Guidelines for the Vibration Design of Structures

ANGLO TECHNICAL DIVISION AA Best Practice Guideline BPG S002

GUIDELINES FOR THE VIBRATION DESIGN OF STRUCTURES AA BPG S002 Issue 2 October 2007

GJ Krige Anglo Technical Division Anglo Operations Limited

ANGLO TECHNICAL DIVISION

CONTENTS 1 2

3

4

5

6

SCOPE ...................................................................................................................... 5 DEFINITIONS, NOTATION AND PROPERTIES....................................................... 7 2.1 Definitions ........................................................................................................... 7 2.2 Notation and Units be Used................................................................................ 8 2.3 Section Properties ............................................................................................ 11 VIBRATION DESIGN PROCEDURE....................................................................... 13 3.1 Step 1: Necessary Data and Information......................................................... 13 3.2 Step 2: Clarify Details of the Structure to be Designed ................................... 13 3.3 Step 3: Build a Computer Model....................................................................... 13 3.4 Step 4: Assess the Results............................................................................... 13 3.5 Step 5: Prepare Structural Design Calculations and Drawings ........................ 14 GENERAL CONCEPTS AND THEORY .................................................................. 15 4.1 Dynamic and Harmonic Loads.......................................................................... 15 4.2 Dynamic Characteristics of Structures.............................................................. 17 4.2.1 The Single Degree of Freedom System. ................................................... 17 4.2.2 Response to Harmonic Excitation ............................................................. 18 4.2.3 Resonance and Tuning ............................................................................. 18 4.2.4 Damping .................................................................................................... 20 4.2.5 Multi Degree of Freedom Systems ............................................................ 22 4.2.6 Mode Shapes and More about Natural Frequencies................................. 22 LOADS..................................................................................................................... 24 5.1 Rotating Unbalance .......................................................................................... 24 5.1.1 Motors and Turbines.................................................................................. 24 5.1.2 Vibrating Equipment .................................................................................. 26 5.2 Loads Applied to the Structure ......................................................................... 27 5.2.1 Data Required From the Equipment Supplier............................................ 27 5.2.2 Calculation of Spring Stiffness................................................................... 28 5.2.3 Example 1.................................................................................................. 30 5.2.4 Example 2.................................................................................................. 31 5.3 Impact Loads .................................................................................................... 32 5.3.1 Types of Impact Loads .............................................................................. 32 5.3.2 Energy Equations ...................................................................................... 32 5.3.3 Moving Mass Hits Stationary Mass ........................................................... 34 5.4 Ground Motion from Blasting and Piling ........................................................... 34 5.4.1 Basic Equation........................................................................................... 34 5.4.2 Blasting...................................................................................................... 34 5.4.3 Piling.......................................................................................................... 35 STRUCTURAL MODELLING AND RESPONSE ..................................................... 36 6.1 Modelling the Structure..................................................................................... 36 6.1.1 Models ....................................................................................................... 36 6.1.2 Degrees of Freedom.................................................................................. 36 6.1.3 Modelling Structural Geometry .................................................................. 37 6.1.4 Modelling Mass.......................................................................................... 38 6.1.5 Modelling Materials.................................................................................... 39 6.1.6 Modelling Connections .............................................................................. 39 6.1.7 Modelling Floors ........................................................................................ 39 6.2 Composite Beams and Floors .......................................................................... 42 6.3 Confirming the Accuracy of the Model.............................................................. 43 6.4 Modelling Machinery on Structures .................................................................. 44 6.5 Calculation of Dynamic Response.................................................................... 46 6.5.1 General...................................................................................................... 46 6.5.2 Significant Modes ...................................................................................... 46 Guidelines for the Vibration Design of Structures – Issue 2 AA BPG S002 2 of 92


6.6 Dynamic Analysis Computer Programmes ....................................................... 47 6.6.1 ROBOT V6 ................................................................................................ 47 6.6.2 PROKON. .................................................................................................. 47 6.6.3 Common Errors with Computer Vibration analysis.................................... 48 7 VIBRATION LIMITS ................................................................................................. 49 7.1 Introduction ....................................................................................................... 49 7.2 Human Sensitivity ............................................................................................. 49 7.3 Equipment and Machine Sensitivity.................................................................. 53 7.4 Structural Sensitivity ......................................................................................... 53 7.4.1 Brittle Finishes ........................................................................................... 53 7.4.2 Fatigue Life................................................................................................ 54 8 DESIGN GUIDANCE FOR SPECIFIC EQUIPMENT AND STRUCTURES............. 55 8.1 Crushers ........................................................................................................... 55 8.1.1 Modelling Crusher Support Structures ...................................................... 55 8.1.2 Loads Applied by Crushers ....................................................................... 55 8.2 Rotating Tubes ................................................................................................. 57 8.2.1 Types of Load Generated.......................................................................... 57 8.2.2 Specific Equipment.................................................................................... 59 8.3 Vibrating Screens and Feeders ........................................................................ 59 8.3.1 Basic Requirements .................................................................................. 59 8.3.2 Design and Use of Sub-frames ................................................................. 60 8.4 Rock Breakers .................................................................................................. 62 8.5 Design of Grizzly Bars ...................................................................................... 63 8.6 Vessel Agitation ................................................................................................ 64 8.6.1 Applied Loads............................................................................................ 64 8.6.2 Design Requirements ................................................................................ 65 8.7 Wood Chippers ................................................................................................. 66 9 PRACTICAL GUIDeLINES FOR FOUNDATIONS................................................... 67 9.1 Traditional Rules of Thumb............................................................................... 67 9.2 Simple Rules..................................................................................................... 67 9.3 Modelling Foundations...................................................................................... 68 9.3.1 Soil Conditions........................................................................................... 68 9.3.2 Simplified Preliminary Calculations ........................................................... 70 9.3.3 Damping .................................................................................................... 72 10 PRACTICAL DETAILS FOR TERTIARY STRUCTURAL ELEMENTS ................ 73 10.1 Individual Members ....................................................................................... 73 10.1.1 Approximate Natural Frequencies of Individual Members ......................... 73 10.1.2 Limitation of Slenderness Ratio to 80........................................................ 73 10.2 Walkways and Hand Railing ......................................................................... 74 10.3 Sheeting Rails ............................................................................................... 75 10.4 Plating on Chutes, Bins and Underpans ....................................................... 75 10.4.1 Natural Frequencies of Rectangular Panels.............................................. 75 10.5 Bracing Systems ........................................................................................... 78 11 PRACTICAL DETAILS FOR CONNECTIONS ..................................................... 80 11.1 Bolted connections........................................................................................ 80 11.2 Welded Connections ..................................................................................... 80 11.3 Beam-to-beam Connections ......................................................................... 80 11.4 Bracing Connections ..................................................................................... 81 12 VIBRATION MEASUREMENTS........................................................................... 82 12.1 What Should be Measured?.......................................................................... 82 12.2 Measuring Equipment ................................................................................... 82 12.3 Recording Measurements ............................................................................. 83 12.4 Relating Measured Displacements to Implied Stresses................................ 83 12.5 Baseline Vibration Measurement Guide........................................................ 84 12.5.1 Baseline Measurements ............................................................................ 84 Guidelines for the Vibration Design of Structures – Issue 2 AA BPG S002 3 of 92


12.5.2 What Baseline Measurements Can and Can’t Do ..................................... 87 13 TROUBLE-SHOOTING AND STRUCTURAL MODIFICATION ........................... 88 13.1 Interpreting and Using Measurements .......................................................... 88 13.2 Changes to Applied Loads ............................................................................ 88 13.3 Structural Modifications................................................................................. 88 13.4 Common Concerns of Site Personnel........................................................... 89 14 BIBLIOGRAPHY................................................................................................... 91 14.1 Standards and Specifications........................................................................ 91 14.1.1 SANS Standards ....................................................................................... 91 14.1.2 AAC Specifications .................................................................................... 91 14.2 Text Books .................................................................................................... 91 14.3 Journal Papers .............................................................................................. 91 14.4 AAC Reports ................................................................................................. 91 15 RECORD OF MODIFICATIONS .......................................................................... 92

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1 SCOPE The purpose of this guide is primarily to assist with understanding the concepts on which structural dynamics is based, and provide guidance in the practical implementation of dynamic design. The following concepts need to be understood: (a) (b) (c) (d) (e) (f)

Natural frequencies, resonance and damping. Mass, force and inertia force. Degrees of freedom and computer modelling. Evaluation of member section properties and end constraints. Dynamic loading, dynamic reactions from vibrating equipment. Assessment of allowable amplitudes and fatigue life. Warning: Do not attempt dynamic analysis and design of any structures unless you understand these concepts. Too often, the powerful computer analysis packages available today are simply “thrown” at a dynamic design problem and it is assumed that a satisfactory structure will simply pop out. This does not happen.

This guide covers the full spectrum of the design procedures to be adopted for general structures carrying equipment that generates dynamic loads. It is not intended to cover design to resist environmental dynamic loads such as wind or earthquake, nor is it intended to cover the design of unusual structures such as tall masts or long bridges. Dynamic analysis and design of structures is aimed at ensuring three important criteria. (a) There should not be resonance. (b) The amplitudes of vibration should not exceed predefined limits. (c) The structure should have an adequate fatigue life. In order to achieve this, the Designer must understand the dynamic behaviour of structures, must know the dynamic loads acting on the structure, must be able to model and analyse the structure, and finally must be able to understand and assess structural behaviour against the predefined limits. The guide is thus divided into the following eight sections: (a)

(b) (c)

(d) (e) (f)

A basic presentation of the theory of the response of structures to dynamic loads. This is not intended as a detailed or comprehensive coverage of dynamic analysis theory, but rather a simple treatment to assist in understanding some fundamental concepts. For a fuller treatment of dynamic analysis theory it will be necessary to consult one of many excellent books on the market, a few of which are listed under the bibliography. Discussion of the dynamic loads that may be applied to industrial structures. Guidance with modelling of structures for computer analysis. This section includes a brief presentation of the computer programs used by ATD Structural engineering for dynamic analysis of structures. This is not intended to replace the respective manuals, but to provide guidance in their appropriate use. Description of the limits placed on vibration severity. A step by step practical guide to the design of some structures that must often be dealt with by ATD Structural Engineering, including supporting structures for crushers, screens, feeders, etc. Some guidance for the appropriate design of tertiary structural members.



(g) (h)

Discussion of how to deal with problems that arise on specific installations. This includes some comments regarding vibration measurements, typical problems that have been experienced, and guidance regarding fixing problems. A bibliography which includes the relevant codes and specifications as well as additional material for anyone wanting more detailed information. Warning: Throughout the document, various warnings are given. It is important to take note of these, as they are areas where experience has shown that mistakes tend to be made.



2 DEFINITIONS, NOTATION AND PROPERTIES The notation used throughout this guidelines document is listed below for ease of reference.

2.1

Definitions

algorithm

Logical arithmetic or computational procedure for solving a problem.

amplitude

The maximum value of any harmonic quantity, ie force, acceleration, velocity, or displacement. It is equal to half the peak-to-peak value.

damping

Dissipation or absorption of energy. Damping is usually assumed to be “viscous damping” which means that it is proportional to the velocity.

dynamic magnification factor

This is the ratio of the dynamic displacement amplitude of a structure to the static displacement if the same structure is subjected to a static load equivalent to the applied dynamic load amplitude.

frequency

This is the rate at which a harmonic quantity varies with time. It is important to distinguish between a cyclical frequency, f, which is measured in cycles/second, and radial frequency, ω, which is measured in radians/second. These are directly related by: ω = 2π f

frequency ratio

The ratio of the frequency of an applied harmonic load to the natural frequency of the structure, ie ωE/ ωN.

harmonic

Any quantity that varies with time according to: Q = Qo sin (ωt +Φ). Many of the loads applied by industrial equipment are harmonic loads.

inertia force

The force required to accelerate the mass of a specified portion of a structure. These terms describe the relationship between two or more harmonic quantities. If the quantities are in the same direction, and attain their maximum or minimum values at the same time, they are said to be inphase, and their phase angles are the same, i.e. Φ1 = Φ2. If the two quantities act in opposite directions and one attains its maximum value when the other attains its minimum value, they are said to be out-ofphase, and their phase angles differ by π, i.e. Φ1 = Φ2 + π.

in-phase out-of-phase

model

A model is a representation of a complex object. In dynamic design a model typically refers to a computer representation of a real structure.

mode shape

A natural shape in which a structure will vibrate. mathematically referred to as an “eigenvector”.

natural frequency

The frequency at which a structure will naturally vibrate in the absence of any applied force.

This is



peak-to-peak

The difference between the maximum and minimum values of a dynamic quantity. When the quantity is harmonic the peak-to-peak value is twice the amplitude. The peak-to-peak displacement of vibrating equipment is sometimes referred to as the “throw”.

periodical

Changing in time according to a regular pattern that repeats itself.

period

The period of a harmonic quantity is the time taken for one complete cycle. Thus: P=

1 2π = f ω

ppv

This is the peak particle velocity, generally used to describe the severity of ground motion generated by blasting or other disturbance.

rms

This is the “root mean square” value of a dynamic quantity. mathematically defined as: a RMS =

It is

1T 2 ∫a ( t )dt T0

When the quantity is harmonic, its RMS value is 0,707 times the amplitude.

2.2

Notation and Units be Used

The recommended units are given below for each symbol. It is highly recommended that these units are used for all calculations, or serious errors can be introduced. Warning: If these recommended units are not used for any reason, it is necessary to reduce all calculations to the basic units of mass, length, and time in order to ensure the accuracy of calculations. This problem usually arises because the relationship between force and mass is determined by the gravity constant. In static design calculations this does not matter, because we are always using forces, never mass, so whatever gravity constant we use we get the same answer as long as we are consistent. In dynamic design calculations we use mass and force, so the correct gravity constant is crucial. If we choose millimetres as our length unit, then the gravity constant is 9810 mm/s2, and not 9,81 m/s2. As one example, consider the elastic modulus of steel. (a)

If we are using metres as our unit of length: E = 200x103 MPa = 200x109 Pa = 200x109 N/m2 = 200x109 kg.m/s2/m2.

(b)

If we are using millimetres as our unit of length, we should use: E = 200x109 kg.m/s2/m2 = 200x106 kg.mm/s2/mm2 But, because we tend to forget the effect our choice of units has on the gravity constant, we are inclined to incorrectly use: E = 200x103 MPa = 200x103 N/mm2 = 200x103 kg.m/s2/mm2. This gives a stiffness which is too low by a factor of 1000, leading to completely wrong results.



Symbol Units a m aM ao bx

m/s2 m/s2 m

by

m

A b B c

m2 m m m

C CC CQ D do dC d1,d2 dw D

N.s/m N/m3 N/m3 m m m m m m

DMF Dv Dx

Nm2/m Nm2/m

Dy

Nm2/m

E E ECstat ECdyn

m N/m2 N/m2 N/m2

F fE

Hz Hz

fN f1 f2 F(t) FA FB FI FM

Hz Hz Hz N N N N Nm

FP Fo FQ

N N Nm

Description Linear dimension, distance from ground impact Linear acceleration Linear acceleration amplitude Spacing of x-direction stiffeners in orthotropic plating (m) Spacing of y-direction stiffeners in orthotropic plating (m) Cross sectional area Linear dimension Deflection of beam or post Axial displacement of steel coil spring under permanent loads only Damping constant Soil uniform compression modulus Soil uniform shear modulus Displacement Amplitude of displacement Depth of beam or slab Width of plates Diameter of steel coil spring wire Diameter of steel coil spring or rubber buffer Dynamic magnification factor Shear rigidity of stiffened plate Flexural rigidity of stiffened plate about x-axis Flexural rigidity of stiffened plate about y-axis Eccentricity of rotating mass Elastic modulus of material Static elastic modulus of concrete Dynamic elastic modulus of concrete General cyclical frequency Exciting frequency, i.e. frequency of applied dynamic force Natural frequency First natural frequency Second natural frequency Applied load Axial load applied by equipment Breaking load on rock breaker Inertia force Bending moment applied by equipment Pushing load on rock breaker Amplitude of applied load Torque applied by equipment

Comment

See SAISC Red Handbook

200x109 N/ m2 for steel 30x109 N/ m2 34x109 N/ m2



FV G G

N m/s2 N/ m2

Shear load applied by equipment Acceleration due to gravity Shear modulus of material

h

m

hCG

m

Ib

kg.m2

Im Ix Iox

kg.m2/m m4 m4

Iy Ioy

m4 m4

J K

m4

k KH

N/m N/m

KV

N/m

KΦ

Nm/rad

L m M MB MR

m kg/m kg kg kg

MS

kg

Free (unloaded) height of steel coil spring. Height through which a falling body falls before impact Height of machine above centre of gravity of base Mass inertia of machine and foundation Mass inertia per unit length Moment of inertia about x-axis Moment of inertia of stiffener and plate about x-axis Moment of inertia about y-axis Moment of inertia of stiffener and plate about y-axis Torsion constant Dieckmann “K” value for determination of vibration severity in terms of human sensitivity Stiffness of structure Horizontal stiffness of steel coil spring, rubber buffer, or machine base Vertical stiffness of steel coil spring, rubber buffer, or machine base Rotational stiffness of machine base Length of the member Mass per metre Mass Mass of machine base Mass of rotating or moving part of machine Mass of stationary part of machine or body impacted by moving body Number Period of harmonic quantity Frequency ratio Radius of gyration about x-axis Radius of gyration about y-axis Time elapsed Thickness of plate Linear velocity Amplitude of linear velocity Peak particle velocity (ppv) Linear velocity after impact between a moving body and a stationary body

N P R rx ry T t V vo vP vS

s m m s m m/s m/s m/s m/s

9,81 m/s2 78x109 N/ m2 for steel 12x109 N/ m2 for concrete

See section properties below See SAISC Red Handbook See SAISC Red Handbook See SAISC Red Handbook

See SAISC Red Handbook

See SAISC Red Handbook See SAISC Red Handbook



w

kg or J

x(t)

m

∆H

m

∆V

m

ξ Φ ρ

rad kg/m3

µ ω ωE

rad/s rad/s

ωN

rad/s

Mass of explosive per delay ground impact energy Displacement of a degree freedom with time Measured horizontal deflection top of portal column Measured vertical deflection centre of beam of slab Damping ratio Phase angle Density of the material

or of at at

7850 kg/m3 for steel 2450 kg/m3 for concrete

Poisson’s ratio for the material General radial frequency Radial frequency of applied harmonic load Radial natural frequency of structure

Note that the values for elastic modulus, shear modulus and density for concrete quoted above are typical values only. The use of specific aggregates may lead to different values. Warning: Drawings, particularly those from European equipment manufacturers, often give dynamic forces in units of kgf. This must be multiplied by 9,81 to convert to units of N, which can then be used in further calculations.

