Tolerance

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MECHANI MECH ANICAL CAL AD ADV VANT ANTAGE AGE

Tolerance Analysis Manual

Mechanical Advantage version 7.0 February 2001

Copyright © Cognition Corporation, 2001 All Rights Reserved

Part Number D0223-1-70

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This document, as well as the software described in it, is provided under the terms of Cognition Corporation’s Program License Agreement (PLA), which sets forth Cognition’s sole warranty. COGNITION CORPORATION DISCLAIMS ALL OTHER WARRANTIES EITHER EXPRESS OR IMPLIED, INCLUDING ALL WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE. IN NO EVENT WILL COGNITION CORPORATION CORPORATION BE LIABLE FOR CONS CONSEQ EQUE UENT NTIA IAL L DAMA DAMAGE GES S EVEN EVEN IF COGN COGNIT ITIO ION N HAS HAS BEEN BEEN ADVI ADVISE SED D OF THE THE POSS POSSIB IBIL ILIT ITY Y OF SUCH DAMAGES. THE WARRANTY AND REMEDIES SET FORTH IN SAME ARE EXCLUSIVE AND IN LEIU OF ALL OTHERS. Except as permitted by license, no part of this document may be reproduced, stored in a retrieval system, or transmitted in any form by any means without the prior written permission of Cognition Corporation. Use Use of this this docu documen mentt is rese reserv rved ed excl exclus usiv ively ely for for Cogn Cogniti ition on’s ’s cust custom omer ers, s, poten potentia tiall cust custom omer ers, s, and and pers person onne nel. l. The The info inform rmat atio ion n and and grap graphi hics cs cont contai aine ned d here herein in are are the the sole sole prop proper erty ty of Cogn Cognit itio ion n and and shal shalll not not be divu divulg lged ed to any any thir third d party without prior written permission of Cognition. The information in this document is subject to change without notice. Cognition assumes no responsibility for errors that may appear in this document, although every effort has been made to ensure correctness. AX, Design Advantage and Draft Note are trademarks of Cognition Corporation. Sketch Note, Solid Note, Math Note, DCAP Note, Program Note and CogniLink are trademarks of Cognition Corporation. Mechanical Advantage, Cost Advantage and MathSolve are registered trademarks of Cognition Corporation. Corporation. ACIS is a registered trademark of Spatial Technology, Inc. Adobe Acrobat is a registered trademark of Adobe Systems Incorporated. Exceed is a registered trademark of Hummingbird Communications, Ltd. HOOPS is a registered trademark of TechSoft America. InstallShield is a registered trademark of InstallShield Corporation. IRIX is a registered trademark of Silicon Graphics, Inc. Microsoft, Windows and Windows NT are registered trademarks of Microsoft. OSF/Motif is a registered trademark of Open Software Foundation. Nutcraker is a registered trademark of Mortice Kern Systems Inc. Solaris is a registered trademark of Sun Microsystems, Inc.

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Mechanical Advantage Tolerance Analysis Manual - February 2001

Contents Chapter 1

Introduction Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 About this Guide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Chapter 2

Introduction Introduction to Tolerance Analysis Tolerance Analysis as a Design Tool . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Theoretical Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Tolerance Deviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Mechanical Advantage Tolerance Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Worst Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Critical Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Designing for Tolerance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 A Conceptual Example. Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Advantage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 The Tolerance Analysis Advantage Default Tolerances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 All Analysis Performed on the Same Model. Model . . . . . . . . . . . . . . . . . . . . . . . . 11 All Analyses Performed Automatically . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Results Displayed in Easy-to-Read Format . . . . . . . . . . . . . . . . . . . . . . . . 11 Easy Contributor Recognition. Recognition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 As an Aid to Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Cost Savings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 As a Guide to a Better Tolerancing Scheme . . . . . . . . . . . . . . . . . . . . . . . . 12

Chapter 3

Tolerance Analysis Quick Tour The Example Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Creating The Sketch Note Model. Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Creating a Primary Datum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


13 14 14 14

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Contents Referencing the Primary Datum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Secondary Datum. Datum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dimensioning the Model. Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Creating the Derived Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Setting Default Tolerances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Analyzing the Derived Dimension. Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tolerance Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Tolerance Analysis Report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Showing Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Interpreting the Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Considering Alternatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alternative 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alternative 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alternative 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Non-linear Worst Case Analysis. Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Monte Carlo Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 4

