

(10.3)

f l

The center frequency within the band is obtained from

f, =

rn

(10.4)

The bandwidth at the center frequency of the one-third octave band becomes Af

=f

, -f l

= 0.231f,

(10.5)

upper frequency of band (Hz) f l = lower frequency of band (Hz) f, = center frequency of band (Hz)

where

f2 =

10.5

DETERMINING THE SOUND PRESSURE SPECTRAL DENSITY

The sound pressure spectral density P, is defined as the sound pressure ( P ) squared and divided by the one-third octave bandwidth:

P' p, = Af

[ (lb/in.2)2/Hz]

(10.6)

SOUND PRESSURE RESPONSE TO ACOUSTIC NOISE EXCITATION

239

The sound pressure P can be determined with the use of Eq. 10.1 in a slightly different form as follows:

or iog,, P

=

dB -log,, 20

io + log,,

2.9 x 10-9

Taking the antilogarithm of both sides results in the sound pressure shown in a slightly different form:

P = (2.9 x 10-9) x I O ~ B / * O

(10.7)

Substitute Eqs. 10.7 and 10.5 into Eq. 10.6:

P, =

(2.9 x 10-9)* x 0.231f ,

iod~/10

[ (lb/in.')*/Hz]

(10.8)

10.6 SOUND PRESSURE RESPONSE TO ACOUSTIC NOISE EXCITATION

In a lightly damped structural system excited by acoustic noise, the RMS sound pressure response, or output (P,,,), can be obtained using the same method as for random vibration, because the two methods of excitation are similar. From Eq. 9.36 PRMS =

{Z

(10.9)

where P, = sound pressure spectral density (SPSD) in (lb/in.*>*/Hz at the resonant frequency f, = resonant frequency (Hz) Q = transmissibility at the resonant frequency (dimensionless) The RMS sound pressure response can be used to obtain the stresses in the structure, and the fatigue life can be obtained by following the Gaussian probability distribution for a single-degree-of-freedom system using Miner's cumulative damage index as shown in Eq. 9.42.

240


Octave band center frequency. Hz

FIGURE 10.4. Acoustic noise qualification test spectrum.

Panel

Bulkheads

I

I

I

I

FIGURE 10.5. An electronic box with a thin flat aluminum side panel.

Sample Problem-Fatigue to Acoustic Noise

Life of a Sheet-Metal Panel Exposed

An electronic box must be capable of withstanding an acoustic noise qualification test spectrum as shown in Fig. 10.4 for a period of 30 minutes. Weight is a problem, so 2024-T3 aluminum side panels with a thickness of 0.040 in. are proposed, as shown in Fig. 10.5. Will the thin side panels survive the acoustic environment? Solution. The resonant frequency of the panel must be determined first to establish the dynamic characteristics, and to find the acoustic noise level acting on the side panels. The panel is considered to be a flat uniform square plate clamped on all four edges. Using the relations shown in Chapter 6 for plates.

(10.10) 10 x l o 6 1b/im2 (modulus of elasticity for aluminum) in. (thickness of panel) p = 0.30 (Poisson’s ratio, dimensionless)

where E h

=

= 0.040


g

241

= 386

in./s2 (acceleration of gravity) y = 0.10 1 b / h 3 (density of aluminum) a = 10.0 in. (length of square side panel) Eh3 (10 x 106)(0.040)3 D= = 58.6 lb in. 12(1- F’) (12) 1 - (0.3)’]

[

yh

(0.10)(0.040)

P = -=

386

g

= 0.00001036

lb . s2/in3

Substitute into Eq. 10.10: (10.11) An examination of Fig. 10.4 shows an acoustic noise level of 150 dB at a frequency of 141 Hz. Deflections and stresses in the side panels will be obtained for this noise level, using the fundamental resonant frequency of the panel as the center frequency of a one-third octave band. Substitute into Eq. 10.8 for the SPSD level: P, =

(lb/in .2 )’ (2.9 x 1 0 - 9 ) ’ ( 1 0 ) ~ ~ ~ ~ = 0.000258 (0.231) (141) Hz

Substituting into Eq. 10.9 for the sound pressure response, where

f,,= 141 Hz (resonant frequency of panel) Q=

fi=

=

11.9 (approximate transmissibility)

we have 77

PRMS =

-(0.000258)(141)(11.9) 2

= 0.824

lb/in.*

(10.12)

The dynamic displacement of the aluminum panel must be determined to see if the structural system is linear or nonlinear. When large displacements occur in the flat panel, the panel will carry some of the load in direct bending and some of the load as a membrane. When the panel is loaded as a membrane, it is possible to carry large loads with a relatively small displacement. When the dynamic displacements in the panel are less than one-third of the panel thickness, linear plate-bending theory can be used to calculate the plate-bending stresses. When the panel deflections are greater than one-third the panel thickness, large-deflection theory must be used to obtain an accurate estimate of the true stress condition. This theory goes into

242


membrane stresses and shows the proportion of the load carried by direct bending and the proportion of the load carried as a membrane, where the load is carried in pure tension. The total RMS acoustic pressure response acting on the panel is determined from Eq. 10.12. Part of this pressure will be carried as a membrane acoustic pressure, and part will be carried as a bending acoustic pressure when the plate deflection exceeds one-third of the panel thickness: qm = membrane acoustic pressure (Ib/in.* RMS) q h = bending acoustic pressure (Ib/in.* RMS)

Deflections resulting from membrane loading can be determined from Timoshenko's formula for a square plate [29]:

q,c 1'3 Y = ~ . N Q c ( ~ ) or

Y 'Eh q,= 0.516C4 (Ib/in

.2)

(10.13)

Deflections resulting from the plate bending due to a uniform load on a square plate with clamped edges can be determined as follows:

Y = 0.224-

4bC4 Eh3

Or

"=

YEh 0.224C4

(lb/in

.2

)

(10.14)

The total acoustic pressure acting on the panel is the sum of the membrane pressure and the bending pressure, which is the acoustic response pressure shown by Eq. 10.12. (Note: C = 10/2 = 5.0 in.).

P,,,

=

( 1 . 9 4+ ~ 4.46 y*

i

(10.15)

In the above equation, everything is known except the deflection Y , so Y can be determined. Substitute Eq. 10.12 into Eq. 10.15: 0.824 = 4.567Y+ 1241.6Y3 Y = 0.0735 in.

(RMS displacement)

(10.16)

Since this displacement is much greater than the panel thickness of 0.040 in., the acoustic stresses will be nonlinear. Membrane and direct bending stresses in the panel will have to be calculated.


243

The portion of the load that is carried as a membrane can be obtained from Eq. 10.13, where

Y = 0.0735 in. (RMS displacement) E = 10 x 106 lb/in.* (modulus of elasticity of aluminum) h = 0.040 in. (thickness) C = 10/2 = 5.0 in. (half of panel dimension) Thus (0.0735)3(10 x 106)(0.040) 9,

=

(0.516)(5.0)4

= 0.492

1b/ina2 RMS

(10.17)

The membrane stress in the panel due to the acoustic noise can be determined from the following expression [29]:

S,

= 0.396(

S,

=

0.396

1

qiEC2 7

i

(10.18)

(0.492)2(10 x 106)(5.0)2 =

( 0.040)2

1318 lb/in.2 RMS (10.19)

The portion of the load that is carried in direct bending can be determined from Eq. 10.14: (0.0735)(10 x 106)(0.040)3 qb

=

(0.224) (5 .0)4

= 0.336

1b/ina2 RMS

(10.20)

The maximum bending moment developed in the square panel with fixed edges can be determined from the following relation [29]:

M

= 0.0513qbn2

M = (0.0513)(0.336)(10.0)* = 1.724 in:lb/in.

RMS

(10.21)

The bending stress is determined from the relation 6M

sb -- - =h2

(6) (1.724)

(0.040) *

= 6465

lb/in.2 RMS

( 10.22)

The maximum total stress at the edge of the panel will be the sum of the membrane stress and the bending stress. Since reversed stresses are involved,

244


fatigue must be considered, so stress concentrations ( K ) must be included in the fatigue life evaluation. Typical stress risers found in electronic structures are about K = 2.0 at holes and corners. Thus

+ S,) (total stress) S, = 2( I318 + 6465) = 15,566 Ib/in.2 S, = K ( S,

(10.23) RMS

(10.24)

Miner's cumulative damage can be calculated to obtain an accurate estimate of the fatigue life expected for the panel. A quicker way to obtain the fatigue life, but with less accuracy, is to consider only the 3 a stresses, which do most of the damage, but occur only 4.33% of the time. The 3 a stresses are as follows: S,

(3)(15,566)

=

= 46,698

lb/in.2

(10.25)

The approximate fatigue life of the 2024-T3 aluminum panel can be determined from the S-N fatigue curve shown in Fig. 10.6. An examination of this curve shows that an approximate fatigue life of about 1.8 X IO4 fatigue cycles can be expected. The time it will take for the panel to fail can be obtained from the relations established for random vibration, as follows:

T=

T= =

N

(10.26)

0.0433f , 1.8 x l o 4 cycles to fail (0.0433) ( 141 cycles/s) (60 s/min)

(10.27)

49.1 minutes to fail

The acoustic noise qualification test only lasts for 30 minutes. At first glance it would appear that the panel could meet the requirements. However, when manufacturing tolerances are considered, and when there is a strong

' ~,

.

:a

x 104

57000

0 -

18C22~-

-

v

I

Q

c

s

I

.L___

i.p--_l13-

106

105 \

106

cycles t o f a

1:-

.c8

I

FIGURE 10.6. Fatigue curve for 2024-T3 aluminum sheet.

245

DETERMINING THE SOUND ACCELERATION SPECTRAL DENSITY

possibility that these tests must often be repeated several times due to calibration and electrical problems, it must be concluded that the panel design is deficient. The panel design must be changed. The performance can be improved by making the panel slightly thicker, or by increasing the damping in the panel to reduce the transmissibility Q to reduce the stresses. These changes will, of course, increase the weight slightly. 10.7 DETERMINING THE SOUND ACCELERATION SPECTRAL DENSITY

An alternate method of solution can be used to solve acoustic noise problems associated with sound pressure acting on flat panels. This is the sound acceleration method, where the acoustic sound pressure is converted into a sound acceleration acting on the panel. The acceleration response of the panel is then determined, which leads to the dynamic load. The dynamic load is then converted back to a dynamic pressure response acting on the panel, The which is the same as the sound pressure response of the panel (PRMS). stresses in the panel are then determined as before. The sound acceleration spectral density A , is defined as the square of the acceleration level expressed in gravity units, divided by the frequency bandwidth across the one-third octave points. This is the same idea that was used before, except that sound acceleration terms are now being used instead of sound pressure terms: (10.28) The acceleration level G, due to the sound pressure P can be determined from the specific weight w of the thin flat panel as follows: (10.29) where P = sound pressure (lb/in.*> from the dB acoustic level (RMS) w = specific weight of the panel (lb/in.*> y = density of the panel ( l b / i ~ ~ . ~ > h = thickness of uniform panel (in.) The RMS acceleration response to the broadband acoustic noise can be determined using the same method previously shown for random vibration with Eq. 9.36 since the methods of excitation are the same: (10.30)

246


Sample Problem-Alternate Method of Acoustic Noise Analysis

Use the sound acceleration spectral density method to solve the previous sample problem.

Solution. Determine the acceleration level due to the sound pressure of 150 dB using Eq. 10.29, where lb/in.* (sound pressure due to 150 dB; see Eq. 10.2) y = 0.10 I b / i r ~ .(density ~ of aluminum panel) 12 = 0.040 in. (thickness of panel) G, = 0.092/(0.10)(0.040) = 23.0 (acceleration due to sound) (RMS)

P

= 0.092

Determine the acoustic acceleration spectral density using Eq. 10.28, where

G,

f,

= 23.0 =

(RMS)

141 HZ

We obtain

A, =

(23 .O)’ (0.231)( 141)

=

16.24 G2/Hz

(10.31)

Substitute into Eq. 10.30 to determine the acceleration response to the sound pressure excitation, where A , = 16.24 G2/H2 (sound acceleration spectral density) f, = 141 Hz = 11.87 (approximate transmissibility) Q= We obtain

G,,,

-/

= 206.6

=

(10.32)

This acceleration response must now be converted back to a sound ). This is obtained from the acceleration force pressure response (PRhIs divided by the area of the panel, using one square inch of panel area:

PRMS =

wp

RMS

(lb/in.* RMS)

(10.33)

A,

. ~0.004 ) lb (panel weight) where Wp = (1 in.)(l in.)(0.04 in.)(O.lO l b / i ~ ~ = A , = 1.0 i n 2 (panel surface area) G,,, = 206.6 (acceleration response of panel)

DETERMINING THE SOUND ACCELERATION SPECTRAL DENSITY

247

We obtain

(0.004) (206.6) PRMS =

1.o

= 0.826

lb/in.*

(10.34)

Comparing the acceleration response result with the sound pressure response result in Eq. 10.12 shows they are the same. The panel stresses due to the acoustic noise can now be obtained for the nonlinear system, starting with the membrane loading condition defined by Eq. 10.13 through Eq. 10.27.

Designing Electronics for Shock Environments

11.1

INTRODUCTION

Shock is often defined as a rapid transfer of energy to a mechanical system, which results in a significant increase in the stress, velocity, acceleration, or displacement within the system. The time in which the energy transfer takes place is usually related to the resonant frequency, or to the natural period of the system. Shock will often excite many of the natural frequencies in a complex structure, which can produce four basic types of failures in electronic systems. These failures are due to (1) high stresses, which can cause fractures or permanent deformations in the structure: (2) high acceleration levels, which can cause relays to chatter, potentiometers to slip, and bolts to loosen; (3) high displacements, which can cause impact between adjacent circuit boards-cracking components and solder joints, breaking cables and harnesses, and fracturing castings; and (4) electrical malfunctions that occur during the shock but disappear when the shock energy dissipates. This last effect can occur in crystal oscillators, capacitors, and hybrids. A large thin hybrid cover can displace and cause a temporary short circuit with the internal die bond wires. The hybrid typically appears normal after the shock exposure, which makes it difficult to find and correct the failure. Fatigue is usually not an important consideration in shock, unless a million or more stress cycles are involved. When less than a few thousand stress cycles are expected, fatigue stress concentrations are ignored because they do not have a great influence on how or when the structure will fail. Stress-concentration factors must be included in any shock evaluation where fatigue effects are expected to accumulate. The impact sensitivity, notch sensitivity, and brittleness of a structure are important, especially when high-strength steels and castings are used to carry high shock loads. When the ductility of any structural load-carrying material is less than about 370, problems can develop, unless the dynamic stresses are carefully evaluated with respect to the anticipated environment. However, in most structural elements the ductility is normally greater than about 5%, so 248

SPECIFYING THE SHOCK ENVIRONMENT Onedegree of freedom

v + : < , ;

249

Twodegrees of freedo-n

s i s s a h [ , , , i ; Ori , : : !

2 K

2 KI

FIGURE 11.1. Lumped mass-spring systems with one and two degrees of freedom.

the limiting factor in the design is usually based on the yield strength of the material. Isolation systems are often used to protect sensitive electronic equipment in severe shock environments. Care must be exercised to allow sufficient sway space around the equipment to prevent impact against other surrounding structures. Shock analysis techniques can become quite complex, unless some simplifying approximations are made. For this reason, sophisticated electronic systems are often simulated using simple masses, springs, and dampers to estimate the dynamic characteristics of the system. Figure 11.1 shows typical one- and two-degree-of-freedom systems used to approximate electronic systems. The single degree can be used to represent a chassis or a PCB. The two degrees can be used to represent a PCB mounted within the chassis. The chassis represents the first degree of freedom, and the PCB the second.

11.2 SPECIFYING THE SHOCK ENVIRONMENT Many different methods have been used to specify shock motion or its effects. The three most popular methods are (1) pulse shock, ( 2 ) velocity shock, and (3) shock response spectrum. Pulse shock deals with accelerations or displacements in the form of well-known shapes such as the square wave, half sine wave, and various types of triangular waves (vertical rise, vertical decay, and symmetrical). Pulse shocks are easy to work with because the mathematics are simple and convenient. However, pulse shocks do not represent the real world. The true shock environment is seldom a simple pulse. Nevertheless, simple pulses are often effective in revealing weak areas in many different types of structures. Some typical pulse shocks are shown in Fig. 11.2. Velocity shock is concerned with systems that experience a sudden velocity change, such as a falling package whose velocity abruptly goes to zero when

250

DESIGNING ELECTRONICS FOR SHOCK ENVIRONMENTS

I(',

Versed slne

101

FIGURE 11.2. Various types of shock pulses.

Vertical "se (bast pulse)

(1

I

Symmetrica trlangular

I f )

Vertical decay triangular

the package strikes the ground. This is a common test that is called the drop shock. Sometimes an inclined plane is used, where a package gains velocity as it slides down the plane and hits a rigid wall. Another type of velocity shock makes use of a heavy hammer that slams into a fixture that supports a test specimen. The hammer imparts a sudden velocity to the fixture and the test specimen. This type of velocity shock test is used extensively by the Navy to

FIGURE 11.3. Lightweight Navy shock machine. with a 400-lb hammer, for shocktesting electronic assemblies that weight less than 250 lb.

PULSE SHOCK

Shock response acceleration

251

iiG I

1

1

fi

f2

f3

fa

f5

f6

Transient shock

input

FIGURE 11.4. Development of a shock response spectrum curve, showing how a

large number of single-degree-of-freedom systems, with different resonant frequencies, are used to describe the dynamic characteristics of a transient wave.

simulate the effects of explosions on ship and submarine equipment. Figure 11.3 shows a typical lightweight shock machine used by the U.S. Navy to test equipment weighing less than 250 Ib. The shock response spectrum deals with the way in which a structure responds to the shock motions, rather than trying to describe the shock motion itself. The spectrum is a plot of the peak acceleration response of an infinite number of single-degree-of-freedom systems to a complex transient wave form. The individual single-degree-of-freedom masses are usually specified as having a transmissibility Q of 10 when excited at their resonant frequency with a sinusoidal vibration input. This method of analysis is more representative of the real world, but the mathematics is far more complex than the mathematics of the simple shock pulse. Figure 11.4 shows how the peak shock response acceleration levels are used to develop the shock response spectrum.

11.3 PULSE SHOCK

Pulse shocks are often specified for testing electronic equipment, and many military specifications such as MIL-E-5400, MIL-STD-810, and MIL-T-5422

252


define different types of shock pulses and detailed methods for testing with these pulses. The half-sine shock pulse is the most common shock pulse used for testing almost any kind of commercial, industrial, or military product. This is because the half-sine shock pulse is the easiest to generate, and because it is easy to analyze and to evaluate. This does not mean that the half-sine shock pulse accurately represents the actual shock expected in the real environment. Impact machines are generally used to generate the shock pulses shown in Fig. 11.2. Some of these machines have rigid tables that drop on shaped lead pellets, some have compressed air cylinders to sharply increase the velocity for a short drop height, and some use a flexible beam or plate to produce a strong rebound that can almost double the velocity change through a change in the direction of motion. Pulse shocks are conveniently measured with an accelerometer, so it is convenient to define pulse shocks in terms of acceleration and time. With this method the area under the shock pulse represents the change in velocity. and the area under the velocity curve represents the displacement. When a drop test is used to generate a shock pulse, the rebound condition must be evaluated. The rebound coefficient can vary from a value of 1.0 for zero rebound, to a value of 2.0 for a full rebound. The velocity change for a drop test can then be related to the coefficient of rebound:

v = c&E

(11.1)

where I/ = velocity change (ft/s) C = coefficient of rebound (1.0-2.0) g = acceleration of gravity (usually 32.2 ft/s’) H = drop height (ft)

11.4 HALF-SINE SHOCK PULSE FOR ZERO REBOUND AND FULL REBOUND The half-sine shock pulse for a drop shock can be generated using any condition between zero and full rebound. The shock pulse will be the same for zero and for full rebound when the acceleration and time are the same. The velocity change will also be the same, since the area under the acceleration curve represents velocity change. However, the velocity curves will look different because there is a change in the direction for full rebound. The displacements produced by zero rebound and full rebound will be different for the same drop heights. Also, the drop heights required to produce the same acceleration levels will be different for the two conditions. These results can be demonstrated with a sample problem.

HALF-SINE SHOCK PULSE FOR ZERO REBOUND AND FULL REBOUND

253

20 5 Full rebound -Time

c 0 0Y 3 .l - c

>

-273 -41 0

I

c

Time

c

E -za z -

n

FIGURE 11.5. Relation between acceleration, velocity, and displacement for a 100-G, 0.020-second half-sine shock pulse for zero rebound, 50% rebound, and full rebound.

Sample Problem-Half-Sine

Shock-Pulse Drop Test

An electronic box must be shock-tested using a 100-G half-sine shock pulse with a 0.020-second time base for a qualification test. Determine the expected velocity changes, the dynamic displacements, and the drop heights required for a zero-rebound and a full-rebound test. Figure 11.5 shows how the acceleration, velocity, and displacements will vary with respect to time for zero rebound, 50% rebound, and full rebound 1441. Solution Zero Rebound The zero-rebound condition will be generated by a drop machine that smashes a lead pellet as shown in Fig. 11.6. When the electronic box is rigidly fastened to the drop table, the shock pulse will be transmitted from the rigid table to the box. The area under the acceleration curve than becomes the velocity change. The area can be obtained using integration methods, as follows:

Y=A,sinArea

ITX

t

= /x=tYdr x=o

(shock pulse)

(11.2) (11.3)

254


'

Drc,p guJde

rods

,

AElectionir box

a

Drop table

I

i n raised position ~

1 1 Displacement

=

0 4 1 ft

Lead pellet for zero rebound

~

Floor line

I !

FIGURE 11.6. Drop shock machine, with a lead pellet, used to produce a zero-rebound shock.

This will result in the following velocity change: Area

= hV=

2A,t

(11.4)

~

x

where A , = maximum acceleration peak for 100 G = lOO(32.2) = 3220 ft/s2 t = 0.020 second = time duration of pulse Thus (2)(3220)(0.020)

AT/=

= 41 .0 ft / S

(11.5)

7T

The area under the velocity curve represents the displacement of the lead pellet when it is smashed. Integrating the area under the velocity curve results in the following displacement: (11.6) (3220)(0.020)'

z=

=

0.41 ft

(11.7)

7T

The displacement can also be obtained from the velocity change: (41 ft/s)(0.020 s) 2

z=

=

0.41 ft

(11.8)

HALF-SINE SHOCK PULSE FOR ZERO REBOUND AND FULL REBOUND

255

Free-Fall Drop Height-Zero Rebound The free-fall drop height required to generate the 100-G half-sine shock pulse can be determined by setting Eq. 11.1 equal to Eq. 11.4. This will result in three expressions as shown below:

&H=-

2A,t 7T

2A2,t2 H=-=-P2

AV2 2g

2t2G2g

(11.9)

-7T2

Using the first expression to find the drop height, (2) (3220 ft/s2 )’( 0.020)’ (32.2 f t / s 2 ) x 2

H=

=

26.1 ft

(11.lo)

Using the second expression to find the drop height,

H=

(41 ft/s)’ (2)(32.2 ft/s2)

= 26.1

ft

(11.11)

Using the third expression to find the drop height,

H=

(2)(0.020)2(100 G)2(32.2 ft/s2)

= 26.1

ft

(11.12)

7T2

Full Rebound The full-rebound condition will be generated by a drop table that hits a spring that bounces the drop table back to its initial starting point, as shown in Fig. 11.7. For the full rebound, the coefficient of rebound is 2.0. However, the velocity change is still equal to the area under the half-sine shock pulse, shown by Eqs. 11.3 and 11.4. Its value was shown by Eq. 11.5 to be 41.0 ft/s.

Electronic

table

4

Drop height

I L---

\

FIGURE 11.7. D r o p shock machine, with a spring, used to produce a full-rebound shock.

256


The area under the velocity curve still represents the deflection of the spring that bounces the drop table back to its original starting position. Integrating the area under the velocity curve results in the following expression:

A,? hVt Area=Z= 7 - - - - displacement Tr 2Tr

(11.13)

The deflection in the spring due to the 100-G half-sine shock pulse with a 0.020-second duration can now be determined from Eq. 11.13:

Z =

(3220 ft/s’)(0.020 s)’ =

0.13 ft

(11.14)

Tr2

The deflection in the spring can also be determined from the second expression of Eq. 11.13:

Z=

(41.0)(0.020 s) 27i

= 0.13

ft

(11.15)

Free-Fall Drop Height-Full Rebound The free-fall drop height required to generate the 100-G half-sine shock pulse can be determined by setting Eq. 11.1equal to Eq. 11.4 when the coefficient of rebound is 2.0. This results in three relations as follows:

Ait’ A V 2 t2G2g H = -- - - -8g 2 ~ ‘ 27i’g

(11.16)

Using the first expression,

H=

(3220 ft/s2)’(0.020 s)’ 2 ~ ’ ( 3 2 . 2ft/s2)

= 6.53 ft

(11.17)

Using the second expression,

H=

(41.0 ft/s)’ (8)(32.2 ft/s2)

= 6.53 ft

(11.18)

RESPONSE OF ELECTRONIC STRUCTURES TO SHOCK PULSES

257

Using the third expression,

H=

(020)2( 100 G)*(32.2 ft/s2) 2r2

= 6.53

ft

(11.19)

Alternate Method of Solution-Full Rebound The dynamic displacement of the spring that returns the drop table back to its starting position can also be determined from the acceleration value in units of G and the frequency of the half-sine shock pulse, as shown below. 9.8 G

z=-

f2

(see Eq. 2.30)

(11.20)

where G = 100 G f = natural frequency of shock pulse = 1/(2)(0.020 S) = 25 HZ Thus (9.8) ( 100)

z=

(W2

=

1.568 in. = 0.13 ft

(11.21)

11.5 RESPONSE OF ELECTRONIC STRUCTURES TO SHOCK PULSES

Various types of shock pulses are often used to excite electronic assemblies to simulate transportation environments, bench handling conditions, and pyrotechnic events used to separate multiple stages on missiles and spacecraft. The manner in which the various electronic components respond to these shocks will determine if the components will survive the environments. It is often convenient to represent various structural elements in the electronic box as simple masses and springs, so the approximate response characteristics can be evaluated quickly and cheaply. This type of analysis yields fast results, but the accuracy is reduced, since it is impossible to accurately represent a complex structure with a few masses and springs. A real electronic system will typically have many major resonant frequencies. The purpose of the simplified analysis is to try to simulate the first few major resonances where most of the damage normally occurs. When approximations are adequate, a lot of time and money can be saved with a small decrease in the accuracy. Plug-in printed circuit boards (PCBs) can quickly be evaluated by approximating them as a single degree of freedom (using a single mass, spring, and damper). This produces good results, since test data on many different types

258


of PCBs show that most of the damage is produced by the fundamental resonant frequency of the PCB. Dynamic displacements, stresses, and accelerations are usually maximum under these conditions. Therefore, it is necessary to understand how single-degree-of-freedom systems respond to various types of shock pulses, in order to determine if these systems will be damaged by shock pulse environments.

11.6 RESPONSE OF A SIMPLE SYSTEM TO VARIOUS SHOCK PULSES

When a simple single-degree-of-freedom system is excited by a shock pulse, the system can amplify the magnitude of the pulse, or attenuate it. The way in which the system responds to the particular pulse will depend on the ratio

R=

natural frequency of the structure

(11.22)

natural frequency of the shock pulse

The response of a simple system to a half-sine shock pulse is shown in Fig. 11.8. Note that damping does not have much effect on the shock amplification A . When the damping is zero, the maximum shock amplification is only 1.76. Also note that for lightly damped systems the shock amplification is about 2R, when R is less than about 0.5. The shock isolation

i 8

16 1 4

s

1 2 10

Input

08 06 Response to half sine shock pulse

04 0 2

0 5 1

Isolation area

2

3

A

5

Frequency ratio R

6 =

7

8

9

10

fstruct/ fpulse

FIGURE 11.8. Response of a single-degree-of-freedom system, with different damping ratios, to a half-sine shock pulse.

RESPONSE OF A SIMPLE SYSTEM TO VARIOUS SHOCK PULSES

Response t o squarewave shock pulse

m

s

259

Input

lsolatlon area

v

1

2

3

5

4

F r e w e n c y ratio R = / & u t i /

/pulse

FIGURE 11.9. Response of a single-degree-of-freedom system, with zero damping to a square-wave pulse.

-

Rc

=

0

/I

Respoise to sawtooth shock Dulse

1

0

o0r

Isolation area

1 1

1

I

2

Frequency ratio R

I 3

= fstruct

1

I

4

5 fDulse

FIGURE 11.10. Response of a single-degree-of-freedom system, with zero damping, to a vertical-decay sawtooth shock pulse.

area lies to the left of the amplification peak, where the ratio R is less than 0.5. In this area the shock response level is less than the shock input level. The responses of a simple single-degree-of-freedom system to other types of shock pulses are also important. Figure 11.9 shows the response of a simple zero-damped system to a square wave. Figure 11.10 shows the response of a simple zero-damped system to a vertical-decay sawtooth shock pulse. Other types of sawtooth shock pulses can be generated, such as the symmetrical pulse and the vertical-rise pulse, which is sometimes called the blast pulse. These pulses will have amplification values slightly greater than the vertical-decay sawtooth pulse.

260

11.7


HOW PCBs RESPOND TO SHOCK PULSES

When PCBs are excited by shock pulses, they will respond by bending initially in the same direction as the pulse. When the pulse diminishes, the PCBs will then resonate at their own resonant frequencies, of which the fundamental, or lowest, resonant frequency is usually the most prominent. Extreme care must be exercised in the determination of clearances for PCBs that must operate in severe shock environments. Sufficient clearances must be provided to account for tolerance accumulations in the thickness of the PCBs, the component sizes, component lead wire protrusions on the back side of the PCBs, location tolerances, and possible displacement amplitudes of adjacent PCBs moving in opposite directions at the same time. It is important to keep the dynamic displacements low, so the dynamic stresses will be low and the chances for impact between adjacent PCBs will also be low. Experience has shown that high shock acceleration levels can result in cracked solder joints and fractured lead wires on large or heavy electronic components. Large components such as transformers, DIPs, capacitors, and motors must be mounted very carefully to avoid failures in the support structure or in the mounting hardware. When large components are mounted on PCBs that exhibit large displacement amplitudes, the relative motion between the component body and the PCB can often produce high forces and stresses. Figure 11.11 shows how the lead wires and solder joints in a large DIP are stressed when a PCB develops large deflections. The greatest damage will occur when the large components are mounted at the center of a plug-in PCB, where the curvature changes and the acceleration levels are the greatest.

11.8 DETERMINING THE DESIRED PCB RESONANT FREQUENCY FOR SHOCK

Large electronic components such as DIPs, transformers, hybrids, leadless ceramic chip carriers (LCCCs), and pin grid arrays are being used more and more in sophisticated electronic systems. At the same time, the vibration and

Relative Strain in

FIGURE 11.11. PCB resonance producing stresses in lead wires, d u e to relative motion between the component body and the PCB.