2.3

Section Properties

Member section properties are generally obtained from standard handbooks of section properties, but some necessary properties are not often listed by these handbooks. The elusive properties are defined here, for simple hand calculation if required. Im is the mass inertia per unit length of the member. This should not be confused with the moment of inertia, nor the bending moment. This value is not always immediately available, but it can be calculated from section properties d2 quoted in the SAISC Red Handbook as:

Im =

m (I + I ) A x y

or

Im = ρ(I x + I y )

t2 d1

For open sections, J may be approximated by:

J=∑

t1

diti3 3



a

For closed rectangular sections with uniform wall thickness, J may be approximated by: b

t

2 2

J=

2ta b a+b



3 VIBRATION DESIGN PROCEDURE 3.1

Step 1: Necessary Data and Information

Get all the relevant data (using any means necessary!). The data must include: (a) (b) (c) (d) (e)

(f)

3.2

Shape, size and mass of the equipment. If the equipment is supported directly on the floor or by means of some kind of springs or dampers. The type, stiffness and damping constant, of any spring supports or dampers. Magnitude of the dynamic loads applied by the equipment. Don’t always just believe the Suppliers on this one. Do your own check calculations. Direction of the dynamic loads applied by the equipment. A screen is generally assumed to apply a vertical load and a horizontal load at each spring. A vertical crusher applies a load that sweeps around 360 degrees in the horizontal plane. The time dependence of the dynamic loads applied by the equipment. By time dependence is meant whether the dynamic load varies harmonically, whether it is suddenly applied and remains constant for a while before it is just as suddenly removed, or whether it is a short duration impact load. If there are any special conditions regarding the allowable amplitudes. For example, is there is any vibration sensitive equipment in the area, or is there an office on a floor structurally connected to the vibrating floor? Ensure that all Suppliers of equipment to be located in the vicinity of vibrating equipment provide written statements of the vibration their equipment can withstand, or ensure that they are informed in writing what vibration their equipment is required to withstand.

Step 2: Clarify Details of the Structure to be Designed

This means both the obvious aspects, such as whether floors will be grating or concrete, the exact positions where columns and beams are required, etc, and the not so obvious aspects, such as where exactly is access necessary preventing the use of bracing. All these may seem simple but it is always surprising how much time is wasted when this type of information has not been obtained from the start. This step also includes getting information about the supports, in order to calculate the spring constants.

3.3

Step 3: Build a Computer Model

Make sure you have read, and digested Sections 6, 8, 9 and 10. You are now ready to build a computer model. Preliminary sizing of members can be done using static design for all members remote from the actual vibrating equipment. For members providing immediate support to vibrating equipment use the equations given in Section 10 to select members having a sufficiently high individual natural frequency. Before proceeding, check that the model looks right. Check the frequencies, check the mode shapes.

3.4

Step 4: Assess the Results

Make sure you have read, and properly understood Section 7 and the implications of (f) above. You are now ready to assess the results being spat out by your computer model.



3.5

Step 5: Prepare Structural Design Calculations and Drawings

Make sure you have read, and can rationally apply Sections 10 and 11. You are now ready to start preparing structural design calculations and approving what has been put onto structural design drawings.



4 GENERAL CONCEPTS AND THEORY 4.1

Dynamic and Harmonic Loads

A load has the right to call itself dynamic if, and only if: (a) (b)

Its magnitude changes in time. The acceleration is big enough to produce significant inertia forces.

FI = Ma M Examples of dynamic loads, shown in Figure 4.1, are: (a) (b)

The load produced by a boulder hitting a grizzly bar. The load applied by a screen on its supporting structure.

Harmonic force

Motor Screen

Height of fall

Springs Grizzly bar

Harmonic Loads Impact Loads

Figure 4.1: Typical Dynamic Loads A dynamic load is called "harmonic" if it varies in time according to:

F = Fo sin(ωt + φ) The values defining this harmonic movement are shown in Fig. 4.2 The relationships between the values of the respective amplitudes used in the analysis of harmonic loads are: Displacement amplitude Linear velocity amplitude Acceleration amplitude Applied load amplitude Inertia force amplitude

do doω ao = doω2 Fo Maoω2

(m) (m/sec) (m/sec2) (N) (N)


Period

RMS

Amplitude

RMS

Time

Amplitude

Peak-to-Peak


Figure 4.2: Harmonic Load Warning: Some people loosely use the term "amplitude of displacement" when they mean peak-to-peak movement (or stroke). Many people also think in terms of “displacement amplitude only”, but amplitude can refer to any harmonic entity. It is prudent to check what the person you are talking to means by the word amplitude. See Figure 4.3 for an example of out of phase movements. (Note that in Figure 4.3 the difference of phase is shown in terms of time).

Difference of phase

Figure 4.3: Oscillations out of phase



4.2

Dynamic Characteristics of Structures

4.2.1 The Single Degree of Freedom System. The most basic dynamic system is one which can move in only one way, or only one direction. This is called a single degree of freedom (SDOF) system. The SDOF system is the only one that we completely understand. However, Murphy's law being what it is, very few of the structures that we analyze can be accurately represented by a single degree of freedom model. See Figure 4.4 for examples of models for any single degree of freedom system (ignoring damping, and replacing mass and stiffness by the real values).

M

M k=

48 EI L3

k k M

These three single degree of freedom models are mathematically equivalent Figure 4.4: Single Degree of Freedom Models We shall establish a series of rules and laws for the single degree of freedom system. Then we will extrapolate these data for multi degrees of freedom systems. The fundamental characteristics of the single degree of freedom system are: Mass, M Stiffness, k Damping, C

(kg) (N/m) (Ns/m)

Then the basic “equation of motion” for the single degree of freedom system can be written as: M&x& + Cx& + kx = F( t )

A structure subjected to an impact and then left alone will vibrate until the cumulative effects of damping will stop it. Without damping the structure will go on shaking until the end of time. This type of movement, vibration when the applied load is zero, is called "free vibrations". If we initially consider “free vibration”, then the equation of motion is: M&x& + Cx& + kx = 0

The solution for this equation will be left to keen students and textbooks. The frequencies of the free vibrations are called natural frequencies. These frequencies are Guidelines for the Vibration Design of Structures – Issue 2 AA BPG S002 17 of 92


what the structure sort of prefers to vibrate at. The basic information to be derived from this equation is the natural frequency, given below for under-damped structures, i.e. where ξ is less than 1,0. ωD =

k C 2 -( ) M 2M

ωD = ω 1 - ξ 2 C C CR

ξ=

The damping ratio, ξ, actually very seldom exceeds 0,1 for normal structures, and it may be as low as 0,01 for fully welded steel structures. When ξ = 0,1 (the maximum likely value), the above equation gives ωN = 0,995ω. This means that for all practical purposes the damped natural frequency may be taken to be the same as the ω. The final important dynamic characteristic of the system, the natural frequency, is thus given by: f=

ω 1 = 2 π 2π

k M

4.2.2 Response to Harmonic Excitation The solution of the equation of motion with zero applied force helps us to understand the dynamic system, but our real aim is to understand how structures respond to applied loads. The simplest applied load is one which varies harmonically. M&x& + Cx& + kx = F0 sin ωt

The final solution is complicated enough, so we will not attempt here to show how it is obtained. The value of x(t) can be shown, (see textbooks) is: x( t ) = e -ξωt (A cos ωt + B sin ωt ) +

F0 sin(ωt - φ) k

(1 - r 2 )2 + (2ξr )2

The first part of this equation is only of interest during start-up or shut-down of the machine because it dies away quickly with time, and we are generally not too worried about these as they generally happen quickly, without causing significant fatigue damage or psychological disturbance to personnel. So what we are interested in is usually only the second part of the equation, i.e.: x( t ) =

F0 sin(ωt - φ) k

(1 - r 2 )2 + (2ξr )2

In most cases, we are only really interested in the amplitude of x, which is: xo =

F0 k

(1 - r 2 )2 + (2ξr )2

4.2.3 Resonance and Tuning In the technical literature one encounters the concept of a dynamic magnification (or sometimes amplification) factor, DMF. An understanding of this factor is the key to grasping the dynamic performance of any structure, even a very complex one. This Guidelines for the Vibration Design of Structures – Issue 2 AA BPG S002 18 of 92


factor is a way of describing the response of the structure to a harmonic applied load. When the system is resonant, i.e. when the exciting frequency, ωE, equals its natural frequency, ωN, the magnification factor may be extremely high (of the order of 40 for welded steel structures, or 12 for concrete structures), whereas it may also drop well below 1,0 when the exciting frequency is much higher than the natural frequency. DMF =

xo = Fo k

1

(1 - r 2 )2 + (2ξr )2

10 9 8 7 6 5 4 3 2 1 0

Dampin g Ratio High tuning

0

0.05 0.08 0.2 0.5

Low tuning

Resonance

Dynamic Magnification Factor

The diagram in Figure 4.5 represents the dynamic magnification factor as a function of the damping and the frequency ratio.

1

2

3

4

Frequency Ratio

Figure 4.5: Dynamic Magnification Factor Tuning is the action by which a structure is designed in a way that would insure that its natural frequency complies with certain conditions. The most usual condition is for the natural frequency to be different from the exciting frequency. If the natural frequency is lower than the exciting frequency, then the system is low tuned, and the frequency ratio is greater than 1,0. A frequency ratio too close to 1,0 still gives a very high dynamic magnification factor, so we generally only talk of a low tuned structure if the frequency ratio exceeds 1,4. If the natural frequency is higher than the exciting frequency, then the system is high tuned and the frequency ratio is less than 1,0. A frequency ratio too close to 1,0 still gives a very high dynamic magnification factor, so we generally only talk of a high tuned structure if the frequency ratio is less than 0,7. The dynamic magnification graph shown in Figure 4.5 shows that for very low tuning the dynamic magnification factor becomes very small. This does not automatically mean that the amplitudes of a low tuned system will be necessarily smaller that the amplitudes of a high tuned system. A low tuned system is a lot more flexible than a high tuned one. The quantity that it amplifies is normally far bigger than in a high tuned system. That quantity can be a displacement, an acceleration, a force, in fact almost anything.



The single degree of freedom system is the only one for which the dynamic magnification factor has an immediate physical meaning. The notion itself can be defined in multi degrees of freedom systems, but its physical meaning will be buried under tons of equations. When the exciting frequency equals the natural frequency the factor above tends towards infinity. This is referred to as “resonance”. The displacements and related values also tend towards infinity. They would reach it, or more probably die trying that is, if not for the effect of damping. Resonance is the situation in which the response of a structure to a dynamic excitation has its maximum value, which for a damped system is inversely proportional to the damping ratio. At resonance r = 1, so that:

DMF =

1 2ξ

The effect of damping at resonance is shown in Figure 4.6. When there is no damping present, the amplitude of vibration response to any applied harmonic load keeps increasing indefinitely. When there is damping present, the amplitude of the vibration response increases up to a specific maximum value, then remains constant at that value. 4.2.4 Damping Damping is the property of materials to absorb energy by internal friction. Contrary to conventional wisdom, damping is not an unconditional blessing; when close to resonance it reduces the dynamic forces, but when far from resonance in the low tuning range, damping actually increases forces. Its effect is like a slight stiffening of the springs on which the equipment is supported. There are many ways to model the damping. None is perfectly accurate. The most popular seems to be to model the damping effect as a force proportional to the linear velocity. This is called viscous damping. Accurate or not, it will have to do. We will use this model from two reasons. First, for our purposes it is accurate enough. Second, using a more accurate damping model would lead to horrifyingly complex mathematical developments. There is another significant value that has to be defined: the critical damping. It means the highest damping value that allows the system to oscillate. At more than critical damping the system becomes so sluggish that it is no more able to follow the oscillations of the exciting force. It just tries to sort of slowly crawl back to its initial position. Critical damping is not very important for us. Steel damping is very much smaller. For a single degree of freedom system the critical damping is: C R = 2 kM

Damping is primarily a characteristic of the material, and the connections. For each type of structure the damping can be expressed as a fraction (or percentage) of the critical damping. For steel the damping is between 1% and 6%. For concrete it is between 6 % and 10%. For rubber it is about 30%, but then structures are not made of rubber.



50 40 30 Response

20 10 0 -10 0

2

4

6

No damping With damping

-20 -30 -40 -50 Time (seconds)

Figure 4.6: Resonance with and without damping A widely spread misconception about damping is to assume that if a structure is made from a material with high damping then that damping will automatically influence all dynamic deflections of that structure. This is not necessarily true, as damping only becomes effective when there is movement. For example, consider a structure consists of a square concrete floor supported by four concrete columns, one in each corner, as shown in Figure 4.7. It can be described as some kind of upside down pendulum, with the columns representing the beam of the pendulum and the whole mass of the floor as a lumped mass at the end of the pendulum. This structure supports equipment that produces horizontal dynamic forces. The horizontal amplitudes are unacceptable. What can be done about this?

Dynamic Force

Concrete Floor

Figure 4.7: Effect of Additional Mass Conventional wisdom would suggest that some concrete could be added to the floor so that the damping characteristics of the concrete can reduce the amplitudes. The part about damping is complete nonsense. Damping is caused by the internal friction within the material. For damping to act the element has to deflect. In the example in Figure 4.7 Guidelines for the Vibration Design of Structures – Issue 2 AA BPG S002 21 of 92


the floor and the additional concrete are moving horizontally like a rigid block. Since there is effectively no deflection of the concrete floor, damping is almost completely inactive, and it can hardly play any role. This example intends just to clarify the concept of damping. It does not matter what action has to be taken to improve the performance of the system nor if increasing the mass has a favourable effect or not. 4.2.5 Multi Degree of Freedom Systems A general structure can move in many different ways, and in many different directions. This is called a multi degree of freedom (MDOF) system. The basic concepts described for SDOF systems can be transported to MDOF systems as well. By writing the equations of motion for each degree of freedom we obtain a system of linear differential equations that describes the movement of the model under harmonic forces, and whose solutions describe the motion at every degree of freedom. If we put the condition that the exciting forces are all equal to zero then we end up with a system of equations giving us the free vibration of the structure. The mathematical condition for this system to have non-trivial solutions is that the determinant of the characteristic matrix be zero. If the number of degrees of freedom is n then we have to solve an equation of the nth degree. The solutions are the n eigenvalues. We replace the eigenvalues in the system and find the n eigenvectors. The eigenvalues are the radial natural frequencies, the eigenvectors are the mode shapes. Remember that the exciting forces have been set to zero. The non-trivial solutions, the eigenvectors, or mode shapes, are ratios of displacements, describing the shapes in which the structure will vibrate when no force is applied. Thus the mode shapes do not give actual displacements in any physical units. There are as many natural frequencies as there are degrees of freedom. In free vibration, the structure will tend to vibrate with the fundamental (i.e. the lowest) natural frequency. If there are some applied dynamic forces then the structure will try to vibrate according to the mode shape whose natural frequency is closest to the frequency of the dynamic forces, and the frequency will always be the frequency of the exciting forces. A structure subjected to a periodical dynamic loading will vibrate with the same frequency as the loading. That frequency is called the exciting frequency, or the forcing frequency. Harmonic loading is the most common example of periodic loading. 4.2.6 Mode Shapes and More about Natural Frequencies It would be "comfortable" to express a MDOF model in terms of a number of SDOF models, which are easy to solve, as shown in Figure 4.8. Then solve them, add up the results and find the solution for the complete structure. This is where the mode shapes come in. Each mode shape is such an imaginary single degree of freedom system. Solve them, add up the results (amplitudes) from each one and you have the total results for your structure! Is it truly that simple? It is basically true but far from being simple. So put your thinking cap on and keep reading. Consider a MDOF system or model. Imagine a set of links that will force the displacements of each node to have always a fixed ratio to the displacements of each other node. For example, if the displacement of node 5 is 2,0 and the displacement of node 7 is 2,5 then the ratio (2 / 2,5 = 0,8) will stay true, no matter what the magnitude of each displacement is. If node 5's displacement is, say, 1,7 then the displacement of node 7 will be 1,7 x 0,8 = 2,13. The displaced position of the transformed system can then be defined by single parameter. As stated earlier, this means that the new system has only one degree of freedom. Guidelines for the Vibration Design of Structures – Issue 2 AA BPG S002 22 of 92


9 8

Height (m)

7 6

Mode 1

5

Mode 2

4

+

+

3

=

Mode 3 Overall

2 1 0 Mode Shape

Figure 4.8: MDOF System Represented as Several SDOF Systems The computer will calculate the ratios between displacements and will print the results. This is done according to certain rules. Those rules are not relevant for us. The result will be a number of shapes equal to the number of degrees of freedom. These are the mode shapes, also known as the eigenvectors. Each mode shape comes with its own unique, personal and confidential natural frequency (also known as an eigenvalue). The intractable multi-degrees-of-freedom system has been magically transformed in a sum of comfortable single degree of freedom systems. The natural frequencies associated to each mode shape are the natural frequencies of the system. The next step is to calculate the responses of each of the single degree of freedom system and to sum them up. The result will be the response of the complete structure. To apply this method manually is not as simple as it seems. However, at this stage the problem is to understand the concepts, not yet to apply them. The figure above is not really correct, since it gives the impression that the amplitudes are the sums of the mode shapes. The components of the mode shapes are not "displacements". They are just non-dimensional ratios. The method just described is called modal analysis. It can be summarized as follows: The big, ugly, mean and hairy MDOF system is broken down into a number of cute SDOF systems. The SDOF systems are solved one by one and the results are summed up. The messy job of calculating eigenvalues and eigenvectors and then summing them, is done by the computer. The designer just types in the data in whatever format the program requires. By comparing the mode shapes with the pattern of the amplitudes one can grasp intuitively which modes are important in the response and which are not. It also becomes clear what shapes must be changed. It is true that a modification to a mode shape will usually influence other modes as well. The structure will be tuned by a series of successive approximations. It still beats groping in the dark.