Using Tolerance Analysis Understanding Tolerance Analysis Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Analyzed Dimension, Force or Moment. Moment . . . . . . . . . . . . . . . . . . . . . . . . . . . Percent in Tolerance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Contributor Mean and Standard Deviation . . . . . . . . . . . . . . . . . . . . . . . . . Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Percent Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Setting Tolerances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Default Tolerances by Dimension Type . . . . . . . . . . . . . . . . . . . . . . . . . . . Default Tolerances for Dimensions by Precision . . . . . . . . . . . . . . . . . . . . Default Tolerances for Position Constraints or Variables . . . . . . . . . . . . . Tolerance for Specific Dimensions and Constraints. Constraints. . . . . . . . . . . . . . . . . . Tolerance Values for Specific Dimensions . . . . . . . . . . . . . . . . . . . . . Tolerance Zones for Specific Constraints . . . . . . . . . . . . . . . . . . . . . . Tolerance for Specific Forces, Moments and Springs . . . . . . . . . . . . . . . . Tolerance Values for Specific Forces or Moments . . . . . . . . . . . . . . . Tolerance Values for Specific Linear or Torsion Springs . . . . . . . . . . Auto Tolerance for Float Pin-in-Hole Constraints . . . . . . . . . . . . . . . . . . .

ii

15 15 16 18 19 20 20 21 23 23 25 25 27 29 31 34 36

37 38 38 38 38 39 39 39 39 40 41 42 44 45 45 48 49 49 50 51


Contents Constraints that do not have Tolerance Zones . . . . . . . . . . . . . . . . . . . . . . 51 Constraints that use Tolerance Zones with Profile Tolerances . . . . . . . . . . 52 Setting Tolerance Values from the Tolerance Analysis Report . . . . . . . . . 52 Tolerance Values for the Analyzed Item . . . . . . . . . . . . . . . . . . . . . . . 52 Tolerance Values for Constraints, Dimensions, Forces, Moments and Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Tolerance Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 Performing the Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 Performing Multiple Tolerance Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 56 Reanalyzing the Analyzed Dimension, Force or Moment . . . . . . . . . . . . . 58 Showing the Analyzed Dimension, Force or Moment . . . . . . . . . . . . . . . . 58 Showing Contributors to the Analyzed Dimension, Force or Moment . . . 58 Showing Contributors to the Dependent Assembly . . . . . . . . . . . . . . . . . . 59 Sorting the Tolerance Analysis Report . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 Tolerance Distribution Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 Preserving the Current Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 Resetting the Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 Printing the Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 Duplicate Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Combining Duplicate Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Separating Duplicate Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 Writing to an ASCII file . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 Worst Case Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 Performing a Linear Worst Case Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 66 Performing a Non-linear Worst Case Analysis . . . . . . . . . . . . . . . . . . . . . . 67 Monte Carlo Analysis. Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 Monte Carlo Distribution Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 Performing a Monte Carlo Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 Contributor Inclusion in Worst Case and Monte Carlo Analyses . . . . . . . . . . . 72 Contributors Included by Default . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 Specifying Contributor Inclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

Chapter 5

Reading the Tolerance Analysis Report Report Header. Header. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Analyzed Dimension, Force or Moment . . . . . . . . . . . . . . . . . . . . . . . . . . . Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Statistical Failure Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sigma Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Design Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


76 76 76 76 76 77

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Contents Percent in Tolerance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Failure Rate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tol Max and Tol Min . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cp and Cpk Indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cpk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Val Max and Val Min . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Types Included . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Worst Case Analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Linear Worst Case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Non-linear Worst Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Monte Carlo Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Contributor Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part Name . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Contributor Name . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Upper or Lower Zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Contributor Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Contributor Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sigma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . MC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Contributor Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Upper Contribution / Lower Contribution . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 6