PCB b e i d i n g

DETERMINING THE DESIRED PCB RESONANT FREQUENCY FOR SHOCK

261

shock environments are becoming more severe. Since one of the greatest problems with large components is large PCB deflections, which can cause rapid lead wire and solder joint failures, it becomes necessary to reduce PCB deflections to prevent these failures from occurring. One of the easiest methods for reducing the deflections of a PCB is to increase the resonant frequency of the PCB. However, if the PCBs are made too stiff, there will be a big size, weight, and cost penalty. Time and money can be saved by relating the PCB displacement to the geometry of the component and to the test or operating environment. The method used here is based on the same test data previously used to establish the desired PCB displacements for the sinusoidal and random vibration environments. In Chapters 8 and 9, it was shown that testing experience could be used to establish desired PCB displacement relations for improving the fatigue life of large electronic components: see Eq. 8.11 and Eq. 9.53. The same method can be used for the shock environment. When less than a few thousand stress reversals are expected in a particular shock environment, and when the structural material is expected to have a ductility of at least 5%, then fatigue is no longer a major factor and stress-concentration effects are sharply reduced. Fatigue factors found in nonferrous alloys, such as aluminum, typically are of the ultimate tensile strength, and stress concentrations typically are about 2.0 for holes and notches. When these two effects are sharply reduced, the effective stress levels for the shock environment can be increased by a factor of 3 X 2, or 6. Applying this correction factor to the shock environment results in the adjusted PCB displacement value shown below:

Z=

0.00132B

(desired PCB displacement)

Chra

(11.23)

The dynamic displacement expected in the shock environment can be obtained from Eq. 2.30 and from the shock amplification factor A shown in Fig. 11.8:

Z=

9.8 G , , A

(11.24)

f,’

Typical values for the shock amplification range from about 0.5 to about 1.5. Combining Eq. 11.23 and Eq. 11.24 results in the desired PCB resonant frequency for shock that will avoid component failures due to excessive PCB deflections:

fd =

(

9.8 GinAChr\iZ 0.00132B

j”’

(11.25)

262


-E

=

h = ?YO093 in

8 0 In-

DIP

1

ioi

PCE

ib( Pulse

FIGURE 11.12. Plug-in PCB subjected to a 100-G half-sine shock pulse.

where A = shock amplification factor, typically 0.5-1.5 G,, = peak acceleration input to PCB (gravity units) C = constant for different types of components (see Eq. 9.50) B = length of PCB edge parallel to component (in.) h = thickness of PCB (in.) L = length of electronic components (in.) r = relative position factor for component on PCB (see Eq. 9.50) Sample Problem-Response

of a PCB to a Half-Sine Shock Pulse

The plug-in PCB shown in Fig. 11.12 must be capable of passing a qualification test consisting of 18 half-sine shock pulses of 100 G with a time base of 0.008 second. Several 40-pin DIPS with side-brazed lead wires are mounted near the center of the 0.90-lb PCB. Determine if the proposed design will be capable of satisfactory operation in the shock environment. Solution

Expected Resonant Frequency of PCB The first item that should be calculated is the expected resonant frequency of the PCB. In a dynamic situation, the resonant frequency will often yield important information that can be used to develop or produce a reliable design. When the four sides of the PCB are supported and the load is uniformly distributed over the surface, the natural or resonant frequency can be determined from Eq. 6.21:

2 X 106 Ib/in.* for plain epoxy fiberglass PCB 3 x 106 Ib/in.* for PCB with several full copper planes h = 0.093 in. (PCB thickness)

where E

=

DETERMINING THE DESIRED PCB RESONANT FREQUENCY FOR SHOCK

263

p = Poisson’s ratio

0.12 for plain epoxy fiberglass 0.18 with several full copper planes W = 0.9 lb (PCB total weight) b = 8.0 in. (PCB length) a = 6.0 in. (PCB width) g = 386 in./s2 (acceleration of gravity)

=(

Consider the PCB with several full copper planes, which will increase the flexural stiffness. Then

D=

Eh

-

q i - p2)

W gab

P=-=

3 x 106(0.093)3 12[1 - (0.18)*]

0.9 (386)(8.0)(6.0)

=

= 207.8

4.86 x

1b.h.

lb.~~/in.~

f,,=?/T 1 [ = 1 4L 1Hz + (11.26) y] 2

4.86

X

10-

(8.0)2

(6.0)

Desired Resonant Frequency of PCB The desired resonant frequency, which will permit the PCB to pass its qualification test, is shown by Eq. 11.25. The one parameter that is unknown at this time is the shock amplification factor A , which is obtained from R = (natural frequency of structure)/(natural frequency of pulse) as shown in Fig. 11.8. As a first approximation use the calculated value of 141 Hz just obtained to find the shock amplification A . If the desired resonant frequency is sharply different from the expected PCB resonant frequency, another iteration may have to be made with the corrected value of the shock amplification A , where Gin= 100 G (peak input acceleration) f n = 141 Hz (structural frequency, assumed to start) f p = 1/(2)(0.008) = 62.5 Hz (pulse frequency) R = 141/62.5 = 2.25 (ratio that determines A ) A = 1.6 (shock amplification from Fig. 11.8) C = 1.26 (constant for a DIP with side-brazed lead wires) B = 8.0 in. (length of PCB parallel to component) r = 1.0 (factor for component mounted at center of PCB) h = 0.093 in. (PCB thickness) L = 2.0 in. (length of 40-pin DIP)

264


Substitute into Eq. 11.25 for the desired PCB resonant frequency: 0 .s

(9.8)( 100) (1.6) (1.26) (0.093) (6) (0.00132) (8.0)

=

157 HZ

(11.27)

A reexamination of the shock amplification with the 157-Hz PCB resonant frequency shows the value of A = 1.6 still valid. Equation 11.26 shows the PCB is expected to have a resonant frequency of 141 Hz. However, the desired resonant frequency was shown by Eq. 11.27 to be 157 Hz. This is fairly close, but these equations are only approximate, being based on single-degree-of-freedom responses. Therefore, in order to be safe, the PCB resonant frequency should be raised to at least 157 Hz. This can be achieved by increasing the PCB thickness slightly, or by adding more solid-copper planes to increase the plate stiffness factor D , or by using polyimide glass, which has a higher modulus of elasticity than epoxy glass, so the PCB will be stiffer. A small stiffening rib might be added if space is available on the PCB. Sometimes it may be possible to move the larger components away from the center of the PCB toward one of the supported edges. This will reduce the value of Y, which will reduce the desired PCB resonant frequency necessary to survive the same shock level.

11.9 RESPONSE

OF PCB TO OTHER SHOCK PULSES

Square-wave and triangular-wave shock pulses can also be examined for dynamic loads on the PCB as shown in Fig. 11.13. Everything stays exactly the same, except the shock amplification factor A , which will change as shown in Fig. 11.9 for a square wave and in Fig. 11.10 for a triangular wave. Since the value of R = 2.25 stays the same, the A value for the square wave becomes 2.0, and the A value for the triangular wave becomes 1.0. Substitute these values into Eq. 11.25, and the desired PCB resonant frequency for the square wave will be 175 Hz. The desired PCB resonant frequency for the triangular wave will be 111 Hz. The conclusion is that a 100-G shock test

FIGURE 11.13. T h e 100-G square-wave and triangular-wave shock pulses with base of 0.008 second.

a time

RESPONSE OF PCB TO OTHER SHOCK PULSES

265

Traisformer 0 2 5 Ib

\% =

Alumlnum bracket

FIGURE 11.14. Transformer mounted on a cantilever aluminum beam (dimensions in inches).

using a square wave would require a PCB with a resonant frequency of 175 Hz or more to be successful. If the same shock test were run with the triangular-wave pulse, a PCB resonant frequency of 111 Hz or more would be adequate. Many structures are stress sensitive, so special emphasis must be placed on limiting the dynamic stresses developed in shock environments. Sample Problem-Shock

Response of a Transformer Mounting Bracket

A small transformer is mounted on a cantilevered aluminum bracket, as shown in Fig. 11.14. The bracket is mounted within a container that will be dropped from a low-flying helicopter. The container may land on different types of sand or dirt, which will generate different types of shock pulses. Drop tests on the various types of soils show that acceleration levels of about 100 G can be expected with a time base of 0.011 second. The different types of sands and soils are expected to generate the different types of shock pulses shown in Fig. 11.15. Determine if the proposed bracket design will be satisfactory.

Solution. The first step is to find the resonant (or natural) frequency of the bracket. A convenient relation for various types of structures is shown in Eq. 2.10, based on the static displacement Tt:

100 G

1

4

0 0 1 1 ;second

Square wave

100 G

h

4

0011 b s e t o n d

Sawtooth

100 G

h

4 0 0 1 1 bsecond Half sine

FIGURE 11.15. T h e 100-G square-wave, triangular-wave, and half-sine-wave shock pulses developed by impact.

266


where g = 386 in./s2 (acceleration of gravity) W = beam plus transformer weight = (6)(0.5)(0.25)(0.1 1b/i11.~)+ 0.25 = 0.325 lb E = 10 X IO6 lb/in.* (aluminum modulus of elasticity) bh3 (0.25) (0.50)' = 0.00260 in.4 (moment of inertia) I = -- 12 12 L = 6.0 in. (length of bracket) (0.325) ( 6 .0)3 WL3 = 9.0 X in. r,, = -3EZ 3(10 x 106)(0.0026) Thus

f

=

-yT 2.rr

9 . 0 x 10-

=

104Hz

(11.28)

The natural frequency of the pulse is obtained as follows: 1 f =-=

2t

1

2(0.011)

= 45.4

HZ

(11.29)

In order to determine the shock amplification for the different shock pulses, the ratio of the natural frequency of the structure to the natural frequency of the pulse must be obtained:

n - 104 R = -f = fp 45.4

=

2.29

(11.30)

The shock amplification factors for the three different types of shock pulses can be obtained from Figs. 11.8-11.10 for systems with zero damping. These values will be slightly conservative, since all real systems have some damping. Square Shock Pulse The dynamic load acting on the cantilevered bracket can be obtained from the following relation: pd = WG,,A

where W = 0.325 Ib (weight) G,, = 100 G (input acceleration) A = 2.0 (see Fig. 11.9; R = 2.29)

(11.31)

RESPONSE OF

PCB TO OTHER SHOCK PULSES

267

Thus Pd = (0.325) (100) (2.0)

= 65 .O

lb

( 11.32)

The maximum dynamic stress in the beam will occur at the support where the bending moment is the greatest. The stress will be in bending with no stress-concentration factors, since there will be less than a few thousand stress reversals. The bending stress at the beam support is determined from the standard bending relation

MC

s, = 7

(11.33)

where L = 6.0 in. (length of beam) M = PdL = (65)(6) = 390 lb in. (dynamic bending moment) C = h/2 = 0.50/2 = 0.25 in. (centroid distance) I = 0.0026 in.4 (moment of inertia) Thus

s, =

(390) (0.25) 0.0026

=

37,500 lb/in.*

(bending stress)

(11.34)

Safety factor The safety factor can be determined for a beam fabricated from 6061-T6 aluminum alloy, which has a tensile yield strength Sty of 33,000 lb/in.2: Sty 33,000 S F = - = -= 0.88 S, 37,500

(11.35)

Since the safety factor is less than one, the design is not satisfactory because the structure will experience permanent deformation. A change will have to be made in the material or in the structure design to permit the system to pass the required acceleration test. A simple change can be made by increasing the height of the beam, which will increase the resonant frequency, decrease the dynamic displacement, and reduce the stress level. Triangular Shock Pulse The same bracket can be examined when it is excited by the sawtooth triangular shock pulse shown in Fig. 11.15. The natural frequency of the bracket will stay the same, as will the shock pulse frequency, so the frequency ratio R = 2.29 shown in Eq. 11.30 will stay the same. The shock amplification factor shown in Fig. 11.10 for the triangular pulse, however, will change to 1.0. Since the dynamic stresses are directly related to the shock amplification, a direct ratio can be used to determine the

268


bending stress in the bracket, based on the square shock pulse, as follows:

S,

=

(37,500)

S,

=

18,750 lb/in.’

A , a u t o o t h pulse A s q u a r e puihe

=

(37,500)(:] (11.36)

The safety factor will be as follows: (11.37) Since the safety factor is greater than 1.0, the design is satisfactory and no permanent distortion is expected. Half-Sine Shock Pulse The same bracket can be examined when it is driven by the half-sine shock pulse shown in Fig. 11.15. The natural frequency of the bracket is the same, and the shock pulse frequency is the same, so the frequency ratio R = 2.29 shown by Eq. 11.30 will be the same. The shock amplification factor from Fig. 11.8 for zero damping will be about 1.7. A direct ratio can be used to determine the bending stress in the bracket, based on the square-wave shock pulse:

5 , = (37,500)

A h a i f sine Aquae

S,

= 31,875

=

(37,500)

[

\\d\e

lb/in.’

(11.38)

The safety factor will be SF=

33,000 ~

31,875

=

1.03

( 11.39)

Since the safety factor is greater than 1.0, the design should be acceptable. Half-Sine Shock Pulse with Damping When damping occurs in a dynamic system, some energy is dissipated, so the shock amplification is reduced. For structural work the damping ratio R , is used, which related damping to critical damping:

(11.40) When the transmissibility Q is known, the damping ratio can be obtained. For example, consider a Q value of 5.0: 1 R , = -= 0.10 ( 2 )( 5 )

(dimensionless)

(11.41)

EQUIVALENT SHOCK PULSE

269

This represents a damping value of 10%. The shock amplification A can now be obtained from Fig. 11.8. For a frequency ratio of 2.29 and a damping ratio of 0.10, the amplification will be about 1.4. The new dynamic load and bending stress can now be determined:

Pd = (0.325)(100)(1.4)

s, =

(45.5) (6.0) (0.25) 0.0026

= 45.5

(11.42)

lb

= 26,250

lb/in.*

33,000 S F = -- 1.26 26.250

(11.43)

(11.44)

The safety factor for the damped system is slightly higher than the safety factor for the undamped system.

11.10


When electronic equipment is shipped, the shipping containers are often provided with some type of foam or other cushioning material to protect the electronics from vibration and shock. The packing material must be selected very carefully to make sure it really protects and does not damage the equipment. It is possible for the electronics to act like a concentrated mass and the foam to act like a spring. This can produce a resonant condition that may be excited by the shipping environment. If internal resonances within the electronics (such as circuit boards) are close to the resonances developed by the foam material, then the electronics will not be protected. Instead, the electronics may be damaged by the shipping environment. Sample Problem-Shipping

Crate for an Electronic Box

A shipping crate must be designed to protect a 10-lb electronic box from a free-fall drop of 5.0 ft. The proposed cushioning material has a linear undamped spring rate of 3300 lb/in., and enough room for a sway space of 0.75 in. Some test data are available for the electronics, which show that one of the internal PCBs has a resonant frequency of 120 Hz, a transmissibility Q of 15, and also a capability of withstanding an input shock pulse of 200 G with a time base of 0.025 second for a half sine wave. Determine if the proposed design will be satisfactory. The proposed system is shown in Fig. 11.16.

Solution. The maximum acceleration level the electronic box will develop in the 5-ft drop can be determined by considering the cushioning material as a

270

DESIGNING ELECTRONICS FOR SHOCK ENVIRONMENTS Cushlon ng material

Electronic chassis

Packing crate

PCB

40 7 5

1'1

dynamic

7 displacement

FIGURE 11.16. Electronic box packed in a shipping container.

spring and the electronic box as a mass [421: (11.45) where H = 5 X 12 = 60 in. (drop height) K = 3300 lb/in. (cushioning material spring rate) W = 10 lb (electronic box weight) Thus (11.46) The acceleration response of the box becomes the input acceleration to the PCB contained within the box, with proper consideration for any amplification or attenuation effects. The maximum dynamic deflection Y that the cushioning material will experience due to the shock will be 2H y= G

where H G,,,

(11.47)

= 60 =

in. (drop height) 199 (acceleration from drop height)

Thus (11.48) At first glance the design might look good. The acceleration level of 199 G is less than the shock pulse value of 200 G. Also, the dynamic displacement of 0.60 in. is less than the maximum allowable displacement value of 0.75 in.


271

The problem with the above solution is that no consideration was given to the possibility of any shock amplification. A closer look at the analysis shows that input acceleration levels were examined, not response acceleration levels. When shock response conditions are used and shock amplifications are included in the evaluation of the PCB, then the results of the evaluation will appear quite different. The original test data showed that the PCB could withstand an input shock level of 200 G, using a 0.025-second half-sine shock pulse. The response of the PCB must be determined from this information. But first the equivalent natural frequency of the pulse must be obtained from the above information as follows:

1 fp =

1

5 = (2)(0.025)

=

20 HZ

(11.49)

Next, find the ratio R of the natural frequency of the structure to the natural frequency of the pulse, as shown in Eq. 11.22:

( 11S O ) It is desirable to consider damping in this solution for improved accuracy. The damping ratio is determined from the transmissibility Q as shown in Eq. 11.40: (11.51) An examination of Fig. 11.8 shows the shock amplification will be about 1.10 when R is 6.0:

A

=

1.10

This means that when the shock test data showed an input level of 200 G, the PCB amplified the shock level so the PCB really experienced an acceleration level of

GPCB=AGin = (1.10)(200) = 220

(11.52)

Therefore, the PCB is really capable of withstanding an acceleration level of 220 G rather than the level of 200 G previously shown. Next, find the acceleration level that will be transmitted to the PCB during the 5-ft drop test. The ratio R of the natural frequency of the structure to the natural frequency of the pulse is required once again. The structural natural frequency is still that of the PCB at 120 Hz. However, the pulse natural

272


frequency becomes the resonant frequency of the electronic box on the cushioning material, acting like a system with a single degree of freedom. This is as follows:

fp =

i"s w

2irk

(see Eq. 2.7)

where K = 3300 Ib/in. (cushioning material spring rate) g = 386 in./s2 (acceleration of gravity) W = 10 Ib (weight of electronic box) Thus ( 11.53)


R = -f "= - 120 fp 56.8

= 2.1

(11.54)

Using Fig. 11.8 with a damping ratio R , of 0.0333 as shown in Eq. 11.51 and a frequency ratio R of 2.1 results in a shock amplification value of A

=

1.5

(11.55)

This means that the 199-G shock resulting from the 5-ft drop is amplified by the cushioning material, so the PCB inside the electronic box will see an acceleration level of Go,, = AG,,

=

(1.5)( 199)

= 298.5

(11.56)

Safety Factor The safety factor can be obtained with the use of Eq. 11.52:

(11.57) Since the safety factor is less than one, the design is not acceptable, and a design change must be made. The PCB is only capable of withstanding a shock level of 220 G. However, the coupling effect of the packing-crate cushioning material forces the PCB to respond at a much higher acceleration level of 298.5 G. Therefore, it is desirable to change the properties of the cushioning material to reduce the resulting acceleration levels on the PCB.


Next Page 273

Successively lower values of K are used in Eq. 11.45 until the amplification A times the input G level is 220 or less: GPCB= GinA

=

(11.58)

220

When the cushioning material spring rate is 2000 Ib/in., the chassis natural frequency, or the equivalent shock pulse frequency, becomes = 44.2

R = -f "= - 120 fp 44.2

HZ

(11.59)

( 11.60)

= 2.7

From Fig. 11.8, using a damping ratio of 0.0333, the shock amplification becomes A

=

1.40

(11.61)

The acceleration level developed by the box in the 5-ft drop is again obtained from Eq. 11.45: (11.62) This shock is amplified by the PCB, so the acceleration level the PCB sees is GPCB= GA

=

(155)( 1.4) = 217

(11.63)

The dynamic deflection expected for the cushioning material is obtained from Eq. 11.47: (11.64) It can also be obtained from the following relation: 9.8 G y = - (see Eq. 2.30)

f2

where G f

155 (input acceleration to box from 5-ft drop) = 44.2 Hz (cushioning material frequency; see Eq. 11.59) =

Thus

Y=

(9.8) (155) (44.2)

= 0.77

in

(11.65)

-

CHAPTER 12

Design and Analysis of Electronic Boxes

12.1 INTRODUCTION

Electronic equipment can be packaged in many different configurations depending on the space and shape of factors available in missiles, airplanes, submarines, trains, autos, and ships. The rectangular type of box is generally the most common, because it is usually much less expensive to fabricate, mounting is simple, and plug-in modules such as printed circuit boards (PCBs) conform readily to this shape. Before an extensive and expensive computerized analysis is made to determine the frequency response characteristics of the chassis, it is desirable to take a quick look at the different modes of vibration. Various bending and torsional modes can be examined to determine whether coupling may occur. Preliminary dynamic loads can also be determined to see whether panels will buckle, rivets will shear, or welds will crack. Torsional vibration modes are often a source of trouble in an electronic chassis. These modes, which are generally ignored or overlooked, can result in low resonant frequencies. This in turn can lead to large deflections and stresses that can substantially reduce the fatigue life of the structure.

12.2

DIFFERENT TYPES OF MOUNTS

The type of mount can affect the response characteristics of a chassis. If quick installation and removal are desirable, the entire chassis can be inserted and removed by means of a single jacking screw located at the front of the chassis. Interface connectors can be placed at the rear of the chassis with floating mounts and alignment pins that will position the chassis and ensure accurate connector alignment (Fig. 12.1). This type of installation is very easy to maintain, but very poor from a structural standpoint. The force required to insert the chassis and engage the rear connectors must be transferred from the rear connectors to the front jacking screw where the load is applied. This means that the entire jacking 300

DIFFERENT TYPES OF MOUNTS

301

load must pass through the chassis structure, which must be made stiff enough to prevent buckling. The vibration response for this type of mount is poor because the entire chassis can rotate about the single jacking screw. Torsional modes will usually couple with bending modes, and this can result in very high transmissibilities. Tests on several different types of boxes, with similar mounts, have shown transmissibilities as high as 20 at the front outside corners, with a 2-G peak sinusoidal vibration input, when the resonant frequencies were above 150 Hz. Another type of chassis mount that provides quick installation and removal is the air-transport rack (ATR). This is a sheet-metal mounting rack supplied in several different lengths and widths in an attempt to standardize ATR equipment boxes. The back part of the rack has two spring-loaded alignment pins to position the box for connector engagement and to hold the box during vibration. The front of the rack usually has swivel screw mounts, which swing up to hold the chassis after it has been installed (Fig. 12.2). The ATR mounting rack is usually used with vibration isolators. When the proper selection is made, the isolators can sharply reduce the dynamic loads in an electronic system. If a proper isolator selection is not made, the dynamic loads can be increased. Quite often the ATR mounting rack will be used without isolators. This may lead to trouble unless the spring rate of the mounting rack is included in the response characteristics and the dynamic loading for the chassis. Since the ATR mounting rack is usually made of sheet aluminum, a high G loading on a heavy chassis can fracture the sheet-metal rack or induce high acceleration forces in the chassis itself. Any vibration level of about 2 G or greater should be examined very carefully when this type of mount is being considered. A center-of-gravity (CG) mount should be used to reduce the possibility of coupling torsional modes with bending modes during vibration. When the CG lies on the mounting plane, torsional modes will still occur, but they will not couple with the bending modes during vibration along the X axis (see Fig.

302

DESIGN AND ANALYSIS OF ELECTRONIC BOXES Spring-loaded rear alignment pins

[Sheet-metal

A

- __

Chassis outline _~

~

--

rack

1

Rear connector cut O U t S A

n

Front swivel screws

f?:,

FIGURE 12.2. Air-transport rack (ATR) for mounting electronic equipment.

12.3). This separation of resonances, or decoupling, will reduce the severity of the resonance, thereby increasing the fatigue life of the structure. A chassis with a high CG and a narrow cross section should have its mounts examined very carefully. This type of geometry can lead to very high dynamic loads in the mounts and adjacent structure. This is because of the severe overturning moments that can develop during vibration in the lateral direction (see Fig. 12.4). There are many cases where the mounting geometry is dictated by the thermal environment. For example, in a space vehicle where vacuum conditions exist, many electronic systems are bolted directly to a cold plate to remove the heat. In order to transfer heat effectively in a vacuum, the interfaces must be flat and smooth. Also, a high interface pressure must be

Mounting surfaces in line with CG

FIGURE 12.3. A n electronic box with a CG mount.

303

PRELIMINARY DYNAMIC ANALYSIS

j .iri-JJ-7 LJY

Y

c

I

High CG

Dynamic load

t1

12,

\LOW

mount

Low mount/ Large reaction",z,

1

A

X

fRz

due to the tall narrow chassis

FIGURE 12.4. An electronic box with a low mount and a high CG.

in a vacuum

FIGURE 12.5. Many mounting lugs on an electronic box.

maintained. One good method for accomplishing this is to provide a stiff section with many bolts. The penalty is paid in the form of extra weight. Although the weight is required for thermal reasons, the dynamic characteristics of such a chassis will usually be quite good. Thick walls and a solid mount will provide the chassis with a high resonant frequency, which will result in small displacements and low structural stresses (Fig. 12.5).

12.3 PRELIMINARY DYNAMIC ANALYSIS

A preliminary dynamic analysis of a complex chassis can very often be made by analyzing it as a simply supported beam. If the cross section of the chassis is relatively constant and the chassis is supported at each end, the simple-beam analysis will often reveal many weak points in the initial design. Design changes can then be incorporated early in the program. This will reduce the amount of time spent in setting up comprehensive computer studies that would give the same results. After the preliminary design changes have been

304

DESIGN AND ANALYSIS OF ELECTRONIC BOXES

Y

? t,

-Z

0 060-in epoxy fiberglass interconnecting circuit board

FIGURE 12.6. Dimensions of a tall narrow electronic box.

made, the computer program can then be used to provide a more accurate and detailed analysis of internal and external dynamic loads. Consider an electronic box that uses a riveted sheet-aluminum chassis with top and bottom covers fastened by screws. The chassis will be fastened to the airframe structure by means of two mounting brackets at each end of the chassis (Fig. 12.6). Since the chassis has a uniform cross section with mounting brackets at each end. a preliminary analysis of the chassis can be made by approximating it as a simply supported beam. The high center of gravity indicates that the bending mode will couple with the torsional mode during vibration in the lateral direction along the X axis. This will tend to lower the fundamental resonant mode in this direction. The moment of inertia of the cross section along the X and Y axes will include the epoxy fiberglass interconnecting board (modulus of elasticity of 2 x lo6, compared to 10 x l o 6 for aluminum). The calculations can be simplified by using an equivalent aluminum plate that has the same stiffness as the epoxy fiberglass plate. This is accomplished by adjusting the epoxy plate area and moment of inertia with the ratio of the modulus of elasticity of the two materials. The aluminum equivalents of the epoxy board area and stiffness are (see also Section 4.3)

(for members with the same height)

BOLTED COVERS

12.4

305

BOLTED COVERS

Bolted covers are used to provide access to the top and bottom sections of the chassis. The general tendency in the electronics industry is to provide convenient access to the electronic components. This usually means quarterturn screws so that the covers can be removed and replaced quickly. Tests on this type of fastener indicate it is only about 10% efficient in its ability to hold two members together during vibration, on a scale that ranges from 0% for a joint that has no connection to 100% for a joint that is welded. The epoxy fiberglass circuit-interconnecting board is also a bolted member. However, since this type of installation is usually of a more permanent type, regular screws, and more of them, are normally used. Therefore, a bolted efficiency factor of 25% is used for this member. The bolted efficiency factor of a heavy flanged cover, with many highstrength cap screws, is about 50%. This type of construction is often used for electronic boxes that are to be used in the hard vacuum of outer space. Since covers are not normally removed for maintenance in a space vehicle, many high-strength screws are often used to hold covers in place. A cross section of an electronic chassis is shown in Fig. 12.7.

Y

I

0 040

Local screw bracket 1 %

'

1.

I

I

692

-

I

7 00

_ _

-

370--

interconnecting circuit board

I

Y

0 060 side wall

-

7.0 040 -

~-

75

\Cable

harness a i d wiring area

FIGURE 12.7. Cross section of a sheet-metal electronic chassis (dimensions in inches).

306


TABLE 12.1

Item

Area, in.'

It,: in.'

AY

Y

c

c2

Ac'

1

6.92(0.060) = 0.415 3.50 1.450

0.06(6.92)' 12

=

1.65 0.36 0.13 0.054

2

6.92(0.060) = 0.415 3.50 1.450

0.06(6.92)' 12

=

1.65 0.36 0.13 0.054

3

0.1(4.0)(0.040) = 0.016 6.98 0.111

4"

0.1(4.0)(0.040) = 0.016 0.02 0.000

0.1(4.0)(0.04)' =0 12 0.1(4.0)(0.04)3 12

=0

3.84 14.8 0.237

3.12 9.75 0.156

5

0.75(0.060) = 0.045 0.07 0.003

0.75(0.06)' 12

=

0

3.07

9.42 0.424

6

0.75(0.060) = 0.045 0.07 0.003

0.75 (0.06)' 12

=

0

3.07

9.42 0.424

7" 0.25(3.7)(0.060)(0.2) = 0.011 1.00 0.011

0.25(3.7)(0.06)'( 0.2) 12

=

0

2.14 4.59 0.050

Total

0.963

3.30

3.028

1.399

"Includes a 10% bolted efficiency factor. "Includes a 25% bolted efficiency factor and a correction factor for epoxy fiberglass

The moment of inertia of the cross section along the Y axis can be determined with the use of Table 12.1: C A Y 3.028 y = -E A 0.963

I,= I,, +Ac'

=

The moment of inertia along the Table 12.2: CAX x=-=--

EA

I,

=

I , +Ac'

3.30

=

3.14 in.

+ 1.399 = 4.70 in.4

(12.1)

X axis can be determined with the use of 1.924 -

2.00 in.

0.963 = 0.045

+ 3.44 = 3.48 in.'

(12.2)

For the preliminary analysis, assume the chassis is equivalent to a simply supported beam with a uniform load. The resonant frequency for vibration

BOLTED COVERS

307

TABLE 12.2 Item

Area, h2

X

AX

1

0.415

0.030

0.012

2

0.415

3.970

1.647

3a

0.016

2.000

0.032

4"

0.016

2.000

0.032

5

0.045

0.435

0.019

6

0.045

3.565

0.160

7b

0.011

2.000

0.022

(6.92) (0.060)3 12 (6.92) (0.060)3 12

c

c2

Ac2

= 0.000

1.97

3.89

1.61

= 0.000

1.97 3.89

1.61

= 0.021

0.00

= 0.021

0.00 0.00 0.00

=0.001

1.57

2.46

0.11

= 0.001

1.56

2.44

0.11

= 0.001

0.00 0.00 0.00

(0.10)(0.04)(4.0)3 12 (0.10)(0.04)(4.0)3

~

Total

I,, in.4

12 (0.06)(0.75)3 12 (0.06)(0.75)3 12 0.25(0.060)(3.7)3(0.2) 12

0.00 0.00

~~

0.963

1.924

0.045

3.44

'Includes a 10% bolted efficiency factor. 'Includes a 25% bolted efficiency factor and a correction factor for epoxy fiberglass.

along the Y axis can then be determined from the standard beam equation,

(12.3)

where E I,

10 X l o 6 lb/in.* (aluminum) = 4.70 in.4 (see Eq. 12.1) g = 386 in./s2 (gravity) L = 26.0 in. (length) W = weight approximated by box volume with an average density of 0.031 1 b / h 3 =

W = (4.0)(7.0)(24.75)(0.031)

= 21.5

lb

(12.4)

308


The natural frequency for vibration along the Y axis is

I

v (10 x 106)(4.70)(386)

fv=

7

(21.5) ( 26.0)3

i

l’?