5 LOADS The dynamic loads applied to structures are sometimes quite easy to obtain, from simple equations or from the equipment Suppliers. Other dynamic loads must be dragged out, kicking and screaming, from the equipment Suppliers or somewhere else. To get the dynamic reactions from a Supplier is arguably the most frustrating action in structural dynamic design. The amount of weird data that sometimes floods us in answer to technical questions is hardly believable. The following conversation is absolutely authentic: Q: What forces does your equipment apply to the supporting structure? A: Well, our machine works very quietly. You can put a glass of water on top of it and you will hardly notice a few ripples on the water surface. Q: So the dynamic reactions are so low that they could be ignored? A: Well, we specify a vertical dynamic force of about 150 kN at 1000 rpm. (It is doubtful that even Fort Knox could withstand this kind of loading. The ripples would be on the surface of the planet, not on the water in a glass) Warning: Do not simply accept the loads specified by Suppliers of equipment. Local Suppliers are often only agents for equipment sourced from overseas, so they may not be Engineers, there may be confusion of units from overseas countries, etc. It is always prudent to check that the loads specified make sense. Only then should they be used as if they are accurate. Some guidance to determination of dynamic loads is thus provided here. The vibration induced by various types of machinery is frequently of concern in the structural design of buildings, in particular industrial and mining buildings. The magnitudes of these vibrations are determined by the nature of the machines themselves and how they are supported on the structure. Both of these influences will be considered below.

5.1

Rotating Unbalance

Rotating machines are designed to run at a constant speed for a long period of time. In the case of some machines the intention is that eccentricity should be eliminated if possible. These include turbines, axial compressors, centrifugal pumps, generators, electric motors and fans. In the case of other machines, eccentricity is deliberately introduced in order for the machine to function. These include vibrating screens, and vibratory feeders. 5.1.1 Motors and Turbines Theoretically, it may be possible to eliminate all unbalance, but in practice it is impossible. Static unbalance occurs when the centre of mass of a machine rotor does not coincide with the axis of rotation. The term "static" refers to the fact that static forces, eg gravity, can pinpoint this condition. Gravity will usually cause the out-of-balance rotor to rotate to a position of static equilibrium in which the "heavy" side of the rotor is at the bottom. Dynamic unbalance occurs when two or more masses in different planes on the rotor, produce a moment when the rotor is rotating. In the simplest case of two masses at 180o to each other, in different planes, the rotor may be statically balanced, but will tend to rock in the bearings when rotating.



Figure 5.1 : Rotor Unbalance to ISO 1940



Table 5.1 : Quality Grades for Different Machines to ISO 1940 Balance Quality Rotor Equipment Types Grade, G ______________________________________________________________________ G 4000 Crankshaft drives of rigidly mounted slow marine diesel engines with uneven number of cylinders. G 1600 Crankshaft drives of rigidly mounted large two-cycle engines. G 630 Crankshaft drives of rigidly mounted large four-cycle engines. Crankshaft drives of elastically mounted marine diesel engines. G 250 Crankshaft drives of rigidly mounted fast four-cylinder diesel engines. G 100 Crankshaft drives of fast diesel engines with six or more cylinders. G 40 Crankshaft drives of elastically mounted fast four cycle engines (petrol or diesel) with six or more cylinders. G 16 Drive shafts (propeller shafts, cardan shafts) with special requirements. Crankshaft drives of engines with six or more cylinders, under special requirements. G 6.3 Marine main turbine gears. Fans. Flywheels. Pump impellers. Normal electrical armature. Individual components of engines under special requirements. G 2.5 Gas and steam turbine, including marine turbines. Medium and large electrical armature with special requirements. Small electrical armature.

Both static and dynamic unbalance manifest themselves as vibration at the running speed of the rotor. The reason for this is simply that in both cases the centrifugal force due to the eccentric mass is rotating at the running speed. The actual amount of unbalance present may be difficult to ascertain, as manufacturers are often reluctant to admit that their machinery has any unbalance. ISO 1940 provides some guidance on the balance quality of rotating machines, giving the residual unbalance mass as a function of speed, as shown in Figure 5.1. Different curves in this figure are appropriate for different quality grades, which apply to different types of machine, as listed in Table 5.1. The eccentricity to be used in a particular design may be obtained from the Supplier of a rotating machine, but failing this, the ISO approach, using Table 5.1 and Figure 5.1, can be used to give an appropriate design value. The forces due to this unbalance are given by: o

Fx ( t ) = Meω 2 sin(Ωt ) 0 transverse to rotor axis

Equation 5-1

Fy ( t ) = Meω 2 cos(Ωt ) 90o transverse to rotor axis

Equation 5-2

These two forces will always act at the same time, as the rotor turns. 5.1.2

Vibrating Equipment Guidelines for the Vibration Design of Structures – Issue 2 AA BPG S002 26 of 92


Vibrating screens and vibratory feeders typically consist of a mesh or solid bed within a rigid frame which is supported on a number of springs. They rely on motion of the bed in order to operate. Eccentricity, which has the same effect as unbalance, is thus deliberately introduced on motors mounted below the bed, usually in the form of two semi-circular masses, as shown in figure 2. These masses can be set at varying positions to give different values of eccentricity, and hence different force amplitudes. Most commonly, two electrically coupled motors are used, running in opposite directions. This has the result that the forces applied transverse to the axis of the machine by the two motors exactly oppose each other, giving a nominally zero resultant. The forces applied parallel to the axis of the machine by the two motors reinforce each other, doubling their effect. Thus: Fl ( t ) = 2Meω 2 sin(ωt )

(longitudinally)

Ft ( t ) = 0

(transversely)

5.2

Figure 5.2 : Eccentric Masses

Loads Applied to the Structure Warning: The loads defined here are in general NOT the loads applied to the structure. They are applied to the machine, which responds dynamically. Go on to the next section before assuming you know all about vibration loads.

5.2.1 Data Required From the Equipment Supplier An assumption as popular as it is wrong is that the dynamic reactions of a screen supported on springs or buffers are parallel to the exciting force produced by the screen's vibrator. This means that if the exciting force is applied at, say, 30 degrees then the dynamic reaction will also be applied at 30 degrees. This is usually not true because the springs supporting the screen have different stiffnesses in the vertical and horizontal directions, leading to different dynamic magnification. Another popular superstition is that the dynamic reactions equal the weight of the machine times some “safety factor” up to 10. This nonsense comes from confusing something called "equivalent static load" with a dynamic reaction. An equivalent static load is used mainly cover up ignorance or laziness, or sometimes to check a foundation for overturning. A dynamic reaction is a periodical quantity, more often than not harmonic. An equivalent static force is an imaginary STATIC loading. So what do we need from the equipment Supplier in order to design the supporting structure?



(a) (b)

Either the direction, size and position of the forces applied by the equipment Or appropriate data to enable us to calculate them.

For (a) above, we need the: (i) (ii) (iii) (iv)

working frequency of the equipment dynamic reactions of the equipment mass of the equipment mass moment of inertia and position of the centroid

For (b) above, we need the: (i) (ii) (iii) (iv) (v) (vi)

working frequency of the equipment mass of the equipment (kg) mass moment of inertia and position of the centroid static loads at each corner (N) spring constants (vertical and horizontal) (N/m) magnitude, position and direction of the exciting force (N)

In both cases, general data for the equipment may be required as well, not necessarily Warning: It is extremely important to have the Supplier approve the loads we have calculated. If the loads we use in design are not approved by the Supplier, then we could be left holding the baby. As soon as that screen as little as hiccups, for whatever reason, everybody could shrug and say "Not my problem! I told you so. What do you intend to do to fix it?" We cannot let them say this. This is our line. for dynamic analysis. 5.2.2 Calculation of Spring Stiffness But sometimes the Supplier is unable to provide the spring constants. They can be calculated from the following equations: (a) Steel springs The axial stiffness for steel coil springs is given by: kV =

Gd 4w

Equation 5-3

8D 3 n

The horizontal stiffness for steel coil springs is given by: kH =

kV h 0,385α[1 + 0,77( ) 2 ] D

Equation 5-4

where: n is the number of free coils as shown in Figure 5.3 α is a coefficient obtained from Figure 5.4 Other symbols are as defined in the symbols list.



p = Pitch, d = Diameter of wire, n = Number of free coils Plain ends Plain and Squared Squared and ground ends ground Total coils n n n+2 n+2 Solid length (n+1)d nd (n+3)d (n+2)d Free length np+d np np+3d np+2d Figure 5.3: Steel Coil Springs

α

c/h=0,5 c/h=0,4

c/h=0,3 c/h=0,2

c/h=0,1

Figure 5.4: Values of α for Equation 5-4 (b)

Rubber buffers buffer height (mm) diameter (mm) characteristics of the rubber used



Note: It is good practice to check rubber buffer characteristics with the Supplier because rubber properties are known to vary widely from batch to batch. The dynamic stiffness of rubber buffers is significantly higher than the static one. A study done by Anglo American Corporation in cooperation with VELMET showed that the dynamic stiffness may be as much as 60% more than the static stiffness. (REPORT RAB/83/02 21 JULY 1983 "DYNAMIC STIFFNESS TESTS ON VELMET SCREEN SUPPORT SPRINGS: VERTICAL STIFFNESS"). The horizontal stiffness of a rubber buffer is typically about one third of the vertical stiffness. (c)

Special springs consult the spring Supplier

It also sometimes happens that the Supplier is unable to provide the magnitude of the exciting force. In this case, the desired dynamic reactions can be calculated as demonstrated in the following examples: Warning: Even when the stiffness of the springs is given by the Supplier, we cannot automatically assume that they are correct. The Suppliers often underestimate the horizontal stiffness of a steel spring. It is prudent to check the information supplied! 5.2.3 Example 1 Information provided by Supplier Torque: T = 21 kgf.m = 206 Nm Mass of screen: M = 3000 kg Weight of screen: W = 3000x9,81 = 29430 N Steel springs, two springs per corner Spring wire diameter: d = 0,020 m, height, h = 0,260 m No of free coils: n = 7,5 Spring outer diameter = 0,144 m Spring diameter: D = 0,144 – 0,020 = 0,124 m Exciting force at 65 º to horizontal (a)

Step 1: Calculate spring constants

The axial stiffness for one spring is: kV =

78 x10 9 x0,02 4 8 x0,124 3 x7,5

= 109 x10 3 N/m

The static compression of a spring is: W /8 29430/8 = = 0,0338 m KV 109 x10 3 c 0,0338 = = 0,13 h 0,260 h 0,26 = = 2,09 D 0,124

c=

The horizontal stiffness for one spring is: kH =

109 x10 3 0,26 2 0,385 x1,5[1 + 0,77( ) ] 0,124

= 43,0 x10 3 N/m



(b)

Step 2: Calculate the stroke, S (the peak to peak displacement of the screen at 65 degrees) S=

T 206 = = 0,007 m W 29430

The amplitude of displacement at 65 º is: The vertical amplitude of displacement is: The horizontal amplitude of displacement: (c)

d0 d0V d0H

= S/2 = 0,0035 m = d0 sin(65) = 0,00317 m = d0 cos(65) = 0,00148 m

Step 3: Calculate the dynamic reactions per corner (2 springs): Rv = 2x109x103x0,00317 = 691 N (vertical dynamic reaction) Rh = 2x43x103x0,00148 = 127 N (horizontal dynamic reaction)

5.2.4 Example 2 Consider the same screen as that in Example 1. Assume that this time we know the exciting force FO, and not the torque T. The other data stay the same. The information from the Supplier is: FO = 90456 N (at 65 º) The exciting screen runs at 900 rpm. ωE = 94,25 rad/sec Spring data is the same as above. (a)

Step 1: Calculate the vertical and horizontal components of FO. FOV = FOsin(65) = 81981 N FOH = FOcos(65) = 38228 N

(b)

(c)

Step 2 : Calculate vertical and horizontal natural frequencies of the screen on springs. Work with total mass of screen and with all 8 springs. Do not attempt to split these – they all form one single screen. KV and KH have been calculated in Example 1 above. ω NV =

8k V = M

8 x109 x10 3 = 17,05 rad/s 3000

ω NH =

8k H = M

8 x 43 x10 3 = 10,71 rad/s 3000

Step 3: Calculate the dynamic reactions on corner (2 springs) RV =

RH =

FOV / 4 ωE 2 2 ) ] [1 ( ω NV FOH / 4 ωE 2 2 ) ] [1 ( ω NH

=

=

81981 / 4 94,25 2 2 ) ] [1 ( 17,05 38228 / 4 94,25 2 2 ) ] [1 ( 10,71

= 693 N

= 125 N

The results are practically the same as the results of Example 1. Note that the angle of the total dynamic reaction (i.e. the resultant of RV and RH) to the horizontal is quite different from the angle of the exciting force to the horizontal, because of the different dynamic magnification in the two directions. Guidelines for the Vibration Design of Structures – Issue 2 AA BPG S002 31 of 92


Angle of exciting force Angle of dynamic reaction

= 65 º = atan(RV/RH) = 79,8 º

What if the Supplier doesn’t know the applied torque, the exciting force, or the mass of the equipment? The Supplier has to know something!

5.3

Impact Loads

Impact loads are defined as those loads which are applied to structures for a short time only, or which are suddenly applied to the structure. 5.3.1 Types of Impact Loads The first possible source of impact is motion. The motion may be the movement of some vehicle, which causes impact for example when a train collides with a station stopping device. Alternatively, it may be due to a mass falling onto a structure below, such as for example, when a conveyor belt breaks and its tensioning counterweight falls. The important variables in determining the magnitude of this force are: (a) (b)

The impact velocity, v. This may be well known in certain instances, but in other cases it may be necessary to make reasonable assumptions. The distance over which the moving body is stopped, D. This is determined either by the spring stiffness of the structure or buffer, or, in the case of plastic deformation of either the structure or the moving body, by the extent of the plastic deformation.

5.3.2 Energy Equations It is usually appropriate to use energy considerations to calculate motion impact forces. The equations are given in Table 5.2. The information in Table 5.2 gives the deflection under impact and the maximum impact force. It is also often convenient to define an impact factor, as: α=

F Mg

If the impact factor, α, is known then the impact force is: F = αMg

Impact energy absorption may be either elastic or plastic strain energy. Where the energy is absorbed by elastic strain energy, there is no permanent deformation of the buffer. Where the energy is absorbed by plastic strain energy, the energy is absorbed mainly as work done in causing the plastic deformation. There will also inevitably be a certain amount of elastic strain energy, but this is usually small enough to be neglected. As the deformation is plastic, it may be assumed that the force remains constant, at the yield strength of the deforming member



Table 5.2: Energy Equations for Impact Equations Motion Impact Impact energy Kinetic Energy = 21 Mv 2

Falling Impact Potential Energy = Mg(H + D)

Elastic Energy Absorption (i.e. no permanent deformation) Absorbed energy Strain Energy = 1 kD 2 2

Conservation of energy during impact event

1 2 2 Mv

=

D=v

M k

Impact force

F = Dk

1 2 2 kD

D=

α=

kD 2

Mg 2kH 1+ 1+ k Mg

F = Dk

= v Mk

Impact factor

1 2

Mg(H + D) =

F = Mg 1 + 1 +

v k g M

α = 1+ 1+

2kH Mg

2kH Mg

= 1+ 1+

2H D

Plastic Energy Absorption (i.e. permanent plastic deformation) Absorbed energy Strain Energy = FD Impact force F is defined by the plastic strength of the buffer Mg(H + D) = FD Conservation of 21 Mv 2 = FD energy during H Mv 2 D= impact event D= F 2F

Impact force Impact factor

Mg

1

F=

Mv 2 2D

F = Mg 1+

α=

F v2 or Mg 2Dg

α=

H D

F H or 1 + Mg D

This impact factor is plotted in Figure 5.5, for different ratios of the drop height to the static deflection, H/D.



6

Impact factor

5 4

Elastic impact Plastic impact

3 2 1 0 0

1

2

3

4

5

Drop height to Deflection ratio, H/S

Figure 5.5: Impact Factors for Falling Bodies 5.3.3 Moving Mass Hits Stationary Mass A final impact problem is the elastic impact force, and the resulting velocity when a moving mass impacts a stationary mass, and both masses continue in motion at a reduced velocity. The new, reduced velocity is: v S = 1,5

Mv M + MS

The force applied at the point of impact is: F = vS

5.4

MS k 2

Ground Motion from Blasting and Piling

5.4.1 Basic Equation The ground motion generated by blasting, piling or other ground impact condition, is approximately described empirically by the following equation: v P = Ca (

W Cb ) a

where: a is the distance away from the point of blast or ground impact Ca is a site constant defined below Cb is a site constant defined below vP is the ground peak particle velocity (m/s) W is the mass of explosive per delay (kg) or the impact energy (J) 5.4.2 Blasting The following values may be used for blasting in typical ground conditions: W Mass of explosive per delay (kg) Ca 1,000 Cb 1,667 Guidelines for the Vibration Design of Structures – Issue 2 AA BPG S002 34 of 92


5.4.3 Piling The following values may be used for piling in typical ground conditions: W Energy of falling pile hammer (J) Ca 0,001 Cb 0,770



6 STRUCTURAL MODELLING AND RESPONSE 6.1

Modelling the Structure

The time has come to introduce a few new concepts related to taking a real, physical continuous structure of steel and concrete, with paint, bolts, spillage and pigeons, and turning it into something that a computer can deal with in discrete chunks. This is generally referred to as “modelling the structure”. 6.1.1 Models The first thing to recognise is that any real structure is an infinite continuum. It could thus take quite a while, i.e. an infinity, to solve for each structure. That's where the model kicks in. A model is a simplified mathematical representation of the real structure. The use of models is not exclusive to dynamic analysis. They are used in static analysis as well. When we calculate the bending moments in a straight beam we assume that the beam only has one dimension, i.e. length. There is no such thing in nature. It is we who create this imaginary entity, the model, in order to approximate stresses, displacements etc in the real structure. The only difference between a static model and a dynamic model is that the dynamic one has to include the effect of mass and sometimes damping. It is a difference of detail, not of essence. The concept is the same. 6.1.2 Degrees of Freedom The deflected state of a structure is defined by certain parameters, usually the displacements and rotations at joints, or nodes in the model. Each of these defined parameters is called a “degree of freedom”. The simplest structure conceivable has one degree of freedom. Real, useful structures may have hundreds or even thousands, of degrees of freedom. The minimum number of independent parameters that completely defines the deflected structure is the number of degrees of freedom. Consider, for example, a pendulum with three masses lumped along its length, as shown in Figure 6.1. If the beam of the pendulum is infinitely rigid then one parameter, the rotation about the pinned support, will completely define the displaced position of all the masses. The correct model will have one degree of freedom. If, however, the beam has a finite stiffness then one parameter is not enough to completely describe the deflected shape. Considering the pendulum in 2D, each mass can move sideways, and vertically, and rotate. Each mass thus has three degrees of freedom, and the full model has nine degrees of freedom. Generally, structural members are far more flexible in bending than axially, so we may decide that for the purposes of analysing this pendulum, we can justifiably assume that the links are infinitely rigid axially, but flexible in bending. In this case, each mass has only two degrees of freedom, sideways movement and rotation, so the complete model will have six degrees of freedom. If we believe that it is necessary to analyse the behaviour of this pendulum in 3D, then each mass has six degrees of freedom (i.e. displacement along, and rotation about, each of the three principal axes). The complete model would then have eighteen degrees of freedom. The first model has one degree of freedom. The second has up to eighteen. Both represent the same structure. Which one to use is a matter of engineering judgement. It is YOU who have to decide. There is no single reliable fool-proof rule that can solve this problem. Having said this, it is only fair to add that in most cases a normal dose of common sense should be enough to solve the problem.