77 77 78 79 79 80 81 81 82 83 83 84 85 85 86 86 86 86 87 88 89 89 90 90 90 91

Sensitivity Analysis Sensitivity in the Design Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 The Importance of Sensitivity in Tolerance Analysis . . . . . . . . . . . . . . . . . . . . 94 Sensitivity Magnitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 Comparing Mixed Unit Sensitivities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 Perturbing Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 Perturbing the Angular Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 Perturbing Parallel Constraint 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 Perturbing Dimension C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 Perturbing Parallel Constraint 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Perturbing Dimension B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Perturbing Dimension A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

iv


Contents Perturbing the Perpendicular Constraint . . . . . . . . . . . . . . . . . . . . . . . . . . 104 The Effect of Angular Dimensions on Sensitivity . . . . . . . . . . . . . . . . . . . . . . 105 Tightening the Lock on the Angular Dimension. . . . . . . . . . . . . . . . . . . . 105 Re-Dimensioning the Angular Dimension . . . . . . . . . . . . . . . . . . . . . . . . 107 Perturbing Parallel Constraint 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 Replacing the Angular Dimension with a Linear Dimension . . . . . . . . . . 110 The Effect of Analyzing Linear Dimension Types . . . . . . . . . . . . . . . . . . . . . 112 Perturbing Parallel Constraint 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 Perturbing Parallel Constraint 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 Perturbing the Perpendicular Constraint . . . . . . . . . . . . . . . . . . . . . . . . . . 115 Perturbing the Angular Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

Chapter 7

Advanced Topics for Monte Carlo Distribution Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Normal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Weibull . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dimensional Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Constraint Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Uniform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Linear and Non-Linear Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Model Building for Non-Linear Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . . Contributors from Multiple Dependent Assemblies . . . . . . . . . . . . . . . . . Equal-Length Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Linked-In Dimensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . User Control of Monte Carlo Solution Failure Limit. . . . . . . . . . . . . . . . . . . . Plotting Monte Carlo results in a DCAP Note . . . . . . . . . . . . . . . . . . . . . . . . . Monte Carlo TOLLOG. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 8

119 120 121 122 123 124 125 126 126 126 126 127 128 129

Model Building Essentials for Tolerance Analysis Datums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Constraints that Reference Datums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Parallel Constraints. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Perpendicular Constraints. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equal-Length and Equal-Radius Constraints . . . . . . . . . . . . . . . . . . . Collinear Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concentric and Float Pin-in-Hole Constraints . . . . . . . . . . . . . . . . . . Position Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


131 134 135 136 137 138 139 140

v

Contents Choosing Datums for Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tolerance Analysis Results and Bad Datums . . . . . . . . . . . . . . . . . . . . . . Constraint Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chained Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Baseline Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Choosing Appropriate Dimension Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . Chained Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Baseline Dimensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Direct Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary of Model Building Essentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 9

Advanced Model Building for Tolerance Analysis Profile Tolerances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Material Modifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Setting the Modifier Type. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Worst Case Application of Material Modifiers . . . . . . . . . . . . . . . . . . . . Statistical Applications of Material Modifiers . . . . . . . . . . . . . . . . . . . . . Position Constraint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Using a Circle as a Datum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Editing a Position Constraint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tolerance for a Position Constraint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Float Pin-in-Hole Constraint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Automatic Tolerance Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Special Considerations for Float Pin-in-Hole . . . . . . . . . . . . . . . . . . . . . . Pass-Through and Locked Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Setting the Tolerance Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Choosing the Tolerance Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equal-To and Scaled-From Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . Product, Sum to and Sum to Zero Constraints . . . . . . . . . . . . . . . . . . . . . Using Dependent Assemblies for Tolerance Analysis . . . . . . . . . . . . . . . . . . . Advantages of Using Dependent Assemblies . . . . . . . . . . . . . . . . . . . . . . Combine Duplicate Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . When not to Combine Duplicate Contributors . . . . . . . . . . . . . . . . . . . . . Model Building with Projection Lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Model Building to Avoid Sensitivity Multiplication . . . . . . . . . . . . . . . . Parallel Projection Line Orientation. . . . . . . . . . . . . . . . . . . . . . . . . . Constraining Projection Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Model Building to Avoid Duplicate Contributors . . . . . . . . . . . . . . . . . .

vi

141 142 144 145 147 149 150 152 154 155

157 158 158 159 160 161 162 163 163 164 164 165 167 168 168 169 173 174 174 175 179 181 182 182 183 184


Contents

Chapter 10

Critical Parameter Analysis Performance Model Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Special Considerations for Critical Parameter Analysis . . . . . . . . . . . . . . . . . Point-to-Point Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Geometric Contributors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Variable Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Performing Critical Parameter Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Saving CPA Contributor Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Specifying Contributor Inclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Excluding Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tolerance Distribution Graph. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 1: Critical Parameter Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Analyzing Performance Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3D Property Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example 2: Testing Deflection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Appendix A