(12.5)

fv = 344 HZ

The natural frequency for vibration along the X axis can be determined from the same standard beam equation, using the moment of inertia along the X axis: I,

= 3.48

ins4 (see Eq. 12.2)


fx = 2

[

(10 X l o 6 )(3.48) (386) (21.5)(26.0)3

fx = 296 HZ 12.5

(12.6)

COUPLED MODES

The chassis mounts are located at the bottom edge of the box, at each end; this results in a high center of gravity. If an imaginary load is applied at the CG in a direction along the X axis, the chassis will tend to bend in the direction of the load. At the same time, the chassis will tend to rotate in the direction of the load as shown in Fig. 12.8. Since the chassis will tend to bend and rotate simultaneously under the action of a single load, it means that the bending mode will couple with the torsional mode during vibration along the X axis. This coupling will reduce the fundamental resonant mode of the chassis.

Y 4

Y

4

FIGURE 12.8. Testing how vibration modes may tend to couple

COUPLED MODES

309

If a similar load is applied at the CG in the vertical direction, along the Y axis, the chassis will tend to move vertically with no rotation. This means that the vertical bending mode will probably not couple with the torsional mode. In order to determine how the bending mode will couple with the torsional mode, the natural frequency of the chassis in the torsional mode will be computed and combined with the natural frequency of the chassis in the bending mode. The natural frequency of the chassis in the torsional mode can be determined from the torsional frequency equation, by assuming the chassis is equivalent to a single-degree-of-freedom system: 1/2

(see Eq. 2.23)

(12.7)

Here K , is the torsional spring rate of the chassis. This can be determined by placing a unit torque of 1 lbein. at the CG of the chassis and calculating the angular deflection that results. When the CG of the chassis is at its center, one-half of the chassis can be considered, as shown in Fig. 12.9. The angle through which the chassis will rotate can be determined as follows:

Ma

/3= -

(12.8)

GJ

where T / 2 = torsional moment on half a chassis a = L / 2 half of chassis length G = shear modulus J = torsional form factor

=M

(12.9)

Y

f /---

YT

I

+ .

__ L

~

__

4

FIGURE 12.9. Torsional deflection mode in a chassis.

310


The torsional spring rate of the chassis is defined as

K

T

=-

B

(12.10)

e


4GJ K,= L where G L J

(12.11)

=4 X

l o 6 Ib/in.’ (aluminum shear modulus) in. (chassis length) = 4A2/$ds/t (torsional form factor) = 26.0

A good approximation for the bolted thin-wall rectangular section is (also see Roark [19]) J=

I,

+I,

~

2

-

3.48 + 4.70 2

= 4.09

in.4


K,

(4)(4 X 106)(4.09) =

26.0

= 2.52 X

l o 6 in. Ib/rad

(12.12)

The mass moment of inertia for the chassis will be taken with respect to its mounting point at the base, where the rotation will occur: I,,,

W

=

-(4b’

12g

+ c’)

(12.13)

where W = 21.5 lb (weight; see eq. 12.4) g = 386 in./s’ (gravity) b = 7.0 in. (chassis height) c = 4.0 in. (chassis width) Substituting into Eq. 12.13, =

I,,

=

21.5 12(386) [4(7.0)‘+ (4.0)’] 0.983 Ib.in:s*

(12.14)

DYNAMIC LOADS IN A CHASSIS

31 1

The rotational natural frequency can be determined by substituting Eqs. 12.12 and 12.14 into Eq. 12.7:

f,

= 255

Hz

(12.15)

Dunkerley's equation can be used to determine the approximate fundamental resonant frequency of the coupled mode when bending and torsion are combined (this method is conservative, since it will result in a frequency that is slightly lower than the true frequency): 1

1

1

- ---+7

f,'

f??

f,

(12.16)

where fx = 296 Hz (bending along X axis; see Eq. 12.6) f , = 255 Hz (rotation about base; see Eq. 12.15) Substituting into Eq. 12.16,

1

1

-=-

f:

(296)2

1

+ -= 0.267 X lo-' (255)'

=

193 HZ

(12.17)

Equation 12.17 shows that the fundamental resonant frequency of a chassis, with a low mount and a high CG, can be much lower than expected if coupling is ignored. A lower resonant frequency means greater displacements and stresses, which will shorten the fatigue life. 12.6


The dynamic loads developed in a chassis can be determined by considering its dynamic response characteristics. If a sinusoidal vibration input is considered, the inertia loads developed at resonance will be maximum at the center of a uniformly loaded chassis and minimum at the supported ends. The transmissibility at the supported ends will be unity if there is no relative motion between the chassis and its supports. A good approximation for the dynamic load can be obtained with a sinusoidal load distribution acting along the chassis (Fig. 12.10). The unit

31 2


P

Dynamic load distribution on the chassis

- P,

t

-Z

L

L

I

FIGURE 12.10. Vertical load distribution in a chassis.

dynamic load distribution on the chassis can be represented by the expression

n-Z p = p o sinL

(12.18)

The differential equation for this loading is dJY

--

dZ4

po . TZ

- -sinEI

L

(12.19)

The boundary conditions for this equation will be satisfied by using the following expression, where A is the deflection form factor: Y=Asin-

n-Z L

(12.20)

Taking the first four derivatives as indicated, dY

n-

n-Z

dZ

L

L

- =A-cos-d'Y dZ2

-- -

d3Y

-- -

n-Z

n-l

-A-sinL2

L

n-z

T3

dZ3

-A-cosL'

d4Y

T4

dZ4

L4

- =A-sin-

(12.21)

L TZ L

(12.22)

313


Substituting Eq. 12.22 into Eq. 12.19 and solving for A , 7r4

A-sinL4

TZ L

=

po . T Z -sinEI L (12.23)

Substituting Eq. 12.23 into Eq. 12.20 gives the deflection in the chassis at any point due to the unit dynamic load:

The maximum deflection due to the unit dynamic load will occur at the center of the chassis, where Z = L/2:

(12.24)

The maximum deflection in the chassis can also be determined for the bending resonant mode by considering it to be equivalent to a single-degreeof-freedom system (Fig. 12.11). Thus it can be determined from Eq. 2.30 as follows: 9.8G yo = f '

(12.25)

Since Eqs. 12.24 and 12.25 both represent the maximum displacement of the

V

FIGURE 12.11. Chassis simulated as a single-degree-of-freedom system.

31 4


chassis, they must be equal:

poL4 ~ E

9.8G

__=-

Po

I f 2 9.8.rr'EIG =

(12.26)

L"2

Equation 12.26 can be used to determine the maximum unit dynamic load action on the chassis. The moment of inertia I can be determined from the natural frequency equation for a uniform, simply supported beam, as shown by Eq. 12.3. The moment of inertia then becomes I=

4f2WL3 ~

r2Eg Substituting this expression into Eq. 12.26, we have for a 4-G peak sinusoidal vibration input to the electronic box

Po=

9.8.rr'EG L'f?

4f2WL3

WG

=7

(12.27)

where G,,, = 4 (peak input along the X axis) Q = 8 (based on test data for this type of chassis and mount, with removable top and bottom covers) G = G,, x Q = 4(8) = 32 (acceleration) W = 21.5 Ib (weight; see Eq. 12.4) L = 26.0 in. (length; see Fig. 12.6) Substituting into Eq. 12.27, the maximum unit dynamic load becomes

=

(21 3 )(32) 26.0

= 26.4

lb/in.

(12.28)

The total dynamic load developed in the chassis during vibration along the X axis can be determined from the sinusoidal force distribution acting on the chassis as shown in Fig. 12.12. The area under the curve represents the total dynamic load acting on the chassis. This load must also pass through the chassis to the supports. The total dynamic load is Pd=

j0Lp d Z

(12.29)


315

P

t fP“

1LL:r”

-2

L

m

FIGURE 12.12. Dynamic load distribution in the chassis.

where p =posin-

7rZ L

(see Eq. 12.18)

7r

2P@L Pd = 7

(12.30)

This conservatively assumes all of the internal masses are in resonance and in the same phase at the same time during vibration along the X axis, where p o = 26.4 Ib/in. and L = 26.0 in. Substituting into Eq. 12.30, the total dynamic load acting at the CG of the chassis becomes

Pd=

2( 26.4) (26.0)

= 438

lb

(12.31)

77

The bending moment for the chassis at any point can be determined from the following expression:

M = -EI-

d’Y

dZ2


POL2

7rz

M = -sin r’

L

(12.32)

316


The maximum moment will occur at the center of the chassis, where Z is L/2: (12.33) where p o = 26.4 Ib/in. (dynamic load; see Eq. 12.28) L = 26.0 in. (length; see Eq. 12.6) Substituting into Eq. 12.33 for the bending moment at the center of the chassis, (26.4) (26 .0)’ Mma, =

12.7

=

x?

1810 Ib in

(12.34)

BENDING STRESSES IN THE CHASSIS

The bending stresses in the side panels during vibration along the be determined from the standard bending-stress equation

sh = -MC

X axis can ( 12.35)

1 ,

where M = 1810 lb/in. (see Eq. 12.34) C = 2.0 in. (see Fig. 12.7) Z,, = 3.48 i n 4 (see Eq. 12.2) Substituting into Eq. 12.35, the bending stress is

s, =

(1810)(2.0) 3.48

=

1045 lb/in.’

(12.36)

This is a low stress level, so it would seem that the design is satisfactory. However, the side panels must be checked to determine whether they can carry this stress load without buckling. The critical buckling stress for the wide walls of the chassis, under a compressive bending load, can be determined from the following equation [13]: (12.37) The factor K is a function of the edge restraints of the side panel subjected to the compressive bending load. A more detailed sketch of this panel is shown in Fig. 12.13.

317

BENDING STRESSES IN THE CHASSIS

Side panel + [

,

Free h

m

I

I1

b = 6.92

1

Flange on bottom edge

FIGURE 12.13. Dimensions of the side panel on the chassis (in inches).

The top edge of the side panel must be analyzed as a free edge, since there is no support. The bottom of the side panel can be considered supported if the bottom cover has even one screw in this area to keep this edge from rotating. The side panels are riveted to the bulkheads, which will prevent translation but not rotation, so these edges must be considered as supported. The side panel will be analyzed as a flat plate, supported on opposite ends, with one side free and one side supported. The length-to-width ratio, or aspect ratio, is a

8.0

b

6.92

-=--

-

1.15

From the curve shown in Fig. 12.14b,

K

=

1.2

(12.38)

The following information is required:

E = 10 x IO6 1b/in2 (modulus of elasticity, aluminum) t = 0.060 in. (thickness) b = 6.92 in. (width of panel) Substituting into Eq. 12.37 for the critical buckling stress,

s,, = 1.2( 10 x 106) S,,

= 902

lb/in.’

(

0.060 6.92

]

-

’ (12.39)

318


8i-

\

\

-1

Four edges

41-

,

’\

I

Ends suppcited sides fixed

Enas supported one side free one side supported

3

2

1

3

alb

(6)

FIGURE 12.14. K factors for buckling due to bending.

12.8

BUCKLING STRESS RATIO FOR BENDING

Stress ratios can be used to determine the margin of safety for buckling in the side panels, due to the compressive bending stress developed during vibration along the X axis [27]. The buckling stress ratio due to bending is defined as sh

Rh= scr

where S,, S,

= 902 =

(buckling stress; see Eq. 12.39) 1045 lb/in.’ (panel bending stress; see Eq. 12.36)

(12.40)

BUCKLING STRESS RATIO FOR BENDING

319

Substituting into the above equation, this ratio is 1045 R,=-902

-

1.16

(12.41)

The margin of safety (MS) for buckling can be determined from the standard equation [271: 1 MS=--l

(12.42)

Rb

1

MS=-1.16

1 = -0.138

(12.43)

The negative margin of safety shows the side panel will buckle during vibration. Obviously, if the side panels will buckle during vibration, they cannot carry the compressive load. This means the moment of inertia of the cross section, shown by Eq. 12.2, is no longer valid. The chassis will therefore have a coupled resonant frequency lower than the 193 Hz shown by Eq. 12.17. Also, the dynamic stresses will increase because the displacement shown by Eq. 12.25 will increase. Increasing the stresses can result in more rapid failures, since the fatigue life depends on the stress level. A simple method for increasing the stiffness of the side panels, as well as the stiffness of the entire chassis, is to form a flange in the top edge of the panel (Fig. 12.15). If the top cover is then fastened at several spots along this flange, the side panel stiffness will be greatly increased. Enclosing a sheetmetal chassis in this manner makes more effective use of the covers as shear panels. This will substantially increase the overall stiffness of the chassis with very little added weight. These top flanges will increase the stiffness of the side panels by making them equivalent to a flat plate that is supported on all

Top cover

--., Top cover screws

Side panel

*'i I

I Original bottom flange

Bottom cover

Bottom cover screws

FIGURE 12.15. Cross section of the chassis with top flanges added.

320


four sides. The K factor for this geometry, as shown in Fig. 12.14, is

K

= 3.7

(12.44)

Substituting into Eq. 12.37 for the critical buckling stress, 0.060 3.7(10 X l o 6 ) 6.92

(

S,, S,,

=

]

'

2780 lb/in.2

(12.45)

Adding the small flange to the top of the side panels, as shown in Fig. 12.15, will have very little effect on the moment of inertia of the cross section of the chassis as is shown by Eqs. 12.1 and 12.2. Conservatively assuming that these values do not change with the addition of a small flange, there will also be no change in the dynamic load and the bending stress shown by Eq. 12.36. The buckling stress ratio due to bending, however, will change. Substituting Eqs. 12.36 and 12.45 into Eq. 12.40, the buckling stress ratio due to bending for the stiffer chassis, is

Rh

S, S,,

1045

=_=--

2780

- 0.376

(12.46)

Since the bending stress ratio for buckling is less than 1.0, it is obvious the side panels will no longer buckle because of an excessive compressive bending stress. There is, however, another stress acting on the side panels, in addition to the bending stress, which can also cause buckling. This is the torsional shear stress developed by the dynamic load during vibration along the X axis, as shown in Fig. 12.8. This torsional shear stress is due to the overturning moments developed by a high CG. The buckling margin of safety for the side panels can be determined by combining the individual buckling stress ratios for bending and for torsional shear. The buckling stress ratio for bending was shown by Eq. 12.46. The buckling stress ratio for shear can be determined by analyzing the dynamic overturning moments and the resulting shear stresses that are developed in the side panels. 12.9 TORSIONAL STRESSES IN THE CHASSIS

During vibration along the X axis, overturning moments will force the chassis to rotate about its base in a torsional mode that will couple with the bending mode. This is due to a high CG, which is the result of low mounting lugs at each end of the chassis. Torsional shear stresses will then develop as the chassis rotates about a longitudinal axis as shown in Fig. 12.16.

TORSIONAL STRESSES IN THE CHASSIS

321

Shear flow due to torsion

I

-4.0

i n 4

FIGURE 12.16. Shear flow in the chassis due to torsion (dimensions in inches).

The cross section of the electronic chassis, as shown in Figs. 12.7 and 12.15, is characteristic of the type of construction generally used in the electronics industry, where thin sheet-metal boxes have covers that are fastened with a few screws. This does not form a rigid box structure. Since the top and bottom covers will slide as the box twists, these covers are obviously not 100% effective in eliminating relative motion. Therefore, the box cannot be analyzed as a closed section. At the same time, the covers do provide some support, so the box cannot be analyzed as an open section. Under these conditions, an efficiency factor can be used for the bolted covers, based on the ability of the bolted covers to resist relative sliding during vibration. This efficiency factor, as explained in Section 12.4, is a function of the number of bolts, bolt size, panel thickness, and so on. With this efficiency factor it is possible to approximate the shear flow and the shear stress in each panel by using the standard shear flow equations for a closed section. Equilibrium equations can then be written for the chassis cross section using the shear center method, with the shear flow in each panel. Consider the cross section of the chassis shown in Fig. 12.16. Due to symmetry, the shear center will lie at the center of the chassis. The shear flow equation can be written in the form [13] (12.47) of the triangles formed by the straight lines joining the shear center and the extremities of the panels A , = 2(+)(7.0)(2.0) = 14.0 in.2 (two sections) A , = 2(+)(4.0)(3.5) = 14.0 in.2 (two sections) q = shear flow in each panel (lb/in.)

where A

= area

322


Pd = 438.0 lb (dynamic load; see Eq. 12.31) d = 4.0 in. (distance from mount to CG) T = Pdd = 438(4.0) = 1752 1 b . h (torque developed by a 4-G peak input along the X axis) Substituting into Eq. 12.47,

1752

41 + q 2 = -- 62.5 lb/in.

(12.48)

2( 14) Note that the epxoy fiberglass interconnecting circuit board was not included in the system (see Fig. 12.7). This is due to the intermittent spacing of the screw-mounting brackets for this epoxy board on the side walls of the chassis. The shear flow in the side walls will be affected only in the local area of the brackets. Ignoring the epoxy board will have very little effect on the shear flow in the system, and it simplifies the problem substantially. A second equation involving 4,and q2 can be established by considering the shear flow in each panel along with the bolted efficiency factor for each panel. Consider the cross section of the chassis in Fig. 12.17. The shear flow will distribute itself in direct proportion to the shear-carrying ability of each panel. The panels that have a high shear stiffness will carry a higher proportion of the torsional load. Since the shear-carrying ability of each panel is related to its cross sectional area and its bolted efficiency factor, the shear stiffness for each panel can be written as (see Fig. 12.7)

K , =Ale, K,

(12.49)

=A , e ,

where A , = (6.92)(0.060)(2) = 0.83 in.' (area of two side panels) A , = (4.0)(0.040)(2) = 0.32 in.2 (top and bottom panel area)

Shear center

g;' -+ i 7

L

41

\

I

!

~

!

FIGURE 12.17. Shear floor pattern in the chassis cross section.

I

-

nz',1L

T

70

Lql

- L --7

-7

42

40

*

323

TORSIONAL STRESSES IN THE CHASSIS

e, = 100% (efficiency for the side panels) e2 = 10% (bolted efficiency for covers on top and bottom) Substituting into Eq. 12.49,

K , = (0.83)(1.00)

=

K,

= 0.032

=

(0.32)(0.10)

0.83

The shear flow in each panel will be proportional to the shear stiffness:

q1 K , _ -- q,

0.83 _-=

K,

0.032

26 .O

or

q1 = 26.0q2

(12.50)

Substituting Eq. 12.50 into Eq. 12.48 and solving, 26.0q,

+ q2 = 62.5

so 62.5

qr = -= 2.32 Ib/in. 27 .O q1 = 26.0q2 = 60.4 lb/in.

(12.51)

This shows that the side panels will carry a much greater shear load than the bolted top and bottom covers. The shear stress in the side panel, induced by the torsional dynamic load, can be determined from the standard stress equation (12.52) where q1 = 60.4 lb/in. (shear flow; see Eq. 12.51) t = 0.060 in. (panel thickness) Then

60.4

s,=-= 0.060

1007 lb/in.2

(12.53)

This is only an approximate value, since it is based on a cross section with sharp corners and bolted members. However, it still provides a means for evaluating the characteristics of the panels, even if approximate, to determine whether they may buckle due to the shear load.

324


12.10 BUCKLING STRESS RATIO FOR SHEAR

The critical buckling stress, due to shear in the side panels, can be determined by considering the four edges of the side panel to be simply supported when there is a flange on the top and bottom edges of this panel. (See Figs. 12.13 and 12.15.) With a flange on the top edge of the side panel, the shear flow will appear as shown in Figs. 12.16 and 12.18. The critical buckling stress for the side walls, due to the action of the shear load, can be taken in the same form as the critical buckling stress for a compressive load. This equation becomes [13] (12.54) The factor K , is a function of the edge restraints and the aspect ratio of the side panel subjected to shear. Figure 12.19 can be used to determine the K , factor for flat sheet-metal panels simply supported on four sides: a

8.0 6.92

_=-=

b

1.15

The value in Fig. 12.19 for this geometry is

K , = 7.4

(12.55)

Also,

E = 10 x lo6 1 b / h 2 (modulus of elasticity for aluminum) t = 0.060 in. (thickness) b = 6.92 in. (shortest dimension of panel)

Slde panel

1 ---__-----)

I L

I

MARGIN OF SAFETY FOR BUCKLING

325

gr-----7 8

0

1

2

3

4

6

5

8

7

9

10

Aspect ratio a l b

FIGURE 12.19. K , factors for buckling due to shear.

Substituting into Eq. 12.54 for the critical buckling stress, S,,,,

= 7.4( 10 X

1

0.060 lo6) 6.92

[

(12.56)

S,,,, = 5560 lb/in.2 The buckling stress ratio for shear due to torsion is

(12.57) where S,,,, = 5560 1b/ine2 (see Eq. 12.56) S, = 1007 lb/in2 (see Eq. 12.53) Substituting into Eq. 12.57, 1007

R,=-- 0.181 5560 12.11

(12.58)

MARGIN OF SAFETY FOR BUCKLING

The margin of safety (MS) for buckling due to the combined action of the compressive bending and shear stresses for the stiffer chassis can be determined from the standard equation [27] MS =

1 ( R i + R,?)1’2

-1

(12.59)

326


where R , = 0.376 bending ratio (see Eq. 12.46) R , = 0.181 shear ratio (see Eq. 12.58) Substituting into Eq. 12.59,

MS

1 =

7

[(0.376)’+ (0.181)-]

1/2

-1

MS = +1.40

(12.60)

The large positive margin of safety shows the side panels, with the flanges, will not buckle during vibration along the X axis with a 4-G peak input.

12.12 CENTER-OF-GRAVITY MOUNT

If a CG mount is used instead of a base mount, the lateral bending mode will not couple with the torsional mode during vibration along the X axis, as shown in Figs. 12.8 and 12.9. Instead, these resonant modes will be independent of each other (Fig. 12.20). The natural frequency in the vertical direction, along the Y axis, will be the same as the frequency shown in Eq. 12.5: 344 Hz. The natural frequency by the lateral direction, along the X axis, will be the same as the frequency shown by Eq. 12.6: 296 Hz. The natural frequency of the chassis in the torsional mode can still be determined from Eq. 12.7, but the mass moment of inertia, as shown by Eq. 12.13, will change. Now the mass moment of

Y

tLr = 4 ,0

t

L -

Mounting surfaces in line with CG

FIGURE 12.20. An electronic chassis with a CG mount.

X

CENTER-OF-GRAVITY MOUNT

327

inertia must be taken through the CG as follows: (12.61)

Substituting Eqs. 12.12 and 12.62 into Eq. 12.7 for the uncoupled rotational natural frequency,

fR =

460 Hz

(12.63)

With a CG mount, the bending resonant frequency and the rotational resonant frequency will be uncoupled. Therefore, the dynamic stresses in the chassis must be examined for the bending resonant mode acting alone and for the rotational resonant mode acting alone. Considering the bending resonant mode first, with a 4-G sinusoidal vibration input and a CG mount, if the transmissibility at resonance is still 8, then Eq. 12.28 stays the same, and the total dynamic load, as shown by Eq. 12.31, is still 438 lb. The bending stress of 1045 Ib/im2, as shown in Eq. 12.36, will still buckle the side panel when there is no flange at the top. This will result in a negative margin of safety, as shown in Eq. 12.43. If a flange is added to the top edge of the side panel, as shown in Fig. 12.15, then the buckling stress ratio due to bending alone is 0.376 as shown by Eq. 12.46. The margin of safety for buckling, due to bending alone, can then be determined as follows: 1 MS=--l

(12.64)

Rb

Substituting Eq. 12.46 into Eq. 12.64, 1 MS=-0.376

1 = +1.66

(12.65)

The large positive margin of safety shows that the side panels, with top and bottom flanges, will not buckle during vibration with a 4-G peak acceleration input. Consider the uncoupled torsional resonant mode next. With the CG mount, the dynamic overturning moment will be determined by the location of the dynamic CG during vibration along the X axis. Since the dynamic CG will probably shift slightly as various elements of the chassis structure become excited during their resonances, a torsional resonant mode will occur. How-

328


ever, unless the dynamic balance is very poor, or unless a relatively large internal mass experiences a severe resonance, the torsional resonant mode of a CG-mounted chassis will be quite minor. Stresses developed by this resonant mode will be low relative to the stresses developed by the uncoupled bending resonant mode. Therefore, the structural design considerations will most likely be based on the bending-mode characteristics of the chassis with a CG mount.

12.13 SIMPLER METHOD FOR OBTAINING DYNAMIC FORCES AND STRESSES ON A CHASSIS

A simpler, more conservative method of analysis is available for obtaining the dynamic forces and stresses acting on an electronic chassis. Instead of using the more accurate, but more complex, sine wave dynamic loading function shown in Fig. 12.10, simply use a uniform dynamic loading function for the chassis as shown in Fig. 12.21. The new dynamic load acting on the box can be found using the values shown in Eq. 12.27, as shown below: Pd = WGi,Q

=

(21.5)(4.0)(8)

=

688.0 Ib

(12.66)

This uniform unit load Po acting on the chassis can be obtained from the dynamic load and the 26-in. chassis length:

Pd 688 Po = - = - = 26.46 Ib/in. L 26

(12.67)

Po Distribution on chassis

1.

Ill1111 i 1111 1 l l i l -z

L-JI

Electron IC chassis

Z

I-

4-

_f

26

FIGURE 12.21. Simpler but less accurate method for evaluating dynamic loads on a chassis.

SIMPLER METHOD FOR OBTAINING DYNAMIC FORCES AND STRESSES

329

The dynamic bending moment will be maximum at the center of the chassis. The value can be obtained from the sum of the moments as shown below:

POL L Mmax =

POL L

jqj

2(51- 2

=

Po(L)2

-

8-

(26.46)(26)*

8

=

2236 lb .in. (12.68)

The dynamic bending stress in the side panels of the chassis can be obtained from the bending stress Eq. 12.35 as shown below:

M,,, sb=--

I,

c

-

(2236) (2.0) 3.48

=

1285 lb/in.*

(12.69)

The bending-stress level using the conservative method shown above is about 23% greater than the more correct bending-stress value of 1045 lb/in.’ shown in Eq. 12.36.

-

CHAPTER 13

Effects of Manufacturing Methods on the Reliability of Electronics

13.1

INTRODUCTION

An examination of most electronic equipment failures shows they are really mechanical in nature. Typical hard failures include cracked solder joints, broken wires, cracked circuit traces, cracked plated through-holes, broken connector pins, broken screws, cracked components, cracked hermetic seals, and cracked silicon chips. Electromagnetic pulse (EMP), molecular migration, and solder creep may be added to the list. High humidity and condensation may be borderline. Radio frequency interference (RFI), electromagnetic interference (EMI), noise, and relay chattering are often considered as soft failures since the system will still work after the disturbance is removed. Some of these failures are due to poor design practices and some of the failures are due to poor manufacturing practices. It is often quite difficult to determine which is the major fault. Sometimes the failures are a result of deficiencies in both areas. Manufacturing costs are often a major factor in the success or failure of a project and often a major factor in the success of the company itself. One big cost driver is tolerance control. Tight manufacturing tolerances can improve the quality and reliability of a product, but it also increases the costs. Therefore, loose tolerances are desirable since they can reduce costs, but loose tolerances can also reduce the effective fatigue life of electronic equipment operating in severe dynamic and thermal environments. When loose tolerances are used to reduce costs, the electronic design must be made more fault tolerant to keep the costs down while keeping the reliability up. This usually means higher safety factors must be used in the designs, or more analysis and more prototype testing must be performed to ensure the reliability of the product. This will end up increasing the design costs. This procedure is usually acceptable since the design function is a one-time cost, but the manufacturing is a recurring cost.

330

TYPICAL TOLERANCES IN ELECTRONIC COMPONENTS AND LEAD WIRES

331

13.2 TYPICAL TOLERANCES IN ELECTRONIC COMPONENTS AND LEAD WIRES

Dimensional tolerance variations are probably the greatest in the electronic components themselves. An examination of the dual inline package (DIP), shown in Fig. 13.1, shows there are large variations in the dimensions of the component body and the lead wires. Large dimensional variations in these areas mean there will be large variations in the fatigue life of these components during operation in severe vibration, shock, or thermal cycling conditions. Variations in the dimensions of just the component lead wires can easily change the fatigue life of many components by a factor of about 100 : 1. Many different companies fabricate components with the same general form, fit, and function. Every component manufacturer has a different set of tools and dies, which are used to produce a particular component. It becomes extremely expensive for different component manufacturers to meet periodically to try to establish and to hold close dimensional tolerances on a wide variety of electronic components. Pity the purchasing managers in large companies who have their own problems, trying to procure a wide variety of components, in reasonable quantities and at reasonable costs. Then they get a request from the mechanical design engineering group to purchase one particular type of component, with a particular geometry, from one particular company, because it is the only type of component that can pass the vibration requirements. The purchasing department says no because they do not like a single source.