Infinitely rigid links. One degree of freedom.

Flexible links. Up to nine degrees of freedom.

Figure 6.1: Simple Pendulum Degrees of Freedom A degree of freedom refers either to a joint displacement or to a joint rotation. If it refers to a displacement then the quantities involved are length and mass. If it refers to a rotation then the quantities are angle and mass moment of inertia. The bigger the mass, the bigger the force required to impose a given linear acceleration. In a similar manner, the bigger the mass moment of inertia, the bigger the moment required to impose a given angular acceleration. There is no mathematical or other difference between these two types of degrees of freedom. They are conceptually identical. Do not confuse moment (force x length) with mass moment of inertia (mass x length2). Although they are almost homonyms, they are also completely different animals. Warning: When modelling any structure, THINK! When you have finished thinking, ask some questions, then THINK AGAIN! 6.1.3 Modelling Structural Geometry Now that we understand what a degree of freedom is, we need to figure out where to put them. Wise men (wise guys?) have conceived many mathematical algorithms and rules to enhance the similarity between model and physical structure, and several packages now automatically arrange the nodes within the structure. These in-built rules are a great help, but they only apply after the Designer has decided upon questions such as which parts of the structure should the model include, and whether to put nodes along the span of each beam. The first thing to remember is that a simple structure is easy to understand, whilst a complex structure is complex. As the power of computers has increased, almost all packages now run 3D models, rather than 2D models, and it is often assumed that this gives better results. In general it doesn’t, because models are inevitably much more complex now, so the predicted behaviour may be difficult to understand. If your package offers the possibility of doing a 2D model, in most cases this will give better results, because you will understand them. The second thing to remember is to sit back, relax, look at the structure and decide how it is going to respond to the applied dynamic loads. If this doesn’t help, then stick to static analysis and design, and find someone who can visualise the response. Any part of the structure that is moving and bending must be modelled as a member with accurate Guidelines for the Vibration Design of Structures – Issue 2 AA BPG S002 37 of 92


mass and stiffness. Any part of the structure that is moving but not bending must be modelled with accurate mass, but stiffness is irrelevant. Any part of the structure that is neither moving nor bending is irrelevant, and it can be ignored, or modelled as a support to the structure if necessary. Generally, cladding and flooring on structures is not modelled, but remember that this has implications. Floor beams and sheeting rails in a physical structure cannot move sideways, because the cladding or the floor prevents this. However, if the cladding or the floor is not modelled, the model allows these very laterally flexible members to flop around at rather low frequencies, the structure cannot be high tuned, and so Designer panic sets in. There is a simple solution to this dilemma. Don’t model sheeting rails or secondary floor beams either, but remember that the cladding, floors and secondary steelwork do have some mass that must be modelled as lumped masses. When creating a 3D model, also remember that a concrete or steel plate floor has very significant diaphragm stiffness, whose omission may allow the model to develop bogus modes of behaviour. It may be necessary to introduce some imaginary cross bracing into the model to protect against this happening. 6.1.4 Modelling Mass There are generally two approaches to modelling mass in structures. The simpler one, the lumped mass method, simply calculates the mass of each element of the model, and puts half of that mass at each end. It is no problem to model the lumped masses that physically appear in the real structure. This is accurate for blobs of material that physically occur at one place, but it is less accurate for mass distributed along the element, mainly because the rotational effect of the mass about the joints is ignored. This leads to the introduction of "consistent mass" method, which is based on mathematical procedures that recognise the actual location of mass throughout the structure, but are beyond the scope of this guide. This generally gives the best model of the real masses in a structure, but it does take more computer memory and more time for the analysis. However, with a modern computer, the time required for the analysis will be similar for each of the three methods. Well, the difference could be 2000%, but this would mean 40 ms (milliseconds) instead of 2 ms. One must really be in a hurry for this to matter. Almost all commercial dynamic analysis packages now use the consistent mass method. The thoughtful Designer will now be realising that the imposed loads generally applied to structures have an associated mass, and will be asking whether this must be included in the model. Absolutely maybe! A sound dose of engineering judgement is required here. First, remember whether the structure is high tuned or low tuned. Where a structure is low tuned, much of the force causing the vibration is being resisted as an inertia force, accelerating the mass. Extra mass is thus beneficial, and will reduce the dynamic amplitudes. However, if the intention is to high tune a structure, extra mass will make it more difficult to achieve high tuning. Then having achieved high tuning, extra mass will move the structure towards resonance, leading to an increase in the dynamic amplitudes. The normal recommendation is that if a structure is high tuned, then lumped masses equivalent to approximately 20 % of the specified imposed load should be used, but if the structure is low tuned, then no additional mass should be added. However, this is not a hard and fast rule.



6.1.5 Modelling Materials The choice of characteristics for the material being used is the next issue to resolve. Steel, being a reasonably agile material reacts fast to stimuli, so its dynamic properties are generally very similar to its static properties. Concrete, on the other hand is in a different category. Concrete is more of a laid-back material, so it takes a harder push to get it going. The dynamic stiffness of concrete is thus higher than its static stiffness, by about 10 % to 20 % for the frequency ranges typically encountered in mining type structures. So always add VAT to the elastic modulus of concrete to get the dynamic elastic modulus to use. E Cdyn = 1,14xE cstat

Equation 6-1

6.1.6 Modelling Connections There are six possible movements at each connection, i.e. displacement along, and rotation about each of the three axes, as shown in Figure 6.2. The fixity of the connection relating to each possible movement must be established. Z -axis

Y -axis

X -axis

Figure 6.2: Possible Movements at Each Connection Generally the dynamic displacements we deal with are small, and under these conditions most connections behave as if they are rigid connections, but not in all cases. Recommendations for different types of connection are given in Table 6.1. Table 6.1: Recommended Connection Fixity Connection type Simple 2 bolt shear connection, with thin end plate free of flanges Simple shear connection with more than 2 bolts, and thin end plate free of flanges Simple shear connection, with stiffened end plate Full moment connection

Displacement along: X Y Z

Rotation about: X

Y

Z

Free

Rigid Rigid Rigid

Free

Free

Free

Rigid Rigid Rigid Rigid

Rigid

Rigid Rigid Rigid Rigid Rigid

Rigid

Rigid Rigid Rigid Rigid Rigid

Rigid

6.1.7 Modelling Floors Three types of approximation are routinely made by practically everyone when designing a steel floor supporting screens, feeders, or other materials handling equipment: Guidelines for the Vibration Design of Structures – Issue 2 AA BPG S002 39 of 92


(a)

(b)

(c)

The stiffness and mass moment of inertia of the underpans, chutes and similar stuff are often ignored, but they should be considered. Logically, these items should be modelled in the same way as other equipment, but this requires awkward calculations such as establishing the mass moments of inertia and stiffness of the chute. An acceptably accurate analysis will result if the chute mass, with or without contents, is modelled as lumped at its centre of gravity and connected to the support points by rigid links, pinned at their ends. If the structure is high tuned, then the chutes and under pans should be modelled with their full operational contents. If the structure is low tuned, then the chutes and under pans should be modelled empty. A floor may be modelled accurately as a plane grid. What a computer program means by plane grid may sometimes be subtly different from what a mere human assumes. Here is a reminder of what the machine is doing: • Any horizontal translations are ignored • No horizontal loads can be modelled • Any moments about the vertical axis are ignored • The supports restrict vertical translations and/or rotations about the horizontal axes The model must use both the elasticity of the columns and the masses outside the model but supported by those columns. The column stiffness (spring constant) is: Warning: As far as the computer is concerned the quantities ignored are NOT zero. They simply do not exist. If a designer thinks that horizontal vibrations are significant for his structure then he must not use a grid but some other type of structure, usually a 3D frame! k = EA / H where:

E is the elastic modulus A is the column area H is the column height

To ignore it is not acceptable, especially when using a modern computer. To decide what masses to consider supported on that column may be quite complex and involve more engineering judgement than rational assessment. For the initial sizing of a beam supporting vibrating equipment, the following formula can serve as a first shot at the depth: L2 L2
where: D is the depth of the beam (mm) L is the span of the beam (mm) Sectional properties are generally calculated as they would be calculated for static analysis. The only exception to this may be where composite sections (concrete floors on steel beams) are used. See Section 8.2.



When building a computer model, the definition of member releases is crucial to its success. Figure 6.3: Typical Floor Grid Connections If two beams 1 and 2 are framed at the same point into another beam 3 using pinned connections as shown in section B-B in Figure 6.3, then only one of the two must be released. The release must be in the direction of the relevant rotation. If no release is

Full deph end plate

3 A

A 1

2 B

Do not release

B

Moment connection Do not release

Partial depth end plate Do not release Release Block rotation of all beams supporting floors

Section A-A

Release one side only Section B-B

given then the machine will assume that the two beams are continuous over the support. If both are released then the machine will assume that the beam they are framing into is not supported against rotation, and it will put a rotational degree of freedom that is not really there. Beams 1 and 2 are physically identical, you may argue, so why is it correct to model beam 2 as simply supported while beam 1 is fixed to beam 3? The answer is that, for open section beams, the torsional moment of inertia is very much smaller than the bending moment of inertia. Therefore the restraint imposed by beam 3 on beam 2 is negligible. However, beam 2 does effectively restrain beam 3 against torsional rotation. It does not matter whether the beam released is 1 or 2. If beam 3 were to be a closed box section, then it has a much higher torsional stiffness, and its effect on the end conditions of beams 1 and 2 would have to be considered. A similar situation occurs when there are nodes placed along the span of a beam (to provide amplitudes of vibration at those points), or when the computer model uses a consistent mass formulation. The computer will insert a rotational degree of freedom wherever it can, and unsupported beams will completely mess up the image of the eigenvectors and eigenvalues, producing phantom modes that actually do not exist. This is not a real situation, because the floor will physically restrain the beam from vibrating in rotation about its own axis. The solution is to define fictional supports, restraining only the direction of the appropriate rotation.



6.2

Composite Beams and Floors

This is an uncertain area, because concrete is not as well behaved as steel, and the connection between concrete and steel may, or may not, transmit vibration stresses. Within ATD Structural Engineering, we use an adaptation of the SANS 10162-1 code of practice, Section 17 “Composite Beams”. This has the advantage of being both userfriendly and accurate enough. Warning: It can be bad news to use concrete in low tuned structures. Concrete work is less accurate than steel. Contractors are required to produce concrete slabs of at least a minimum thickness, and having a cube strength of at least a certain amount. The mass increases in proportion to the thickness, but the stiffness increases in proportion to the cube of the thickness. The elastic modulus of concrete, unlike steel, increases with increasing cube strength. At the low amplitudes typical of industrial vibration, the friction between the concrete and the steel beams is likely to transmit vibration strains, even in the absence of shear connectors, leading to an effective stiffness that may be well above what was predicted and used in the computer model. These factors mean that the actual frequency of the asbuilt beast may be well above the neat computer prediction. Think and plan your modelling carefully! The components of a general composite structure are shown in Figure 6.4

Internal slab W1 W2

T A

X

C1

Edge slab

C2

C3

S

Figure 6.4: General Composite Structure In the analysis of a structure with composite beams, we first need to assess the effective section properties to be used. This is done by the following three steps: (a)

Determine the width of concrete that will act compositely with the steel. One of two cases will apply:

Case 1 – Internal Slab (slab extends on both sides of the steel beam) (i) The active slab width should not be taken to be more than a quarter of the beam span (i.e. S/4) (ii) The active slab must not extend more than half the distance between the steel beams on each side of the beam being considered (i.e. C1/2 or C2/2). Case 2 –Edge Slab (slab extends on one side of beam only) The active slab width should not taken to be more than one tenth of the beam span (i.e. S/10) (iii) The active slab must not extend inwards by more than half the distance to the next steel beam (i.e. C2/2) Guidelines for the Vibration Design of Structures – Issue 2 AA BPG S002 42 of 92


(b) Determine the effective thickness of the concrete slab. The slab thickness is taken as the overall slab thickness T, provided that: (iv) The slab has a flat underside, or (v) The slab has corrugated steel forms where the height of corrugations does not exceed ¼ of the slab thickness, or (vi) Where ribbed slabs are used, the rib width W1 is at least 125 mm, the rib height A does not exceed 40 mm or 0,4T, and the width between ribs W2 does not exceed 0,25T or 0,2W1. In all other cases, the slab thickness is taken as the depth of the slab minus the height of the ribs (i.e. T-A) (c) Calculate the section properties for the composite section. ROBOT and PROKON can both calculate the section properties for sections of a single material, but not a composite material. In order to calculate the composite section properties a spreadsheet has been written called “Composite Beam Properties”. This spreadsheet is located on the network at G:/ENGINEERING/se/PRODUCTS AND SERVICES/Design aids. The spreadsheet only gives section properties about the X-X axis, because a beam supporting a concrete slab will not vibrate laterally (i.e. about the Y-Y axis) nor will it vibrate torsionally. Any assumed large values of IY and J should thus be entered. The spreadsheet also assumes full shear connection between the concrete and the steel beam, meaning that there is no slip at all. Where the composite slab is low tuned, it is recommended that full shear connection should always be assumed, even if very few shear connectors are used. Where the slab is high tuned, a reduced value of the moment of inertia IXE should be calculated for the composite section, as specified in SANS 10162-1 Section 17.3.1 (a), i.e.:

IXE = IS + 0,85p0,25 (IX − IS ) where: IS is the moment of inertia of the steel beam only IX is the moment of inertia of the composite beam as calculated from the spreadsheet above p is the fraction of full shear connection being used This is a conservative assessment of the moment of inertia of the composite section, as it will give a high estimate of the frequency where the floor is low tuned, and a low estimate of the frequency where the floor is high tuned.

6.3

Confirming the Accuracy of the Model

How do we know that the response of the model to dynamic loadings will be reasonably close to the response of the real structure? We don't! On the well established principle of “garbage in, garbage out” a computer package will give us whatever follows from our input. There are, fortunately some aids to help the careful Designer determine whether the results from the model look reasonable or not. (a)

Look at the first few mode shapes. Do they make sense, or are there members flying off into space, at all sorts of crazy angles? If you understand what you are looking at, the mode shapes will tell you most of what you want to know about the performance of your model. Guidelines for the Vibration Design of Structures – Issue 2 AA BPG S002 43 of 92


(b)

A useful approximate method of obtaining the fundamental natural frequency of any structure is the so-called Rayleigh’s method. In this method, the deflections obtained by applying loads equivalent to the self weight of the structure, applied in an appropriate direction (i.e. vertically for floor vertical vibration, horizontally for overall building sway) are calculated. W1=m1g The loads are: The deflections are: W1 = m1g y1 W2 = m2g y2 W2=m2g . . . . Wn = mng yn The fundamental frequency is then given by: N

ω1 =

g∑ Wi y i i =1 N

∑ Wi y i

W1=m1g

2

W2=m2g

W3=m3g

W4=m4g

i =1

(c)

Calculate some key values by hand, or with a static structural analysis package you understand well. Yes, believe it or not, in today’s computer era there is still value in doing some simple hand calculations! The first key value you can calculate is the ratio of dynamic displacement to static displacement. If the structure is high tuned (frequency ratio less than 0,7) then this ratio should be between 1,0 and 2,0. If the structure is low tuned (frequency ratio more than 1,3) then this ratio should be greater than zero, and less than 2,0. The second key value you can calculate is the inertia force amplitude on the portion of the structure directly supporting the machine. This is calculated by the formula: FI =d o ω E2 M

Equation 6-2

If the structure is high tuned, the inertia force amplitude should generally be quite small, and it must be less than the amplitude of the applied dynamic force. If the structure is low tuned the inertia force amplitude should approach the amplitude of the applied force, and it must be greater than the amplitude of the applied dynamic force. Warning: Build a computer model, and look at the mode shapes before going any further. Mode shapes, with understanding, will tell you more about the accuracy of the model than any other factor. Many a stupid slip in modelling would have been identified early had the interpretative value of the mode shapes been realised and utilised.

6.4

Modelling Machinery on Structures

A question frequently encountered when modelling structures supporting vibrating machinery is whether the mass of the vibrating machinery itself must be included in the model. The answer to this question is based on - you guessed it - frequency ratios. The first frequency required may be called the machinery frequency, ωM. This is the lowest



natural frequency at which the equipment sitting on its springs or mounting pads will vibrate, if its springs are rigidly supported. The second frequency required is the lowest natural frequency of the structure without the equipment on it, ωS. The important frequency ratio is the ratio between these two frequencies, RF = ωM / ωS. Warning: Where machinery is supported on structures do not forget that it has a significant mass. Where necessary, this mass must be considered in the structural model.

Three different ranges of the ratio RF must be considered. The modelling of these three ranges is illustrated in Figure 6.5 (a) RF < 0,25 In this low range, the machine has a low frequency relative to the supporting structure. This means that the machine will tend to move on its supporting springs independently of vibration of the structure. In this case the mass of the machine can safely be omitted from the structural model, and the model is simply subjected to the loads applied through the supports of the machine. Generally, where machinery relies on vibration for its function, such as vibrating screens, vibratory feeders, etc, the supporting springs are generally very flexible, so that this frequency ratio condition is easily met. (b) 0,25 < RF < 1,50 In this intermediate range, the machine has a frequency similar to that of the supporting structure. This means that there is a dynamic interaction between the machine and the structure. In this case the mass of the machine must be modelled as one or more separate degrees of freedom, connected to the structure through its mounts. Generally, this condition does not occur because machinery is either mounted on flexible springs (case (a)) or it is almost rigidly fixed to the structure (case (c)). (c) RF > 1,50 In this high range, the machine behaves as if it is essentially rigidly fixed to the structure. This means that the machine will tend to oscillate together with the structure with very little relative movement. In this case the mass of the machine can safely be added to the structural model as a lumped mass, and the model is then subjected to the loads applied on the machine. Generally, where machinery does not rely on vibration for its function, such as would be true for pumps, winders, crushers, etc, there is practically no flexibility in the supports, so that this frequency ratio condition is easily met. Warning: Where we are working in this high range, and machinery is treated as a lumped mass, it is important to understand the influence of the geometric location of the centre of gravity of the machine. Where the centre of gravity is above the supporting structure, as is almost always the case, the lumped masses must include lumped mass inertias to account for the height of the centre of gravity, or the lumped mass must be added at a node at the centre of gravity which is then connected to the structure by means of rigid links pinned at their ends.