185 186 186 186 186 187 188 190 190 190 191 191 198 204

Computing Tolerance Analysis with RSS Probability Calculations Derived Dimensions, Forces, and Moments . . . . . . . . . . . . . . . . . . . . . . . . . . A-1 Sensitivity Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-2 Mean and Standard Deviation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-3 Dimension, Force, Moment and Variable Contributors . . . . . . . . . . . . . . A-3 Contributors with Tolerance Zones. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-3 Analyzed Object . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-4 Probability in Tolerance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-5 Worst-Case Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-6 Contributors and their Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-7 Linear dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-7 Point-to-Point Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-8 Point-to-Line Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-9 Line-to-Line Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-10 Gridded Point-to-Point Dimension. . . . . . . . . . . . . . . . . . . . . . . . . . A-11 Radial Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-11 Fixed-Length Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-11 Fixed-Radius Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-11 Angular Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-12 Line-at-Angle Constraints. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-16 Parallel Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-17 Perpendicular Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-18 Equal-Dimension Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A-19


vii

Contents Equal-Length Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equal-Radius Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concentric Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Position Constraint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fixed-Point Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Collinear Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equations (Variables) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Appendix B

A-19 A-20 A-21 A-21 A-25 A-26 A-29 A-29 A-30

Incorporation of Material Condition Modifiers into RSS Statistical Analyses Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-1 Position-Size Dependence versus Independence in Manufacturing . . . . . . . . B-1 Tolerance Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-2 RSS Calculations with Bonus Tolerances. . . . . . . . . . . . . . . . . . . . . . . . . B-2 Determining the Appropriate Bonus Tolerance for RSS Analysis . . . . . . B-7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B-10

viii


Chapter 1

Introduction Mechanical Advantage Tolerance Analysis is a design tool that is used by mechanical engineers in the conceptual design and analysis of manufactured products. Tolerance Analysis uses probability calculations, worst case analyses and statistical random sampling to analyze the design model for failure rates, probability in tolerance, percent contribution and sensitivity.

Assumptions The Mechanical Advantage Tolerance Analysis Manual assumes you are a current Mechanical Advantage user, and have created engineering models with Sketch Note. If you are not yet familiar with Sketch Note, you should refer to the Mechanical Advantage Training Manual before using Mechanical Advantage Tolerance Analysis. Tolerance Analysis also assumes that you have a basic understanding of the ASME Y14.5M-1994 dimensional and geometric tolerancing standard.


1

Introduction

About this Guide The Mechanical Advantage Tolerance Analysis Manual provides instructions for building a properly dimensioned sketch model and for performing a Tolerance Analysis and interpreting the Tolerance Analysis Report. • Chapter 1 - Introduction. Provides general information about assumptions and conventions used in this manual. • Chapter 2 - Introduction to Tolerance Analysis . Provides an introduction to the terms and concepts used in the Mechanical Advantage Tolerance Analysis Manual. • Chapter 3 - Tolerance Analysis Quick Tour . The quick tour is a step-bystep tutorial that provides a general overview of the Mechanical Advantage Tolerance Analysis. • Chapter 4 - Using Tolerance Analysis. Reviews the terminology used in the Tolerance Analysis Report and provides instructions for performing tasks associated with Tolerance Analysis, Monte Carlo and Worst Case analyses. • Chapter 5 - Reading the Tolerance Analysis Report . Provides instructions for reading and interpreting the results in the Tolerance Analysis Report. • Chapter 6 - Sensitivity Analysis. Provides examples that demonstrate the relationship between sensitivity and the behavior of your design. • Chapter 7 - Advanced Topics for Monte Carlo. Provides instructions for the advanced user to investigate the impact of non-normal statistical distributions. • Chapter 8 - Model Building Essentials for Tolerance Analysis . Describes model building techniques that are essential to properly dimension and constrain your model. • Chapter 9 - Advanced Model Building for Tolerance Analysis . Describes advanced model building techniques that influence Tolerance Analysis results. • Chapter 10 - Critical Parameter Analysis . Describes concepts and provides step-by-step instructions for performing a Critical Parameter Analysis. • Appendix A - Computing Tolerance Analysis with RSS Probability Calculations . Describes the RSS calculations that are used to derive Tolerance Analysis results. • Appendix B - Incorporation of Material Condition Modifiers into RSS Statistical Analyses. Describes a method whereby MMC and LMC, under certain circumstances, may be included in RSS analyses.