-t---A

332

EFFECTS OF MANUFACTURING METHODS ON RELIABILITY

What happens if the single source goes on strike, or if they burn down, or if there is an earthquake? The mechanical design group must make the design more fault tolerant to accommodate the component variations available from different manufacturers. A design that is more fault tolerant is a design that has higher safety factors to compensate for the higher stresses that can be produced by geometry changes. Higher safety factors result in higher weights, larger sizes, and increased costs. Component manufacturers are often willing to manufacture special components to close tolerances, with reduced prices, when there are large quantities involved. Many large companies, with many different manufacturing divisions, take advantage of this policy by combining their purchasing requirements under one head so they can purchase larger quantities at reduced prices. Another area where large dimensional tolerances can cause reliability problems is in the PCBs themselves. The most critical tolerance is usually the thickness, which has a normal manufacturing tolerance of kO.0070 in. for multilayer PCBs. Typical PCBs have a thickness of about 0.062 in. The total thickness variation due to these tolerances can change the PCB natura1 frequency by about 40%. The displacements and the stresses are inversely related to the square of the PCB natural frequency. The fatigue life of the lead wires is related to an exponent of 6.4. When these two conditions are combined it shows that the full tolerance range from the minimum to the maximum PCB thickness can change the vibration fatigue life by a factor of about 27 : 1. Sample Problem-Effects and Fatigue Life

of PCB Tolerances on Frequency

A 0.062-in. thick, plug-in type PCB has a natural frequency of 150 Hz. The production PCBs are coming in at the lower limit thickness of 0.055 in. Assuming a negligible weight change, determine how this will affect the following: (a) PCB natural frequency. (b) Dynamic displacement and stress. (c) Expected fatigue life. Solution. (a> When the PCB length, width, and weight stay the same, the natural frequency will depend on the thickness only. The part of the frequency equation that relates to the thickness is shown below:

E

f n = 1 5 0 1=150

=

(l50)(0.835)

=

125 Hz

(13.1)

(b) When the input acceleration G level does not change here, the dynamic displacement (or stress) will be related to the frequency and trans-

PROBLEMS ASSOCIATED WITH TOLERANCES ON PCB THICKNESS

333

missibility as shown in Eq. 2.33:

(c) Equation 13.2 shows that the dynamic displacement and the stress level will increase by a factor of 1.314 when the natural frequency is reduced from 150 to 125 Hz. Increasing the dynamic stress will result in a decreased fatigue life. The fatigue life for the lead wires is related to the slope of the fatigue curve, where b = 6.4 as shown below: Life 1

--

Life 2

-

1

(1.314)6.4

= 0.174

(13.3)

The resulting low ratio means that the fatigue life of the lead wires will be reduced by 82.6% when the PCB thickness is reduced to 0.055 in., as shown below: Life reduction

=

(1.00 - 0.174)

= 0.826

or 82.6%

(13.4)

Another way of saying the same thing is that the fatigue life will change by a factor of 5.74 when the PCB thickness is reduced from 0.062 to 0.055 in. as shown below: Fatigue life ratio change

=

(1.314)6

= 5.74

( 13- 5 )

When the full tolerance range from 0.055 to 0.069 in. is considered, the natural frequency will change from 125 to 176 Hz, which is about a 40% change in the natural frequency. Using these two frequency values in Eq. 13.2 will result in a displacement ratio of 1.671. The fatigue life ratio over this PCB thickness change will be Fatigue life ratio change

=

( 1.67If4

=

26.7

(13.6)

13.3 PROBLEMS ASSOCIATED WITH TOLERANCES ON PCB THICKNESS

Problems frequently occur during vibration that are related to the way in which the PCBs are dimensioned and fabricated. This typically happens when prototype models are built and subjected to vibration tests, which are very successful. The system is released for production. The first production units are very successful. As more units are produced and shipped, a rash of vibration-related failures suddenly show up in the component lead wires and solder joints. Everyone gets into the act to solve this emergency. The

334


materials people suspect the wrong materials are being used for the component lead wires. The shipping people suspect the shipping department is not packaging the product properly. The environmental stress screening (ESS) department is accused of improper accelerometer calibration that results in excessively high vibration levels, which are using up too much of the fatigue life. The production assembly people are accused of forgetting to torque the wedge clamp screws that clamp the PCBs rigidly to the chassis, so the PCBs experience higher acceleration levels. The manufacturing drawing dimensions are checked and rechecked to make sure everything is within the proper dimensions and tolerances. Everything seems to be correct, but the field failures continue. One day, quite by accident, someone measures the thickness of the bare fiberglass board material by itself, and finds that it is under the lower limit of the normally accepted tolerance variation associated with the basic material. How is this possible? The dimensions of all the production hardware pieces were examined closely and were found to be in compliance with the production detailed drawings. Obviously another drawing check was required to clear up the confusion. The overall thickness of the board, when it was measured over the copper circuit traces and the solder plating matched the production drawings. However, there were no dimensions called out for the thickness of the board itself. This is a common practice among many electronics manufacturers. The thickness of the PCB is measured over the copper circuit traces and the solder plating. When this thickness is under the lower tolerance limit, the PCB is simply sent back to have more solder added to meet the required value shown on the drawings. The circuit board now meets the drawing requirements, but the thinner board produces a lower natural frequency. This results in higher dynamic stresses and displacements, which reduces the fatigue life of the components mounted on the PCBs. All existing drawings and all new drawings must be changed to show the bare board thickness, as well as the thickness over the copper and solder traces to prevent future problems. 13.4 EFFECTS OF POOR BONDING METHODS ON STRUCTURAL STIFFNESS

Structures often require the addition of local stiffeners to increase the natural frequency. A higher natural frequency decreases displacements and stresses, which increases the fatigue life. Many different structural adhesives are available for bonding metals and plastics. Two of the most popular bonding materials used in the electronics industry are polyurethane and epoxy. Polyurethane is not quite as stiff or as strong as epoxy, but polyurethane is slightly more flexible, especially at low temperatures, so it is less likely to crack under a load. The epoxy is rigid and strong, but it tends to be brittle at

SOLDERING SMALL AXIAL LEADED COMPONENTS ON THROUGH-HOLE PCBS

335

lower temperatures, so it will often crack when it is subjected to alternating loads. Both of these materials work very well when they are used to bond plastic parts together. These materials may not work very well when they are used to bond metals to plastics or metals to metals, unless special precautions are observed. A good rule to follow for bonding any metal parts is if it doesn’t grip, it will slip. Consider a mountain climber going up a steep slope. The climber must punch holes in the side of the mountain to get a good grip, or risk slipping and falling down. Consider a spider in a bathtub trying to climb up the steep wall to escape. It can get about halfway up the steep vertical wall before slipping and falling back down again. Consider a production engineer who must bond a thin, stiff, smooth metal shim under a large critical component on an epoxy fiberglass PCB to prevent it from failing in vibration. The engineer is warned in advance that epoxy will not adhere to a hard smooth surface because there is no grip. The smooth surface must be roughened and several holes must be drilled through the shim. This allows the epoxy to ooze through the shim to form epoxy rivets. These epoxy rivets must be forced to shear before the shim will separate from the PCB. The production engineer claims there is not enough time for all the extra work and besides, there are no process specifications that describe the methods, materials, hand pressure, and number of strokes that must be used to roughen a smooth metal part before it is cemented in place. The engineer simply cleans the shim carefully, applies a thin coat of epoxy on both mating surfaces, clamps them together, and places the assembly in an oven for a rapid cure. After curing, the PCB is vibration-tested to see if the stiffening shim cured the problem. The part fails. An examination of the PCB assembly shows the stiffening shim is missing. The shim is found on the floor. It fell out because it was not properly bonded to the PCB. The simple fix is rejected, all of the existing parts are scrapped, and an expensive redesign of the PCB is ordered to fix the problem. The situation described above is true. It has been repeated literally dozens of times at dozens of different companies. Experience is a great teacher. It is often difficult to appreciate the magnitude of the forces that can be generated in different thermal cycling and dynamic environments by someone who has very little testing experience. 13.5 SOLDERING SMALL AXIAL LEADED COMPONENTS

ON THROUGH-HOLE PCBS Problems are often encountered with the solder joints on small axial leaded components when they are wave soldered on through-hole PCBs. The small diameter component body results in a short vertical wire leg when the component body is mounted flush onto the PCB surface. The flush mounting

336


provides a good heat transfer path for the conduction-cooled component. The solder tends to wick up very high on the vertical leg of the lead wire directly into the wire bend radius, when the ratio of the plated through-hole diameter to the wire diameter is not properly set. This structurally short circuits the wire stain relief. The thin vertical leg of the wire is very flexible, compared to the stiffness of the component body and the PCB. The thin lead wire will bend and act like a strain relief any time the PCB deflects during vibration, shock, or thermal cycling. Since the lead wire is very flexible it can displace through relatively large amplitudes with a very small force. Small forces result in small stresses, which result in a long fatigue life. When the solder wicks up the vertical leg into the bend radius, the strain relief is gone. The resulting structure is now very stiff. Very small displacement amplitudes can now generate very high forces and stresses, which can easily crack the solder joint or the component body. Raising the component body about 0.050 in. above the PCB surface will result in a longer vertical wire leg, so the solder will not be able to wick up into the bend radius. A small kink can be placed in the vertical legs of the lead wires to raise the body of the component higher above the PCB. This will still allow the component to be auto-loaded with pick-and-place machines. Speed bumps can also be used to raise the component body above the PCB. Speed bumps are small cylindrical disks about 0.090 in. in diameter and 0.050 in. high. Pick-and-place machines can be used to pick up one disk with a small vacuum tube. The disk can be touched down on a table surface covered with a thin liquid adhesive film, then deposited on a PCB so it will be positioned to raise the component 0.050 in. above the PCB. The new longer vertical wire legs will prevent the solder from wicking up into the bend radius. The temperature rise of the component must be checked to make sure it will not overheat in the new raised position. 13.6 AREAS WHERE POOR MANUFACTURING METHODS HAVE BEEN KNOWN TO CAUSE PROBLEMS

Manufacturing and assembly processes and procedures must be carefully controlled to ensure the reliability of the electronic equipment. Many early field failures in military electronic equipment have been traced back to manufacturing methods that are not normally evaluated in the preliminary design and analysis evaluations. Some of these documented failures are described below. 1. Conformal coatings are typically used to protect the electronics from the adverse effects of moisture, humidity, salt, damaging vapors and liquids, fuels, and even rough handling. They are usually sprayed on in a thin coating to obtain this protection. Sometimes the coatings are extra thick when they are applied. The coatings can then fill in under small ceramic chip resistors

AREAS WHERE POOR MANUFACTURING METHODS CAUSE PROBLEMS

337

and capacitors and fill in the strain relief on different types of through-hold axial leaded components and large fine-pitch surface mounted components. After a few months of operation in harsh environments records show that the chip resistors and capacitors crack, the axial leaded component solder joints crack, and the solder joints on the fine-pitch components crack, due to thermal expansion differences in thermal cycling conditions. 2. The wave soldering operation has been known to lift some components up from the PCB. Sometimes this produces tilted axial leaded components, where one vertical lead is very short and one vertical leads is very long. The solder then wicks up the short lead into the bend radius of the wire. This solder wicking structurally short circuits the shorter wire strain relief. This increases the forces and stresses in the solder joint, which reduces the operating life. 3. Lead forming dies are often used to quickly bend the component lead wires to fit the solder pads. Sometimes the operators are careless and sometimes the dies are worn or they are not aligned properly. This can result in sharp-bend radii or in deep cuts or scratches in the lead wires. High stress concentrations are developed at these defects, which result in high stresses and a reduced operating life. 4. Machined parts are often used in electronic assemblies. There have been many cases where extra sharp corners are machined in critical structural elements that are required to carry alternating dynamic loads such as vibration. Sharp corners produce high stress concentrations and high stresses, which have been known to reduce the operating life. 5 . Many through-hole components, such as transformers, have lead wires that extend from the flat bottom surface of the component. These types of components must not be flush mounted on PCBs, so there is no air gap between the PCB and the transformer. There are two reasons for this. The first reason is that cleaning the solder paste and flux out from under the component is very difficult. When this is combined with high humidity and a little electric current, it can promote dendritic growth under the component. This growth is a transparent semiconductor, similar in appearance to lacquer or shellac, with a high electrical impedance so low impedance circuits are not affected. High-impedance circuits can be affected by this growth. The second reason is that flush mounted components can prevent the venting action of the hot gases escaping from the plated through-hole during the wave soldering operation. This builds up the air pressure in the plated through-hole, which reduces the wicking action of the solder. Xrays of the plated throughholes show they are only half-full of solder. This has caused premature solder joint failures in field operations. 6. Tantalum capacitors have the highest vibration failure rate of any electronic part. The component body is very heavy and the two support wires are very thin. When the component is not attached to the PCB properly, relative motion of the component body breaks the wires. There is a plastic

338


covering on the metal component body to prevent short circuits with the circuit traces on the PCB. The component body is usually cemented to the PCB for vibration protection. The cement is usually applied at the base of the component where it contacts the PCB. The cement therefore contacts the outer plastic covering on the component body and not the component itself. Over a period of time the plastic stretches so it is no longer snug against the body of the component, so the component body is loose. During vibration the loose component moves around enough to fatigue the lead wires, which break rapidly. The body of the capacitor, not the outer plastic insulator, must be cemented directly to the PCB. The only areas of the capacitor body that are directly available for cementing are the two ends at the lead wires. These are the areas that must be cemented directly to the PCB to support the component properly against vibration. Sometimes the plastic insulation around the body covers the ends of the component. This plastic at both ends must be cut away to provide direct access for cementing the ends of the component directly to the PCB. Care must be used so the cement does not fill in under the wires and structurally short circuit the wire strain relief. It is also a good idea to place a bead of cement completely around the circumference of the component body, over the plastic insulator and down to the PCB, near both ends of the component body for extra protection. Some epoxy cements tend to get thin and run when they are heated for a more rapid cure. The cement can be made more thixotropic, so it will not run, by simply adding microballoons, which are small hollow glass spheres. This results in a mixture with the consistency of toothpaste, which will not run.

7. The copper layers must be located uniformly through multilayer PCBs to prevent them from warping during the lamination process and during the soldering process. Special fixtures may have to fabricated to prevent the PCBs from warping. High stresses can be developed in the component lead wires and solder joints if the PCBs are allowed to warp, which can lead to premature field failures.

13.7 AVIONIC INTEGRITY PROGRAM AND AUTOMOTIVE INTEGRITY PROGRAM (AVIP)

Manufacturers of military electronics and automotive electronics have introduced programs aimed at dramatically improving the reliability of their electronic equipment through a better understanding of the failure mechanisms. Most electronic equipment failures are mechanical in nature. They can usually be traced back to poor design practices and procedures and to poor manufacturing practices and procedures. The Air Force has outlined their avionic integrity program specifications in the document MIL-A-87244.

AVIONIC INTEGRITY PROGRAM AND AUTOMOTIVE INTEGRITY PROGRAM (AVIP)

339

TABLE 13.1. Environmental Failures Outlined in MIL-A-87244 A. Fatigue Failures Due to Vibration, Shock, and Thermal Cycling

Electrical Lead Wires Plated Through Holes

Solder Joints Component case

B. High Operating Temperatures C. Aging Effects

Corrosion Embrittlement

Water vapor Electrical drift

This document describes three environments that are known to produce many electronic failures, as shown in Table 13.1. In all programs the idea is to ensure the electronics will be able to achieve a minimum failure-free operating period (FFOP) over the life of the equipment. The new Air Force F-22 aircraft electronic system had a FFOP requirement of 10,000 hours. It was designed and tested to achieve a FFOP of 2 x 10,000 hours or 20,000 hours to make sure it will meet its requirement. The intent is to achieve a large improvement in the overall reliability while holding down the acquisition and support costs. The AVIP philosophy is to examine the geometric factors that have the greatest influence on the fatigue life of the electronic equipment. These geometric factors are called integrily driuers. A total of 10 integrily drivers are identified and defined as shown below.

1. Component Lead Wire Length. A short wire is stiffer than a long wire, so short wires generally produce higher stresses in vibration, shock, and thermal cycling conditions. The exception is very long, fine-pitch wires on large surface mounted components. These can have low natural frequencies, which can be excited in vibration and experience fatigue failures. 2. Component Lead Wire Diameter. A large wire diameter is stiffer than a small wire diameter, so large wire diameters generally produce higher stresses in vibration, shock, and thermal cycling conditions. 3. PCB Thickness. A thin PCB will have a lower natural frequency than a thick PCB, when everything else is equal. A lower natural frequency results in a larger displacement with higher stresses, which typically results in a shorter fatigue life. 4. Large Component. Large components will generate larger relative displacements between the component and the PCB as the PCB bends during vibration. Larger relative displacements will increase the strain and the stress in the lead wires, resulting in a shorter fatigue life. 5. Heauy Components. Heavy components such as transformers, inductors, and relays generate higher inertia forces in vibration and shock. Higher

340


stresses are generated in the lead wires and solder joints, which reduces the effective fatigue life. 6. Plug-in PCB Edge Supports. Plug-in PCB edge supports are generally classified for three conditions. In decreasing order of stiffness they are clamped, hinged, and free. The clamped edges produce the highest natural frequency, when everything else is the same. Higher frequencies have smaller dynamic displacements, which increase the fatigue life. 7 . Component Mounting on PCB. Components similar to transformers with lead wires extending from a flat bottom should not be flush mounted on a through-hole PCB, where the hard flat bottom surface of the transformer sits on the hard flat surface of the PCB. Thermal expansions in the transformer body and in the PCB itself, in a direction perpendicular to the plane of the PCB, can break the lead wires and solder joints in thermal cycling conditions. 8. Sharp-Bend Radius in the Lead Wire. High stress concentrations can be produced in lead wires that have a very sharp-bend radius. High stresses will reduce the fatigue life in the wire. 9. Conformal Coating Filling Under Small Suq5ace Mounted Chip Resistors and Capacitors. High 2 axis coating thermal expansions can crack these brittle ceramic components. 10. Solder Wicking Up into the Bend Radius on Small Axial Leaded Through-Hole Parts. The compliant vertical leg of the lead wire acts like a strain relief. When the solder wicks up into the bend radius it structurally short circuits the stain relief, resulting in more rapid fatigue failures in the lead wires in vibration and thermal cycling conditions.

13.8 THE BASIC PHILOSOPHY FOR PERFORMING AN AVlP ANALYSIS

Products that are produced by combining several different raw materials will always have some variations in their physical properties. When the materials are purchased from several different sources, greater variations can be expected. Manufacturing machining processes will always involve variations in the physical sizes of structural parts, because it is impossible to fabricate many parts that have exactly the same dimensions. This means that structural members that appear to be identical are often significantly different. When they are exposed to the same alternating stress cycles, they will all show large differences in their fatigue lives because there are large differences in their physical properties. Analysis methods are typically based on the average physical properties obtained from many different samples of the same structural materials. These samples are exposed to varying alternating stresses

THE BASIC PHILOSOPHY FOR PERFORMING AN AVIP ANALYSIS

341

until they fail. The tests always show there is a large amount of scatter in fatigue data. Therefore, one of the best methods for assuring the survival of a product subjected to alternating stresses is to use safety factors, sometimes called factors of ignorance. A typical value of 2.0 is often applied to the product life. For example, when the desired product life is 10,000 hours, the product will be designed to have a minimum life of 2 X 10,000 or 20,000 hours. This improves the chances of achieving a product life of 10,000 hours. The product life for the AVIP philosophy is based on the fatigue damage accumulated in the most critical structural members during exposure to the most critical environments. The most critical structural members will usually be the electrical lead wires and the solder joints on the largest and the heaviest electronic components. (In some systems the cables, electrical connectors, vibration isolators, and mounting flanges on the electronic chassis may be the critical structural members that are most likely to fail.) The most critical environments are usually vibration and thermal cycling. Shock and acoustic noise may be included in some programs. Miner’s cumulative fatigue damage ratio can then be used to estimate the fatigue life from the sum of all of the fatigue damage accumulated in the most critical lead wires. This can also be applied to the damage accumulated in the most critical solder joints. This must be done for all of the critical environments. And this must also be done for the most critical components mounted on all of the various PCBs. This can result in a large and expensive analysis program. Some organizations have set up computer programs that are capable of evaluating every electronic component on every PCB for every environment. However, the usual practice for hand calculations is to narrow the selection down to the three most critical components, involving the three most critical integrity drivers (usually tolerance variation effects in the lead wire length, lead wire diameter, and PCB thickness) on only three of the most critical PCBs, for all of the critical environments. Miner’s cumulative fatigue damage ratio is then used to estimate the resulting fatigue life. Miner’s cumulative fatigue damage ratio is based on an analysis method that sums up a ratio of the actual number of fatigue cycles ( n ) accumulated in a specific structural element, in several different environments, divided by the number of fatigue cycles (N)required to produce a fatigue failure in that same specific structural element in the same environment. as shown below:

(13.7)

Miner’s cumulative damage ratio is based on the idea that the accumulation of damage is linear and is a simple function of the load, and that failures are

342


not related to the loading sequence. It is also based on the damage accumulated by each individual load being applied separately, and not in combination with any other load. For example, Miner’s damage ratio cannot be used to estimate the fatigue life of a system where vibration and thermal cycling are applied at the same time. This combination of environments would be very difficult, time consuming, and very expensive to evaluate even on the newest and fastest computers available at this time. The AVIP analysis philosophy starts by evaluating the displacements, forces, stresses, and fatigue life of the critical lead wires and solder joints based on the nomind dimensions of the three integrity drivers-lead length (or height), lead diameter, and PCB thickness. The sensitivity analysis is made next. This examines the change in the fatigue life compared to the change in the part size as specified by the manufacturing tolerance. Miner’s cumulative damage ratio (CDR) is obtained for each condition. The desired fatigue life often has a safety factor (also called a scatter factor for fatigue) of 2.0 added to ensure a good fatigue life for the AVIP analysis based on a Miner’s damage ratio of 1.0. Any value greater than 1.0 for two lifetimes means the part may not make its required failure-free operating period (FFOP) so a change has to be made to improve the fatigue life. The sensitivity analysis evaluates the CDR by changing the tolerance on one part at a time to measure its effect on the fatigue life. It is used to find the most efficient method for increasing the fatigue life with a minimum change. These methods of analysis are outlined in Tables 13.2 and 13.3. The typical results of an AVIP sensitivity analysis are shown in Tables 13.4 and 13.5 for the most critical lead wire and the most critical solder joint of a high-voltage module, component reference 2110,mounted on a dual-purpose power supply PCB.

TABLE 13.2. Basic Starting Analysis Calculation Matrix Lead Length

Lead Diameter

PCB Thickness

Nominal

Nominal

Nominal

TABLE 13.3. Fatigue Sensitivity Analysis Calculation Matrix Lead Length

Lead Diameter

PCB Thickness

Minimum Nominal Nominal

Nominal Minimum Nominal

Nominal Nominal Minimum

DIFFERENT PERSPECTIVES OF RELIABILITY

343

TABLE 13.4. Dual-Purpose Power Supply Sensitivity Analysis Summary for Lead Wire on High-Voltage Module ZllO

CDR

Condition

Minimum

Minimum

Minimum

Nominal

Lead Height

Lead Diameter

PCB Thickness

0.183 0.043

0.614 0.144

0.906 0.213

0.000

0.000

0.000

0.000

0.383 0.003

1.276 0.011

1.003 0.016

0.372 0.003

0.612

2.045

2.138

0.596

0.000

0.000

0.000

0.000 0.000

0.000 0.001

0.000 0.000

0.000 0.000 0.000 0.003

0.000 0.000 0.000

0.000 0.002 0.002

0.005

0.010

0.006

0.003

0.006

0.014

0.008

0.615

2.051

2.152

0.604

32,415

9.751 1.40

6,596 2.39

33,113 - 0.35

Thermal CumulativeDamage Ratio

Power supply burn-in System burn-in Bay temperature diurnals Mission 5 temperature cycles Power-on temperature cycles Thermal CDR subtotal

0.179 0.042

Dynamic CumulativeDamage Ratio

Power supply burn-in vibration System burn-in vibration 7.77 G RMS + 128 dB OASPL 5.83 G RMS + 125 dB OASPL 3.89 G RMS + 122 dB OASPL 1.94 G RMS + 116 dB OASPL 8.60 G RMS + 147 dB OASPL Dynamic CDR subtotal Total: Thermal + Dynamic CDR Fatigue life, hours A life/A driver

0.000 0.000

0.001 0.001

0.000 0.000

13.9 DIFFERENT PERSPECTIVES OF RELIABILITY

Reliability means different things to different people. In electronics it refers to the failure rate of some type of equipment. The dictionary defines reliable as dependable and trustworthy. A reliable electronic system would then be a dependable and trustworthy system. A system with these properties would be expected to provide trouble-free service for a long time period. It is probably safe to say that a long time period would be many years of trouble-free service. Can a reliable electronic system be described as a system that has an acceptable number of failures during a specified operating period? This does not really make good sense. Yet this is the way that most military, and many commercial, organizations choose to define the reliability of their products in terms of mean time between failures (MTBF). The MTBF is defined as a probabilistic method for evaluating the reliability. Methods for calculating

344


TABLE 13.5. Dual-Purpose Power Supply Sensitivity Analysis Summary for Solder Joint on High-Voltage Module Z l l O CDR

Condition

Nominal

Minimum Lead Height

Minimum Lead Diameter

Minimum PCB Thickness

Thermal CLimulative Damage Ratio Power supply burn-in System burn-in Bay temperature diurnals Mission 5 temperature cycles Power-on temperature cycles Thermal C D R subtotal

0.363 0.422 0.990 6.207 2.000

0.553 0.645 1.507 9.466 3.042

0.203 0.236 0.553 3.469 1.116

0.451 0.523 1.230 7.722 2.485

9.982

15.213

5.577

12.411

0.000 0.000

0.000 0.000

0.000

0.000 0.000 0.000

0.000

0.000

0.000 0.000

0.000 0.000

0.000

0.000 0.000

0.000 0.000

0.000 0.000

0.000

0.000

0.000

0.000

9.982

15.213

5.577

12.411

2.004

1,315 0.69

Dynamic Cumdative Damage Ratio Power supply burn-in vibration System burn-in vibration 7.77 G R M S 128 d B O A S P L 5.83 G R M S 125 d B O A S P L 3.89 G R M S 122 d B O A S P L 1.94 G R M S 116 d B O A S P L 8.60 G R M S 147 d B O A S P L

+ + + + +

Dynamic C D R subtotal Total Thermal

+ Dynamic C D R

Fatigue life, hours 1l i f e / l driver

0.000 0.000

0.000 0.000 0.000 0.000

3,586 - 2.37

0.000

1,611 2.02

the MTBF of different types of electronic systems operating in different types of vehicles are outlined in government document MIL-HDBK-217. Computer programs are available that will compute the MTBF values based on the failure rate data for different components obtained directly from this military document. This makes the calculation very simple, like working from a cookbook. Very little original thinking goes into the analysis and there is no need to have a technical degree to perform the calculations. The typical comments are that everyone is using the MTBF method so why bother doing anything different. The problem is that the MTBF method is not an accurate gage of the life of the equipment. This method gives the mean time between failures. The mean time is not an accurate measure of the time when a failure might occur. When the button is pushed in an airplane to fire a missile at the enemy and the electronics malfunctions at that instant, the battle can be lost, Or take the case where the power in a city goes out and

DIFFERENT PERSPECTIVES OF RELIABILITY

345

the hospital emergency lights do not work because of an electronic malfunction; the patient on the operating table can be lost. Electronics systems can now be designed and manufactured to provide a known failure-free operating period (FFOP). Scheduled maintenance can then be performed to replace a critical part before a failure occurs. This ensures continuous reliable operation for a specified period of time. A much better method for evaluating the reliability of an electronic system would be to use a deterministic approach. A system is reliable when it does not malfunction for a specified period of time. Another way of saying the same thing would be to specify the reliability in terms of a failure-free operating period (FFOP), which is the theme of the AVIP integrity program shown in the previous section. This is a whole different approach to the evaluation of reliability. Much more data involving the fatigue life properties of all the load-carrying materials in various thermal cycling, vibration, and shock environments are required. A great deal of fatigue data are already available for a wide variety of materials. However, new material combinations are constantly being developed so new fatigue data are always required. The total damage accumulated in each critical load-carrying member can be obtained using Miner’s cumulative damage ratio, which is shown in Eq. 13.7. This is a well-documented method for evaluating the fatigue life of structures operating in many different environments. This method calculates a cumulative damage ratio (CDR) considering only one environment at a time. The total damage in any structure is the sum of the individual damages accumulated in the individual environments. Many new computer programs have already been developed and are available to perform these new and more accurate reliability evaluations. Hand calculations can also be used to perform these new reliability evaluations, but it takes a more experienced engineer who is familiar with thermal stress, vibration stress, and fatigue to perform the required calculations.

-

CHAPTER 14

Vibration Fixtures and Vibration Testing

14.1 VIBRATION SIMULATION EQUIPMENT

Vibration frequencies for high-speed vehicles, such as missiles and airplanes, generally extend to 1000 Hz and often up to 5000 Hz. Most specifications for these vehicles require tests that range from about 4 to 2000 Hz. In order to produce harmonic motion over such a broad frequency range, electrodynamic shakers are generally used. These shakers, or vibration machines, are very much like a loudspeaker that has a moving coil. Instead of connecting to a speaker cone, the moving coil connects to an armature, which joins onto the shaker head that simulates the desired harmonic motion. The armature has a driving coil that is cylindrical in shape, and the shaker head is usually an extension of this shape. The shaker head itself must be very rigid in order to control the displacement amplitudes at high frequencies. Electrodynamic shakers that are capable of producing frequencies up to 2000 Hz normally have their shaker-head resonances above 3000 Hz. In order to provide a pure translational motion for the shaker head, various types of flexures or springs, metal and rubber, are often used between the shaker head and the support frame. Even with these devices, many rotational modes often develop in the shaker head during severe resonant conditions. Vibration machines are generally rated in terms of the peak force in pounds, based on sinusoidal wave motion. These machines are available with various force ratings that range from 25 to 25,000 lb. The choice of the vibration machine depends on the maximum weight of the system to be tested and the maximum acceleration force required by the test. Most vibration tests on electronic units require acceleration inputs along each of the three mutually perpendicular axes of the system. In order to provide this type of adaptability, the vibration machine usually has the shaker head mounted on a trunion that permits the head to be rotated and locked in different positions (Fig. 14.1).

346

VIBRATION TEST FIXTURES

347

Shaker head in

vertical position

frame

FIGURE 14.1. A typical vibration machine.

14.2

MOUNTING THE VIBRATION MACHINE

Vibration machines are usually mounted in one of two different ways, depending on their function. If a mobile test system is required and the shaker must be moved to different locations frequently, the vibration machine should be mounted on vibration isolators. Large masses cannot be vibrated effectively with large G forces using this type of installation. In order to keep the shaker from dissipating a great deal of energy shaking itself, the vibration machine should be mounted on a large concrete block at least ten times heavier than the machine. This large mass should be isolated from the building structure because low-frequency resonances may damage the building.

14.3 VIBRATION TEST FIXTURES

The shaker head on a vibration machine usually has some form of a hole pattern that permits the installation of machine screws. These holes can often be used to mount small electronic components for vibration testing. Large electronic boxes require some sort of mechanical adapter that will permit the shaker head to transfer vibrational motion to the electronic box. This adapter, or vibration test fixture as it is commonly called, should really be an extension of the armature in the form of a very rigid structure that can transfer the required force at the required frequency. An optimum fixture would have its lower natural frequency about 50% higher than the highest required forcing frequency in order to avoid fixture resonances during the test. For example, if the vibration test requirements have a range of 5-500 Hz, the vibration test fixture should have its fundamental resonant mode at about 750 Hz when the effective mass of the test specimen is included in the loading on the test fixture. Since most electronic

348

VIBRATION FIXTURES AND VIBRATION TESTING

boxes are not very heavy, a vibration fixture resonance of 750 Hz is not too difficult to obtain. However, if the vibration test must go to 2000 Hz, the desirable fixture resonance would be about 3000 Hz. This may be very difficult to obtain unless the test specimen happens to be quite light, probably less than about 20 Ib. If the test specimen is relatively heavy, perhaps greater than 50 lb, a test fixture with a resonance of 3000 Hz may be so massive that the force required to vibrate the test specimen may exceed the force rating on the available vibration machine. Under these conditions, a compromise has to be made. Either reduce the required force input to the test system, or reduce the weight of the fixture and try to live with the resulting resonances that may develop in the fixture. Very often severe fixture resonances can be reduced by introducing a highly damped fixture structure. This may be in the form of laminated structures where energy is dissipated at several interfaces. Even laminated wooden fixtures have been used successfully. Highly damped castings such as zirconium-magnesium are often used for structures that require high damping and stiffness with light weight. If there is any doubt as to why it is desirable to keep the natural frequency of a fixture at least 50% higher than the highest forcing frequency, remember that a resonance can magnify acceleration forces. If an improperly designed vibration fixture is used to support a sensitive electronic component during a 5-G sinusoidal vibration test, it is possible for this component to receive 100 G if the fixture has a transmissibility of 20 at its resonance. If the fixture were not properly monitored with accelerometers, a casual observer could conclude that this component failed at 5 G. Before any vibration fixtures are designed, it is necessary to understand the fundamentals of vibration. This requires a familiarity with the natural frequency formulas for simple systems such as beams, plates, and multiple spring-mass systems. Without this knowledge even the best designer will be groping in the dark. The end result may be an inadequate fixture that must be redesigned or modified. Rigorous mathematical solutions are not necessary to solve for the natural frequencies of various types of structures. If the natural frequency of a particular structure cannot be found in a reference book, it may be possible to derive the necessary equation using approximate methods such as the Rayleigh method or work and strain-energy methods. 14.4

BASIC FIXTURE DESIGN CONSIDERATIONS

Sharp changes in the cross section of any fixture should be avoided. Sharp changes in the cross section result in a reduction of the effective spring rate without a proportional reduction in the mass. This will result in a lower natural frequency for the fixture.