Mass of machine ignored

Mass of machine modeled on supports

Mass of machine added to structure

Low range

Intermediate range

High range

Figure 6.5: Modelling the Mass of Machinery

6.5

Calculation of Dynamic Response

6.5.1 General Having built the model of the structure, it is generally necessary to use it to calculate the response of the structure to one or more applied dynamic loads.

The required dynamic response is generally two different quantities. (a) Displacement or acceleration amplitudes are required to assess human and machine sensitivity levels. (b) Stress amplitudes are required to calculate the fatigue life of the structure. 6.5.2 Significant Modes The most important question faced by the Designer is how many modes should be included in the response calculation. The default answer would typically be “All of them”, but there may be times when the use of fewer modes gives a quicker result, which is sufficiently accurate. The decision, as with much of vibration analysis, depends on frequencies. Assuming that an accurate calculation of response is required, the minimum number of modes that can be used is determined by including all modes with natural frequencies up to at least 1,5 times the operating frequency of any vibrating equipment supported on the structure.

Where a structure is high tuned, all natural frequencies are higher than the operating frequency of the equipment supported on the structure. So, when a structure is high tuned, a good prediction of the dynamic response will be given if only a small number of modes are used in the analysis. However, if a structure is low tuned, there may be many modes whose frequencies are lower than the operating frequency of equipment supported on the structure. If the equipment is a screen, say, running at 16 Hz, and the natural frequency of the 10th mode is, say, 9 Hz, then the use of 10 modes in the response analysis will give completely wrong results. The wrong results will always predict too low a response, so the error is Guidelines for the Vibration Design of Structures – Issue 2 AA BPG S002 46 of 92


dangerous, and not conservative. For a low tuned structure with a screen operating at 16 Hz, say, it may be necessary to use 20, or 30, or 100 modes to ensure that all modes with frequencies up to at least 24 Hz are included in the analysis.

6.6

Dynamic Analysis Computer Programmes

This section provides some advice regarding the use of the dynamic analysis programmes available within ATD. This is not intended as a manual – all the commercially available packages have manuals, but rather it provides some pointers regarding what works and what doesn’t work, and what the programmes can, and can’t, do. 6.6.1 ROBOT V6 This is generally a very good program. Warning: We have been advised by ROBOT V6 programmers that we could modify the density of the elements supporting a distributed mass in such a way as to include the added distributed mass.

Example. If a 406x178x60 UB vibrates together with a distributed mass of 100 kg/m then the total mass per metre would be 59,8 + 100 = 159,8 kg/m. The density of steel is 7850 kg/m3. To get the new mass of 159,75 kg/m with the same area 7,611x10-3 m2 requires a modified density of 20990 kg/m3 for this particular beam. One minor problem with this method: it does not work! The program does not seem to understand it, and the results are erratic. Lumped masses. In ROBOT the lumped masses are input in force units. The program will make the necessary transformations. It does not matter if it is logical or not (it probably isn’t!), but we are stuck with this approach whenever we use ROBOT. Just make sure that the forces (that are really masses) are given in all the directions in which the respective joint can translate or rotate in the real structure, otherwise the result could be wrong. Distributed masses. ROBOT does not yet have the capacity to handle additional distributed masses on the span of a beam. 6.6.2 PROKON. PROKON is the one of the more common packages used within South Africa at present. PROKON is written using several defaults or computer-specific settings, which must be understood and altered if necessary. The following defaults must be noted:

(a) Application of Defaults PROKON uses defaults set in the computer, not in the specific programme file. This means that defaults set for a particular run will not be transferred if the data file is sent to another computer, or even if it is later brought back into the same computer after the defaults have been altered for different requirements on another project. Always check settings when using PROKON. (b) Mass The default is that load case number 1 is taken as the self weight of the structure. Load case 1 is thus assumed by default as the load case defining the structural mass. Any lumped masses included in other load cases must be indicated. Guidelines for the Vibration Design of Structures – Issue 2 AA BPG S002 47 of 92


(c) Modes used to Calculate Response By accepting the default settings, only the first 10 modes are used to calculate the structural response. This is probably sufficient for high tuned structures, but it is very Warning: The list below records common errors encountered when using computer packages for vibration analysis. Take note! Be warned!

1. Too many member/joint releases. If too many releases are specified, a member or a node may end up unrestrained in a particular direction. For example, if both ends of a member have rotation about the member axis released, then the member is free to spin. Or if all members framing into a particular node have their end rotation about any global axis released, then the node is free to spin about that axis. These will lead to zero divisions in the solution, which mathematically is not a nice thing to do. The programme may refuse to work, or you may get very strange results. Check your member and node releases. 2. Wrong shear resistance. Commonly, frame analysis packages are used where the structure has concrete floors or shear walls. Because the package does not have any finite elements to handle this slab construction, it is modelled as several lumped masses. This is fine as far as mass is concerned, but these slabs have very high shear stiffness. Do not ignore this. Wherever slabs are modelled as lumped masses, some phantom stiff bracing members must also be added in the appropriate directions to ensure adequate shear stiffness. 3. Too few modes in response calculation. Several computer packages allow the user the right to define how many modes are used to calculate the structural response to any applied dynamic load. If, say 10 modes are specified, only the lowest 10 modes will be considered in the analysis. This is tempting, as it can speed up the analysis significantly. Allow yourself to be tempted, but with care! ALWAYS USE ALL MODES WITH FREQUENCIES UP TO, AND WELL BEYOND, THE HIGHEST OPERATING FREQUENCY OF ANY EQUIPMENT ON THE STRUCTURE. 4. Model too complex A computer model that is too complex is confusing. It has too many modes, many of which are irrelevant, but which are calculated anyway. The Engineer must always stay in control of the analysis, not let the computer take over. likely to be insufficient for low tuned structures. Refer to section 5.4.2, and adjust this default accordingly. 6.6.3

Common Errors with Computer Vibration analysis

Warning: The files for a structural model may sometimes have to be transferred from one computer to another for some reason. BEWARE when this is done with PROKON. All default parameters revert to their default values when files are transferred to another computer, because default values are a function of the computer settings, and not the individual model. All default values must be checked and reset where necessary.



7 VIBRATION LIMITS 7.1

Introduction

Setting appropriate limits to vibration is one of the most vague and uncertain parts of dynamic analysis and design. We need to consider how people respond to vibration, how equipment and machinery are effected by vibration, and how the structure itself is likely to suffer under the influence of continued vibration. The greatest degree of uncertainty lies in the vibration limits which people can tolerate. It does not refer much to the fatigue calculations, although they do have a high level of conservatism built into them, so it is quite likely that a structure having a calculated fatigue life of, say, 5 years, will survived unscathed for 10 years or more. When considering machines, the greatest uncertainty is again people, this time how people forget to specify things, and then duck and dive looking for scapegoats when something goes wrong.

7.2

Human Sensitivity

Human tolerance to vibrations varies not only from person to person, but the same person may today be quite happy with a situation, complain bitterly tomorrow and wonder the day after tomorrow why this structure is so bulky and heavy, since it does not vibrate at all. The tolerance of the owner of a building who must foot the bill for remedial work, or someone who thinks there must be an insurance claim, or a worker who is disgruntled because salaries are too low, or a Consultant who can get paid for fixing the problem, are all very different. Check out BS 6611 or SANS 2631 (ISO 2631), in which a “Motion Sickness Dose Value” is defined for tolerance to low frequency vibration. This is related to the percentage of people who will get seasick and vomit. The Designer’s decision is whether that percentage of people vomiting is acceptable. The codes of practice of the V.D.I. (Union of German Engineers) deny that there is such a thing as allowable amplitudes and define something that represents the perceptibility of the vibrations: almost perceptible, clearly perceptible etc. However, we do need some criteria to determine the allowable amplitudes, so here goes. There are three significant limits in human reaction to vibrations. These were defined by earlier versions of ISO 2631 (which were probably more useful than the current version) as: 1.- Limit of comfort – the “Reduced Comfort Boundary” 2.- Limit of efficiency – the “Fatigue Decreased Efficiency Limit” 3.- Limit of health and safety – the “Exposure Limit” The Fatigue Decreased Efficiency Limits for vertical and horizontal vibrations are shown in Figure 7.1 and 7.2 respectively. In both Figure 7.1 and 7.2 the limits are shown as a function of frequency in (a), and as a function of exposure time in (b). In all cases, the Reduced Comfort Boundary is obtained by dividing the Fatigue Decreased Efficiency Limit accelerations by 3,15. The Exposure Limit is obtained by multiplying the Fatigue Decreased Efficiency Limit accelerations by 2,0.



RMS Acceleration (m/s2)

100

10

1 minute 1 hour 4 hours 8 hours 1 day

1

0.1 1

10

100

Frequency (Hz)

(a)


100

10

1 Hz and 16 Hz 2 Hz and 11 Hz 4 Hz to 8 Hz 50 Hz

1

0.1 0.01

0.1

1

10

100

Exposure time (hours)

(b) Figure 7.1: Fatigue Decreased Efficiency Limit for Vertical Vibration




100

10

1 minute 1 hour 4 hours 8 hours 1 day

1

0.1 1

10

100

Frequency (Hz)

(a)


100

10 1 Hz to 2 Hz 4 Hz 12,5 Hz 50 Hz 1

0.1 0.01

0.1

1

10

100

Exposure time (hours)

(b) Figure 7.2: Fatigue Decreased Efficiency Limit for Horizontal Vibration These limits are expressed in different ways by different codes. It is possible to become very sophisticated about evaluating human sensitivity, as many of the more recent codes do, but it is doubtful whether this is actually useful. ISO 2631 (earlier versions) and BS 6472 give specific numerical guidance regarding acceptable vibrations. Other earlier codes use a factor typically called "K". K takes into account the direction (horizontal or vertical), magnitude and frequency of the vibrations, position of the body and other variables. The K factor is a function of the RMS value of the acceleration. K can be expressed as a function of the amplitude (of displacement, acceleration etc). The Specification AAC114001 adopts this K value approach. Table 7.1 gives the K values used by Specification AAC114001. These limits are indicated in the spreadsheet “Design Aid DA6 Vibration Limits”, located at G:/ENGINEERING/se/DESIGN AIDS.



Table 7.1: K value Definition adopted by Specification AAC114001 Frequency (Hz) Horizontal Vertical 1 to 2 28aH 10aV√f 2 to 4 56aH/f 4 to 8 20aV 160aV/f 8 to 80 In the above Table 7.1, aH and aV are the horizontal and vertical mm/s2, and f is the cyclical frequency in Hz.

Undetermined 28a 33,5af0.25 160a/f RMS accelerations in

Table 7.2 shows the K interpretation (limits of comfort) based on VDI 2057 Part 2 (1987) Table 7.2: Interpretation of K Values According to: ISO 2631 (updated in 1982) & VDI 2057 Part 2 (1986) Reduced Comfort Boundary 1 minute …………………………….. 16 minutes ………………………….. 25 minutes ………………………….. 1 hour ……………………………….. 2.5 hours…………………………….. 4 hours ..……………………………. 8 hours ……………………………… 16 hours……………………………… 24 hours……………………………..

K

According to VDI 2057 Part2 (1987) Degree of perception

17.7 13.4 11.1 7.5 6.3 4.4 3.2 2.0 1.6 1.3 0.9 0.4 0.1

Very strongly perceptible

Strongly perceptible Very well perceptible Just Perceptible Under limit of perception

The Specification AAC114001 allows the following K values: (a) (b) (c)

Less than 4 hours exposure, i.e. where access is only required for short periods, K ≤7 Up to 8 hours exposure, i.e. where access is required for an entire shift, K ≤ 4,2 Up to 12 hours exposure, K ≤ 3,5.

An EXCEL spreadsheet giving a graphic representation of these limits is available at g:/ENGINEERING/se/DESIGN AIDS “Vibration limits”. Where there is vertical and horizontal vibration simultaneously, these must be combined using the following equation: K EQUIV = K H 2 + K 2V

where: KV = the K value for vertical vibration KH = the K value for horizontal vibration KEQUIV is then checked using the limits for vertical vibration. As a matter of policy, if an outside Contractor feels that the allowable K should be increased or decreased then he must contact ATD Structural Engineering for approval. Guidelines for the Vibration Design of Structures – Issue 2 AA BPG S002 52 of 92


7.3

Equipment and Machine Sensitivity

Believe it or not, equipment and machines are also sensitive to vibration. An interesting fact of life seems to be that when discussing a purchase with Suppliers, the equipment can accommodate almost any vibration that will be thrown at it, but once a purchase has been made, and there is any malfunction of the equipment in service, the same Suppliers claim that it is of course the ambient vibration that has caused the problem. Warning: When negotiating the purchase of any equipment or machinery that will operate in an environment that includes vibrating equipment, ALWAYS insist that potential Suppliers specify (in writing, before any contract is signed) the level of vibration that their equipment can tolerate. This must then be checked against structural vibration analysis prior to any construction work commencing. It can also be used afterwards if there is any dispute regarding performance of the equipment.

In our experience, the most sensitive equipment likely to be located in a vibration environment, is electrical switchgear, particularly “soft start” units, and area lighting. Sensitivity of Spring Mounted Equipment Experience has demonstrated that equipment supported on springs, such as vibrating screens and vibratory feeders, may be sensitive to lateral vibration of the supporting structure. The equipment is typically not designed to withstand lateral motion or lateral forces, so fatigue cracking may result if the structure vibrates laterally. Typically, the lateral vibratoin induced by this equipment is limited, because there is nominally no lateral force. However, where the supporting structure is not symmetrical, or other vibrating equipment also operates on the same structure, a lateral component of vibration may be introduced. In order to ensure that there is little likelihood of damage to this equipment, the lateral vibration must not exceed 10 % of the in-line vibration.

7.4

Structural Sensitivity

There are two aspects of structural sensitivity that require the Designer’s attention. The first is brittle construction materials or finishes, where vibration at higher frequencies can lead to cracking or dislodging of the material. The second aspect is the possibility of fatigue damage or even failure due to the high number of stress cycles. 7.4.1 Brittle Finishes Brittle finishes are not generally our concern in the mining environment. They include things like tiled floors and walls, glazing, and poorly constructed brickwork. Research into the likelihood of damage occurring to brittle finishes has tended to concentrate on the ground motion leading to damage to building finishes. This work has determined that the likelihood of damage is more closely related to ground velocity than to either ground acceleration or ground displacement. A general, conservative rule of thumb, applied by some design codes is that the ground velocity should not exceed 5 mm/s. This is conservative, so if the ground velocity is less than 5 mm/s, think no further. Brittle finishes will not be damaged. For more specific guidance, although still somewhat conservative, use Figure 7.3.

The structural vibration velocities at which initial damage to brittle finishes may be expected collected from various sources, and are given in Table 7.3. It should be noted that observations of damage vary very widely. Damage is unlikely at velocities below those in Table 7.3, but in many cases where the velocity was more than double these Guidelines for the Vibration Design of Structures – Issue 2 AA BPG S002 53 of 92


values there was no observed damage. These values should thus not be taken as a hard-and-fast rule, but as general guidance.

Peak ground velocity (mm/s)

30 25 20 15 10 5 0 10

20

30

40

50

60

70

80

90

Frequency (Hz) General masonry - Impact Freestanding and brittle masonry - Impact General masonry - Continuous Freestanding and brittle masonry - Continuous

Figure 7.3: Ground Velocity Limits for Brittle Structures Table 7.3: Structural Velocities at which Damage to Brittle Finishes may Occur Type of Structure Damage Peak Structural Velocity (mm/s) Brick building Appearance of first cracks 50 Brick building Deep cracks 150 Brick building Falling plaster 200 Concrete structure Appearance of first cracks 200 Any Appearance of cracks in tiles 30 7.4.2 Fatigue Life Fatigue life is calculated in terms of structural design standards. This guide will not attempt to teach the reader how to use SANS 0162 or BS 7608. This just serves as a friendly reminder that, like it or not, fatigue exists. Wherever vibration is encountered, the fatigue life of the structure must be calculated, to ensure survival. Vibration stresses are generally low, but stress cycles quickly mount up to huge numbers.

There is just one word of warning! Computer analysis, and for that matter hand analysis, give the maximum stress in one direction. Under vibration conditions, the vibration stress varies between a positive maximum, and a negative minimum with the same absolute value. The fatigue stress range is thus twice the stress calculated. Warning: The fatigue stress range is usually twice the maximum vibration stress calculated.



8 DESIGN GUIDANCE FOR SPECIFIC EQUIPMENT AND STRUCTURES Warning: The loads given in this Chapter are given for guidance only. Loads must always be obtained from Suppliers. If what you are given looks suspicious, get the Supplier to check and explain!

8.1

Crushers

There are crushers and then there are CRUSHERS. Some produce negligible dynamic loads. Others produce loads so big that it is impossible to support them on a conventional structure. It is a big and crushing world out there. 8.1.1 Modelling Crusher Support Structures Very few crushers are supported on springs. Most are supported directly on a suspended floor or a concrete foundation. Others are supported not on proper springs or buffers but on some funny little pieces of rubber or stuff. The single most important difference between a crusher support model and a conventional model is that the mass of the crusher must be included in that model, with mass moment of inertia and all the trimmings.

The simplest way to model a crusher for a dynamic program is to define an element that would have the same outline as the machine (normally a cylinder or a box-shape) and to determine the mass such that the full crusher element would have exactly the same mass as the real crusher. This automatically takes care of all the mass moments of inertia. 8.1.2 Loads Applied by Crushers The loads applied by crushers to the structures allocated the hazardous job of supporting them depend on the type of crusher.

(a) Cone Crushers Cone crushers consist primarily of a cone rotating and tilting about a vertical axis inside a cylinder. See Figure 8.1. The cone is mounted eccentrically with respect to the cylinder, so that as it rotates it crushes rock falling between it and the cylinder. The rotation speed is generally slow, in the region of 2 Hz to 4 Hz, but the cone mass is relatively high because the cone is heavily constructed, and the eccentricity is quite high or rock will not be crushed. The centrifugal load will be more-or-less horizontal and sweeping 360 degrees. The simplest (and acceptably accurate) way to handle it is to analyze the structure under two separate horizontal harmonic loads, applied at 90 degrees from each other. The results will then be evaluated using engineering judgement. It is important to recognise that the centre of gravity of crushers is above their support point, so these forces will inevitably and unavoidably induce dynamic moments (reflected as vertical support forces) as well. Remember!