2


Conventions

Conventions “MA” is an abbreviation for and is used interchangeably with the full product name, Mechanical Advantage™. Certain printing conventions have been used in this guide to provide visual cues to help you find and use the information you need. Example

Description

Bold type

Bold type indicates a Mechanical Advantage command you must select. For example: • Choose Copy from the Edit menu. or

• Click OK to confirm and close the window. Bold type is also used for emphasis.

Courier

Monospaced type indicates a file name, directory name, script name, or type displayed by the system in a UNIX or Windows NT window.

Bold courier

Bold, monospaced type indicates text you must type as shown.

Italics

Italic type indicates a Mechanical Advantage term, and is used for short explanations in discussions of terminal sessions. Italic type is also used for emphasis.

RETURN

Small capitals are used for keys that you must press. Note that the RETURN key is labeled ENTER on some keyboards.

Note:

Additional information about how Mechanical Advantage operates, or an important consideration or instruction.

The following is a sample terminal session according to typeface conventions: These are short explanations you type this the computer displays this


3

Introduction

4


Chapter 2

Introduction to Tolerance Analysis Mechanical Advantage Tolerance Analysis uses Root Sum Squared (RSS) probability calculations, sensitivity and worst case analyses, and statistical random sampling to assist engineers in the design of products that consistently function properly and are relatively cost efficient to manufacture.

Tolerance Analysis as a Design Tool As a design tool, Tolerance Analysis enhances the performance and reliability of mechanical equipment by improving the process for design, development and manufacture. Tolerance Analysis helps engineers to locate and correct problems during the design of the product, thereby reducing the cost and time required to manufacture and bring it to market. Tolerance Analysis uses RSS probability calculations and statistical random sampling to set limits on manufacturing variation and to determine the effect of variation on design performance. Consider the simple example of a machinist drilling a hole in a manufactured part. The machinist uses a 1/4 inch drill bit to create a 1/4 inch hole. As the drill bit is worn down, the hole size decreases. At what point is the hole size too small? The accumulated effect of one or more deviations in a design can lead to a malfunctioning product or one that cannot be manufactured. Tolerance Analysis calculates the accumulated effect of manufacturing deviations and assists you to determine an acceptable range of variation.


5

Introduction to Tolerance Analysis

Theoretical Overview Tolerance Analysis is used during product design as a measure of quality control and to improve design efficiency. One way to ensure quality in a manufactured product is to reduce the amount of variation ( tolerance) between each part. For example, a machine has many parts that work together to create a whole. If one, or more, of these parts does not function as specified, the whole process can be affected. Dimensions. A measurement of distance (linear dimension, including arc length and spline length), angle (angular dimension), radii of circles and circular arcs (radial dimension), or diameters of circles (diameter dimension). In Mechanical Advantage, dimensions are either locked or derived. For more information about dimensions, see the Mechanical Advantage Sketch Note Manual. Constraints. A relationship applied between geometric objects which causes one to behave in a specified way when the other changes (parallel, perpendicular, equal-length, etc.). For more information about constraints, see the Mechanical Advantage Sketch Note Manual . Contributors . Dimensions or constraints that directly affect the amount of variation in the analyzed dimension, force or moment. Tolerance. A specification of how far a dimension, constraint, force, variable or moment may vary from the nominal.

Tolerance Analysis predicts the behavior of the manufactured product based on an understanding of how the contributing dimensions, forces, moments, variables and constraints behave. The behavior of the contributing parts can then be adjusted to cause the product to function as desired. The ability to identify and assign tolerances to contributors helps to ensure that design criteria can be satisfied within specified manufacturing limits. If the manufacture of a product is off by small measurements in even a few parts, the accumulated effect of those deviations can lead to a malfunctioning product or to one that cannot be assembled. Consider the following approaches for improving critical design parameters. One approach is to tighten the tolerance on the part and force it to function properly. This option is usually very costly to manufacture. An alternative is to reduce the affect of contributors on the final product by modifying the dimension scheme. Early identification of critical design parameters is paramount to design efficiency and can greatly reduce the number and magnitude of problems resulting from an incorrect tolerancing scheme. A well thought-out design can reduce the need for restrictive, expensive tolerances on individual parts or dimensions.