BASIC FIXTURE DESIGN CONSIDERATIONS

1

349

h

Fixture designs should be kept simple, since this keeps costs down and permits a more accurate analysis to be made with standard handbook equations. Always consider the stiffness-to-weight ratio of the fixture to make maximum use of the fixture mass. Understand the characteristics of the basic frequency equations to know what factors affect the resonant frequency. For example, the natural frequency of a uniform cantilevered rod during vibration along its longitudinal axis (Fig. 14.2) is

f” where A E

=-[-I

1 AEg 4 WL

1’2

(14.1)

= cross-sectional

area (in.*) of elasticity (lb/in.2) g = acceleration of gravity (in./s’) W = weight (lb) L = rod length (in.) = modulus

An examination of Eq. 14.1 shows that if the cross-sectional area A is doubled, the weight W will also be doubled, so there is no increase in the resonant frequency. If the rod material is magnesium, aluminum, or steel, there will be no difference in the natural frequency. This is because the ratio of the modulus of elasticity to the density of these three materials is just about the same (see Chapter 4, Sample Problem-Natural Frequencies of Beams). If the cantilevered rod in Fig. 14.2 is vibrated in the vertical direction, its natural frequency is (14.2) where I = moment of inertia of the cross section = bh3/12. An examination of Eq. 14.2 shows that if the beam depth is doubled the weight is doubled, but the moment of inertia of the cross section increases eight times, so the natural frequency is doubled.

350


rBolts,

/Bolted interface

-L-

/Bolted interface

1 0 - 4

FIGURE 14.3. A simply supported beam with two bolted sections.

FIGURE 14.4. The equivalent bolted beam considering a 25% bolted

efficiency factor.

--lo

-

Fixtures should be kept as small as possible to keep masses low and spring rates high. It is better to make fixtures out of castings, solid plates, or welded assemblies than of bolted assemblies. At high frequencies, bolted assemblies tend to slide and separate so that the calculated stiffness does not really exist. If a large increase in stiffness is not required, bolted assemblies provide a substantial amount of damping during resonant conditions that effectively reduces transmissibilities. If bolted assemblies are also cemented with epoxy cements, a very rigid fixture can be fabricated. Vibration test data on bolted assemblies indicate that the typical efficiency of a bolted joint is about 25%. This will vary, of course, depending on the relative stiffness of the structure as well as the size, spacing, and number of bolts. With many bolts this factor may go as high as 5096, but very few designers use enough properly spaced, large bolts to reach that efficiency. A convenient method for determining the moment of inertia for a bolted assembly is to reduce the effective width of the bolted member to 25% of its original width. A simply supported beam made up of two bolted sections 1 in. wide is shown in Fig. 14.3. The effective moment of inertia of the bolted assembly can be approximated by reducing the width of the top member to 0.25 in. when the bolted efficiency factor is 25% (Fig. 14.4).

14.5 EFFECTIVE SPRING RATES FOR BOLTS Bolts are generally threaded into tapped holes to a depth of about two diameters, so there appear to be many threads retaining the bolt. However. a close examination of the engaged threads reveals that, for a normal class 2 fit, only a few threads are actually holding the bolt. If these threads happen to be near the tip of the bolt, the effective length of the bolt can extend well into the tapped hole. When bolts are loaded in an axial direction and the

EFFECTIVE SPRING RATES FOR BOLTS

351

spring rate of the bolt must be determined, it is suggested that the effective length of the bolt should be considered as extending at least one diameter into the tapped hole. All bolts should be installed with a predetermined torque value, which depends on the bolt material and the function. The torque should be checked periodically on the bolts that can influence the vibration characteristics of the system, because bolts that are inserted and removed often tend to loosen more easily during vibration. If a bolt is assumed to be similar to a rod subjected to an axial load (see Fig. 14.5), the spring rate is

A€ K=L where A E L

(14.3)

area of cross section modulus of elasticity = length (extending one diameter into tapped hole) = =

Equation 14.3 shows the spring rate is not affected by the bolt torque. The torque determines the preload on the bolt; this preload must be exceeded (for metal-to-metal interfaces) by the external dynamic load before the bolted interfaces will separate. Although Eq. 14.3 shows that the preload torque does not have an effect on the spring rate, an examination of the load-deflection curves with an without a preload, as displayed in Fig. 14.6, shows the preload tends to increase the effective spring rate. (The preload will produce a nonlinear system that is beyond the scope of this book.) The spring rate hP/AX is the same for both systems shown in Fig. 14.6. However, the apparent spring rate, which is the slope along line AB, is higher for the preloaded spring. The natural frequency is slightly higher if a resonance occurs with a preloaded spring than with no preload. The transmissibility with a preloaded spring is much lower than the transmissibility of a spring without a preload. This is due to the additional energy dissipated by the slapping action of the interfaces on the preloaded bolts. There are times when it is desirable to increase the spring rate of a bolted system that has a fixed number of bolts and a fixed thread size. The spring

1

Block with +,/clearan; hole Bolt l n!Bolted ---,--interface\ I, ' I

n

1

4-L

L

I---?) rdl -7

\Tapped

%\'

Y

hole

Ik D (bolt diameter)

FIGURE 14.5. Effective length of a bolt should b e considered as extending a distance of o n e diameter into the tapped hole.

352


J

-2

Preload

5

IP

Load

AP

P

A

A

Deflection

Deflection

-X

(b)

(a)

FIGURE 14.6. Spring rate ( a ) without preload and ( b )with preload.

rate of the bolts can often be increased by making the bolts out of beryllium, which has a modulus of elasticity of 42 X lo6. Sometimes shoulder bolts can be installed where the body of the bolt above the thread is larger than the thread diameter. Long bolts are generally poor for vibration, since they can “wind up” when they are installed. Under vibration these bolts can “unwind,” so they may become loose very quickly.

14.6

BOLT PRELOAD TORQUE

The relation between the applied torque and the resulting preload in a bolt can be determined from a simple approximate equation [91 T

= 0.2DP

(14.4)

where T = torque (in: Ib) D = outside diameter of thread (in.) P = axial load induced in bolt (Ib) 0.2 = constant for most bolt and bolt materials Only about 10% of the applied torque goes into tightening a bolt; about 90% is lost in friction: about 50% of the torque under the bolt head, and about 40% in the bolt threads. Tests on socket-head cap screws [lo] show that a heavy lubricant has a noticeable effect on the torque-tension curve in the lower range but has negligible effect under severe stress. These tests also showed that tension is an approximate straight-line function of the applied torque up to the yield point.

353

ROCKING MODES AND OVERTURNING MOMENTS

The ideal tightening torque should stress the bolt up to the elastic limit of the material. Because this condition is very difficult to obtain during mass production on an assembly line, the bolt torque should be limited to a value that will stress the bolt material to about 80% of the yield point. (See Chapter 1, Section 1.8 for recommended tightening torques.) Sample Problem-Determining

Desired Bolt Torque

Determine the desired installation torque for a 300-series stainless-steel bolt, size 6-32. Assume cut threads, since rolled threads will work-harden the bolt and increase the yield strength. Solution. The desired installation torque can be determined from Eq. 14.4, where Sty = 33,000 Ib/in.* (tensile yield strength of 300-series stainless steel) A = 0.0090 in.* (stress area of bolt) P =AS,, = (0.009)(33,000) = 297.0 Ib (tensile load in bolt) D = 0.138 in. (diameter of bolt)

We obtain

T = 0.2(0.138)(297)

= 8.2

lbsin.

(14.5)

This represents the net torque to tighten the bolt. If a locking device is used, the torque required to overcome the locking torque must be added to the initial tightening torque.

14.7

ROCKING MODES AND OVERTURNING MOMENTS

Severe rocking modes will often develop in the shaker head during vibration in the vertical direction if the vibration fixture or the electronic box is either very tall or very broad (Fig. 14.7). These are usually caused by an unbalanced condition or by a shift in the center of gravity during major resonant conditions. Balancing a large fixture is very important, especially if the fixture is bolted directly to the shaker head. The balancing should be done with the actual electronic box or a dummy load attached to the fixture. Balancing weights, if used, should be made a permanent part of the fixture, or they may soon be lost, and forgotten until the shaker head is damaged. The importance of balancing every vibration fixture cannot be overemphasized, yet it seems that many electronic vibration testing laboratories never bother to balance their fixtures. This poor practice has led to many short circuits in

354


’*

)-c + -z4.= !

Electronic

~

Vi bration

I’

direction

--tL

~-~

‘ /

\---C--J I

\ L

+-----.--A

(b)

(a)

FIGURE 14.7. Rocking modes that may occur during vibration tests

Combined CG of / fixture a n d box

-.,,l~~+~] Vibration

Unsymmetrical vibration fixture

t

Electronic box

direction i

-

FIGURE 14.8. An unsymmetrical vibration fixture can develop rocking modes.

armatures that have had their electrical insulation rubbed off by large overturning moments. Rocking modes can also develop in relatively small fixtures that are unsymmetrical (Fig. 14.8). Although the combined center of gravity of the fixture and the electronic box is exactly on the centerline of the shaker, it will not stay there at high frequencies. The effective mass of the electronic box is smaller at high frequencies, after passing through major resonance points, while the stiff unsymmetrical fixture maintains its effective mass. This results

OIL-FILM SLIDER TABLES /Eiectronic

355

box Vibration

I

i

direction

vibration fixture

FIGURE 14.9. A symmetrical vibration fixture.

in a dynamic shift of the CG, which results in rocking modes. There are fewer problems if symmetrical fixtures are designed with a low center of gravity (Fig. 14.9). Overturning moments in the shaker head on a vibration machine can also be due to nonuniform flexures or springs that support the head. A great deal of time can be saved by knowing the response characteristics of the bare shaker head, which may be out of balance. The vibration characteristics of the bare shaker head should be recorded over the normal frequency bandwidth, to provide a permanent record that can be checked from time to time to determine if any deterioration has taken place. 14.8 OIL-FILM SLIDER TABLES

Most vibration test laboratories make use of oil-film slider tables when vibration tests must be run in the horizontal plane. An oil-film slider, as the name implies, consists of a large flat plate that slides on a film of oil. The plate usually rests on a very rigid foundation of concrete, steel, or granite, and one edge of the plate is bolted to the shaker head (Fig. 14.10). Shaker head-\

Electronic box

--

Test fixture Slider p l a t e l

, Oil film

b Rigid foundation

---__

Vibration machine

FIGURE 14.10. Vibration machine with an oil-film slider plate.

356


The test specimen, which consists of the vibration fixture and a test package, is normally bolted to the slider plate. The test specimen can usually be rotated 90" without changing the slider plate to permit vibration along both horizontal axes. Oil-film slider tables are capable of very pure translational motion with very little cross talk (rotational modes in either the horizontal or vertical planes). This is due to the viscous nature of the oil film acting on a very large surface area. Overturning moments in the vertical plane and rotational moments in the horizontal plane can be effectively damped out in most cases. If an attempt is made to vibrate a very tall test specimen with a high CG, the surface area of the plate may have to be increased and a thicker slider plate may have to be used in order to prevent lifting and slapping of the slider plate. counterweights may also be required. Before slider plates were used, horizontal vibration tests were run using flexure tables and suspension systems. These systems are very difficult to control during resonant conditions, and their use is not recommended if an oil-film slider can be used instead. 14.9 VIBRATION FIXTURE COUNTERWEIGHTS

Oil-film slider plates can eliminate many difficulties related to overturning moments developed during vibration in the horizontal plane. However, when tall masses with high C G must be vibrated, counterweights may have to be used to lower the CG. Counterweights are normally made of a dense material, such as steel or lead, to keep the overall size down. counterweights may be good for the static balance on a vibration system but they may not be good for the dynamic balance, due to the lack of dynamic similarity to the test specimen. Consider a tall nose cone that must be vibrated in a direction perpendicular to the axis of the cone. A severe resonance in the nose cone could shift the CG so that a counterweight would Nose cone

l -

\Nose cones back-to-back (b)

FIGURE 14.11. ( a ) Statically balanced system and ( b ) dynamically balanced system.

SUSPENSION SYSTEMS

357

not be able to compensate for the overturning moment. Under these conditions, dynamic similarity could be obtained if two nose cones were vibrated back to back simultaneously. Any dynamic change in one nose cone would be duplicated in the opposite nose cone so that severe overturning moments can be reduced (Fig. 14.11). 14.10 A SUMMARY FOR

GOOD FIXTURE DESIGN

Vibration test fixtures will provide adequate performance if a few simple rules can be followed:

1. Understand the test specimen and the test specification. 2. Analyze preliminary designs and try to keep the lowest natural frequency about 50% higher than the highest forcing frequency. 3. Avoid sharp changes in the cross section. 4. Consider the stiffness-to-mass ratio for optimum design. 5. Keep fixtures as small as possible. 6. Avoid bolted fixture assemblies, except where ribs may be required for stiffness and damping. 7. Keep fixture designs simple. 8. Design symmetrical fixtures. 9. Design for dynamic similarity. 10. Consider the effective length of the bolt thread engagement when calculating the effective bolt spring rate. 11. Torque all bolts. 12. Use fine threads, instead of coarse threads, on bolts wherever possible. 13. Balance fixtures with the test specimen. 14.1 1

SUSPENSION SYSTEMS

Vibration tests are often required on large electronic control consoles that are used on ships and submarines. These units may weigh several thousand pounds; this makes them very difficult to handle without special equipment. Very often a special facility must be designed and fabricated just to perform vibration tests. Oil-film slider tables may not be practical in this case, so a suspension system might be considered. A special A frame may have to be designed to provide a rigid structure to support the console during vibration. Special vibration holding fixtures may also have to be provided to shake a large electronic console, which is generally fabricated of thin (12- to 18-gage) sheet metal. The vibration holding fixture is normally fastened to the console at the same points where the console is fastened to the ship’s structure.

358


ii Flectronic

control

console

FIGURE 14.12. Vibration machine with a large suspended electronic console.

The console-and-holding fixture assembly is suspended in the A frame by wire cables (Fig. 14.12). A mechanical fuse is advisable for this type of installation. Since there is very little damping in a wire suspension system, overturning moments may become quite severe. Large angular displacements in the electronic console can force the shaker head to rotate enough to permit the electrical coil windings in the shaker head to rub against the stationary field windings; this would eventually cause an electrical short circuit in the system.

14.12 MECHANICAL FUSES A mechanical fuse is a safety device that permits vibrational motion to be transferred from the shaker head to the test specimen unless severe overturning moments are developed. These moments then break the fuse before the shaker head is damaged. A typical fuse is shown in Fig. 14.13.

/Fuse (with flats for installation)

Righthand t h r e a d 1

~~

w

f

-

\Lefthand ~ thread

FIGURE 14.13. Enlarged view of a typical mechanical fuse.

-

DISTINGUISHING BENDING MODES FROM ROCKING MODES F(xture-ard-1 - 1 elec!rontci control-conscie weigbt

1 1

F~~~

b/h//\/\hfl

,1

359

aker-head

Sh weight

K

FIGURE 14.14. Simulating the suspension system as a two-mass system

The spring rate of the fuse can be determined by considering it to be equivalent to a rod in tension and compression. Equation 14.3 can then be used directly. When an overturning moment is applied, the bending stress in the fuse is

Mc

s, = r

(14.6)

moment (Ib . in.) moment of inertia (in.4) c = radius of the fuse section (in.)

where M I

= bending =

The natural frequency of the suspension system can be approximated by considering it as a two-mass system with a spring between the masses (Fig. 14.14). Using the methods shown in Chapter 2 the natural frequency equation is (14.7) If the weight of the console, W,, is very large with respect to the weight of the shaker head, W,, then the equation becomes (14.8) This is similar to the single-degree-of-freedom equations shown in Chapter 2 . 14.13 DISTINGUISHING BENDING MODES FROM ROCKING MODES

There are occasions when it is rather difficult to determine what is taking place during a vibration test unless proper instrumentation is provided. The case of a large rigid vibration fixture during vibration in the vertical direction, with one accelerometer placed at each end, is shown in Fig. 14.15. With an input of 2 G, both end accelerometers read 10 G. What is the vibration mode of the fixture?

360


Accelerometer

V i bration

A

Vibration

t

machine

direction

-

/

\

,\-\'<,,~'

,

,

FIGURE 14.15. Vibration machine and test fixture with only two accelerometers.

L--.:I

-~

Tran>Iatior i (I

)

r ,+:/ ~

/""

Rc'o'cn f

bJ

.,

-

.

.-

e \,

v,-3. b>

Bend 16

(0

FIGURE 14.16. T h r e e different vibration modes on a test fixture.

There are three different vibration modes that can have these characteristics: pure translation, rotation, and bending (Fig. 14.16). Unless more accelerometers are used or more points are monitored, it is very difficult to tell what is happening without the use of a strobe light.

14.14 PUSH-BAR COUPLINGS Oil-film slider plates are usually bolted directly to the shaker head. This is generally adequate if the test frequencies are below about 500 Hz and if the test specimen is light enough. When the test frequencies go up to 2000 Hz and the test specimen becomes very heavy, the few bolts holding the slider plate to the shaker head may not have a spring rate high enough to produce the required force at the high frequencies. What often happens is that the shaker head has an output of 30 G but the slider plate receives only 3 G because of a soft bolt-spring system. In order to increase the stiffness of the bolt system, a push-bar coupling is often used. The coupling permits the use of many more bolts at the junction. although another bolted interface is added, the increase in the number of bolts will increase the overall stiffness of the system.

PUSH-BAR COUPLINGS

(Slider plate

361

I

u

FIGURE 14.17. View of a slider plate bolted to a shaker head

tap in

a \ ; : J

shaker head

$

/

,

\

,

16.0 dia.

FIGURE 14.18. A typical bolt pattern in a shaker head.

Consider the vibration system shown in Fig. 14.10. If the slider plate is bolted directly to the shaker head, the interface would probably appear as shown in Fig. 14.17. Considering the bolt-hole pattern shown in Fig. 14.18 only five b o l t s could be used in t h e slider p l a t e . The bolted interface system shown in Fig. 14.17 is nonlinear, since a pulling force on the slip plates places the bolts, which have a small cross-sectional area, in tension. A pushing force, however, acts directly on the slider plate and places that member, which as a large cross section, in compression. A load-deflection diagram for this type of system is similar to that shown in Fig. 14.19. Nonlinear systems do not have the same response characteristics as linear systems, and the natural frequency equations are not the same. However, if the nonlinear system is approximated as a linear system, then linear-system equations can be used to estimate the probable performance characteristics, as shown in the following section.

362


1+

/i

Load

Sdder plate in .o-?pre;sion

- Loaa

FIGURE 14.19. A load-deflection curve for a nonlinear spring.

300 ib Bolts

FIGURE 14.20. Simulating the slider plate as a two-mass system.

H-

1

100 Ib

-

I\/Wii~ I 1

6

-

1

The natural frequency of the vibration system shown in Fig. 14.10 can be approximated by lumping the masses into a two-mass system with one spring where the bolts act as the spring (Fig. 14.20). Weight W , is as follows: Slider plate Vibration fixture Electronic box

150 lb 90 60 W , = 300 lb

(14.9)

Weight W , is the shaker head. Assuming an MB vibration machine Model C-126, which has a rating of 9000 force pounds, the shaker head weighs Wz = 100 lb

(14.10)

The spring rate of the bolts can be determined from Eq. 14.3:

where E = 30 x lo6 lb/in.’ (steel bolts) L = 0.625 1 bolt diameter (see Fig. 14.17) L = 0.625 + 0.375 = 1.00 in. (length) A , = 0.0876 in., (stress area for a in.-diameter fine thread on one bolt)

+

PUSH-BAR COUPLINGS

363

For five bolts, the spring rate is

K=

5(0.0876)(30 X l o 6 )

13.10 X 106 Ib/in.

=

1.oo

(14.11)

Substituting Eqs. 14.9-14.11 into Eq. 14.7, (13.1 X 106)(3.86 X 10')

1/?

3.00 x 10' (14.12)

f, = 1310 HZ

Since the system is really nonlinear, the natural frequency would be somewhat higher than this. However, this system would not be adequate to perform vibration tests as high as 2000 Hz if high G forces are required. Since the spring system is relatively soft, a high G force in the shaker head produces a relatively small G force, at frequencies well above the resonance of the bolted system, in the slider plate. Even if the spring rate of the bolts indicates that a resonance might occur in the vibration oil-film-slider system, a high preload in the bolts may not permit a resonance to occur. In order for a resonance to occur, the dynamic loading on the oil-film slider plate must exceed the preload in the bolts. If steel bolts with a yield strength S, of 35,000 Ib/in.? are used to fasten the slider plate directly to the shaker head, then the maximum allowable preload in each bolt is P = A S S ,= (0.0876)(35,000)

=

3060 Ib

(14.13)

The installation torque required on each bolt can be determined from Eq. 14.4:

T = 0.2DP = 0.2(0.375)(3060) = 230 in:lb

(14.14)

The total preload on five bolts is P, = 5(3060)

=

15,300 Ib

(14.15)

The dynamic load acting on the slider plate can be approximated from the input G force, the weight, and the transmissibility at resonance: Pd

Q

= G,"Wi

(14.16)

Assuming a 10-G peak sinusoidal vibration input with a transmissibility of about 6, the dynamic load is Pd = 10(300)(6) = 18,000 lb

(14.17)

364

VIBRATION FIXTURES AND VIBRATION TESTING /Push-bar

‘I

coupler using 11 bolts,

St a f i e r h e a d

S h a - e r ‘Ieid bc t pattern

FIGURE 14.21. A push-bar coupler with 11 bolts to increase the stiffness.

The dynamic load shown by Eq. 14.17 is greater than the total preload on the bolts shown in Eq. 14.15; this means the bolt preload is not high enough to prevent a resonant condition from occurring at about 1310 Hz, as shown by Eq. 14.12. A push-bar coupling could be designed to pick up 11 bolts on the shaker head as shown in Fig. 14.21. (For maximum rigidity, the slider plate should be welded to the push-bar coupler.) All bolted interfaces should load the bolts in tension and not in shear, to obtain the maximum effective stiffness. Bolted interfaces loaded in shear can slip and slide with respect to each other at high frequencies. Even if several large press-fit dowels are used in the interface to prevent relative motion, it can still occur at high G loads. Therefore. the bolts should be loaded in tension for the best results. 14.1 5

SLIDER PLATE LONGITUDINAL RESONANCE

A slider plate can develop resonances during vibration in the longitudinal direction if the plate is long enough. The natural frequency for a structure under these conditions can be determined from Eq. 14.1. Since the width of the plate is not important during longitudinal vibration, consider a unit width of 1.0 in. and a length of 30.0 in. (Fig. 14.22). The following information is

FIGURE 14.22. A unit width o n a n oil-film slider plate (dimensions in inches).

365

ACCELERATION FORCE CAPABILITY OF SHAKER

then required to determine the resonant frequency: 1.0 in.* (cross-sectional area) 10.5 x l o 6 lb/in.* (modulus for aluminum) = 386 in/s2 (gravity) W = (1)(1)(30)(0.1) = 3.0 lb L = 30 in. A

=

E g

=

Substituting into Eq. 14.1, the longitudinal natural frequency becomes (1.0)( 10.5 X 106)(3.86 X l o 2 )

(3 .o>(30)

f, = 1680 HZ

(14.18)

This represents the natural frequency of the bare slider plate. If the masses of the fixture and electronic box are included, the natural frequency will be lower. A more comprehensive analysis of the slider plate, vibration fixture, electronic box, push-bar coupling, and shaker head could treat this combination as a multiple spring-mass system or as a distributed system using finite element modeling methods (FEM) with a computer. 14.16 ACCELERATION FORCE CAPABILITY OF SHAKER

Vibration machines are rated in peak force (pounds) based on sinusoidal vibration, which is determined from the maximum allowable power that can be dissipated in the machine. In order to calculate the maximum G force possible in a system, the force rating of the machine must be known along with the entire weight of the moving mass. The equation then becomes

Gpeak

=

rated force shaker head + slider + specimen

(14.19)

The peak G-force capability for the system shown in Section 14.14 can be determined for the MB C-126 vibration machine, which has a peak rating of 9000 force pounds. Substituting Eqs. 14.9 and 14.10 into Eq. 14.19, the peak G-force capability becomes

9000 Gpeak =

100 + 300

= 22.5

(14.20)

366

14.17


POSITIONING THE SERVO-CONTROL ACCELEROMETER

The G force transmitted from the shaker head to the vibration test specimen can be controlled directly from the shaker head or from an accelerometer placed near the test specimen. When the G force is controlled from the shaker head, a constant G force can be maintained on the shaker head over the entire frequency band. If a resonance should develop in the slider plate or the vibration fixture, the G forces will build up at resonance so that the input to the test specimen will be far greater than the output of the shaker head. At frequencies above the resonance the opposite situation occurs, and the input to the test specimen will be far less than the output of the shaker head. When the G force controlled from an accelerometer placed near the test specimen, the vibration machine can vary the output of the shaker head to hold a constant G force on the control accelerometer over the entire frequency band. If the control accelerometer is placed on a resonant structure, the output of the shaker head will be reduced when the structure passes through its resonance. At frequencies above the resonance, the output of the shaker head will be increased to hold a constant G force on the control accelerometer. Standard piezoelectric accelerometers, the same types that are generally used to monitor and record acceleration data, are normally used as servocontrol accelerometers. These small devices are usually equipped with screw studs to fasten them to the test equipment. Many testing laboratories use adhesives such as dental cement, Eastman 910 quick-drying cement, and even double-backed tape to mount accelerometers. These methods all work quite well on accelerometers that are used to monitor acceleration data, but they should never be used to fasten a control accelerometer to a vibrating system. The servo-control accelerometer actually controls the G-force output of the shaker head. If the servo-control accelerometer should fall off during the vibration test, the shaker head will lose the feedback provided by this accelerometer. This results in a rapid buildup of the acceleration force until the limit displacement switch of the shaker head cuts off the power. By this time the damage has already been done: very high G forces can be developed very rapidly. Once an expensive electronic system has been subjected to acceleration loads that far exceed its specifications, there is no good way to determine the actual and potential damage. In most cases where this has happened, and it has happened at many places, the electronic equipment has to be rebuilt because the customer will not accept an item that may fail shortly after it is put into service. Since the position of the servo-control accelerometer can determine the actual G force received by the tested specimen, the location selected for this accelerometer can be quite important to the success of a vibration test. A resonant fixture on a slider plate is shown in Fig. 14.23.

MORE ACCURATE METHOD FOR ESTIMATING TRANSMISSIBILITY Q IN STRUCTURES

367

Resonant fixture

,X\\\\K

Electronic box

\'~\\\,,\\\\'~\\\\'~\ \ \ \ \ \\ \\ \\ \'\\\\\\\\,

FIGURE 14.23. A resonant fixture on an oil-film slider plate.

If the top of the resonant fixture has a transmissibility of 10, then it will see 20 G at its resonant peak when the control accelerometer is held at 2 G. This means the electronic box will probably see an average of about 11 G at the fixture resonance point instead of the required input of 2 G. If the control accelerometer is mounted at the top of the resonant fixture, the bottom of the fixture will see only 0.2 G at the resonant peak. This means that the electronic box will probably see an average of about 1.1 G at the fixture resonance point instead of the required input of 2 G. The best solution to this problem, of course, is to design resonance-free fixtures. Since this is not always possible, a compromise is the only possibility, so the control accelerometer will probably be placed halfway up the fixture.

14.18 MORE ACCURATE METHOD FOR ESTIMATING THE TRANSMISSIBILITY Q IN STRUCTURES

Damping in a vibrating structure plays an important part in establishing the magnitude of the transmissibility Q that will be developed when the structure is excited at its natural frequency. Dynamic coupling and phasing relationships can also play an important part. When the damping is increased, more kinetic energy is converted into heat so there is less energy available to do work on the structure, which results in a decreased transmissibility. When the damping is decreased, less kinetic energy is converted into heat so there is more energy available to do work on the structure, which results in an increased transmissibility. This can also be related to the dynamic displacements and stresses. Higher values produce more damping, which decreases

368


the transmissibility. Lower values produce less damping, which increases the transmissibility. The input acceleration G levels play an important part because high input acceleration G levels result in high stresses and high damping, which reduces the transmissibility. Low input acceleration G levels have the opposite effect. An examination of Eq. 2.30 shows that when the G level is held constant, the displacement Yo is inversely related to the square of the frequency. When the frequency is doubled, the displacement is reduced by a factor of 4. Reducing the displacement reduces the stress, which decreases the damping so the transmissibility is increased. The above information leads to two very important conclusions: 1. Increasing the natural frequency will increase the transmissibility. 2. Increasing the input acceleration G level will decrease the transmissiblilty.

The above conclusions were combined with several years of accumulated test data on different types of electronic support structures that were broken down into three basic groups: beams supporting some electronics, PCBs with some perimeter supports, and small enclosed electronic boxes. The data were plotted on log-log paper to see if some general type of straight-line curve, similar to a S-N curve, could be developed. After churning a large number of approximations, the general equation shown below was finally selected: (14.21)

where A A A

1.0 for beam-type structures = 0.50 for plug-in PCBs or perimeter-supported PCBs = 0.25 for small electronic chassis or electronic boxes f, = natural frequency (Hz) G,, = sinusoidal vibration input acceleration G level (dimensionless) =

The above equation relates to electronic assemblies, so a beam structure is expected to hold several electronic components with some interconnecting wires or cables. The PCB are expected to be well populated with an assortment of electronic components. The small chassis is expected to be between about 8 and 30 in. in its longest dimension, with a bolted cover to provide access to various types of electronic components such as PCBs, harnesses, cables, and connectors. Sample Problem-Transmissibility

Expected for a Plug-in PCB

Determine the approximate transmissibility expected for a plug-in PCB with a natural frequency of 280 Hz using peak sine vibration input G levels of 0.2 G, 2 G , 5 G , and 8 G.

CROSS-COUPLING EFFECTS IN VIBRATION TEST FIXTURES

Solution. Use Eq. 14.21 with the value A input level.

[

=

1

280 Q=O.50 (0,210.6

369

0.50 starting with the 0.2 G

0.76

=

75’4

Peak Sine Input Level Gin

Approximate Q Expected

0.2 2.0 5.0 8.0

75.4 26.4 17.4 14.0

(14.22)

The above data correlate fairly well with vibration test data performed on a wide variety of PCBs. Many PCB sample problems in this book use an approximate transmissibility Q value based on the square root of the natural frequency, which results in a value of 16.7. This is close to the value for the 5-G peak sinusoidal vibration input level shown above.