Figure 8.1: Cone Crusher



Example Crusher power: Throughput: Feed size: Discharge size: Total crusher mass: Cone size:

520 kW 500 to 600 tons/hour - 250 mm - 25 mm 68 000 kg 3,1 m high x 2,0 m diameter

Forces: m1 15111 kg m2 474 kg m3 122 kg r1 0,013 m r2 0,0318 m r3 0,4286 m Speed 220 rpm, i.e. 220/60 = 3,667 Hz Speed (ω) 3,667x2π = 23,04 rad/s F1 15111x0,013x23,042 = 104280 N F2 474x0,0318x23,042 = 8001 N F3 122x0,4286x23,042 = 27757 N F1 + F2 - F3 Total horizontal = 104280 + 8001 – 27757 N force = 84524 N Distances below bearing: Crusher 2566,5 mm support m1 1085,3 mm m2 2784,8 mm m3 2717,3 mm Total F1(2,5665-1,0853) + F2(2,5665overturning 2,7848) - F3(2,5665-2,7173) Nm = 104280(2,5665-1,0853) + moment 8001(2,5665-2,7848) – 27757(2,5665-2,7173) Nm = 156899 Nm Figure 8.2: Cone Crusher (b) Jaw Crushers Jaw crushers consist primarily of a fixed steel plane and a moving steel jaw. See Figure 8.3. The moving jaw is pivoted at its base, and is thrust towards and away from the fixed steel plane by an eccentric mass or an eccentric shaft. The motion is primarily horizontal. Rock falls between the jaw and the fixed plane as the jaw moves away from the fixed plane, and it is crushed as the jaw moves back towards the fixed plane. The speed is generally slow, in the region of 1 Hz to 4 Hz, but the moving jaw mass is relatively high because it is heavily constructed, and the eccentricity is quite high or rock will not be crushed. The dynamic load generated by the action of jaw crushers is essentially horizontal, and may be idealised as a single harmonic load, applied in the direction of jaw motion. Figure 8.3: Jaw Crusher Guidelines for the Vibration Design of Structures – Issue 2 AA BPG S002 56 of 92


(c)

Roller Crushers and Mineral Sizers

Roller crushers and mineral sizers consist primarily two contra-rotating rollers, with rotation about a horizontal axis. See Figure 8.4. Rock falling between the rollers is crushed as it passes through the narrowest passage between the two rollers. The rotation speed is generally slow, in the region of 0,5 Hz to 3 Hz, but the roller mass may be relatively high because the rollers are heavily constructed. The rollers are nominally concentric to their axes.

Figure 8.4: Roller Crusher

The dynamic load applied by a roller crusher to its supporting structure is small, because the operation of the crusher does not rely on any eccentric motion of heavy components. Because the rollers are contra-rotating, any dynamic loads from the two rollers will tend to compensate horizontally, but be additive vertically. A relatively small vertical dynamic load should be anticipated on the support structure for roller crushers. It may conservatively be assumed that the eccentricity due to construction tolerance and wear will be of the order of 1 % of the roller radius. This results in a dynamic load amplitude that is generally less than 10 % of the weight of the rollers. (d) Flywheels Remember also that crushers use heavy flywheels. These are rotating masses, where the intention is that the mass is concentric. However, due to manufacturing tolerances, this will not be the case, so flywheels should be treated in the same way as motors and turbines. (e) Dynamic Loads due to Breaking Rocks Breaking rock causes additional random loads to be applied to the crusher supporting structure. However, because these loads are random, and they are generally relatively small, they are usually not considered.

8.2

Rotating Tubes

A range of equipment is essentially composed of rotating tubes, supported by rollers at each end or along the sides. The equipment includes mills, scrubbers, trommels, cement kilns, and peletizers. 8.2.1 Types of Load Generated In the ideal world (wherever that might exist!) these pieces of equipment are circular, and just turn neatly about their axis, without imposing any vibration loads onto their supporting structure. However, in the real world, things are never so simple. There are several sources of vibration loads that may be applied by rotating tubes.

(a) Ovalling of the tube Where the construction of the rotating tube is fairly light, and allows a small distortion of the circular tube into an oval or some other shape, there is a vibration load applied as the longer axis transfers the weight of the equipment and contents from one roller to the other, as shown in Figure 8.5. Guidelines for the Vibration Design of Structures – Issue 2 AA BPG S002 57 of 92


The frequency of this vibrating load is determined by the number of lobes in the imperfect shape multiplied by the rotational speed of the equipment. See Figure 8.6. Generally, the number of lobes will be determined by the construction details of the tube. The amplitude may be as much as the nominal reaction at the relevant roller, because in the extreme the load may vary from zero up to twice the nominal reaction. Our experience, however, suggests that a realistic assumption is that the load varies between 50 % and 150 % of the nominal constant load. This leads to a dynamic load with an amplitude of 50 % of the nominal load. It must be remembered that the load applied to the support rollers must go through the centre of the bearings, so it has a vertical component and a horizontal component. Don’t think that just because this is a dynamic load it will let you off easily! Resolution of forces still applies. If the loads on both rollers are equal, the net horizontal load is zero. However, if the two loads are not equal, there is a resulting horizontal dynamic load app-lied to the supporting structure.

Figure 8.5: Ovalling of the Tube

4 lobes

2 lobes

3 lobes

Figure 8.6: Different Numbers of Lobes on the Tube (b) Material falling off lifters Some mills and trommels have “lifters” mounted on their inside surface to lift and mix the contents thus ensuring adequate processing. As material falls off the lifters, there is some tendency to generate oscillatory loads on the tube. The frequency of this load is well defined by the number of lifters and the rotation speed of the tube. The amplitude of the load is more difficult to define. However, unless there is resonance between the lifting frequency and some natural frequency, this is unlikely to be a problem load, so avoiding resonance is the key design consideration. (c)

Misalignment of girth gear or cutting errors in girth gear or drive gear



Misalignment of the girth gear or the motor drive shaft, or poor cutting of the teeth on either the girth gear or the drive gear may lead to vibration at the frequency of gear teeth intersections. This is commonly in the frequency range of 25 Hz to 60 Hz. 8.2.2 Specific Equipment (a) Mills Mills are typically heavy pieces of equipment that are usually mounted directly onto a heavy concrete foundation, composed of a thick base slab and a thick plinth at each end to support the mill.

Experience shows that as an approximate general rule of thumb, the fundamental natural frequency of pitching (i.e. rocking in the mill axis longitudinal direction) and of rolling (i.e. rocking transverse to the mill axis direction) should both not be less than 3,0 Hz, when calculated with the mill carrying its normal full load of material.

8.3

Vibrating Screens and Feeders

8.3.1 Basic Requirements Vibrating screens and vibratory feeders are simple eccentric mass machines. They are supported on flexible springs to beams below, or on cables and flexible springs to beams above, as they rely on vibration to function, and flexible springs enable vibration without imparting large forces to their supporting structures. This means that vibrating screens and vibratory feeders are low tuned, leading to large displacements when they are shut down. The shut down displacements should be obtained from Suppliers, and adequate clearances must be provided to avoid vibrating screens and vibratory feeders striking surrounding objects. The clearance should never be less than 100 mm. Warning: Always check shut down displacements with Suppliers and ensure adequate clearance around vibrating screens and vibratory feeders. Don’t believe they can’t give it to you!

As a rough guide, the peak-to-peak displacement of vibrating screens is typically in the range from 6 mm to 10 mm, a little less for vibratory feeders. This means that the vertical amplitude of motion is about 2 mm to 3 mm, and the horizontal amplitude is about 2 mm to 4 mm. The total dynamic load (sum of loads at all four corners) applied by vibrating screens and vibratory feeders to supporting structures is typically a few percent of the weight of the screen or feeder. If the total dynamic load is given as less than 1 %, don’t believe it. If the Supplier shows that this is correct, then check the static deflection of the springs, because they will have to be VERY flexible. If the total dynamic force is greater than 10 % the screen or feeder will shake the teeth out of the structure too quickly, although with rubber blocks or buffers the total dynamic force may actually approach this level. When working with vibrating screens or vibratory feeders, the following aspects may cause difficulties, and should be checked: (a)

The dynamic loads given on drawings of screens or feeders are usually given per spring, or per corner. Check that you are satisfied which has been specified. If there is any doubt, check with the Supplier. Small screens and feeders typically have one spring at each corner. Large screens or feeders may have two, three, or even more springs at some corners. Frequently, on large screens and feeders, there are more springs at the feed end, where material drops onto the screen or feeder, than at the discharge end. Guidelines for the Vibration Design of Structures – Issue 2 AA BPG S002 59 of 92


(b)

The excitation load to screens and feeders can usually be adjusted, by setting the eccentricity of the eccentric masses. Increasing this load is a possible way of improving throughput, or setting it correctly may be overlooked during commissioning, so excessive vibration may result from incorrectly adjusted eccentric masses. Rubber blocks typically last much longer than steel coil springs, so coil springs may be replaced by rubber blocks. As rubber blocks are much stiffer than steel coil springs, this leads to much higher dynamic loads being applied to the structural supports. On numerous occasions, this has been found to be at least in part responsible for reported high vibration levels.

(c)

8.3.2 Design and Use of Sub-frames (a) General A sub-frame is a mechanical device that absorbs part of the energy transmitted from the equipment to the supporting structure. The reason for using sub-frames is to reduce the dynamic reactions. A typical sub-frame looks more or less as shown in Figure 8.7. The physical shape of the sub-frame is often determined by process considerations, such as underpans and chutes. This reduces the options of structural optimization.

Screen outline dotted

Sub-frame

Springs supporting sub-frame on structure Fig 8.7: Schematic Layout of Typical Sub-frame (b) Reactions On the Sub-frames and the Structure The screen reactions on the sub-frame and the sub-frame reactions on the supporting structure are determined by the dynamic behaviour of the screen and sub-frame assembly. A model of the screen and sub-frame must be created to determine this dynamic behaviour. The approach for this is generally the same as for a conventional structure. (i)

Use ROBOT or PROKON. The model must contain the screen, the springs between the screen and the sub-frame, the sub-frame, and the springs between the sub-frame and the support structure. Guidelines for the Vibration Design of Structures – Issue 2 AA BPG S002 60 of 92


(ii)

(iii)

The springs can be modelled as spring elements in ROBOT, or as bar elements in PROKON. To model the springs as bar elements, the effective properties must be calculated either by hand or using the spreadsheet “Design Aid DA11 Equivalent Spring Dimensions for Modelling Screens” which may be found at G:/ENGINEERING/se/DESIGN AIDS. Use of the spreadsheet is selfevident. In order to model the screen itself one must: Either approximate the position of its centre of gravity, and lump the whole mass of the screen there, then calculate the mass moment of inertia of the entire screen about its centre of gravity. A single node at the centre of gravity of the screen is then given this mass and mass moment of inertia, and is connected to the springs by using rigid links pinned at their ends. Warning: There are some restrictions in the use of rigid links. The restrictions depend upon the specific program. Consult the manual!

Or the screen can be modelled (using some good deal of engineering nouse, i.e. good sense and judgement) as a grid of beams connected to the springs. This grid of beams must be braced to represent the diaphragm action of the screen, and their mass and mass of moment of inertia must accurately represent the whole screen structure. Warning: Check the implications of tolerances and spillage on the operation of sub-frames. Get it wrong, and a sub-frame may lead to the vibration loads applied to the supporting structure being substantially larger, not smaller.

The forces in the springs between the sub-frames and the supporting structures are the loads finally applied to the structure. When using sub-frames, it is crucial to investigate the influences of tolerances and other effects. The steel from which sub-frames are generally constructed has rolling tolerances of up to 4%. There is also a strong likelihood of a certain amount of spillage accumulating on the sub-frame quite quickly. The springs supplied have tolerances in their stiffnesses. Now, the point is this. The effective operation of sub-frames depends all of these factors. It is recommended that the Designer should check the effects of the following “what ifs”: (i) (ii)

Mass of the sub-frame oversize and accumulated spillage increasing the subframe mass by 20 %, and the springs having a stiffness of 10 % less than specified. Mass of the sub-frame is unlikely to be undersize, so do not reduce the subframe mass, but use springs having a stiffness of 20 % more than specified.

The structure must then be designed for the highest loads arising from the nominal design conditions or either of these “what if” scenarios.



8.4

Rock Breakers

Rock breakers are mean beasts! The loads they can exert are big. Their main purpose is to break rocks that are too big to fit through grizzlies, but their Operators will use them to push large rocks around on the grizzly as well. The Operators also try to push large rocks into a corner of the bin, so that they can be broken by hammering them where they can’t escape! It is thus important to know the magnitude of the loads (FB the breaking force, FP the pushing force, and FQ the slewing torque) that they can apply in the various different directions shown in Figure 8.8. The breaking load FB is also made up of two different components, a quasi-static load and a hammer load, as shown in Figure 8.9. (Note that the actual magnitude of the loads given in Figure 8.9 only applies to one specific rock breaker. The actual values must be established in each particular case).

FQ

FB

FP

Figure 8.8: Loads Applied by a Typical Rock Breaker

70

Rock breaker force (kN)

60 50 40

Quasi-static force Ham m er force Total breaking force FB

30 20 10 0 0

0.5

1

1.5

2

Tim e (seconds)

Figure 8.9: Breaking Load FB So, when working with rock breakers, it is important to talk in detail to the Supplier. At least the following information must be obtained and included in design considerations: (a) (b)

The hammer load. The maximum push that can be exerted vertically and horizontally, from which the quasi-static component of the breaking load and the push load can be obtained. If the Supplier cannot provide this information, it can be calculated from the Guidelines for the Vibration Design of Structures – Issue 2 AA BPG S002 62 of 92


(c)

maximum thrust of the hydraulic cylinders controlling the boom, which the Supplier should be able to provide. The maximum slewing moment.

It is not generally the Structural Designer’s task to check or ensure integrity of the rock breaker, but all of these loads induce reactions onto the supporting structure, whose integrity is the Structural Designer’s responsibility.

8.5

Design of Grizzly Bars

The function of grizzlies is to keep oversize material out of crushers and off conveyor belts. Grizzlies are one of the structures Structural Designers have to cope with that really take a pounding. Large rocks (the “how big” question is often not satisfactorily answered) fall from the back of haul trucks or LHDs (Load Haul Dumpers used underground), a few metres above the grizzly with frightening amounts of energy. This Section describes an approximate evaluation of the maximum stress produced by a mass that hits one beam at mid-span with a known speed. Figure 8.10: Large Rock destined for Grizzly!! The stress is calculated by assuming that the kinetic energy of the hitting mass is transformed into strain (deformation) energy. By equating the two energies it is possible to calculate the force that, if applied statically at the point of impact, would produce strain energy equal to the kinetic energy of the hitting mass. See Section 5.3. An EXCEL spreadsheet is available at G:/ENGINEERING/se/DESIGN AIDS “Grizzly design” to perform the necessary calculations. The operation of the spreadsheet is selfexplanatory. This spreadsheet allows the use of billets, or other structural sections for the grizzly bars. It also allows for loss of energy due to fracturing of the rock during impact. The spreadsheet allows any amount of energy loss, but it is recommended that this should never be set at more than 10 % energy loss. The most important thing with the design of grizzlies is that a lot (A LOT!) of engineering judgement must be used when evaluating the results. The following assumptions are made in the spreadsheet “Grizzly design”: (a)

(b) (c)

The equations used for the programs assume that the falling mass is completely stopped by the grizzly bar, unless the grizzly is angled at more than 45º above the horizontal. This is obviously not true, as can easily be proved by watching a grizzly at work during about 10 seconds. The grizzly bars do not collapse when the yield stress is reached. The Designer must thus check the compactness of whatever structural section is chosen. No overall lateral torsional buckling is possible under this type of loading. This is generally true when the grizzly has a rectangular grid of bars, but may not be true if the bars run in one direction only. The Designer must ensure an adequate structural design.



(d)

The conditions are known. There is always the requirement for interaction between the Structural Designer and the Mine to get as close as possible to this assumption, but there are still difficulties. Questions that still arise include: (i) What is the height of the lip of the haul truck bucket when rocks fall? Is it the closed height or the fully tipped height, or somewhere in between? (ii) How big are the biggest rocks? (iii) Are rocks generally dumped onto the clear grizzly, or onto a pile of other rock on the grizzly?

8.6

Vessel Agitation

8.6.1 Applied Loads Agitation of fluids in circular vessels is used in numerous processes in mining and paper production. Flotation cells in mining applications are a typical example. The rotation speed of the agitators is usually quite slow, of the order of 60 rpm (1 Hz) or even less. However, experience in ATD has shown that the need for various services below the vessels leads to little bracing being used in support structures which are thus rather flexible laterally. This, linked to the relatively high mass of the vessels contents leads to quite low natural frequencies, and the distinct possibility of resonance. Light walkways are also often provided for access to machinery above the vessels, which may well also have quite low natural frequencies. Warning: When taking measurements of vibration induced by vessel agitation watch the minimum frequency range of the instruments used. The RION VA10 instrument available in ATD Structural Engineering does not give accurate measurements below 3 Hz because of built in high pass filters. The RION SA78 instrument available in ATD Structural Engineering measures down to 1 Hz with accuracy.

The agitators frequently have three or four blades (or paddles) and there are vanes, or baffles around the perimeter of the tank. This may lead to a higher “vane passing” frequency, which is given by the agitator rotation frequency multiplied by the number of vanes. Experience in ATD, however, suggests that this higher frequency is seldom the culprit in vibration problems related to vessel agitation. A typical agitator shaft has a torque and an axial force applied, and it may well also have a bending moment applied. All of these forces should be obtained from the Supplier of the agitator, but the description below allows approximate values to be determined if necessary. (a) Torque The torque FQ arises from the need to swirl and mix the liquid in the tank. Normally, it is derived from the power of the drive motor. The maximum value of torque is about three times the motor power divided by the agitator speed ωE, because electrical motors do strange things on start-up. FQ ≈ 3

Power Power Power ≈ 29 =3 ωE 2 πf E rpm



LB

β

Two blades

Figure 8.11: Vessel Agitation (b) Axial Load The axial force FA arises from the length and angle of the blades, the applied torque and turbulence in the fluid. The axial force will be directed upwards or downwards, depending on the blade angle and the direction of rotation. An approximation for the axial force may be obtained from the equation: FA =

FQ L B tan β

where:

LB is the length of the blade β is the angle of twist of the blade

(c) Bending Moment and Shear Load The bending moment FM applied by agitators to their support structure is difficult to determine, because it is determined by the very complex fluid-blade interaction. Also, broken or damaged blades, and misalignment of the agitator shaft will tend to significantly increase the bending moment. Typically, the bending moment is a similar order of magnitude to the applied torque. The shear force FV is equal to the bending moment divided by the vertical distance between the blades and the underside of the gearbox. 8.6.2 Design Requirements The loads defined above are the maximum quasi-static loads to be used for ensuring adequate strength of the agitator support structure. However, there are also varying loads due to operation of the agitator. ATD has experience of several cases of fatigue damage induced by these loads, and some experience of unacceptable low frequency vibration of the tanks on their supporting structures.