6


Tolerance Analysis as a Design Tool

Tolerance Deviations Tolerance Analysis identifies the accumulated effect of tolerances in one part of a design on those in another. For example, if a part consists of five assemblies, each of which has a seemingly acceptable tolerance of .005 inch, the effect on the whole assembly can be a dramatically unacceptable deviation of .025 inch.

.005

.005

.005

.005

.005

total deviation = .025 While this simple dimensional tolerance example is easily understood, most tolerance effects are not so clear. Angular dimensions contribute to tolerance in unexpected ways, and geometric tolerances are even more problematic. For example, if two lines are supposed to be parallel, how far off from true parallel can they safely be, and what is the effect of that deviation on other aspects of the design?


7


Mechanical Advantage Tolerance Analysis Tolerance Analysis uses a variety of methods, including Worst Case, Monte Carlo and Critical Parameter Analysis, to analyze the design model for failure rates, probability in tolerance, percent contribution and sensitivity. The results of a Tolerance Analysis are displayed in an easy to read report that identifies whether the design will function within the specified parameters. The report identifies which parts have the greatest influence (contribution) on the outcome. With this information you can revise the design. Failure Rates. Based on the design model and tolerancing scheme, the system calculates the number of times per million the analyzed dimension, force or moment will be out of tolerance. Probability in Tolerance . Based on the design model and tolerancing scheme, this is the likelihood that the analyzed dimension, force or moment is within tolerance. Percent Contribution . The amount of variation a particular dimension, constraint, force, variable or moment contributes to the probability that the analyzed item is in tolerance. Sensitivity. The rate of change of the analyzed dimension, force or moment with respect to a contributor. Mathematically, it is the first partial derivative of the analyzed item with respect to a contributor. It is sometimes described as a measure of the leverage of a contributor.

Worst Case Worst Case calculates the tolerance limits of the analyzed dimension, force or moment when the contributors are at their maximum and minimum tolerance limits. MA provides both a Linear Worst Case and, optionally, a Non-linear Worst Case. With every Tolerance Analysis you perform, the system automatically calculates a Linear Worst Case scenario and displays the information in the Tolerance Analysis Report. When performing a Linear Worst Case, the system estimates the minimum and maximum amount of variation in the analyzed item. The resulting values for the analyzed item are displayed almost immediately. When tolerance problems are very critical, you can perform a Non-linear Worst Case analysis for more accurate results. With Non-linear Worst Case, the system sets the values of each dimension to their maximum and minimum tolerance limits and calculates the resulting values for the analyzed item. The geometric contributors are then estimated and applied to the analyzed item.

8


Designing for Tolerance

Monte Carlo Monte Carlo is an optional statistical technique that uses random sampling to calculate values for mean, standard deviation and probability. To more accurately model the manufacturing environment, Monte Carlo analysis allows you to specify a sample size as well as modify distribution types for contributors, choosing from Normal, Uniform and Weibull. This approach can produce results typical of those found in the manufacturing process. Modifying Monte Carlo distribution types is an advanced capability that requires manufacturing process distribution information.

Critical Parameter Critical Parameter Analysis applies the power of Tolerance Analysis to nongeometric parameters, enabling users to rapidly analyze and gain insight into the factors that affect performance and reliability in their designs. Critical Parameter Analysis aids in the identification of contributors that cause variation in performance. For example, some critical parameters that influence performance include response time, torque, current, force, stress, weight, velocity, inertia and temperature.

Designing for Tolerance Designing for tolerance is a widely recognized practice for quality engineering. The amount of variation in the final product is directly influenced by the geometry of parts, how the parts are designed to fit together and how parts are designed for manufacturing. The design phase of a project is the first and best opportunity you have for controlling tolerance and reducing variation. The quality of your design, combined with Tolerance Analysis, will help you avoid the manufacturing expense of a product that relies on restrictive tolerances. When a product is designed properly—with attention given to fit and function from its inception—the impact of the tolerances can be kept sufficiently small and the product will typically be easier to manufacture.


9


A Conceptual Example To introduce you to the Mechanical Advantage Tolerance Analysis and Critical Parameter Analysis, consider a simple and easy to follow example: the design of a pump that delivers solution at a constant rate of 50 ml/min. An acceptable amount of variation in the design allows the flow rate to fluctuate between 49 and 51 ml/min. Consider the following: 1.

The flow rate of the pump is a variable that needs to be controlled. Flow rate is the output.

2.