VIBRATION TESTING CASE HISTORIES 14.19 CROSS-COUPLING EFFECTS IN VIBRATION TEST FIXTURES

The best way to start this section is to describe the events associated with the vibration testing of a small electronics module. An electrodynamic shaker was set up to vibrate in the vertical direction. The vibration test fixture was in the shape of a cube frame that was assembled by welding together the ends of 12 aluminum bars each measuring 1.0 in. square and 9.0 in. long as shown in Fig. 14.24. The fixture was bolted to the top of the shaker head and the small

Frame vibration fixture

9.0

I l,O-ln, square aluminum bars

FIGURE 14.24. Poor vibration fixture using 1.0 in. square aluminum bars welded together.

370


electronics module was bolted to the top of the fixture. A small accelerometer was fastened to the top of the fixture, adjacent to the test module, to control a 5.0-G peak sinusoidal vibration input level in the vertical direction. The electronics module being tested was rated at 25 G peak. The sine sweep started at 25 Hz with the module operating electrically. Somewhere in the area of about 200 Hz the module failed. Everyone was upset because they did not expect any problems with a 5-G test on a module that was rated at 25 G. A call was placed to the module vendor complaining about the false advertised rating of 25 G. The module vendor again verified and guaranteed the 25-G rating and suggested they hire someone to look at their problem. An outside consultant came in and requested that two additional accelerometers be mounted in the two horizontal axes to monitor the magnitude of the cross coupling. The program managers denied the request saying there was no horizontal cross coupling on the fixture. The consultant asked them if they had test data that verified the lack of horizontal cross coupling. The program managers replied that cross-coupling test date were not necessary and that it was a waste of time. They claimed that when the system is being vibrated in the vertical direction it can only move in the vertical direction and no other direction was possible. The consultant now insisted that two more horizontally oriented accelerometers be attached to the fixture to examine cross-coupling responses. Again the program managers denied the request. This standoff went on for about an hour until the program managers allowed two more accelerometers to be mounted on the fixture in the horizontal direction. The test data showed the horizontal response was 104 G peak for a 5-G peak vertical input. It was obvious to the consultant that the frame fixture had a low natural frequency in the horizontal direction. The consultant recommended the frame fixture be replaced with a small solid aluminum block fixture that would have a natural frequency well above 2000 Hz, and the problem was solved. Nature tends to be very efficient in our world of dynamics. The theory of deformation says that structures under load will deform in a pattern that keeps the strain energy at a minimum. Another way of saying the same thing is that any time a structure is being vibrated, it will respond by moving in the direction that generates the minimum strain energy, which always results in the lowest natural frequency. The direction of maximum response is often in a direction that is different from the input direction. PROGRESSIVE VIBRATION SHEAR FAILURES IN BOLTED STRUCTURES 14.20

A rugged wheel axle on a piece of farm machinery was bolted to a rigid frame with many large bolts, as shown in Fig. 14.25. Every few years the wheel axle would become loose because many of the large bolts had sheared off. The farmer kept replacing the bolts when they broke to keep the machinery

VIBRATION PUSH-BAR COUPLERS WITH BOLTS LOADED IN SHEAR

371

Bolts loading in shear Vibration

FIGURE 14.25. Progressive shear in bolted joints subjected to alternating dynamic loads.

working. A dynamic stress analysis was made of the bolts using data from accelerometers that were attached to the axle. The analysis assumed that half of the bolts would carry the alternating shear loads. This analysis showed the bolts should not fail so the farmer thought the bolts were defective. Higherquality and higher-strength bolts were installed. They lasted a little longer but they also failed. Bolted joints require large holes in the mated parts to allow easy assembly for mass produced parts with standard machining tolerances. A close examination of bolted joints under an alternating load shows that the bolted part will move slightly during vibration since friction is trying to prevent relative motion between the two bolted parts. Friction and vibration are like oil and water-they do not mix. The bolted part will move until one bolt picks up the load. The first bolt that picks up the load will become the pivot point and the part will continue to move until the second bolt picks up the load. This loading will alternate between the two bolts until one of the bolts fractures so another bolt picks up the load. This process will be repeated again and again until the bolted assembly falls apart. Only two bolts will cany the alternating load in a bolted assembly when loose tolerances are used in mass produced machine parts, no matter how many bolts are used. If bucked rivets are used instead of bolts, the rivets expand to fill the holes so relative motion between the riveted parts is not possible. In this case all of the rivets will carry some part of the alternating load. If rivets cannot be used, then two (or more) large press-fit dowel pins should be added with a shear area large enough to carry the alternating shear loads. The standard practice is that bolts should be loaded in tension and dowel pins should be loaded in shear.

4.21 VIBRATION PUSH-BAR COUPLERS WITH BOLTS LOADED IN SHEAR

Engineers with limited vibration testing experience are often requested to design special fixtures for testing electronic equipment. Bolted joints are often employed because they are quick and easy to install and remove. A

372

VIBRATION FIXTURES AND VIBRATION TESTING Shaker head

Bolts loaded in shear

Vibration direction

Bolts loading in tension

FIGURE 14.26. Poor arrangement for oil-film slider vibration fixture using a push-bar coupler with attachment bolts loaded in shear.

typical installation of this type for an aluminum oil-film slider plate and a bolted coupler is shown in Fig. 14.26. When the test specimen is heavy, the vibration machine cannot transfer high acceleration G levels to the slider plate at high frequency because there is too much slippage at the slider plate bolted interface. This type of installation relies on friction at the bolted interface to transfer the load from the shaker to the slider plate. This system works well for forcing frequencies below about 800 Hz. With a shaker forcing frequency of about 1000 Hz and a 10-G peak sinusoidal vibration input, only about 2 G peak is transferred to the slider plate because of the slippage at the bolted joint. When clearances are provided in the fixture bolt holes to allow easy assembly, the rigid connection from the shaker to the slider plate is lost because the interface friction at the bolted joint cannot carry the high forces without slipping. The bolt tightening torque can be increased, or a couple of extra bolts can be added, or larger bolts can be used, but these changes will not improve the high-frequency testing performance very much. In the previous section it was pointed out that bolts should be loaded in tension and dowel pins should be loaded in shear, so dowel pins are proposed. Three 1.0-in. diameter holes are drilled and reamed for closely machined press-fit dowel pins. Three hard steel dowel pins are placed in a cold chamber before they are press-fit into the fixture to ensure a good tight fit as the pins return to room temperature and expand. The test engineers feel confident that the dowel pin fixture will solve their high-frequency testing problem. The tests are resumed with a shaker forcing frequency of 1000 Hz and a 10-G peak sinusoidal vibration input. The slider plate accelerometer only shows a value of 3 G peak instead of the expected 10 G peak. The test engineers think the slider plate accelerometer is defective. It is not defective. What happened? The hard steel dowel pin acting against the softer aluminum causes the aluminum to brinell locally at the corners in about 4 seconds, as shown in Fig. 14.27. With 1000 cycles per second, in 4 seconds there are 4000 stress reversals, which loosens the dowel pins. One good way to solve this problem is to weld a slider plate to a push-bar coupler, which

BOLTING PCB CENTERS TOGETHER TO IMPROVE THEIR VIBRATION FATIGUE LIFE

373

Brinelling corners

FIGURE 14.27. Loss of effectiveness for press-fit dowel pins in push-bar couplers d u e to rapid brinelling of the softer aluminum corners during high-frequency vibration.

Shaker head

i

1

1 --&e\ l-dfl Push-bar coupler

Bolts loaded in tension

FIGURE 14.28. Good arrangement for vibration fixture with push-bar coupler welded to slider plate and many mounting bolts loaded in tension.

has all of the attachment bolts to the shaker head loaded in tension as shown in Fig. 14.28.

14.22 BOLTING PCB CENTERS TOGETHER TO IMPROVE THEIR VIBRATION FATIGUE LIFE

Plug-in type PCBs usually have relatively low natural frequencies because their centers are unsupported so they tend to “oil can.” As a result, the transmissibility Q values increase, producing more rapid component lead

374

VIBRATION FIXTURES AND VIBRATION TESTING Perimeter flange

Piug-in Cover

"'\

Vi bration

direction

,

I,

,

lj

V

I

0.25-in. diameter steel rod

Spacer between each PCB

through every PCB

'Slider

piate

FIGURE 14.29. Long steel rod through the center of every plug-in PCB for improved support to increase PCB natural frequency. It did not work. High dynamic loads destroyed the chassis.

wire and solder joint fatigue failures. When vibration tests are run on individual plug-in PCBs that have their centers supported, there is a big increase in the natural frequency and a big increase in the fatigue life. A prototype aluminum dip-brazed chassis with 0.090-in. thick walls was fabiicated to hold 20 dummy plug-in type PCBs, weighing 1.0 lb each, to study this idea. The chassis used a 0.25-in. diameter steel rod, threaded at both ends, that passed through a hole in the center of each PCB. Hollow aluminum cylinder tube spacers with close tolerances were placed between every PCB so the centers of every PBC would be locked together when the steel bar was bolted and clamped at both ends of the chassis as shown in Fig. 14.29. In order to avoid any possible vibration problems, a rugged reinforcing rib 1.0 in. long by 0.38 in. thick was dip-brazed around the perimeter of the chassis. The steel rod through the chassis was placed under, but close to, the reinforcing perimeter rib. Doublers were used at both ends of the chassis to make sure the 1.0-in. long rib would carry, the dynamic load instead of the thinner 0.090-in. thick aluminum wall. The assembled chassis with the 20 dummy PCBs was instrumented with several small accelerometers and bolted to an oil-film slider plate, which was bolted to the head of an electrodynamic shaker for a 5.0-G peak sinusoidal vibration test from 20 to 2000 Hz. The vibration input was along the longitudinal axis of the chassis, perpendicular to the plane of the PCBs. The forcing frequency was increased slowly while several X-Y plotters printed the responses of several accelerometers mounted within the test chassis. As the forcing frequency approached the 425-Hz natural frequency of the chassis, there was a sharp increase in the noise level similar to a railroad train passing through the test area. This was quickly followed by what can only be described as an explosion, as the rugged dip-brazed electronic chassis with the stiff perimeter reinforcing rib literally tore itself apart.

VIBRATION FAILURES CAUSED BY CARELESS MANUFACTURING METHODS

375

Hindsight is always very clear. It should have been obvious that when all of the PCBs were locked together they would all have the same natural frequency. Unless someone has had some personal testing experience, it is very difficult to imagine the amount of kinetic energy that is available when 20 lb of mass goes into a resonant condition. With only a 5-G peak sinusoidal vibration input acceleration level, a 20 lb mass with a transmissibility Q of about 20 will produce a dynamic load of 20 X 20 X 5 = 2000 lb acting on two reinforcing ribs, front and rear, This results in a dynamic force of 1000 lb on one rib. With a chassis width of 8.0 in. the approximate bending moment on one reinforcing rib will be 500 X 4 = 2000 lb ‘in. The moment of inertia for one rib will be 0.0316 in.4. The dynamic bending stress on one rib will be approximately

Mc

sb -- - =I

(2000) (1.0/2) 0.0316

= 31,645

lb/in.’

(14.23)

The approximate fatigue life of the 6061 T-4 aluminum dip-brazed chassis can be obtained from Eq. 8.2 using an ultimate tensile strength of 36,000 for the dip-brazed aluminum and a fatigue exponent of 6.4 as shown below:

zr

1

36,000 6.4 N , = N 2 - =(1000) 31,645

( ]

(

=

2282 cycles to fail

(14.24)

The time to fail can be estimated from the chassis natural frequency of 425 Hz as follows: Time to fail

2282 cycles to fail =


= 0.089 minute = 5.4 seconds

(14.25)

14.23 VIBRATION FAILURES CAUSED BY CARELESS MANUFACTURING METHODS

A small shaft on a spinning gyro for an inertial navigation system was failing during a flight acceptance test of 3.5 G RMS. A dynamic stress and fatigue life analysis of the shaft, including a generous stress concentration factor showed that it should not fail. The shaft was fabricated by machining the end of a 0.25-in. diameter dowel rod down to a smaller diameter of 0.065 in., with a very small radius at the step in the shaft. The failures occurred at the step in the shaft. There were some questions as to just how the machining operation was being performed. If a lathe with a single cutter was being used, there was a possibility that the cutting action on a small diameter could generate a bending stress high enough to initiate a small crack. This crack

376

VIBRATION FIXTURES A N D VIBRATION TESTING

might then propagate during the vibration acceptance test, producing the observed series of failures. A visit was made to the manufacturing area and a discussion was held with the shop supervisor. The shop supervisor was very upset by the idea that he would allow such a machining operation to be performed in his machine shop. The supervisor claimed that he had more knowledge of fatigue fractures than most of the engineers working on the program. He insisted that all of his workers were instructed to use three cutters on the shaft, spaced at 120" apart so there would be no possibility of any crack initiation in the shaft. His parting words were that the engineers had a lot of nerve to blame their poor designs on his careless manufacturing methods. The engineering dynamic loads and stresses were checked and rechecked with the same conclusion. The vibration level was not high enough to cause a shaft failure. Another visit was made to the manufacturing area, but instead of talking to the supervisor again the individual machining areas were inspected. Sure enough, in one of the areas it was obvious that the machinist was using only a single cutter on the shaft. The supervisor was then called over to examine the machining setup. His face turned white when he saw that only one cutter was being used to machine the shaft. When he asked the machinist why a single cutter was being used when the use of three cutters was standard practice, the machinist said that it took too long to set up and cut with three cutters so the daily quota could not be met. The single cutter was much faster, and it was obvious to the machinist that the single cutter did the job just as well as the three cutters. The above circumstances are a classic case of giving instructions with no follow-up to make sure the instructions are being followed. Some supervisors are too lazy to follow up on their instructions while other supervisors feel embarrassed when they have to verify that their instructions are being followed. In any case these people will always assume that they know what is going on in their areas, when they really do not know. They often pass on incorrect information in critical investigations, which makes it very difficult to get good data that can be used to make intelligent decisions. 14.24 ALLEGED VIBRATION FAILURE THAT WAS REALLY CAUSED BY DROPPING A LARGE CHASSIS

A group of four mechanical engineers sought out the advice of a vibration specialist regarding what they called a vibration failure in a large expensive inertial navigation cast aluminum chassis. The 0.25-in. thick casting had a deep V shaped crack where each leg of the V shape was about 2 in. long. The crack penetration was about 0.50 in. deep into the chassis so the thick walls of the chassis were clearly visible in both legs of the V crack. The vibration specialist took one look at the deep concentrated crack penetration in the thick casting and declared that the crack was not a vibration failure. The crack was caused by a high impact, probably from dropping the chassis onto a

METHODS

FOR INCREASING THE VIBRATION AND SHOCK CAPABILITY

377

hard floor, The four engineers insisted that they had proof that the chassis was not dropped. When they were asked what their proof was, they replied that they asked everyone on the program if the chassis was dropped and everyone said no. Since the chassis had just completed a vibration test, they concluded that the only possible cause of the crack had to be the vibration. The vibration specialist suggested that no one in their right mind would ever admit to dropping a chassis that was worth at least $150,000. The sad part of this incident is that four graduate mechanical engineers could not tell the difference between an obvious impact crack and a vibration-induced crack. The lesson learned in this incident is that people will lie when they are afraid of losing their jobs. Again it shows that it is often very difficult to obtain the correct information concerning failures, especially when people are afraid they may lose their jobs.

14.25 METHODS FOR INCREASING THE VIBRATION AND SHOCK CAPABILITY ON EXISTING SYSTEMS

Military procurement agencies recently decided to scrap many popular documents that related to the procurement of military electronic equipment because of increased expenses. They have turned to best commercial practice electronic equipment to reduce these costs. Commercial electronic equipment does not have the required high-temperature reliability normally needed for military programs. This means that better cooling must be supplied for the commercial equipment when it is used in military programs. This can often be achieved by using a more powerful fan for improved cooling, or by adding larger heat exchanges, or by adding larger convection cooling fins. Improving the vibration and shock capability on a commercial program, for possible use in a military program, will usually require some structural adjustments to existing PCBs and to existing chassis enclosures. The PCBs can often be made more vibration and shock resistant by semi-encapsulating the PCBs with a semisoft silicone or some type of foam. This will increase the fatigue life but it will make repairs very difficult. Snubbers have been used successfully in converting commercial hardware for use in military programs. Snubbers work by striking each other when the natural frequency of the PCB is excited. This impacting action significantly reduces the dynamic displacement, thus improving the fatigue life. Epoxy glass rods about 0.25 in. in diameter work very well when they are cemented near the centers of adjacent PCBs with a small clearance between each pair of snubbers. It may be difficult to find good open spaces between the components near the center of the PCB. Good snubber locations can often be found by making transparent copies of each side of each PCB. These can then be grouped together in the proper sequence and held up to a light to find the best locations for snubber pairs.

378


One quick-fix approach that has been used to improve the vibration and shock capability of an electronic chassis full of plug-in PCBs is to fill the chassis with small rubber balls or small hollow lightweight spheres similar to table tennis (Ping-Pong) balls. When the chassis is full of these spheres the PCBs cannot deflect very much so the vibration performance and the shock performance are substantially improved. This system will work very well in a chassis that is cooled by conduction from the PCBs to the side walls of the chassis. This system is also very easy to repair. The chassis cover is unbolted and the balls are spilled out and saved. This gives free access to the PCBs for any required maintenance. The balls are then replaced and the cover is replaced to complete the maintenance or repair. Sometimes the chassis structure itself requires some reinforcement. This can often be done with the screw-and-glue method. Reinforcing angles or doubler side panels can be glued to critical sections of the chassis, adding small screws if necessary for electrical continuity or for extra reinforcement. These types of quick fixes work very well when only a few systems have to be reinforced. When several hundred systems have to be modified, it may be cheaper to redesign the entire system for the new applications.

Environmental Stress Screening for Electronic Equipment (ESSEE)

15.1

INTRODUCTION

A couple of decades ago manufacturers of electronic equipment used burn-in procedures to eliminate defective electronic components. New electronic components were placed in a chamber set at some elevated temperature between 55°C and 71"C, and electrically operated for several days. The components that survived this process were used in the electronic systems that were sold to the consumers. This practice eliminated many weak components, which resulted in a more reliable product. As the electronic products became more complex and more sophisticated, the burn-in procedures did not work as well and the failure rates began to increase once again. Some new screening processes and procedures were required. A 2.2-G peak sinusoidal vibration stimulus was added to the high-temperature soak condition, using a low input frequency that did not excite any natural frequencies within the electronic system being screened. Eventually this procedure added broadband random vibration and temperature cycling to further improve the screening effectiveness. This vibration and temperature screen was often called a shake-and-bake test. This upset many people who insisted that the screen was an extension of the manufacturing process, and not a test. The argument given was that failures are not desired in a test, but failures are desired in the screen. The purpose of the screening is to get rid of the infant mortality groups of early failures, as shown in Fig. 15.1. The latent manufacturing defects are precipitated into hard electronic failures so they can be repaired at the factory, before the product is delivered to the consumer. The consumer is very happy with the more reliable product. The news of the good quality product spreads rapidly so more are sold and everyone is happy.

15.2

ENVIRONMENTAL STRESS SCREENING PHILOSOPHY

Environmental stress screening can be defined in many different ways, depending on the objectives of the program. One simple definition relates to 379

380

ENVIRONMENTAL STRESS SCREENING FOR ELECTRONIC EQUIPMENT

Time

FIGURE 15.1. Bathtub reliability curve showing infant mortality, useful life, and

wear-out.

its purpose, which is to provide a cost-effective method for improving the reliability of the electronic equipment. Another popular definition states that the purpose of the screen is to stimulate the electronics in some way that will force hidden flaws into hard failures, so the product can be repaired at the factory before it is shipped, so the consumer receives a reliable product. The idea behind the screen is that hidden defects and flaws are related to the manufacturing and the assembly processes and procedures, and not to the basic design of the product. However, experience with various types of screens has shown that they can reveal poor manufacturing practices as well as poor design practices. A careful examination of the screening data will often reveal failure patterns on component parts: improper processing procedures may show up, circuit design errors may become obvious, or careless manufacturing methods may be revealed. A large number of failures during the screening process, or very few failures during the screen, may be a clue that the screening process itself is deficient. A very effective document on the stress screening of electronic equipment can be found in the Navy’s NAVMAT-P-9492, issued in May 1979 under the title of Navy Manufacturing Screening Program [471. The final objective of the screening process is to eliminate its requirement. The screening process should eventually enable the engineering and manufacturing groups to constantly improve their techniques. The constant improvement process in the design, the hardware, and the manufacturing quality should eventually reach a point where a high quality is achieved with a low cost, so the screening is no longer required. If there is no continuous improvement in the hardware quality over time. it should become obvious that the screening process is not effective and requires a change, otherwise it is a never-ending exercise that ends up costing the company extra money.

SCREENING ENVIRONMENTS

381

15.3 SCREENING ENVIRONMENTS

The most common screens are usually based on the actual operating environments. The most popular screens are thermal cycling and some form of vibration. Experience has shown that some form of thermal cycling screen and vibration screen can be effective for revealing manufacturing flaws. Screens are also effective for electronics that are designed to work in very quiet and stable air-conditioned offices, where thermal cycling and vibration conditions are never expected. Military electronics experience has shown that thermal cycling screens are more effective than vibration screens in revealing manufacturing flaws. However, the use of both types of screens is strongly recommended for improved results. These can be done separately, or they can be combined and done simultaneously, electrically operating or nonelectrically operating. It is easier to observe the failures when the system is operating electrically. However, this screen method is more expensive. A few successful screens have used shock pulses and some screens have used acoustic noise. Time is money, so there is a strong desire to reduce the screening time to save money. Some early thermal cycling screens used thermal shocking to reduce the time it takes to ramp the temperature up and down. Two different chambers were set next to each other, either vertically or horizontally. One chamber was set at + 100°C and the other chamber was set at - 40°C. A piston was used to carry the electronic unit from chamber to chamber in about 5 seconds. The dwell time at each extreme temperature was based on the size of the electronic unit. This method was scrapped. Many more failures were produced in the screen than were ever experienced in the field. The ramp rate for the temperature cycle was investigated and tested in an attempt to find a rate that provided good reliability results in the field with an acceptable cost basis. Ramp rates from 2°C per minute to 30°C per minute were investigated. The ramp rate favored was 20°C per minute for seven thermal cycles from -55°C to +71"C, with power on during the increasing temperature cycle and power off during the decreasing temperature cycle. A few companies have found their screens work best with a temperature ramp rate of 2°C per minute. Most companies appear to be using ramp rates between about 10°C and 20°C per minute. The larger, more massive electronic components and the PCB itself should use thermocouples to find the approximate time it takes to reach a stable condition at the high and low temperatures. This information will be helpful in establishing the dwell time at the high and low temperatures. Different forms of vibration exposure have been very successful in finding latent defects in electronic systems. Sinusoidal vibration sweeping from 20 Hz to 2000 Hz to 20 Hz at a rate of half an octave per minute has some benefits. Broadband random vibration for several minutes covering a frequency range from 20 to 2000 Hz appears to be the most effective form of vibration based on the successful screening data obtained from many different companies. Vibration is usually applied for a period of about 5 minutes in each of three

382


major axes, for a total of 15 minutes. When all the PCBs are installed in the same direction, then the vibration is typically applied for 10 minutes along the axis perpendicular to the plane of the PCBs [501. When the PCBs are installed in two different directions, then simultaneous shaking along two different axes perpendicular to the PCBs is often used. The typical installation uses two different shakers with an oil-film slider plate shaking in the horizontal plane set 90” to each other. This is an expensive operation. The lower-cost compressed-air shakers with piston-driven thumper vibrators shake along all three axes and also provide rotation about all three axes, simultaneously exciting six degrees of freedom. The thumpers usually have a provision for performing thermal cycling in combination with the vibration. This is a big advantage for screening. The disadvantage is that there are no peak-notch filters available to flatten out any high resonant transmissibility spikes that have the potential for damaging good electronic components. A single electrodynamic shaker excites only one axis at a time. This is a disadvantage. These shaker systems often have the peak-notch filters, which is an advantage. A popular random vibration profile is shown in Fig. 15.2. Military programs have been successful using an input acceleration level of about 6.0 G RMS on small electronic boxes, where the power spectral density (PSD) is 0.04 G’/Hz from 80 to 350 Hz. Small is defined here as having a maximum dimension of about 24 in. Random vibration can be very dangerous, especially on large boxes or cabinets, if it is not used properly. A good understanding of structural dynamics and cumulative fatigue damage is required to ensure a successful program. High vibration acceleration levels even for short periods of time can generate a large amount of damage. Broadband random vibration can be developed in several different ways. The most accurate control method is with an electrodynamic shaker, which is very expensive. Some electrodynamic vibration machines can use closed-loop tapes to perform a lower-cost random vibration. Several companies offer low-cost pneumatic piston thumper quasirandom vibration machines that are combined with thermal cycling capability packaged in a relatively small enclosure.

i

0.04 G2/Hz

20

80

3 50

2000

FIGURE 15.2. Proposed ESS random vibration PSD profile from NAVMAT-P-9492.

383

THINGS AN ACCEPTABLE SCREEN ARE NOT EXPECTED TO DO

Random vibration levels of 6.0 G RMS can be very damaging on large cabinets. This can be demonstrated by comparing the U.S. Navy shock requirement MIL-S-901 and vibration MIL-STD-I 67 requirements for a 6-ft cabinet weighing 500 lb. The Navy uses a 3000-lb swinging hammer to impact a fixture that supports the cabinet for a shock test. This produces an input shock level of about 150 G. The electronic equipment inside the cabinet usually sees a level of about 200 G. These cabinets almost always pass this shock test. However, the same cabinet very seldom passes the 1.0-G peak input sinusoidal vibration test for a period of about 1 minute, so a waiver is often requested and granted. This means that a 1.0-G peak input sine vibration test for 1 minute generates more damage than a 150-G peak input shock test. The purpose of this comparison is to demonstrate that vibration can excite natural frequencies, which can be very damaging to large structures. A good preliminaly rule to follow for a random vibration screen is that the longest dimension ( L ) of an electronic box, in feet, multiplied by the G RMS input level should not exceed a value of about 6.0, as follows:

( L ) x ( G RMS) 5 6.0

(15.1)

This means that a 6-ft tall electronic cabinet should not be exposed to a random vibration screening level of more than about 1.0 G RMS without extensive proof-testing. A 1.0-ft long electronic box should not be exposed to a screening level of more than about 6.0 G RMS without extensive prooftesting. A 3.0-ft electronic box should not be exposed to a random vibration screening level of more than about 2.0 G RMS without extensive prooftesting.

15.4 THINGS AN ACCEPTABLE SCREEN ARE EXPECTED TO DO

1. Provide a cost-effective screening process. 2. Prescipitate hidden flaws rapidly so they can be detected. 3. Not use up too much life of the system. 4. Not exceed the design capability of the system. 5. Precipitate a large percentage of the hidden flaws. 6. Increase the reliability of the hardware in the field. 7. Provide some means for measuring the effectiveness of the screen. 15.5 THINGS AN ACCEPTABLE SCREEN ARE NOT EXPECTED TO DO

1. Generate high damage levels that cause good parts to fail. 2. Generate high damage levels that produce additional flaws.

384


3. Become the basis for establishing the design requirements. 4. Try to use the measured operating environments for the screen. 5 . Use up more than about 5 1 0 % of the effective fatigue life.

15.6 TO SCREEN OR NOT TO SCREEN, THAT IS THE PROBLEM

Many companies do not believe that the screening process is effective in improving the reliability of their products in the field. In fact, they have data that show that the screening process actually reduces the reliability of their products in the field. An equally large number of companies believe the screening process substantially improves the reliability of their products in the field, and they have actual case histories that support their positions. Both groups are right in their conclusions. The success of any screening program must be tailored to the specific requirements for that program. Every company wants to save money. Some companies treat a screening program like a pair of stretch socks, where one size fits all. Their sister company across the river spent a lot of time and money establishing their screen, which appears to be very successful. Why not simply use their successful screen and save a lot of time and money? This screen is implemented and it turns out to be a disaster. This is strong evidence in the minds of the people responsible for the poor screen that screens do not work. There is strong evidence from hundreds of companies, however, that screening does work [48].