(a) The following quasi-static loads should be considered for strength design: Torque: Maximum load specified by Supplier Bending Moment: Maximum bending moment specified by Supplier Lateral Load: Maximum bending moment divided by length of shaft from gearbox to blades Axial Load: Maximum load specified by Supplier Guidelines for the Vibration Design of Structures – Issue 2 AA BPG S002 65 of 92


(b) The following fluctuating loads should be considered for fatigue design: Torque: Actual peak startup torque, or 50 % of maximum load specified by Supplier, applied once for each startup Bending Moment: 50 % of maximum bending moment specified by Supplier, applied once for each rotation of agitator shaft Lateral Load: Bending moment divided by length of shaft from gearbox to blades Axial Load: Operating axial load specified by Supplier, 50 % of maximum load specified by Supplier, applied once for each startup (c) The following dynamic loads should be considered for vibration assessment: Bending Moment: Load amplitude equal to 50 % of maximum bending moment specified by Supplier, applied at the rotation frequency of agitator shaft Lateral Load: Load amplitude equal to bending moment divided by length of shaft from gearbox to blades, applied at the rotation frequency of agitator shaft

8.7

Wood Chippers

An early part of the processing of logs into paper consists of reducing the logs to small chips. See Figure 8.12. This is done by means of wood chipper machines. Wood chippers are rotating machines with fairly heavy chipping heads which double as flywheels, so there will be a component of dynamic excitation related to their rotation, as for any other rotating machine. However, there is an additional impact force component as the blades strike the log to reduce it to chips.

Log being chipped

Figure 8.12: Schematic End View of Wood Chipper Head



9 PRACTICAL GUIDELINES FOR FOUNDATIONS This section deals with the design of concrete block foundations, used for vibrating equipment mounted at, or near, ground level. Typical equipment in this category includes large electricity generators, compressors and crushers.

9.1

Traditional Rules of Thumb

The most common traditional “rules of thumb” used to design the concrete foundations for small equipment simply require that the concrete block foundation has a mass of more than 10 times the machine mass for a reciprocating machine, and more than 5 times the machine mass for a rotating machine. These simple rules may be used for small machines, with a mass of up to 500 kg and a power output not exceeding 50 kW, but are not good enough for larger machines.

9.2

Simple Rules

If it is assumed that the soil supporting a machine foundation is very flexible, so that it provides very little resistance to small amplitude vibration motion, then the dynamic forces generated by the machine only accelerate the mass of the machine and foundation. Under these conditions, the amplitude of base motion in various different places and directions may be described by the simple equations given in Table 9.1. In these equations symbols are as defined in Figure 9.1 and e is the machine eccentricity, MR is the mass of the moving portion of the machine, MS is the mass of the static portion of the machine, MB is the mass of the foundation, M is the total mass (i.e. MR + MS + MB), and Ib is the mass moment of inertia of the machine and foundation. These equations give conservative predictions for foundation motion provided the foundation is not in resonance with the machine speed. Table 9.1: Displacements Based on Simple Foundation Motion Equations Location and direction Displacement Equation Vertical or horizontal linear motion M d= R e M Vertical at edge of base due to rotation M h b d = R CG e 2Ib 2 Horizontal at centre of gravity of machine due to rotation MRhCG d= e Ib

h d

a hCG b

Figure 9.1: Schematic of Machine on Foundation



In addition, it is generally recommended that the width of the base should not be less than 1,5 times the height of the centre of the machine, i.e.: b ≥ 1,5(h + d C )

9.3

Modelling Foundations

Equipment mounted on concrete block foundations, but larger than that covered in Section 9.1 or where resonance is a possibility so that the equations given in Section 9.2 cannot be used, must be designed giving due consideration to the mass distribution, and the underlying soil stiffness. This should be done using a finite element programme, but a first approximation may be obtained using simplified calculations based on the dynamic behaviour of the machine on its base. 9.3.1 Soil Conditions Any adequate model of the machine foundation must use as good a representation of the underlying soil as possible. Soil, whether sand, clay or even soft rock is a granular material, rather than a homogeneous continuum.

The analysis requires the soil stiffness properties as inputs. Two important parameters must be distinguished. These are the elastic modulus E of the soil, and the modulus of uniform compression CC of the soil. A finite element analysis will generally use E, whereas most simplified equations (including the ones used here) use CC. These are theoretically related by the equation: C C = 1.13

1 E 2 1− ν A

Warning: Understand the parameters used. E is not equal to CC, they are not even equivalent, they just sound similar.

The soil properties are generally obtained from one of the following two procedures: (a) Plate bearing test In this test, a plate of a specified size is placed on the soil and a specified load is applied. The settlement of the plate is measured as the load is applied. The soil compression stiffness CC is then calculated directly from the plate area, the load and the measured displacement. A typical plate bearing test result is shown in Figure 9.2. In this test several cycles of loading, to increasing maximum load, were applied. It can clearly be seen that there is both an elastic component and an inelastic component of the settlement. Vibration characteristics are determined by the elastic component of the settlement only. Thus, all the soil stiffness values that are used for vibration analysis are derived from this elastic component. Bearing tests are typically carried out using plates with an area of the order of 0,2 m2 to 1,5 m2, otherwise the loads that must be applied become huge.



Load (kN)

200

Elastic

150 100 Inelastic

50 0 0

2

4

6

8

10

Settlement (mm) Figure 9.2: Typical Plate Bearing Test Warning: Check, then recheck, soil properties with the Geotechnical Engineer. Geotechnical Engineers usually think simple total long-term settlement, not such fancy intricacies as vibration.

(b) CSW (continuous shear wave) test In this test, an impact is applied to the soil, and the speed of transmission of the resulting shear wave through the soil medium is measured. The shear modulus of the soil G is then calculated from the shear wave transmission speed. This test gives the elastic stiffness of the soil, generally quoted as the shear modulus, G. The CSW test involves very small shear strains. A higher level of soil strain is typically experienced below completed structures. Under these higher strain conditions, the actual shear modulus of granular soils is reduced. Figure 9.3 shows the shear modulus reduction typically experienced for completed structures on granular soils. The shear strain encountered will usually be in the range 0,01 % to 0,1 %. The appropriate shear modulus for soil assessment will thus usually be in the range of 75 % to 50 % of the CSW value.

Figure 9.3: Shear Modulus Variation with Strain



However, where the structure foundation is located on soft rock, this may be reversed. The CSW shear modulus obtained for soft rock is likely to be determined primarily by the shear modulus of granular soil lenses within the rock. As the nominal shear strain increases, these lenses are likely to be compressed, so that the actual behaviour of the soft rock is representative of a higher shear modulus. However, little is known of how much this increase is likely to be. So, CSW shear modulus values reduce with increasing shear strain in granular soils, but are more likely to increase in soft rock. (c) Soil Characteristics to Use The soil characteristics must be obtained from a geotechnical report for the site, prior to completing the vibration analysis for any foundation. However, as a preliminary approximation, the values given in Table 9.2 may be used for soil stiffness. Table 9.2: Approximate Soil Modulus of Uniform Compression Values for AT = 10 m2 Soil type Allowable bearing Soil modulus of uniform pressure (kPa) compression, CC (kN/m3) Firm clay or sandy clay. 100 25 000 Medium dense sand or silty sand. Stiff clay or sandy clay. 200 50 000 Compact poorly graded gravel. Dense sand or silty sand. Very stiff clay or sandy clay. 400 100 000 Compact well graded gravel. Very dense sand or silty sand.

9.3.2 Simplified Preliminary Calculations The simplified preliminary calculations are based on the following procedure:

(a) The soil uniform compression modulus is calculated. The soil uniform compression modulus is calculated from the test results, using the equation: C C = C Ctest

A test A

where: CCtest is the soil uniform compression modulus from the geotechnical test Atest is the area of the geotechnical test A is the area of the machine foundation being designed (b)

The soil uniform shear modulus and uniform rotation modulus are calculated.

The ratio between the soil uniform shear modulus CV and the soil uniform compression modulus of the soil CC, the shear stiffness ratio, is primarily a function of the Poisson’s ratio ν of the soil, and to a lesser extent also of the geometry of the foundation. Based on the assumption of elastic continuum behaviour of the soil, the ratio αQ between the soil uniform shear modulus CQ and the uniform compression modulus CC is given as listed in Table 9.3. The soil uniform shear modulus is given by:



CQ = α QCC The value of Poisson’s ratio for soil is generally approximately 0,2. Table 9.3: Ratio αQ between CQ and CC Poisson’s ratio ν a/b = 0,5 0,1 0,96 0,2 0,91 0,3 0,85 0,4 0,78

a/b = 1,0 0,95 0,89 0,82 0,75

a/b = 2,0 0,94 0,87 0,80 0,72

Due to the fact that under conditions of rotation of the foundation the soil pressures are not uniform, but vary linearly across the base, an adjusted soil uniform rotational modulus must be calculated. Based on the assumptions of a rigid foundation and elastic continuum behaviour of the soil, the ratio αφ between the soil uniform rotational modulus Cφ and the uniform compression modulus CC is given as listed in Table 9.4. Table 9.4: Ratio αφ between Cφ and CC a/b = 0,5 a/b = 1,0 1,58 1,88

a/b = 2,0 2,31

The soil uniform rotational modulus is given by: Cϕ = α ϕCC (c) The stiffness for vertical motion is calculated. This is given by:

K C = CC A (d) The stiffness for horizontal motion is calculated. This is given by: K Q = CQ A (e) The stiffness for rotational motion is calculated. This is given by: a 3b 12 ab 3 K ϕ = C ϕI = C ϕ 12 K ϕ = C ϕI = C ϕ

or

for rotation about an axis parallel to side b for rotation about an axis parallel to side a

(f) The system is analysed to determine to frequencies. For a symmetrical arrangement of machine and foundation, vertical vibration can be approximated as a single degree of freedom system. Horizontal and rotational vibration will always be coupled, because the centre of gravity of the foundation and machine will always be well above the soil-foundation interface. This can be idealised as two, two degree of freedom systems.



The calculation procedure outlined above has been programmed in the EXCEL spreadsheet “Design Aid DA13 Machine bases”, located at G:/ENGINEERING/se/DESIGN AIDS. 9.3.3 Damping There are two damping mechanisms applicable to soils. The first is “structural damping” which is similar to the damping that occurs within any material. Internal friction between the grains of soil causes energy losses in much the same way as internal friction between molecules in other materials. However, because the soil extends more-or-less infinitely laterally and downwards away from the foundation under consideration, there is also “dispersion damping”. Here energy is lost to the finite system considered because oscillations get the travel bug, and head off into the far distance.

Unfortunately, very little information is available regarding the magnitude of soil damping. Generally, the design approach is to avoid resonance, in which case damping has little influence, so little effort has been expended in determining this difficult quantity. A conservative design will assume zero damping. A more realistic design will assume modal damping of, say, 5 %.



10 PRACTICAL DETAILS FOR TERTIARY STRUCTURAL ELEMENTS 10.1 Individual Members The local vibration of individual members of structures is often not very well predicted by computer models of the entire structure, so it is not uncommon to experience local lateral or torsional vibrations of individual members in structures. When assessing structural vibrations, it is thus necessary to evaluate the frequencies of individual members within structures. This is probably more easily and more accurately done by hand using simple formulae, rather than by using a computer package such as ROBOT or PROKON. 10.1.1 Approximate Natural Frequencies of Individual Members The lowest two natural frequencies, in Hz, of individual structural members may be estimated from the following formulae. A spreadsheet to calculate these frequencies is located at g:/ENGINEERING/se/DESIGN AIDS “Design Aids DA12 Beam Frequencies”.

(a) Flexural frequencies Members may be either simply supported, fixed ended, or cantilevered. (a) f1 =

1,57 EI L2 m

(b) f1 =

Simply supported beams 6,28 EI m L2

Fixed end beams

3,56 EI m L2

(c)

f2 =

f2 =

9,82 EI m L2

f2 =

3,51 EI L2 m

Cantilevers

0,56 EI f1 = 2 m L

(b) Torsional frequencies Members are assumed to be fixed against rotation about the axis of the member at both ends. f1 =

0,54 GJ L Im

f2 =

1,13 GJ L Im

(c) Axial frequencies Members are assumed to be fixed against axial movement at both ends. f1 =

0,54 EA L m

f2 =

1,13 EA L m

10.1.2 Limitation of Slenderness Ratio to 80 It has been fairly common practice to limit the local vibration of steel members by simply ensuring that the slenderness ratio does not exceed 80. The flexural frequencies above can be re-written for steel members to give:



(Simply supported) f1 = = = =

1,57 2

L

EI ρA

1,57 E L2 ρ 1,57 L2 7925r

(Fixed ends) f1 =

I A

=

200 x10 9 r 7850

= =

L2 7925 = L L r

3,56 2

L

EI ρA

3,56 E ρ L2

I A

200 x10 9 r 7850 L2 17969r 3,56

L2 12578 = 0,7L L r

If the slenderness ratio is limited to not more than 80, these equations can be written as: (Simply supported) 7925 f1 ≥ 80L 99 ≥ L

(Fixed ends) 12578 80L 157 ≥ L

f1 ≥

This procedure thus gives fairly high frequencies, provided the individual member lengths are fairly short. Generally, vibrating screens and other vibrating equipment operate at frequencies up to about 20 Hz, so natural frequencies of 30 Hz and higher are desirable to avoid resonance problems. If the slenderness ratio is kept below 80, the natural frequencies will be above 30 Hz where a simply supported member is less than 3,3 m long, or where a fixed ended member is less than 5,2 m long. This method is simple to apply, because PROKON (and other packages) can perform a design based on limiting the slenderness ratio, but PROKON does not calculate the natural frequencies of individual members. However, for longer members, it may not avoid resonance, so it should be used with care.

10.2 Walkways and Hand Railing A frequent phenomenon in any structure with vibrating machinery of any type, is to see, the hand railing shaking, or hear it rattling. This is due to the fact that hand railing is usually attached to light walkway, or platform stringers, often only a 180 mm or 200 mm deep channel section. These stringers have very little torsional stiffness, so with the hand railing protruding upwards by 1 m or so, the whole arrangement twists very easily. Typical hand railing layouts have a fundamental natural frequency in the range between 8 Hz and 16 Hz, which leads to frequent problems of unacceptable vibrations due to resonance. There are two requirements for ensuring that hand railing has a sufficiently high natural frequency to avoid resonance. These are: (a) (b)

Hand railing standards should be larger than the normal 48 mm diameter, preferably not less than 70 mm. Torsional stiffness must be provided to the stringers to which the hand railing is attached. This can be done either by boxing the stringers, or by ensuring adequate stiffness of the cross members between walkway stringers. Details that may be used are suggested in Figure 10.1. Guidelines for the Vibration Design of Structures – Issue 2 AA BPG S002 74 of 92


Provide access holes for bolting

10.1(a): Possible Stringer Sections

10.1 (b): Possible Cross Member D t il Figure 10.1: Possible Hand Railing Details An EXCEL spreadsheet located at g:/ENGINEERING/se/DESIGN AIDS “Hand railing dynamic design” is available to check the approximate fundamental natural frequency of proposed hand railing layouts, or PROKON or ROBOT can be used to create a simple, local model for the hand railing.

10.3 Sheeting Rails Sheeting rails should be treated as isolated beams, as described in Section 10.1, with allowance made for the mass of the attached sheeting. The sheeting is, however, flexible, so its mass will not all move exactly with the sheeting rail. It is recommended that two cases be considered to provide upper and lower bounds on the natural frequency of sheeting rails. (a) (b)

Lower bound on frequency. Add the entire mass of the sheeting associated with the relevant sheeting rail in the frequency calculation. Upper bound frequency. Add 30 % of the mass of the sheeting associated with the relevant sheeting rail in the frequency calculation.

An important practical detail is that sheeting in vibrating structures must be fixed to the sheeting rail in every trough, not in every second trough as is usually done.

10.4 Plating on Chutes, Bins and Underpans 10.4.1 Natural Frequencies of Rectangular Panels

(a) Single Flat Plate Panels The fundamental natural frequency f1 (Hz) of a single flat rectangular steel panel where all four edges are simply supported is given by Szilard as: f1=

1 Et 3 π 1 ( 2+ 2 ) 2 a b 12(1-µ 2 )m



The fundamental natural frequency f1 (Hz), of a single flat rectangular steel panel where all four edges are fixed is given by Szilard as: f1≈

12 2π

1 Et 3 7 1 4 1 ( 4+ + 4 ) 2 2 7a b 2 a b 12(1-µ 2 )m

where: a and b are the two panel dimensions as shown in Figure 10.2 (m). E is the plate elastic modulus (Pa). t is the plate thickness (m). µ is the plate Poisson’s ratio, usually taken as 0,3 for steel. m is the mass of the plate and liners (kg/m2).

a (2)

a (3)

b (2)

b (1)

a (1)

Figure 10.2: Typical Plate with Several Panels As discussed earlier in this guide, in order to avoid resonance, the natural frequency of a structure or component of a structure should be at least 1,5 times the operating frequency of the equipment supported. Assuming equipment with a maximum frequency of 16 Hz, the natural frequency required to avoid resonance is thus about 25 Hz. The maximum plate panel sizes to ensure natural frequencies of the plating exceeding 25 Hz are shown in Figures 10.3 for simply supported plate panels, and Figure 10.4 for fixed edge plate panels.