The flow rate (output) is influenced by variables that contribute to the pumps ability to produce an output. These variables include: • Diameter of the piston. • Stroke length. • Motor speed. • Viscosity of the solution.

3.

Flow rate varies because the piston radius, stroke length, motor speed and viscosity may vary. Variations in the inputs are transmitted to the output.

The design scheme for the pump is dimensioned and constrained using Mechanical Advantage Sketch Note. A Tolerance Analysis and Critical Parameter Analysis are performed from the sketch note to analyze the impact of variability on flow rate. Tolerance Analysis and Critical Parameter Analysis simulate the behavior of the inputs and predicts the impact on the output. The results are displayed in a report that identifies the contribution of dimensions and constraints to the analyzed dimension. Not only does the report provide a percentage rate for the analyzed dimension being in tolerance, but also provides information to improve the results: • Depending on how you designed the model, you may find out that certain dimensions and constraints do not influence the flow rate. • The tolerance report lists which contributors have the greatest impact on the output and, thus, helps you to determine which tolerance to tighten and by how much. • This information also assists you in modifying the design so the sensitive contributors will have less of an influence on the output. Since Tolerance Analysis and Critical Parameter Analysis are performed from your sketch note, all adjustments to the design can be quickly reanalyzed.

10


The Tolerance Analysis Advantage

The Tolerance Analysis Advantage Default Tolerances Using Tolerance Analysis, you can set default tolerances for any type of dimension in a design. The default values can be overridden for any specific dimension. For example, you can specify a default tolerance of ± 0.5 ° for all angular dimensions. If you have a specific angle that requires a ± 0.2 ° tolerance, you can set the value for the specific angle and override the default. Once tolerances have been applied, you can analyze any derived dimension, force or moment in the design. A full Tolerance Analysis is performed by choosing a command from the sketch note. The system then performs in seconds or minutes a complex analysis that could take days or weeks if performed manually.

All Analysis Performed on the Same Model All analysis is performed on the same Mechanical Advantage sketch model that you created with sketch note. When you analyze the sketch model for tolerance and sensitivity you significantly enhance your productivity by ensuring accuracy and consistency throughout the development process. In addition, if the design is changed for functional reasons, you can immediately and quickly reanalyze the design to determine the impact of the changes.

All Analyses Performed Automatically To analyze a design for tolerance and sensitivity, just select a derived item as a focus of analysis and choose the command. The analysis is performed automatically and results are displayed in seconds. Automatic analysis contributes greatly to increased productivity by avoiding the time-consuming and error-prone calculations of manual analysis.

Results Displayed in Easy-to-Read Format Mechanical Advantage displays the results of an analysis in an easy-to-read report. You can write the report to an ASCII text file that can be exported to other applications. The report includes a statistical evaluation of the analyzed item and a worst-case summation of all dimensional contributors. It also lists the contributions resulting from the geometric tolerances you applied to the design.

Easy Contributor Recognition The quick and easy approach to contributor identification is perhaps the most time-saving feature of the Tolerance Analysis Report. The report displays all contributors in a tabular format, and highlights the contributing dimensions on the sketch note, thus helping you avoid the common problem of overlooking a contributor.


11


As an Aid to Design The sensitivity data displayed in the Tolerance Analysis Report tells you the rate and in what direction a small change in a contributor will affect the analyzed item. By eliminating the typical guesswork of what to change, by how much, and in what direction, this information allows you to make cost-effective changes in order to arrive at a desired condition. You can also use sensitivity information as a design tool to study motion amplification or attenuation, velocity, and acceleration.

Cost Savings Mechanical Advantage helps to reduce costs in the development process by performing Tolerance Analysis much faster and more accurately than other methods. Because you can identify all tolerance-related problem areas during the design phase, Tolerance Analysis can reduce the number of prototypes and mockups required to complete a design. The benefits of Tolerance Analysis are not limited to product design. After a design is released to production, Tolerance Analysis can be used for in-process inspection of parts to determine their acceptability for assembly, eliminating costly interruptions on the production line and greatly reducing the time-tomarket.

As a Guide to a Better Tolerancing Scheme In a mechanism or assembly, the sensitivity of the variation in one dimension to the variation in another can be a strong indication of the stability of the design for both functional and manufacturing considerations. Frequently you can adjust the dimensional and geometric tolerancing scheme in order to improve undesirable situations. Sensitivity information derived by Mechanical Advantage Tolerance Analyses is not easily arrived at by other methods.