15.7 PREPARATIONS PRIOR TO THE START OF A SCREENING PROGRAM

Electronic products are often like children, where everyone has a different personality. When they are not treated properly they put up a fuss and they misbehave. There must be an understanding of the character of the child or the character of the product, in order to come up with recommendations for an effective screen. The physical properties of the electronic hardware should be known before any screening program is implemented. This involves an understanding of the thermal, dynamic, and fatigue properties of the individual electronic components and materials used in the system. This is usually done in two ways, by analysis and by test. Analysis alone may not be accurate enough to establish an effective screen, depending on the knowledge and experience of the analyst. Testing alone may not be enough, depending again on the knowledge and experience of the testing personnel. Another factor here is the type of equipment that is available. Temperature cycling chambers, automatic temperature recorders, electrodynamic vibration machines with random capability, oil-film slider plates, many small accelerometers,

PREPARATIONS PRIOR TO THE START OF A SCREENING PROGRAM

385

many recording channels, and strobe lights are required to perform an adequate screen. All this equipment is very expensive. Many small organizations do not have the funds available. Some may already have vibration machines with sinusoidal vibration capability only. Or they may already have the lower-cost pneumatic thumper type of shakers. Companies with no thermal or vibration capability often use outside vendor subcontractors with these facilities to perform their thermal and vibration screens. Vibration screens can be very dangerous and can cause the most problems if they are not carefully controlled. The natural frequencies and transmissibility Q values of various structural elements can be obtained by making a sinusoidal vibration survey. Real hardware or mock-up hardware can be used but it must be well instrumented with small accelerometers. An input level of about 2.0 G peak should be used to obtain good test data. A sweep rate of about 1 octave per minute should be used, from 10 Hz to 2000 Hz back to 10 Hz. The sweep should be made in both directions, up and down, to make sure they look similar. If the Q plots are sharply different, it is a sign of a nonlinear system. The higher Q values should be used to evaluate the random vibration response. This type of information must be known or obtained before a good screen can be recommended. Another piece of information that is very valuable to have before the screen is recommended is the fragility level of the electronic hardware. This can be obtained by analysis or by test. The best data are obtained by testing real hardware or mock-up hardware. The hardware must be well instrumented with small accelerometers and tested with increasing vibration acceleration G levels, using sinusoidal resonant dwells or broadband random vibration, until a failure occurs. If the system is not working electrically, it may be difficult to know if or when a failure has occurred. The test may have to be stopped every few minutes to check various cables, connectors, lead wires, and solder joints for failures. Plug-in types PCBs have high transmissibility Q values at their center with high failure rates, so these areas should be checked carefully, especially if there are large components surface mounted or through-hole mounted near the center of the PCB. When a failure is discovered it should be recorded, the test should be stopped, and repairs should be made. The test should then be continued using increasing vibration acceleration G levels until another failure occurs. This process of increasing testing G levels, failure, repair, and retest should be repeated until it is obvious that most of the life has been used up [49]. Some companies with random vibration capability in electrodynamic shakers use the notching filter capabilities to reduce the effective transmissibility Q levels at the centers of the most critical plug-in PCBs from values of about 18 down to values of about 2. This reduces the large differences in the acceleration levels developed at the center of the PCB compared to the edges of the PCB. This process flattens the response acceleration levels across the face of the PCB, resulting in a better screen, according to some ESS personnel. The other side of this coin is that the real world does not have

386


peak-notch filters, so the centers of the plug-in PCBs in the real world will always have higher acceleration levels than the edges of the PCBs. But this is another area where there are many opposing opinions as to whether the actual operating environment should have any influence on the screening methods proposed. It would be nice if the budget could support a repeat of the vibration fragility tests on a couple of different samples because the failures are always different on different samples. This is due to the differences in material properties and differences in the manufacturing tolerances on the physical dimensions of the parts being tested. See Chapter 13, Section 13.8, for more on the sensitivity analysis, which shows how small tolerance differences can turn into big differences in the fatigue life in the lead wires and solder joints. Engineers should be thankful if they are allowed the luxury of even one prototype test model. The attitude in many organizations is that prototype test models are no longer necessary because all of the required information is in their computers. So, when a test model is available, it is important to make careful notes of the physical dimension for future use to help define internal forces and stresses as well as the required tolerances on manufactured parts. Prototype or production hardware is also recommended for performing thermal cycling tests to understand the failure mechanisms and the fatigue properties of the lead wires and solder joints for different thermal cycling rates and different maximum and minimum temperatures. Solder joint failures in thermal cycling tests are not as dramatic as in vibration tests. Thermal cycling damage is accumulated very slowly while vibration damage is accumulated very fast. Good test results can be obtained using mock-up solder joints that are produced by the same methods that will be used in the production hardware, The approximate number of thermal cycles desired in a screening program is often based on the number of component parts in the system as shown in Fig. 15.3. This shows only about 8 to 10 thermal cycles are adequate for screening a system with 4000 component parts. Research shows that many military suppliers use 10 to 20 thermal cycles for large complex electronic assemblies and 20 to 40 thermal cycles for individual electronic component parts. A very successful solder joint fragility test, or proof of life test, was developed for one of the new Air Force fighter electronic systems. This test was run on a series of PCBs with a variety of surface mounted and throughhole mounted components. The purpose of these tests was to prove that the PCB design and manufacturing methods could provide a service life with at least a 10,000-hour failure-free operating period (FFOP). The system was designed to provide a FFOP of 20,000 hours, which means that a safety factor of 2.0 was used on the design life to ensure it could meet the original operational life requirements. The manufacturing group used the production vapor phase soldering methods to assemble the test PCBs, so the test results would accurately reflect production methods. A total of 1000 thermal cycles were used in the test. Two different test groups were used with 500 thermal

THERMAL CYCLING, RANDOM VIBRATION, ELECTRICAL OPERATION

W

4-

P

387

o"T----7

0.6

Complexity of equipment

0.4

0.1 1

2

3

4

5

7

6

8

9 1 0 1 1 1 2

Temperature cycles

FIGURE 15.3. Proposed number of thermal cycles based on the complexity of the

equipment.

cycles each. The first group was 500 cycles from -50°C to +50"C and the second group was 500 cycles from 0°C to 100°C. This covered the complete temperature range of 150°C without extreme temperature changes. Both groups used a temperature ramp rate of 20°C per minute, and a dwell period of 30 minutes at each extreme temperature. The units were not working electrically. The definition of a solder joint failure was any visible crack using a 35 power microscope. It was not possible to examine the solder joints while they were in the temperature cycling ovens. Every 25 cycles that test PCBs were removed from the oven at room temperature and examined with a microscope. Several people were used to perform the visual examinations because after about 150 thermal cycles the solder becomes wavy. It is very difficult to distinguish the difference between a crack and a wave with a shadow, because a shadow on a wave looks like a crack. Tests were instrumented with thermocouples to record the temperature change rate for different critical components and PCBs. This information was used as a guide for selecting the temperature cycling rates and peak temperature for the proposed thermal cycling screening program.

+

15.8 COMBINED THERMAL CYCLING, RANDOM VIBRATION, AND ELECTRICAL OPERATION

Experience and research on stress screening show that combining random vibration, thermal cycling, and electrical operation all at the same time precipitates the most latent defects, but at a high relative cost. The pneumatic thumper piston shakers, with their integral thermal cycling enclosures, are very convenient for this type of screen. The equipment cost is low but the

388


time and the personnel requirement costs are high. Many different technical groups are required to coordinate, control, and record the screening results under these conditions. The typical screen starts with an electrical checkout of the system. This process very seldom goes smoothly due to problems with the large electrical control consoles usually required to operate the system electrically. Moving large electronic control consoles across rough floors to screening rooms in remote areas often causes internal electrical problems due to vibration that can shake wires loose, loosen screws, and disengage electrical connectors. The electrical engineers that oversee and monitor the electrical system very seldom know anything about the design of the test control consoles. These are two different functions that are normally performed by two different groups. Everyone associated with the screening then sits down and waits for the test control console engineer to show up to repair the test equipment. People become bored so they go back to their offices, or take a coffee break. Another two hours are lost trying to get the group together again. These are real problems that waste a lot of time and money. It is necessary to coordinate the work effort so these types of problems are solved in advance. This is another reason that many organizations choose not to operate their systems electrically, because of the problems associated with this function. The normal screening procedure using combined thermal cycling and vibration environments is to run about half of the required thermal cycles with no vibration. This is followed by 10 to 15 minutes of combined random vibration with the thermal cycling, then finishing up the remaining thermal cycles with no vibration. The combined thermal cycling and vibration is easy to run when the environmental screening system is located in a single pneumatic piston thumper enclosure. The random vibration is often split up so some of it is imposed at the high-temperature part of the thermal cycle and some of it is imposed at the low-temperature part of the thermal cycle. The combined thermal cycle and vibration screen is more time consuming to run and more expensive with an electrodynamic shaker system because the screening sample has to be installed, removed, and installed again. The electronic system screen will still start with about half of the required number of thermal cycles run in a special thermal chamber. The electronic system is then removed from the thermal cycling chamber and mounted on the vibration shaker. A special plywood box is often placed over the electronic assembly. This box has hoses attached to a local or to a portable thermal cycling chamber. The ambient air around the electronic system can then be temperature cycled while the electronic assembly is exposed to the random vibration at the same time for 10 to 15 minutes. The electronic assembly is then removed from the vibration shaker and placed back into the thermal cycling chamber to complete the remaining thermal cycles. Everyone prays that the electronic system will not be dropped and damaged during the process of installing, removing, and transporting it from one laboratory to another for the various parts of the screen. This has happened before and it will probably happen again somewhere. Several companies manufacture

IMPORTANCE OF THE SCREENING ENVIRONMENT SEQUENCE

389

temperature cycling chambers that sit on the top of an electrodynamic vibration machine for combining vibration with thermal cycling. These assemblies usually permit the vibration machine to move up and down only along the vertical axis. These combination chambers are not normally used for vibration along the horizontal axis, where an oil-film slider plate must be used for a large electronic system. It is easy to see why very few companies want to bother with combining thermal cycling, vibration, and electrical operation at the same time. Even when screening data show this method provides the best results, the complex procedure, the personnel required, and the time involved to achieve these reliability improvements are often not cost effective. 15.9 SEPARATE THERMAL CYCLING, RANDOM VIBRATION, AND ELECTRICAL OPERATION

Separate screening environments with thermal cycling, random vibration, and electrical checkout are the choice of most organizations. Separate screening environments may not be as good as combined environments, but they are much easier and much less expensive to implement. The normal procedure is to check the system first electrically to verify that it is working properly. The system is then subjected to about half of the required thermal cycles not electrically operating. The system is checked electrically after the initial thermal cycling to again verify that it is working properly. Any intermittent failures that occurred in the initial thermal cycling phase will probably not be noticed at this point. It is strongly recommended that during the electrical checkout after thermal screening that the system be dynamically excited somehow, to possibly reveal any intermittent failures that do not show up in a static condition. The easiest type of dynamic excitement that can be used is to rapidly and firmly rap the electronic system in different areas with the eraser part of an ordinary pencil, while the system is being checked electrically. This may not be a well-controlled dynamic condition, but it often reveals intermittent failures. The next phase involves the random vibration exposure for 10 to 15 minutes along one or more axes. Many companies choose to run this phase without any electrical operation. However, this is a very good place to electrically operate the system to watch for any intermittent failures. Hard failures can be observed under any static electrical checkout, but intermittent failures normally require some type of dynamic stimulus to uncover a fault. The electronic system is now returned to complete the remaining thermal cycles without electrical operation. 15.10 IMPORTANCE OF THE SCREENING ENVIRONMENT SEQUENCE

The sequence of the screen does not appear to have a significant effect on the overall results, according to the data obtained from many organizations

390


involved in screening programs. It does not appear to matter a great deal if the thermal cycling is done first or last, or if the random vibration is done at the beginning, in the middle, or at the end of the screening program. Are there differences in the results when there are differences in the screening sequence? The answer is yes, but they are not dramatic. The important requirement is to impose some type of thermal cycling and some type of broadband random vibration. These two environments have demonstrated a very high success rate for rapidly uncovering latent defects in production electronic equipment. This applies to commercial, industrial, or military equipment. The number of thermal cycles, the temperature ramp rates, and the random vibration all have to be tailored to fit each type of electronic unit. This is not like a pair of stretch socks, where one size fits all. A screen for an individual component part is bound to be substantially different from a screen for a large sophisticated electronic assembly. The success or failure of any screening program is decided by the failure rate of the production units in the field. A high field failure rate is usually an indication of a poor screening program. It is also a sign that some changes must be made in the program to reduce the field failure rate. A low field failure rate is usually a sign of a good screening program. 15.11 HOW DAMAGE CAN BE DEVELOPED IN A THERMAL CYCLING SCREEN

Differences in the thermal coefficient of expansion (TCE) between the PCB and the various components mounted on the PCB can produce deflection differences during thermal cycling conditions. These deflection differences may result in high stresses in the component body, the lead wires, the solder joints in the lead wires, and the cables and harnesses. These factors depend on the geometry of the different types of leaded or leadless components that can be surface mounted or through-hole mounted on the PCB, the strain relief in the cables and harnesses, as well as the thermal cycling conditions. Solder can cause a lot of problems during thermal cycling conditions. This is due to its elastoplastic and creep properties at temperatures near 100°C. Displace a cantilevered bar of solder rapidly through an amplitude A at 100°C and assume it generates an initial stress of about 1000 Ib/in.*. Hold that amplitude A constant for awhile. The solder will start to creep and relieve the initial stress. In a period of about two hours the solder stress will be close to zero. Now bring the bar of solder rapidly back to its original starting position through the same amplitude A. The solder stress level will now be 1000 Ib/in.’ compared to the zero stress at the original starting position. If the rapid displacement of the solder bar is continued in the opposite direction through a negative displacement amplitude A , the solder stress will be 2000 lb/in.’. The creep and stress-relieving properties of solder in this case resulted in the doubling of the solder stress. When solder is

HOW DAMAGE CAN BE DEVELOPED IN A THERMAL CYCLING SCREEN

391

displaced and held in the displaced condition for long periods at high temperatures, the creep effect can relax the stresses down to a near-zero condition. When the solder is then displaced in the opposite direction through the same amplitude, the solder stress can be doubled. Higher stress levels will shorten the effective fatigue life of the solder. Solder has a short-term ultimate strength of about 6500 lb/in.* at room temperature. Extensive testing and analysis experience with solder creep has shown that it is a good practice to keep the solder joint stress down to a value of about 400 lb/in.’ to ensure a long failure-free operating life in thermal cycling conditions [54]. The increased solder stress conditions due to creep described above can occur during thermal cycling events at the high point of a long temperature dwell, when there is a temperature reversal of a similar magnitude. High solder joint stresses can reduce the expected fatigue life of the solder very rapidly. When creep in the solder joint can relax the original stress level 100% to a near-zero condition, a thermal cycling event can increase the original stress level in the solder 100%. When the creep in the solder joint can relax the original stress level 5096, a thermal cycling event can increase the original solder joint stress 50%. When the creep in the solder joint can relax the original stress level 25%, a thermal cycling event can increase the original solder joint stress 25%. When there is 0% creep in the solder joint and there is 0% stress relaxing, a thermal cycling event will increase the original solder joint stress 0%. The standard method for evaluating the fatigue life of a structure is to examine the single-amplitude stress compared to the number of cycles required to produce a failure. In a typical sine wave the single amplitude refers to the displacement from zero to the peak. This is the same as half of the double amplitude, which is half of the peak-to-peak amplitude. The same philosophy is applied to thermal cycling. The single-amplitude stress will then be related to the single-amplitude temperature cycle. This is half of the double amplitude or half of the peak-to-peak temperature cycle. These analysis methods are based on linear structures. But solder is not a linear material: it can creep extensively, especially at high temperatures above about 100°C. Therefore, the methods shown above are intended to provide a more accurate method for evaluating the approximate fatigue life of solder when some creep is included. There are many arguments related to the analysis methods for evaluating solder joint strains and stresses. Should the peak-to-peak temperature range be used in the analysis of a thermal cycling event or should half the peak-to-peak temperature cycle range be used for the analysis? In a rapid temperature cycle event, where the solder does not have a chance to creep, the solder should be treated like a linear material. In this case one-half of the peak-to-peak temperature cycle range should be used. In a condition where there is extensive creep in a thermal cycling event, the peak-to-peak temperature range should be used for the analysis. When there is no information

392


available as to the use of the electronic equipment, or when a conservative design is desired, the peak-to-peak temperature cycle range might be considered. If the information on solder creep presented above appears to be too complex, there is nothing wrong with performing a conservative analysis by using the double amplitude or the peak-to-peak temperature cycling range. This is equivalent to using a safety factor of 2.0 on a rapid temperature cycle where little or no creep will occur in the solder joints. The fatigue exponent (b) for solder in thermal cycling conditions is about 2.5. When a safety factor of 2.0 is used in the fatigue life calculation, it will reduce the calculated fatigue life by a factor of about 5.6. 15.12 ESTIMATING THE AMOUNT OF FATIGUE LIFE USED UP IN A RANDOM VIBRATION SCREEN

Random vibration for a period of about 10 minutes along one axis has proved to be an effective screening tool for locating latent defects. It is not as effective as imposing several thermal cycles on the electronic assembly, but it does help to find additional potential failures. Random vibration is effective because it is broadband so it can excite many different natural frequencies and higher harmonics at the same time in the chassis support structure, in the various internal PCBs, and in the interconnecting cables and harnesses. Natural frequencies of small (less than 30 in.) electronic chassis with PCBs typically have natural frequencies over 100 Hz. This means that every second of vibration the structure accumulates 100 stress reversals. Every stress reversal uses up some part of the fatigue life of the system. When all of the life is used up the system will fail. The most critical structural elements will fail first. A chain is only as strong as its weakest link, so the weakest links will fail first. It is impossible to estimate how much fatigue life is being used up in a vibration screen, without knowing the actual fatigue life of the weakest link of the system. Once the approximate fatigue life of the weakest link is a particular electronic design has been established, the question becomes how much of that fatigue life should be used up to obtain a cost-effective screen? This is a tough question to answer, because everyone seems to have a different idea or feeling in this area. Some people feel it is acceptable to use up about 5% of the fatigue life of an electronic system to achieve an acceptable screen. They may be reminded that this situation is similar to the purchase of a new car with a 100,000-mile warranty. Their new car has 5000 miles showing on the odometer, which was required to achieve the 100,000-mile warranty. This represents only about 5% of life used up, but the idea of a new car with 5000 miles now appears to be different because the idea is unacceptable. Surveys made among engineers on the subject of how much fatigue life should be used up in an acceptable

ESTIMATING FATIGUE LIFE USED UP IN A RANDOM VIBRATION SCREEN

393

screen, varied from zero fatigue life used up to a maximum of 10% of the fatigue life used up. Another question that comes up concerns the disposition of an electronic system that keeps failing in successive screens, until about 25% of the life is used up. Should this system be shipped? Should the customer be informed that 25% of the life was used up? What happens if the customer rejects the system, should it be scrapped? The answers to these questions depend on the price of the electronic system. If the system cost is less than about $50, it will be scrapped. If the system cost is more than about $1000 it will be shipped even if an analysis shows that about 50% of the life has been used up. This is really a very simple problem. The typical procedure for the expensive electronic system will be a meeting with the company vice president, who happens to be an accountant or an attorney or a business major who has a good knowledge of business but not engineering. The vice president will arrange a meeting with the engineer who calculated the 25% or 50% life number and request a copy of the calculations. The meeting will be very short. The vice president will glance at the detailed analysis and quickly thumb through the pages pretending to understand what is being read. After less than one minute, the vice president will challenge the analysis. The engineer will be asked to positively guarantee the accuracy of the analysis on that one system without any additional testing. Fatigue has such a wide amount of scatter that it is impossible to calculate the precise amount of fatigue life used up in one particular system. It is possible to estimate the fatigue life probability on a statistical basis when many systems are involved, but not on a single system. The engineer has to answer that the accuracy cannot positively be guaranteed on that single system without any additional testing. The vice president will claim the engineer’s analysis is flawed and the engineer will be directed to have the system shipped. This has happened many times and will happen again and again because no one want to lose any money when hard evidence is lacking. Two giant aerospace companies were recently involved in an ESS dispute where a $150,000 electronic assembly failed the random vibration ESS environment about 20 times. It finally passes and was shipped to the customer who rejected the unit because it had too much life used up. The system manufacturer claimed that an analysis showed there was enough life left to complete the equivalent of about five missions. The customer did not trust the analysis and demanded more proof that the system had a very high probability of performing its required mission. The manufacturer was not able to provide any dat that would satisfy the customer. There was a standoff and neither side would give in. Finally, the decision was made to split the cost of the unit equally between the two companies and to impose the operational random vibration environment continuously, with the unit operating electrically, until it failed. Either by luck or by good engineering calculations, the fatigue life test showed the system did have enough life left

394


to complete five full missions. Both companies agreed to use the results of this destructive test to settle any future problems associated with excessive ESS failures. Many companies try to avoid the problem of using up too much fatigue life by reducing the vibration and thermal cycling exposure after each failureand-repair cycle. The vibration acceleration G RMS level may be reduced 50% or the vibration duration may be reduced 50% after each failure-and-repair cycle. This reduction ratio is often applied to the number of thermal cycles required after each failure-and-repair cycle. Methods for calculating the approximate vibration fatigue life can be obtained from a database developed from a combination of extensive vibration tests and finite element methods (FEMs). Several prototype models should be used to run vibration tests with increasing acceleration levels until failures occur. One model is not enough because there is so much scatter to the fatigue test data that substantial errors could occur. The test units should be fabricated by the manufacturing group, using production-type manufacturing methods. These prototype models must be carefully instrumented with many small accelerometers properly placed in critical areas. Sinusoidal vibration sweeps should be made first to find the natural frequencies and the transmissibility Q values in critical areas. A strobe light should be used to examine the mode shapes at the various natural frequencies to obtain a better understanding of the failure mechanisms. The next step is to run broadband random vibration using a white noise (flat-top PSD) starting with an input level of about 5 G RMS. Hold this level for about 20 minutes. If there are no failures, increase the input level in increments of 2 G RMS and hold each level for about 20 minutes until a failure is observed. Record, repair, and photograph each failure and continue increasing the G RMS until there is virtually no more life left in the system. Perform a computergenerated structural FEM analysis of the system using the observed and recorded average test fatigue life for various structural members to obtain physical constants from the FEM analysis. This type of data can be used to calculate the approximate fatigue life of similar structural members in other electronic systems subjected to similar random vibration conditions. The methods for evaluating the fatigue life characteristics of an electronic assembly described above will work well. The problem will be to convince the program manager or the vice president of engineering to provide the funds, the personnel, and the time to run such a series of tests. The typical answer to the request for funds to run such a program is a flat no. The reason given is that the company has provided the engineers with millions of dollars worth of fast new computers with software to perform these tasks so building and testing prototype models is no longer necessary. After spending all that money, many company executives do not want to hear any requests for prototype model building and testing because, in their minds, it would mean that the money was wasted.


395

Experience has shown that the PCBs are the most critical parts of the assembly, so they will often fail first. Before any type of vibration or shock dynamic analysis can be made, or before any ESS program can be recommended, it is important to know how the chassis assembly was designed. Random vibration is broadband so it can excite every natural frequency of the chassis and the PCBs within the frequency band. This means that it may be possible for a chassis resonance to couple with and amplify a PCB resonance, which can result in a very rapid PCB failure. The dynamic coupling properties between the chassis and the various PCBs must be known before an individual PCB dynamic analysis can be made. However, if the octave rule has been followed, the dynamic coupling between the chassis and the PCBs will be sharply reduced, so the coupling effects can be ignored with very little error. Octave means to double. The natural frequency of the chassis and the PCB must be separated by a ratio of more than 2 : 1. In addition, the weight of any one PCB must be less than 10% of the weight of the chassis. With these two conditions, the dynamic coupling between the chassis and the PCBs will be small. Therefore, the dynamic input to the PCBs will be approximately the same as the dynamic input to the chassis. Sample Problem-Fatigue Thermal Cycling Screen

Life Used Up in a Vibration and

Find the approximate amount of fatigue life used up in a random vibration and thermal cycling screen of an electronic system that has several 5.0 X 7.0 X 0.070-in. thick plug-in epoxy fiberglass PCBs mounted in a chassis. The proposed vibration screen is 5.0 minutes per axis for three axes, for a total of 15 minutes with a white-noise PSD input of 0.060 G2/Hz. The octave rule was followed so the dynamic input to the PCBs will be about the same as the input to the chassis. The proposed thermal cycling screen is 15 cycles from 88°C to - 36°C at a rate of 20°C per minute with a 1.0-hour dwell period at each temperature extreme. One of the most critical components for vibration and thermal cycling is expected to be a 1.2-in. long hybrid shown in Fig. 15.4. The component is mounted at the center of the PCB parallel to the 5.0-in. edge. The lead wires extend from the bottom surface of the component, which can be through-hole or surface mounted on the PCB.

+

Solution A Fatigue Life Used Up in the Raridom Vibration Screen. The natural frequency needed for the PCB to achieve a fatigue life of about 20 million stress cycles in the lead wires and solder joints can be obtained from Chapter 9 random vibration as shown below:

fd =

[

2 9 . 4 C h r d m 0.00022 B

)'"

(15.2)

396


-I

l-

I

0.070in

PCB FIGURE 15.4. Differences in the PCB and component thermal coefficients of expansion (TCE) force the lead wires to bend in thermal cycling environments.

where C = 1.26 (component type lead wires extending from bottom surface) h = 0.070 in. (PCB thickness) Y = 1.0 (relative position factor, for component mounted at center of PCB) P = 0.060 G’/Hz (PSD input to PCB) L = 1.20 in. (length of critical hybrid component) B = 5.0 in. (length of PCB edge parallel to length of critical component)

/ (29.4)(1.26)(0.070)(1.0)J(

~ / 2 ) ( 0 . 0 6 ) (1.

The approximate fatigue life of the component lead wires and solder joints will be about Life

20 x 10‘ cycles to fail =

(208 cycles/s) (3600 s/h)

= 26.7

hours to fail

(15.4)

A 15-minute (0.25-hour) random vibration screen will use up the following amount of life:

Life used up

=

0.25 hour 26.7 hours

= 0.0094 = 0.94%

(15.5)

The random vibration screen will use up about 5 % of the life if it is imposed five times.


397

Solution 8 Thermal Cycling Screen Effects on the Lead Wires. Thermal expansion differences X between the component and the PCB will take place with respect to centroid of the component. The maximum distance between the lead wires will be the diagonal dimension of the component body. The temperature difference used to find the expansion difference will be from the middle (or neutral temperature) point to the maximum and minimum temperatures. This will be half of the peak-to-peak temperature range. Note that the expansion difference X determines the relative bending displacement and the bending stress in the lead wire. It also determines the shear stress in the solder joint when there are no solder creep effects. Solder normally will not creep during a very rapid thermal cycle. At a temperature of 125°C and an initial solder stress of 3000 1b/im2, the creep effects in solder will reduce the stress level about 50% in about 45 seconds. In a thermal cycling event, this can increase the effective joint stress by 50%. When the solder creep effects reduce the stress level 25%, a thermal cycling event can increase the effective solder joint stress by 25%. Solder creep can increase the effective solder joint stress and cause more rapid fatigue failures. The solder creep factor is based on the stress level, the temperature, and the dwell time at the high temperature. The subscripts P and C refer to the PCB and the component, respectively.

X = ( a P- a,-)L(A t ) where a p = 15.0 x aC = 6.0 x

id(

( 15.6)

in./in./"C (TCE of epoxy fiberglass in X-Y plane) in./in./"C (TCE of ceramic component body)

1.20)2+ (0.50)2 = 0.65 in. (diagonal body length from centroid to end) A t = +[(88) - (-3611 = 62°C (half the peak-to-peak temperature range) L

=

X = (15.0 - 6.0) = 3.627 X

X

10-6(0.65)(62)

l o w 4in.

(relative displacement)

(15.7)

The component lead wires are very compliant compared to the stiffness of the PCB and the electronic component body. Therefore, all of the deflection can be assumed to take place in the bending of the lead wires with very little error. The end lead wires will experience the most bending because they are the farthest away from the centroid. The length of the lead wire does not stop at the interface of the component body or at the interface of the PCB. FEM analysis and test data show the effective length of a wire in bending extends about one diameter into the component body and about one diameter into the PCB solder joint for a through-hole mounting condition. The bending deflection of the lead wire can be obtained by considering the wire

398


to be a uniform beam fixed at both ends, with lateral motion possible. The deflection equation can be obtained from a structural handbook as shown below:

(15 -8) where X = 3.627 X in. (see Eq. 15.7) E , = 20 X lo6 lb/in.2 (kovar wire modulus of elasticity) d, = 0.018 in. (wire diameter) i n 4 (wire moment I , = (~r/64)(d,)~ = ( ~ / 6 4 ) ( 0 . 0 1 8 )=~ 5.15 x of inertia) L , = 0.060 + 2(0.018) = 0.096 in. (effective length of wire in bending includes one wire diameter into component and one into the PCB)

P,

(12)(20 x 106)(5.15x 10-9)(3.627 x =

=

( 0.096)3

0.507 lb (15.9)

The bending moment in the wire can be obtained from the sum of the moments:

M,=--

P,L,

-

2

(0.507)(0.096) 0.0243 Ibsin. 2

(15.10)

The bending stress in the wire can be obtained from a structural handbook as follows:

su

M,C, ,=--

-

I,

(0.0243) (0.018/2) 5.15 x i o p 9

=

42,466 lb/in.*

(15.11)

The thermal cycle fatigue life of the kovar wire can be obtained from Fig. 3.1 using the fatigue equation. N,

=N,

[2

=

(1000)

[ 42,4661 84,000

= 78,691 wire

thermal cycles to fail (15.12)

The electronic system will never be able to accumulate this many thermal cycles in its life, so the probability of a lead wire failure is small, as long as there are no sharp and deep cuts or scratches in the wire to produce a high stress concentration.


399

Solder shear tearout

FIGURE 15.5. Shear tearout stress pattern in a through-hold solder joint produced by lead wire bending during thermal cycling.

Solution C Thermal Cycling Screen Effects on the Solder Joints. The overturning moment in the lead wire that forces it to bend will produce a shear tearout stress in the solder joint, as shown in Fig. 15.5. The magnitude of this stress can be obtained from the following [54]: (15.13) where M ,

lb .in. (overturning moment in solder joint; see Eq. 15.10) h , = 0.070 in. (solder joint height assumed to be the same as the PCB thickness) d, = 0.018 in. (wire diameter) d P T H= 0.030 in. (plated through-hole diameter) d,, = (d, + d,,, )/2 = (0.018 + 0.030)/2 = 0.024 in. (average solder shear area diameter) A , = (7r/4)(da, = ( ~ / 4 > ( 0 . 0 2 4 )=~4.52 X i n 2 (solder shear tearout area) = 0.0243

>’

SST =

0.0243 (0.070)(4.52 X lo-‘)

= 768

Ib/in.*

(nominal shear tearout stress) (15.14)

400


I

9-

!

a-

I

Solder

-

Initial stress, 6 89 MN/m2 (1000 PSI! I

I

!

1

The solder is expected to creep and strain relieve itself at a temperature of 88°C. The amount of creep after one hour is expected to be about 25% as shown in Fig. 15.6. The thermal cycling event is expected to increase the shear tearout stress level 25% as shown below. Adjusted S,,

=

1.25(768)

=

960 Ib/in.’

(15.15)

The approximate thermal cycle fatigue life of the solder joint can be obtained from Fig. 3.2 with the fatigue equation. N2 = N ,

(2] ’

=

(80.000)

[ 960 1 200

2.5

=

1585 solder thermal cycles to fail (15.16)

The amount of fatigue life that would be used up by 15 thermal cycles will be as follows: Life used up

=

15 cycles 1585 cycles to fail

= 0.0095 =

0.9572

(15.17)

The thermal cycling screen will use up about 5% of the life if it is imposed five times.

-

BIBLIOGRAPHY

1. Stephen H. Crandall, Random Vibration, Technology Press, Wiley, New York, 1958. 2. Charles E . Crede, Vibration and Shock Isolation, Wiley, New York, 1957.

3. Edward J. Lunney and Charles E. Crede, The Establishment of Vibration and

Shock Tests for Airborne Electronics, Wright Air Development Center, Technical Report 57-75, January 1958, ASTIA Document AD142349. 4. Arthur W. Leissa, Vibration of Plates, National Aeronuatics and Space Administration, 1969. 5. Norman E . Lee, Mechanical Engineering as Applied to Militaly Electronic Equipment, Coles Signal Laboratory, Red Bank, NJ, June 22, 1949. 6. Geoffrey W. A. Dumrner and Norman B. Griffin, Environmental Testing Tech-

niques for Electronics and Materials. 7. Charles E. Crede and Edward J. Lunney, Establishment of Vibration and Shock Tests for Missile Electronics as Derived from the Measured Environment, Wright Air Development Center Technical Report 56-503, December 1, 1956, ASTIA Document AD118133.

8. Rough road tests, performed at A.C. Spark Plug, Division of General Motors, Milwaukee, WI, July 15, 1959. 9. W. C. Stewart, “Determining Bolt Tension,” Mach. Des. Mug., November 1955. 10. R. W. Dicely and H. J. Long, “Torque Tension Charts for Selection and Application of Socket H e a d Cap Screws,” Mach. Des. Mag., September 5, 1957. 11. Kent’s Mechanical Engineer’s Handbook, Design and Production Volume, Wiley, New York, 1950. 12. B. Saelman, “Calculating Tearout Strength for Cantilevered Beams,” Much. Des. Mag., January 1954. 13. E. F. Bruhn, Analysis and Design of Aircraft Structures, Tristate Offset Co., Cincinnati, OH, 1952.