Simply Supported Edges 2.2 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2

Aspect ratio b/a 1 without liner 2 without liner 10 without liner 1 with 8 mm liner 2 with 8 mm liner 10 with 8 mm liner 3

4

5

6

7

8

9

10

11

12

Plate thickness (mm)

Figure 10.3: Maximum Simply Supported Plate Sizes for 25 Hz Natural Frequency

Fixed Edges 2.2 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2

Aspect ratio b/a 1 without liners 2 without liners 10 without liners 1 with 8 mm liners 2 with 8 mm liners 10 with 8 mm liners 3

4

5

6

7

8

9

10

11

12

Plate thickness (mm)

Figure 10.4: Maximum Fixed Edge Plate Sizes for 25 Hz Natural Frequency (b) Stiffened (Orthotropic) Plating The equations for stiffened plates (often referred to as orthotropic plates) are a little more complex, but still manageable by hand for the simply supported case. The lowest natural frequency for simply supported stiffened plating is given by: Dx f1≈

π 2

a

4

+

2D v 2 2

a b m

+

Dy b4

where: Dv =

Et 3 2

+

G Jx Jy ( + ) is the shear rigidity of the stiffened plating 6 a b

12(1 - µ ) EI x is the flexural rigidity of the stiffened plating about the x-axis Dx = bx



Dv =

EI y by

is the flexural rigidity of the stiffened plating about the y-axis

a and b are the two overall panel dimensions (m) as shown in Figure 10.5. bx and by are the spacing of the stiffeners in the x- and y directions (m) as shown in Figure 10.5. E is the plate elastic modulus (Pa). G is the plate shear modulus (Pa) t is the plate thickness (m). µ is the plate Poisson’s ratio, usually taken as 0,3 for steel. Ix is the moment of inertia of the x-direction stiffeners including the plating (m4) Iy is the moment of inertia of the y-direction stiffeners including the plating (m4) Jx is the torsion constant of the x-direction stiffeners excluding the plating (m4) Jy is the torsion constant of the y-direction stiffeners excluding the plating (m4) m is the mass of the plate and liners (kg/m2). A spreadsheet, “Design Aid DA14 Plate vibration”, to facilitate calculation of the frequencies for plating is available at G:/ENGINEERING/se/DESIGN AIDS. Use of the spreadsheet is self-explanatory.

X Y A N ax ay by by

A

bx N

Figure 10.5: Typical Arrangement of Stiffened Plating

10.5 Bracing Systems Structural bracing, particularly tension bracing members, may be long and slender, and thus be prone to vibrating. Tubular sections are quite popular as they have the lowest slenderness for a given cross-sectional area of bracing members, and thus a higher natural frequency than other section shapes. Because bracing members run diagonally, they also run past other members such as girts or other bracing members. This may lead to high noise levels if vibration causes the bracing and other members to “rattle” against each other. The following steps will ensure that bracing in dynamically loaded structures remains trouble free.



(a)

The natural frequency of bracing members must be high enough to avoid resonance. See Section 10.1. (b) End connections for bracing members must be welded (but check for fatigue), or made using slip-resistant connectors. (c) Bracing members must be detailed so that: (i) either they are at least 20 mm clear of other members they pass. (ii) or they are flush with, and bolted to, other members they pass.



11 PRACTICAL DETAILS FOR CONNECTIONS 11.1 Bolted connections Where bolts are used for joining members in a structure carrying vibration loading, the following points should always be observed: (a) (b)

Never use ordinary bolts in tension. Bolts carrying tension must always be tensioned, so either high strength friction grip bolts, or swage lock fasteners should be used Bolts carrying shear should preferably be high strength friction grip bolts or swage lock fasteners. If ordinary bolts are selected, they must use nuts that will prevent loosening.

11.2 Welded Connections Welded connections should be avoided as far as possible on structures carrying vibration loading. Where welding is necessary, the following points must be observed: (a) (b)

Never use intermittent welding. Rather use a smaller continuous weld. Watch fatigue details. The fatigue life of a welded connection may be reduced spectacularly by poor welding details. If you don’t properly understand the effects of welding on the fatigue life of structures, then ask someone who does!

11.3 Beam-to-beam Connections Experience has shown several commonly used connections to be bad news when they are used in a structure carrying vibration loading. These connections are shown in Figure 11.1.

Warning: Never use the connections shown in Figure 11.1 in a structure supporting vibrating loads. They will turn around and bite you every time.



Cracks from weld if plate girder Cracks from radius

Cracks from bottom of T

Full depth T OK

Figure 11.1: Bad Commonly used Beam-to-Beam Connection Details

11.4 Bracing Connections The most important aspects of bracing connections in structures supporting vibrating equipment are: (a) (b)

The bolts must be slip-resistant bolts, i.e. either friction grip bolts or swage lock fasteners. Care must be exercised in detailing, to ensure that gussets are properly anchored to the main structural members. This is illustrated in Figure 11.2.

Cycling load leads to web cracking

Figure 11.2: Some Bad Bracing Connections



12 VIBRATION MEASUREMENTS 12.1 What Should be Measured? Vibration amplitudes may be described by acceleration, velocity, displacement, or even strain amplitude. Usually the easiest parameter to measure is acceleration, which can be integrated over short durations to provide velocity or displacement. Simple transducers can be attached to any point on a structure and provide an absolute measure of acceleration. Displacement can quite easily be measured, but it is generally not possible to measure absolute displacement. The displacement transducer must be mounted somewhere and measures the change of displacement to the desired point, so the displacement obtained is a relative displacement between the mounting point and the desired point on the structure. We generally don’t have fixed mounting points around structures, because the whole structure is floating about in space. Strain is not often measured, as placement of strain gauges on structures is time consuming, and dynamic strains are often small. When accelerations are measured, velocities can easily be calculated by integrating the accelerations with respect to time, and displacements can be calculated by integrating the velocities with respect to time. Generally, the equipment used for vibration measurements has the capability of performing these integrations automatically, so that accelerations, velocities or displacements can be displayed. If the vibration is harmonic in nature, there is a simple relationship between the amplitudes of acceleration, velocity and displacement. This is: aM = ωE v = ωE 2 d v = ωE d or d=

aM v = ωE ωE 2

v=

aM ωE

where: aM is the linear acceleration amplitude (m/s2) v is the linear velocity amplitude (m/s2) d is the displacement amplitude (m/s2) ωE is the radial frequency of the vibration (radians/s)

12.2 Measuring Equipment In selecting measuring equipment for vibration measurements, it is important to ensure that certain characteristics of the equipment are appropriate. Generally, two factors must be considered: (a) (i)

Frequency Range Minimum value. Some transducers can measure from 0 Hz (i.e. constant acceleration such as gravity) upwards. Other types of transducers only measure from a defined minimum frequency upwards. Where the operating frequency is low, such as is likely to be true of floatation cells and crushers, it is possible that the minimum frequency range may be a problem. For example, the RION Vibration Analyser used in ATD Structural Engineering, the minimum frequency is 3,0 Hz. This means that vibration measurements on float cells operating at 2,0 Hz will not be correct. Guidelines for the Vibration Design of Structures – Issue 2 AA BPG S002 82 of 92


(ii)

Maximum value. All transducers will have their own natural frequencies, and will thus experience resonance at some fairly high frequency. Most transducers measure accurately up to at least 100 Hz, whereas in structural engineering we are generally only concerned about frequencies up to about 50 Hz. This is thus seldom a problem in structural engineering measurements, but it must be borne in mind.

Many measuring systems also incorporate filters. A “low pass filter” eliminates the high frequency content of measurements, and allows the low frequency content through. Conversely, a “high pass filter” eliminates the low frequency content, and allows the high frequency content through. All electrical measuring systems have some drift, by which is meant they shift with time. In order to eliminate the effects of this drift, many systems use a low pass filter set at between 1 Hz and 3 Hz. This will have to same effect as a non-zero minimum measuring frequency, so users must be aware of what filters are used in vibration measuring equipment. The RION Vibration Analyser VA10 used in ATD Structural Engineering, has a low pass filter that can be set at 3 Hz or 10 Hz. It should always be set at 3 Hz for structural engineering measurements. The RION Spectral Analyser SA78 used in ATD Structural Engineering may be used to frequencies as low as 1,0 Hz (b) Sampling Rate The sampling rate is the rate at which the analogue measured signal is converted into digital numbers for computer storage and analysis, or display. In order to obtain good measurements, the sampling rate must be at least six times the highest frequency of interest. Thus, if a maximum frequency of 25 Hz is expected, the sampling rate must be at least 150 Hz, i.e. samples must be read at intervals not exceeding 0,006 seconds.

12.3 Recording Measurements There are numerous different aspects to be considered when vibration measurements are made, and these should all be recorded to ensure complete records. (a) (b) (c) (d) (e)

Measurements may be RMS or peak-to-peak. Acceleration, velocity, or displacement may be measured and recorded. The filter may be set at various frequencies. Measurements may be taken in any one of three different directions. The key frequencies in any measurement should always be noted.

A sample measurement sheet is shown in Table 12.4.

12.4 Relating Measured Displacements to Implied Stresses An important aspect of interpretation of measurements is to know what stresses are implied by measured displacements. A simple, conservative method (i.e. a method giving a high estimate of the implied stress) is to ignore the damping, giving the same relationship between displacement and stress as for static load. This gives the equations in Table 12.1 to obtain the stress σ (N/m2) for beams, in Table 12.2 to obtain the stress for portal columns (i.e. where bracing is not used), and Table 12.3 to obtain the stress for slabs.



Table 12.1: Conversion from Measured Deflection to Implied Stress for Beams Simply Supported Fixed Ended Beam Beam Centre of Beam End of Beam 6Ed S ∆ V 12Ed S ∆ V 12Ed S ∆ V Beam with High σ = σ = σ = 2 2 Central Mass L L L2 Beam with Uniformly Distributed Mass

σ=

4,8Ed S ∆ V 2

L

σ=

8Ed S ∆ V

σ=

2

L

16Ed S ∆ V L2

Table 12.2: Conversion from Measured Deflection to Implied Stress for Portal Columns Fixed one end, pinned other end Fixed both ends σ=

3Ed S ∆ H 2L

σ=

2

3Ed S ∆ H L2

Table 12.3: Conversion from Measured Deflection to Implied Stress for Slabs Simply Supported Fixed Edge Slab Slab Centre of Slab End of Slab 6Ed S ∆ 12Ed S ∆ 12Ed S ∆ Slab with High σ = σ = σ = 2 2 Central Mass L L L2 Slab with Uniformly Distributed Mass

σ=

4,8Ed S ∆ 2

L

σ=

8Ed S ∆ 2

L

σ=

16Ed S ∆ L2

In Tables 12.1 to 12.3: dS is the depth of the beam, portal column, or slab on which the measurements are taken (m) E is the elastic modulus of the material of which the beam is made (N/m2) L is the length of the beam on which the measurements are taken (m) ∆H is the measured horizontal deflection at the top of the portal column (m) ∆V is the measured vertical deflection at the centre of the beam (m)

12.5 Baseline Vibration Measurement Guide From time-to-time it is necessary to obtain “baseline vibration measurements”, by which is meant ambient vibration measurements on an existing structure, for later comparison with other vibration measurement data. The most common reason for wanting to obtain baseline vibration measurements is to define a contractual baseline against which vibration can be checked following commissioning of a new building, or modification of an existing building. 12.5.1 Baseline Measurements (a) The first consideration when planning baseline measurements is what measurements are necessary. The fundamental requirements are the vibration magnitudes and the major frequencies.

(i)

(ii)

Vibration Magnitudes Baseline measurements will typically be accelerations, unless there is a specific Client request, or some other good reason, for using another parameter. Vibration magnitudes may be described by peak values, or averaged values (usually RMS values are used for averaging), or both. Consideration must be given to the time-varying nature of the vibration in defining the most appropriate way to describe the vibration magnitude. Frequencies Guidelines for the Vibration Design of Structures – Issue 2 AA BPG S002 84 of 92


Human sensitivity to vibration, and the sensitivity of equipment to vibration, tends to be frequency-dependent. It is thus necessary to include frequencies in baseline measurements. (b)

The second consideration when planning baseline measurements is where, and in which direction, measurements should be taken. This is dependent on the reason for requiring the vibration baseline measurements. If they are required because a machine is to be replaced, and vibrations due to the new machine are to be checked against vibrations caused by the existing machine, then measurements should clearly be taken on the closest structural members that will remain in place. However, if it is desired to ensure that vibrations in the vicinity of sensitive equipment of office areas do not increase, then the measurements should clearly concentrate on these areas.

(c)

When baseline measurements are taken, it is important that the actual conditions must be recorded. This must include: (i) (ii) (iii)

What vibrating equipment is operating, and what equipment is standing. The throughput in the equipment that is operating. Other ambient conditions, such as the extent of spillage on floors and adhering to the inside of chutes.

A typical form that can be used for recording vibration baseline measurements is shown in Table 12.2.



Table 12.4: Typical From for Recording Vibration Measurements or Baseline Measurements Mine Area

Anglo Technical Division Date

Building Measuring device Sampling speed

Filter setting Other

Machines running Throughput Sketch of Structure

Measurement Frequency location (Hz)

P-P Accelerations (m/s2) X Y Z

RMS Accelerations (m/s2) X Y Z



12.5.2 What Baseline Measurements Can and Can’t Do

(a)

What they can do (i) (ii) (iii)

(b)

Define ambient existing vibration under actual existing conditions and at specified locations. Provide a comparative baseline against which other vibrations can later be evaluated or compared. Be used to give a good indication of whether existing vibration is within serviceability and strength limits for the structure.

What they cannot do (i) (ii) (iii)

Define worst case vibrations under operational conditions not existing when measurements are taken. Predict the structural behaviour under new equipment running at different frequencies to existing equipment. Predict the structural response at locations other than where measurements are actually taken.



13 TROUBLE-SHOOTING AND STRUCTURAL MODIFICATION 13.1 Interpreting and Using Measurements (a) Implication of Frequencies The following information can generally be obtained from the measured frequencies: (i) (ii) (iii)

Frequencies equal to the operational speed of equipment, identify that equipment as the source of the vibration, because structural vibration will always be at the exciting frequency. Frequencies much higher than the operating speed of equipment may indicate misalignment or other problems with gearboxes, motors, or tooth meshing of ring gears. If the vibration loading has a significant impact component, then measured frequencies are likely to show natural frequencies of the structure.

(b) Implication of Amplitudes The amplitudes, together with the frequencies, enables assessment of the structural integrity by checking whether serviceability, strength or fatigue life limits are exceeded.

13.2 Changes to Applied Loads Experience shows that vibration problems may result from modifications which lead to changes in the applied vibration loads. Typical examples of this are the following: (a) (b)

Setting of the eccentric masses on vibration screens or vibratory feeders may be changed in an effort to increase production throughput. This leads to a proportional increase in the applied vibration loads. Steel coil springs often have only a fairly short operational life. Frequent replacements may cause Site personnel to replace coil springs with rubber blocks, which generally have a significantly longer life. What may not be realised, is that rubber blocks are much stiffer than steel coil springs, particularly in the vertical direction. See Section 5.2. This means that significantly higher excitation is required to obtain the specified throw of the screen or feeder, leading to significantly higher loads being applied to the structure. These loads may be as much as five times higher than where steel coil springs are used.

Where structural vibration results from changes to the applied loads, it is often possible to simply, and very easily, revert to the original conditions. Where this is not possible, a re-design of the structure using the new loads is necessary, leading to structural modifications.

13.3 Structural Modifications Where a structural vibration problem is identified, the preferable solution is generally to increase the stiffness of relevant structural members. If the structure is already high tuned, this is the only rational solution, and is probably fairly easily achieved. Where the structure is initially low tuned, increasing the stiffness may worsen the problem, as it will move the structure closer to resonance. Extra mass may be added, but this impacts negatively on the static design of the structure. A large increase in stiffness to change the structure to a high tuned structure may be possible, but this is likely to involve major structural modifications. This is a much more difficult problem than the high tuned structure, and it may require a combination of extra mass and extra stiffness.



Warning: Careful analysis and thought must go into modifications to solve vibration problems in low tuned structures.

13.4 Common Concerns of Site Personnel Experience has shown that there are some common concerns expressed by Site Personnel. These include: (a)

Beat phenomenon. This occurs in every installation where separate pieces of equipment having similar operating speeds are located close together. The most common example of this is where a number of parallel process streams are used, requiring several nominally identical screens, feeders, or other vibrating equipment next to each other. Although the equipment runs at nominally the same speed, unless there is electrical or mechanical coupling, there is always a small difference in speed. This leads to two pieces of equipment running in phase for a short while with a reinforcing effect on resulting vibration, then running out of phase for a while with a destructive effect on resulting vibration. This is shown in the vibrations in Figure 13.1, which shows the interaction of two machines which nominally run at 16 Hz, but where one runs 2½ % fast, and the other runs 2½ % slow.

(b)

Rattling, or large amplitude vibration of light fittings, sheeting, or other finishes. It is impossible to model all of these minor elements in a structural vibration model. Each case must be treated on its own merits. It may be possible to easily move the fitting to a less sensitive location. Local stiffening may be possible. Insertion of rubber isolation mounts may solve the problem.



15,6 Hz vibration 1.5 1 0.5 0 -0.5 0

5

10

15

20

-1 -1.5 Time (seconds)

16,4 Hz vibration 1.5 Amplitude 2

1 0.5 0 -0.5 0

5

10

15

20

-1 -1.5 Time (seconds)

2.5 2 1.5 1 0.5 0 -0.5 0 -1 -1.5 -2 -2.5

5

10

15

20

Time (seconds)

Figure 13.1: Vibration Histories showing the Beat Phenomenon



14 BIBLIOGRAPHY 14.1 Standards and Specifications 14.1.1 SANS Standards SANS ISO 4866:1990 “Mechanical Vibration and Shock – Vibration of Buildings – Guidelines for the Measurement of Vibrations and Evaluation of their effects on Buildings.” First Edition 1999. SANS ISO 2631-1:1997 “Mechanical Vibration and Shock – Evaluation of Human exposure to whole-body Vibration. Part 1: General Requirements.” First Edition 1997. ISO 2631-1:1985 “Mechanical Vibration and Shock – Evaluation of Human exposure to whole-body Vibration. Part 1: General Requirements.” 14.1.2 AAC Specifications AAC 114/1 “Design Criteria for Steel Structures.” Issue 9 1998.

14.2 Text Books Barkan DD “Dynamics of Bases and Foundations”, McGraw-Hill, 1962. Irish K and Walker WP “Foundations for Reciprocating Machines”, Concrete Publications Limited, London, 1969. Paz M “Structural Dynamics. Theory & Computation”, Van Nostrand Reinhold. Szilard R “Theory and Analysis of Plates. Classical and Numerical Methods”. Prentice Hall. Warburton GB Library.

“The Dynamical Behaviour of Structures”, Pergamon International

14.3 Journal Papers Sen GS “Blasting Vibration and Structural Damage”, Civil Engineering, September 1981, pgs 42-44.

14.4 AAC Reports Report RAB/83/02 "Dynamic Stiffness Tests on Velmet Screen Support Springs: Vertical Stiffness", 21 July 1983. Report 2005-S-25 “Anglo, Base – Skorpion Mine. Agitator Support”, 26 December 2005.

Leach and Neutralisation Tank



15 RECORD OF MODIFICATIONS Date Jan 2006 Oct 2007

Description Publication of Edition 1.0 Symbols updated for consistency Loading information for crushers and agitators added Publication of Issue 2 Some editorial errors corrected Section 8 revised to include more information on mills Section 9 expanded to provide more detailed guidance

Author G.J. Krige G.J. Krige


Guidelines for the Vibration Design of Structures

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