14. M. R. Achter, Effects of High Vacuum on Mechanical Properties, US. Naval Research Laboratory, Washington, DC, 1961. 15. McClintock and Argon, Mechanical Behavior of Materials, Addison-Wesley, Reading, MA, 1966. 16. A. M. Freudenthal, Fatigue in Aircraft Structures, Academic Press, New York, 1956. 17. Marks Handbook, McGraw-Hill, New York, 1951. 18. R. E . Peterson, Stress Concentration Design Factors, Wiley, New York, 1959. 19. R. J. Roark, Formulas for Stress and Strain, McGraw-Hill, New York, 1943.

401

402

BIBLIOGRAPHY

20. S. L. Hoyt, Metals and Alloys Data Book, Reinhold, New York, 1943. 21 C. Lipson and R. Juvinal, Handbook of Stress and Strength, Macmillan, New York, 1963. 22. H. J. Grover, S. A. Gordon, and L. R . Jackson, Fatigue of Metals and Structures, Bureau of Aeronautics, Department of the Navy, 1954. 23. M. A . Miner. “Cumulative Damage in Fatigue,” J . Appl. Mech., 12, September 1945. 24. Reynolds Metals Co., Structural Aluminum Design Handbook, 1968. 25. F. B. Stulen, H . N. Cummings, and W. C. Schulte, “ A Design Guide Preventing Fatigue Failures,”Mach. Des. Mag., Part 5. June 22, 1961. 26. F. R. Shanley, Strength of Materials, McGraw-Hill, New York, 1957. 27. MIL-Handbook-SA, Metallic Materials and Elements for Aerospace Vehicle Structures. Department of Defense, Washington, D C . 28. J. W. S. Rayleigh, The Theoy of Sound, Dover Publications, Mineola, N Y , 1945. 29. S. Timoshenko and S. W. Krieger. Theor?,of Plates and Shells, McGraw-Hill, New York, 1959. 30. R. W. Little, Master‘s Thesis, University of Wisconsin, 1959. 31. G . B. Warburton, “ T h e Vibrations of Rectangular Plates,” Proc. Insr. Mech. Eng., 168 (121, 1954.

32. F. J. Stanek, Uniformb Loaded Square Plate with No Lateral or Tangential Edge Displacements, Ph.D. Thesis, University of Illinois, 1956. 33. P. A. Laura and B. F. Saffel, Jr., .‘Study of Small Amplitude Vibrations of Clamped Rectangular Plates Using Polynomial Approximations,” J . Acoust. SOC. Amer., 41(4), 1967. 34. M. Vet, “Vibration Analysis of Thin Rectangular Plates,” Mach. Des. Mag.. April 13. 1967. 35. Road Shock and Vibration Environment for a Series of Wheeled and Track-Laying Vehicles, Report DPS-999, March-June, 1963. 36. Navships 900, 185, A Design of Shock and Resistant Vibration Electronic Equipment

for Shipboard Use. 37. D. S . Steinberg, “Circuit Components vs. G Forces,“ Mach. Des. Mag., October 14. 1971. 38. Steinberg & Associates, Assessment of Vibration on Avionic Design, prepared for Universal Energy Systems, Dayton. OH, August 1984. 39. R . N. Wild, Some Fatigue Properties of Solders and Solder Joints, IBM Report 74Z00044S, July 1974.

40. Werner Engelmaier, Effects of Power Cycling on Leadless Chip Carrier Mounting

Reliabiliv and Technology, Bell Laboratories, November 1982. 41. S. 0. Rice, “Mathematical Analysis of R a n d o m Noise,” Bell Sys. Tech. J., July 1944. 42. Raymond D. Mindlin, “Dynamics of Package Cushioning,” Bell Sys. Tech. J., pp. 353-461, July-October 1945. 43. D. S. Steinberg. “Snubbers Calm PCB Vibration,” Mach. Des. Mag., March 24. 1977.

BIBLIOGRAPHY

403

44. M. McWhirter, “Shock Machines and Shock Test Specifications,” Sandia Corp. Lecture Series, 1970. 45. J. Sugamele, Evaluation of Acoustic Environmental Effects on Flight Electronic Equipment, Boeing Airplane Co., 1966. 46. R. W. Hess, R. W. Fralich, and H. H . Hubbard, Studies of Structural Failures Due to Acoustic Loading, NACA T.N. 4050, 1957. 47. NAVMAT P-9492, Nu y Manufacturing Screening Program, Department of Navy, May 1979. 48. Institute of Environmental Science, 940 E. Northwest Highway, Mt. Prospect, IL 60056. 49. W. Silver, Proposed Recommended Practice in Applying Broadband Vibration Screening to Electronic Hardware, IES Shock and Vibration Committee, February 1981. 50. A. J. Curtis and R . D. McKain, A Quantitative Method of Tailoring Input Spectra for Random Vibration Screens, Hughes Aircraft Co., June 1987. . September 25, 51. Martin Dynamics Manual, Section 4, “Vibration of B e a m s , ” ~ 102, 1957. 52. R . A. Wilk, “ W e a r of Connector Contacts Exposed to Relative Motion,” ZES Proceedings, Random Vibration Seminar, Los Angeles, March 25-26, 1982. 53. MIL-HDBK-304, Militaly Standardization Handbook Package Cushioning Design, Department of Defense, Washington, DC, November 1964. 54. D. S. Steinberg, Cooling Techniques for Electronic Equipment, Second Edition, Wiley, New York, 1991.

INDEX

Index Terms

Links

A Acceleration, relation to frequency sample problem, transformer bracket Accelerometer positioning

26 31 106

370

106

366

367

370 servo control

367

Acoustic noise

234

air jets

237

cumulative damage

244

damage to panels

236

fatigue life

240

generating methods

236

membranes

242

microphonic effects

235

nonlinear stress

242

one-third octave bands

238

sample problem, thin chassis panel

240

siren

237

sound acceleration spectral density

245

sound pressure level

234

sound pressure spectral density

236

Air transport rack (ATR) Alignment pins, electronic box

12 301

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Amplification

31

shock

258

vibration

261

32

Aspect ratio, side panel

317

AVIP

338

B Beams

19

37

56

bending displacements

20

28

56

bending moments

29

bending stresses

30

beryllium

61

352

cantilever, see Cantilevered beams chart, natural frequencies

63

clamped (or fixed)

62

natural frequency equations

62

nonuniform cross section

64

polynomial deflection equations

62

sample problem, beam natural frequency

60

simply supported

27

trigonometric deflection equations

57

Bending moments sample problem, beam bending moments

57

29

30

210

267

Bents, see Frames bents and arcs Bolted covers sample problem, bolted covers on a chassis Bolted efficiency factors

306 305 137

Bolts, see Screws


Index Terms

Links

Buckling, critical stress

318

sample problem, chassis wall in bending

316

sample problem, chassis wall in shear

324

C Cantilevered beams

10

19

63

208

nonuniform cross sections

64

sample problem, stepped beams

66

Castigliano’s strain energy theory

90

frames and bents

80 302

chassis mount

302

overturning moments

309

Chassis

300

bolted covers

305

critical panel buckling stress

318

dynamic coupling with PCB

227

sample problem, chassis coupling with PCB natural frequency sample problem, nonuniform chassis

303

309

305

226 307

311

327

68

response to random vibration

206

torsional mode

309

sample problem, chassis torsion frequency

68

80

circular arcs

Center of gravity

62

311

torsional stresses

321

transmissibility Q,

314

trigonometric load distribution

312

368

sample problem, trigonometric load on


Index Terms

Links

Chassis (Cont.) trigonometric load distribution (Cont.) a chassis uniform load distribution sample problem, uniform load on a chassis Circuit boards

bolted installation component mounting

315 328 328 103

107

116

122

127

145

105

107

145 49 336

conformal coatings

104

105

337

damping

159

162

163

desired natural frequency

172

216

217

120

127

116

122

141

260 random vibration, sample problem

218

shock, sample problem

262

sinusoidal vibration, sample problem

174

different types of supports

111 146

displacements

111 130

dynamic stress sample problem, PCB dynamic stress

131 131

edge conditions

112

117

electrical connectors

105

223

223

224

epoxy fiberglass boards

121

131

fatigue life

174

219

174

219

sample problem, connector fatigue life

sample problem, PCB fatigue life

134


Index Terms

Links

Circuit boards (Cont.) heat sinks

103

105

kinetic energy in bending

113

119

laminated construction

103

105

loose edge guides

104

154

sample problem, loose edge guides

330

Miner’s damage ratio

212

341

multilayer

103

110

mounting methods

104

107

natural frequencies for different supports

127

145

circular plate

148

octagonal plate

146

rectangular plate

127

triangular plate

147

sample problem, relays plug-in types

110

182

manufacturing tolerances

octave rule

110

150

146

154

156 103

120

154

156

159

300

374

385

120

132

strain energy in bending

112

119

stiffening ribs

132

sample problem, natural frequency

sample problem, natural frequency with ribs stiffening ribs fastened with screws

132 137

sample problem, natural frequency ribs with screws surface mounted components testing to destruction with vibration

137 39

49

385


Index Terms

Links

Circuit boards (Cont.) through-hole mounted components

49

97

sample problem, relative PCB displacement

50

torsion with stiffening ribs

135

transmissibility Q

108

109

173

385

effects of input acceleration G level wedge clamps sample problem, PCB with wedge clamps Circular arcs

368 177 179 90

clamped ends, strain energy analysis

92

displacements with a concentrated load

94

hinged ends, strain energy analysis

90

spring rate with a concentrated load

92

strain relief, component lead wire

95

sample problem, offset fatigue life

110

97

Cold plates

11

103

110

Component mounting

75

97

170

173

176

216

101

174

208

212

219

260 fatigue life

location effects

175

orientation effects

176

size effects

173

surface mount

39

49

through-hole mount

49

97


Index Terms Composite structures

Links 69

72

133

305

71

72

74

71

73

74

circuit boards

133

138

electronic chassis

305

Computer applications

144

Conformal coatings

337

Connectors, electrical

105

223

edge support properties

106

107

effects of acceleration G levels

108

effects of random vibration

223

fatigue life

224

beams, stiffness sample problem, natural frequency

172

105

216

224

sample problem, random vibration fatigue life 224 fretting corrosion

223

intermittent electrical signals

223

Coupled vibration modes

6

37

308

311

sample problem, chassis and PCB

226

bending and torsion

311

226

sample problem, chassis in bending and torsion definition test for

308 37 308

Covers on chassis bolted efficiency factors

305 137

305

sample problem, natural frequency with bolted covers

306


Index Terms

Links

Cumulative fatigue, see Fatigue, cumulative damage Curves

32

152

258

283

153

shock amplification, single degree-of-freedom

258

shock amplification, two degree-of-freedom

283

284

velocity shock, shipping crates

285

287

288

17

30

32

134

258

268

281

367

as a function of acceleration G level

34

108

368

as a function of frequency

33

35

105

108

368

108

368

30

34

constrained layer

163

164

conversion of kinetic energy into heat

268

276

26

159

electronic chassis

361

368

low levels

267

367

ratio

31

33

relation to transmissibility

33

108

viscous

30

vibration amplification

32

vibration transmissibility

32

D Damping

circuit boards coefficients

effects of displacement and stress

348

374


Index Terms Decibel

Links 197

234

acoustic noise

234

235

random vibration

197

198

3

4

18

20

26

28

111

116

159

191

209

217

254

261

27

56

58

62

64

266

sample problem, concentrated load on a beam 19

27

265

27

28

26

29

173

218

254

261

111

116

122

126

173

217

256

261

Degrees of freedom Displacement

beams

bending

19 57

dynamic

plates

261 relation to frequency shock

26 254 266

sample problem, pulse shock vibration

sample problem, beam displacement Dunkerley coupling equation

262

265

20

26

28

111

116

173

217

261

27

266

311


Index Terms

Links

Dynamic bending stresses

29

132

211

243

267

316

29

209

267

26

29

173

218

254

261

27

56

58

62

64

266

circuit boards

172

217

261

chassis

313

321

Dynamic loads

29

132

211

243

267

316

235

239

241

209

267

beams sample problem, simply supported beam circuit boards sample problem, dynamic load on a PCB chassis

29 132 131 317

sample problem, dynamic stress on a chassis 311 Dynamic displacements

beams

acoustic noise

sample problem, acoustic pressure on a panel 235 beams

29

bolts

360

chassis

317

sample problem, dynamic loads on a chassis

311

circuit boards

132

electrical connectors

223

electrical lead wires

49

fasteners

360

pyrotechnic shock

290

328

292

296


Index Terms

Links

Dynamic loads (Cont.) random vibration

193

197

199

207

209

216

223

227

shaker head

361

sine vibration

170

311

323

328 spring members

18

38

transformer bracket

27

265

velocity shock

285

Edge guides, loose

104

154

Electrical connectors

105

223

E

Electrical wire harness Electronic chassis

11 300

area moment of inertia

305

bolted cover

305

sample problem, chassis with bolted covers

182

305

buckling panels

317

center of gravity mount

302

coupling bending and torsion

311

high center of gravity mount

301

mass moment of inertia

310

natural frequency

307

octave rule

150

perimeter stiffening rib

374

shear flow

321

shear panel

324

303

309

308

322

324


Index Terms

Links

Electronic chassis (Cont.) simply supported ends

302

single jacking screw

301

tall narrow cross section

303

testing to destruction

385

torsional resonance

309

Electronic components

cementing to circuit boards

308

39

76

170

176

217

220

379

391

396

338

dynamic loads

40

46

lead wires

39

94

97

40

49

53

97

396

5

398

43

48

54

174

212

214

392

396

forces and stresses

41

46

sample problem, wire fatigue life

43

54

174

212

214

396

solder joint stresses

44

100

399

spring rates

87

92

94

98

100

396 as beam members

effective length fatigue life

strain relief mounting considerations

97 76

97

170

173

176

216

260 This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Electronic components (Cont.) mounting considerations (Cont.) location on circuit board

176

orientation on circuit board

176

surface mounting

39

49

through hole mounting

49

76

97

49

76

97

relative motion on vibrating circuit boards

216 sample problem, wire forces due to relative motion vibration testing to failure End support conditions

beams

circuit boards

circular arcs

50

97

385 39

94

97

111

127

130

147

396

19

27

63

68

111

120

128

147

90

92

95

94

97

57

127

97 component lead wires

39 396

electronic chassis

68

301

309

frames and bents

78

81

84

rectangular plates

111

120

127

346

354

358

367

369

372

146 vibration fixtures


Index Terms Endurance limit

Links 42

44

48

211 aluminum alloys

211


42

48

fatigue life relations

39

42

45

167

168

174

44

48

137

305

219 relation to ultimate strength

42 211

Environmental stress screening bathtub reliability curve

379 380

combined thermal, vibration, electrical operation 387 damage potential

390

effects of temperature on solder creep

400

estimating fatigue life used up in screen

392

NAVMAT-P-9492

380

philosophy

379

random vibration environments

382

sample problem, fatigue life

395

screening preparations

384

separate thermal, vibration and electrical operation 389 thermal environments

381

F Fasteners

7 350

cut threads or rolled threads loosening progressive shear failures

353 8 370

371


Index Terms

Links

Fasteners (Cont.) sample problem, preload torque on a screw

352

selection criteria

8

9

350

torque requirements

9

43

48

54

99

174

166

212

392

396

AVIP

338

341

cumulative damage

168

169

vibration fixtures Fatigue life

350

343

344 electrical lead wires

343

sample problem, fatigue in lead wires

168

solder joints

344

169

cycle ratio

343

344

damage

212

341

definition

39

endurance limit

42

48

life

39

330

connectors, fretting fatigue

223

224

effects of manufacturing tolerances

330

estimating, methods

340

341

212

341

Miner’s cumulative damage ratio sample problem, estimating the fatigue life S–N curves

aluminum

212 42

44

167

211

48

211

copper wires

48

kovar wires

42


Index Terms

Links

Fatigue life (Cont.) S–N curves (Cont.) fatigue exponents

167

sample problem, fatigue life of kovar wires

40

sample problem, solder shear fatigue life

43

solder

44

Flanges on chassis panels

305

Foam rubber supports for PCB

107

test data, PCB with foam rubber supports Forced vibrations amplification curve

106 30 32

circuit boards

106

counterweights

356

dynamic shift in the center of gravity

364

dynamic similarity

356

enclosed chassis

304

resonant vibration fixtures

348

rocking modes

354

single degree-of-freedom with damping

31

transmissibility curve

32

Frames bents and arcs

75

circular arcs

90

displacement with concentrated load

90

365

305

93

component lead wires

97

deflections

80

82

83

87

92

94

96 fixed ends circular arcs, strain energy

83 90


Index Terms

Links

Frames bents and arcs (Cont.) fixed ends (Cont.) rectangular bent strain energy

81

wire arc strain relief, strain energy

95

hinged ends

78

circular arc strain energy

90

rectangular bent strain energy

81

rectangular bent superposition

78

Free body diagram

Free vibrations

31

38

79

81

84

88

17

with damping

30

multiple mass system, no damping

25

36

359

4

21

24

56

59

112

117

123

beams

56

59

definitions

56

deflections

56

62

111

116

122

124

111

116

122

124

126

pendulum rotating vector

26

Fretting corrosion, see Wear, connectors Fuse, mechanical

358

G Geometric boundary conditions

125 plates


Index Terms

Links

Geometric boundary conditions (Cont.) slope

57

112

117

123 Geometric stress concentrations G forces

beams sample problem, G levels on beams circuit boards

167

168

10

13

15

108

131

140

174

208

218

253

258

262

27

208

265

28

265

13

108

171

174

214

215

131

174

262

300

310

311

315

320

328

312

321

328

362

365

369

372

374

260 sample problem, G levels on PCBs electronic chassis

sample problem, G levels on chassis relation to frequency

26

trucks

15

vibration fixtures

H Half power points

186

Harmonic modes

5

Heat exchangers

11

Hertz, definition

2


Index Terms

Links

I Inertia

area moment

20

23

28

60

74

123

138

209

265

20

28

60

74

123

138

265 average boxes beams

64

68

305 20

28

209

265

72

73

74

forces

11

209

374

lumped masses

23

226

265

composite

74

310 mass moment

23

310

plates

123

138

polar

23

310

57

59

112

57

59

112

114

119

beams

57

59

plates

112

114

Integration methods random vibration PSD area sample problem, area under PSD curve shock pulse area strain energy

Isolators chassis

198 199 253

119

275 277


Index Terms

Links

Isolators (Cont.) heat development

276

sample problem, isolators for vibration and shock 278 shock

280

vibration

278

wire cable considerations

11

J J factor, torsion rectangular chassis round shaft sample problem, torsion J factor for a chassis

21

22

310 22 310

K K factor, buckling

318

bending

317

sample problem, buckling, chassis side walls

318

shear

324

325

Kinetic energy

58

113

beams

58

uniform loads

119

58

circuit boards

113

119

circular arcs

91

93

frames and bents

81

84

multiple mass systems

36

one mass system

18

relation to work

18

torsion system

21


Index Terms

Links

L Leadless ceramic chip carrier (LCCC) LCCC component factor for PCB vibration Lead wires, electrical

component factor for PCB vibration effects of PCB displacements

217 217 40

49

76

97

170

173

216

221

217 76

97

170

173 fatigue life

97

174

219

force developed

76

95

99

170

216

83

87

92

94

96

99

spring rate

100 strain relief sample problem, wire strain relief Line replaceable unit (LRU) Load path through chassis Lock washers Loose PCB edge guides sample problem, PCB with loose edge guides Lugs, chassis mount

97 97 12 300

301

9 182 187 229

302

303

19

21

24

36

37

172

183

208

309

19

208

304 Lumped mass systems

beams


Index Terms

Links

Lumped mass systems (Cont.) chassis

309

313

321

chassis and PCB

172

215

226

PCB

183

215

sample problem, coupling chassis and PCB kinetic energy

226 18

21

36

330

332

334

M Manufacturing problems

338 AVIP program

339

effects of tolerances

331

failure free operating period (FFOP)

345

MTBF

343

sample problem, effects on PCB frequency

332

Margin of safety, chassis

319

buckling due to bending

317

combined bending and shear

325

shear

324

torsion

320

Membranes, acoustic noise

237

242

MIL specifications

175

185

Miner’s cumulative fatigue damage

212

341

definition

212


343

344

fatigue cycle ratio

212

341

random vibration

212

251

398

sample problem, fatigue life using Miner’s method

212


Index Terms Mode shapes Moment of inertia, area

beams

nonuniform, average

Links 5 20

23

28

60

74

123

138

209

265

20

28

74

209

265

64

68

bolted efficiency factor

138

electronic chassis

305

sample problem, chassis with bolted covers electrical lead wires sample problem, wire displacement stresses ribs on PCB sample problem, ribs on a PCB torsion, thin wall Moment of inertia, mass electronic chassis

304 40 50

97

133

138

133

138

310 23 310

base mount

310


327

round flat disk sample problem, disk and shaft torsion Mounting lugs on a chassis

310

22

23

22

23

229

302

19

22

25

28

35

37

83

87

N Natural frequency, simple systems

beams, nonuniform

64

beams, uniform

63

bents and frames

80


Index Terms

Links

Natural frequency, simple systems (Cont.) circuit boards

103

120

127

145

154

156

159

300

374

385 bolted ribs

138

change in edge condition

106

circular shape, various supports

148

octagon shape, various supports

146

polynomial functions

116

rectangular shape

127

ribs in two directions

141

triangular shape, various supports

147

trigonometric functions

111

circular arcs

90

fixed ends

92

hinged ends

90

108

145

92

composite beams

69

damped vibration

17

definition

17

effects on transmissibility Q

31

172

368

369

300

303

electronic chassis

94

33

173

307

309 bending mode

303


302

326

coupled mode

308

311

high center of gravity mount

308

309

torsion mode

309


Index Terms

Links

Natural frequency, simple systems (Cont.) fundamental

17

Rayleigh method

56

57

58

59

111

114

116 cantilever beams clamped rectangular plates clamped uniform beams supported rectangular plates supported uniform beams

62 122 62 111 57

static displacement method

19

torsion system

21

NAVMAT-P-9492 environmental stress screening

116

301

309

152

154

379

combining thermal, vibration and electrical operation

387

estimating amount of fatigue life used up

392

preparations prior to the screen

384

proposed number of thermal cycles

387

random vibration power spectral density curve sample problem, fatigue life used up Nonuniform cross sections sample problem, average moment of inertia

382 395 64 68

O Octave rule, chassis with PCBs

150 225

forward rule

155


Index Terms

Links

Octave rule, chassis with PCBs (Cont.) natural frequency ratio sample problem, relay chattering problem

152

153

156

potential problems with reverse rule

155

156

reverse rule

155

transmissibility ratio

152

153

weight ratio

152

153

Oil film slider plates

355

356

361

364 bolts loaded in shear

350

371

372

bolts loaded in tension

352

361

364

dowel pin applications

372

373

2

18

99

100

103

120

154

156

159

300

374

385

34

172

368

369

P Periodic motion Plated through holes Plates, natural frequency

effects on transmissibility

circle, hexagon, rectangle, triangle

145

polynomial series equation derivation

116

sample problem, plate natural frequency

120

trigonometric series equation derivation

111

with bolted ribs

138

with rigidly attached ribs

133

173

sample problem, PCB natural frequency with ribs

133


Index Terms

Links

Power spectral density (PSD)

192

195

effects of PSD levels on acceleration G levels

207

222

effects of transmissibility

207

210

finding PSD and frequency break points

200

sample problem, finding PSD break points input area under PSD curves

201 195

196

sample problem, finding area under PSD curve196 integration methods for finding areas

198

integration methods for finding PSD values

201

PSD response values

207

Preloaded bolts sample problem, torque required for bolt preload Printed circuit boards (PCBs)

351

352

361

352

353

103

106

108

111

116

127

217

261

218

261

145 fretting corrosion problems on plug-in connectors 223 sample problem, connector fatigue life

224

maximum allowable dynamic displacement

172

random vibration

217

shock

261

sinusoidal vibration

172

minimum desired natural frequency

173

random vibration equation derivation

218

sample problem, random natural frequency

218

sample problem, shock natural frequency

262

sample problem, sinusoidal natural frequency 174 shock frequency equation derivation

261

sinusoidal vibration equation derivation

173


Index Terms Probability distribution functions Gaussian instantaneous distribution

Links 202

205

202

quick method, random fatigue life

213

sample problem, random fatigue life

208

three band technique random fatigue life

204

Rayleigh peak distribution

205

Push bar couplers, vibration testing

355

Pyrotechnic shock

289

364

372

346

348

353

357

367

acceleration levels

290

effects on electronic component

296

hybrid design

297

shock response spectrum

292

sample problem, response of chassis and PCB 292 simulations with hammer impacting a plate,

291

typical frequencies and acceleration G levels

290

Q Qualification tests

design equipment to survive 5 qualification tests

224

R Rack, air transport (ATR)

12

301

chassis mount

301

326

Random vibration

188

190

bending stress

209

cumulative damage

212

damage

213


218

210


Index Terms

Links

Random vibration (Cont.) failure modes

188

189

fatigue

211

input curves

194

195

life

212

214

392

395 multiple degree-of-freedom

224

octave rule

225

PCB as a single degree-of-freedom system

215

positive zero crossings

231

sample problem, positive zero crossings quick respond analysis method sample problem, quick response analysis

233 213 214

response characteristics

189

single degree-of-freedom response

224

three band life technique

204

sample problem, three band life analysis time history Rotating vector describing vibration Rotational vibration modes

Rough road tests Rubber, PCB support

225

225

208 204 3 21

301

309

321

308

15 107

S Screws

7

138

350

352 selection criteria

8

cover attachment

305


Index Terms

Links

Screws (Cont.) effective length

351

efficiency factors

137

friction

352

loosening

352

preload effects

351

spring rates

351

torque and preload relations

352

352

9

Shaker head force

362

365

Shear flow, torsion

321

322

Ship and submarine vibration shock, velocity change

350

8

lubrication

torque values, recommended

138

13 252

254

285

248

258

296

affects on components

291

296

amplification

258

259

chassis response

269

282

cushioning material

270

287

desired PCB resonant frequency

260

261

286 Shock

sample problem, PCB resonant frequency

262

equivalent shock pulse

269

full rebound

252

half sine pulse

253

area under half sine pulse

258

253

hammer impact

291

isolators

275

sample problem, selecting shock isolators

294

278


Index Terms

Links

Shock (Cont.) mounts for ships

13

nonlinear

286

octave rule

284

PCB response

291

pulse shock

250

253

258

259 sample problem, full rebound pyrotechnic sample problem, chassis and PCB

258 289 292

response spectrum

251

square wave

259

stresses

267

triangular pulse

250

two degree-of-freedom

282

response curves for different damping

283

velocity shock

285

zero rebound

253

Simple harmonic motion Sinusoidal vibration allowable PCB displacement component mounting

290

259

284

2

18

166

171

172 76

97

173

176

damage estimate

166

167

desired PCB natural frequency

173

170

sample problem, desired PCB natural frequency lead wire strain relief sample problem, strain relief fatigue life

174 94

96

97


Index Terms

Links

Sinusoidal vibration (Cont.) loose PCB edge guides reasons to avoid loose PCB guides

182

183

182

sample problem, PCB frequency with loose guides qualification tests

sine sweep through a resonance sample problem, sweep through a resonance wedge clamps on PCBs

185 346

348

357

367

353

186 186 177

sample problem, natural frequency, wedge clamps wedge clamp efficiency factor Slider plates, oil film

Suspension systems Solder

179 178 355

356

361

367

372

373

44

104

387

395

44

100

358 43 399

alternating stress

44

bonding PCB stiffening ribs

104

component size, effects

396

creep rate

400

fatigue

43 390

physical properties

44

400

plated through holes

43

99

100

sample problem, solder stress

47

101

399

shear tearout stress

43

399

surface mounted components

217


Index Terms

Links

Solder (Cont.) temperature effects Springs chassis torsion spring rate

400 23 310

coil springs

25

damping

31

deflection

26

dynamic loads

18

36

38

effective, for bolts

351

352

nonlinear effects

352

362

parallel

24

PCBs

183

preload effects

352

series snubbers

24 164

static deflection

28

strain energy

18

torsion

21

work

18

Square plates

6

129

240

19

20

27

28

78

209

133

138

Static deflection

36

265 Stepped beams Stiffeners

64 107 141

Strobe light Superposition, bent sample problem, bent deflection

106 78 78


Index Terms Surface mounted devices

Links 217


173

Suspension vibration systems

358

Sweep through a resonance

186

218

261

138

350

T Tee sections

132

beams

350

stiffening ribs

132

138

141

345

346

354

355

357

358

Test fixtures

bolted assemblies bolt torque

350 9

352

dynamic shift in center of gravity (CG)

354

355

fastening accelerometers

360

366

367

mechanical fuse

358

oil film slider

355

356

367

364

372

354

360

21

22

301

308

321

326

closed sections

301

308

combined with bending

308

311

372 positioning accelerometers

366

preloaded bolts

352

push bar couplers

355 373

rocking modes

353

summary for good design

357

suspension systems

358

Torsion


Index Terms

Links

Torsion (Cont.) electronic chassis

301

308

321

322

mass moment of inertia

23

310

327

rectangular cross section

310 21

22

309

133

138

141

resonant frequency ribs

309

single degree-of-freedom

21

spring rate

23

310

strain energy

21

58

80

82

83

90

112

118

Transformer bracket

27

265

Transmissibility

28

35

41

368 beams

28

41

circuit boards

35

104

108

368 damping relations

30

dynamic loads

46

311

314

368

function of frequency

28

34

general equation

33

368

mathematical model

31

single degree-of-freedom

32

transformer bracket

47

electronic chassis

vibration fixture

348

41

370


Index Terms Trigonometric functions

Links 3

26

56

62

111

125

140

177

312

beams

56

58

62

boxes

308

312

327

62

combined with polynomial

62

displacements

56

58

dynamic loads

14

29

kinetic energy

40

58

natural frequency

58

60

63

108

120

127

111

125

177

111

116

122

125

130

177

Rayleigh method

56

139

140

slopes

57

59

112

18

22

57

80

83

90

94

112

21

135

321

1

17

150

188

300

346

106

360

367

145 plates and PCBs displacements

117 strain energy

Torsion natural frequency

309

Trunion, vibration machine

347

V Vibration

accelerometers


Index Terms

Links

Vibration (Cont.) airplanes and missiles

10

automobiles, trucks and trains

156

bending coupled with torsion

308

counter weights

356

die bond wires

297

fixtures

347

design summary

357

wood laminations

348

forced

30

free

17

harmonic modes isolators

300

5 275

277

301

sample problem, selecting vibration isolators 278 linear systems mechanical fuse

2 358

modes

5

nodes

5

nonlinear systems

342

nose cones

356

oil film slider

355

351

356

364

168

175

189

224

250

367 qualification tests

random representation

188 3

servo control accelerometer

366

effects of location

367


Index Terms

Links

Vibration (Cont.) shaker machine

347

bolt patterns

361

mounting

347

ships and submarines sinusoidal sources

13 166 1

suspension systems

358

unsymmetrical fixtures

354

variable cross section

364

64

beams

64

electronic chassis

68

66

67

W Wear, connectors

223

224

Wedge clamps

177

179

percent fixity Work

178 21

22


VIBRATION ANALYSIS FOR ELECTRONIC EQUIPMENT

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