12


Chapter 3

Tolerance Analysis Quick Tour This chapter uses a step-by-step tutorial to provide a general overview of the Mechanical Advantage Tolerance Analysis. The tour assumes you are a current Mechanical Advantage user, and have created engineering models with Sketch Note. The Quick Tour also assumes you have Mechanical Advantage running and are using it in conjunction with these instructions. If you are not yet familiar with Sketch Note, you should refer to the Mechanical Advantage Training Manual before beginning the Tolerance Analysis Quick Tour. The Quick Tour also assumes that you have a basic understanding of the ASME Y14.5M-1994 dimensional and geometric tolerancing standard. In the Quick Tour, you will follow step-by-step instructions to solve a tolerance problem using Tolerance Analysis. The Tour also examines several alternative design strategies which are much less tolerance sensitive.

The Example Problem The Quick Tour uses the simple example of a linkage in a mechanism. The critical functional dimension for this design is the distance between the centers of the circles. Assume the part is machined from sheet metal stock. The design engineer must build a sketch note model that reflects the critical functional criteria, and dimension it in a manner that makes sense to the machinist.


13

Tolerance Analysis Quick Tour

Creating The Sketch Note Model Mechanical Advantage makes performing a Tolerance Analysis easy, but relies heavily on a “well-built” sketch note model. To achieve accurate Tolerance Analysis results, your model should reflect the critical functional criteria of the design, and simulate the way a part will be made, assembled, and measured. The ASME Y14.5M-1994 dimensioning and tolerancing standard prescribes the use of datums on drawings. Datums are a point of reference from which you should constrain your model. Proper datum specification is important when you build sketch note models since this will reduce the amount of variation in the design and improve your Tolerance Analysis results. The dimensions and constraints in the model should reference common datums. The number of datums in a design model is a direct result of the dimensioning scheme used to constrain the geometry. The following examples demonstrate proper dimensioning schemes that reduce the number of datums used in the design.

Getting Started 1.

From the New menu of the Mechanical Advantage desktop, create a new sketch note.

2.

Open the sketch noteandbeginyour model bycreating the lowerandleft edges of the linkage, as shown in the example. The linear dimensions are point-to-line dimensions.

Creating a Primary Datum The horizontal line is the primary datum for the model since its location is anchored with fixed-point and line-at-angle constraints. 3.

14

Select the horizontal line. From the Constrain menu, choose the Line-atAngle and Fixed-Point commands. Add the dimensions as shown in the example.


Creating The Sketch Note Model

Referencing the Primary Datum Use a perpendicular constraint to constrain the vertical line so that it references the primary datum. 4.

Select the vertical line. From the Constrain menu, choose the Perpendicular To command. Click on the horizontal line that you designated as the primary datum line.

A small arrow is displayed on the perpendicular constraint and points to the horizontal line, thus indicating the constraint references the primary datum.

The Secondary Datum The vertical line is a good choice for the secondary datum since it is dimensioned off of a fixed-point and uses the perpendicular constraint to directly reference the primary datum. Whenever possible, use a dimensioning scheme that constrains the geometry in reference to the primary or secondary datums.


15


Dimensioning the Model

16

5.

Create the dimensions and constraints for the linkage as shown below. The linear dimensions shown are point-to-point dimensions. Take care that your parallel and perpendicular constraints point to the lines as shown.

6.

From the Connections menu, choose the Show Underconstrained Items command to verify the model is fully constrained.


Creating The Sketch Note Model 7.

Create the circles and dimension them as shown.

8.

Choose the Show Underconstrained Items command to verify that the model is fully constrained. Tolerance Analysis requires that your model is fully constrained.


17


Creating the Derived Dimension 9.

Complete the sketch by creating the dimension named CFD, which is the critical functional dimension for the design. CFD is a derived dimension since the placement of the holes is fully constrained by existing constraints and dimensions.

10. Optional. Name the dimensions as shown above (select each dimension individually, and choose Options from the Edit menu). Naming dimensions is not necessary, but makes them easier to reference later in the example.

18


Creating The Sketch Note Model

Setting Default Tolerances Before you perform a Tolerance Analysis for the first time, set the default tolerances. These defaults will be used in the tolerance calculations. 1.

From the Analysis menu, choose the Set Default Tolerances... command. In the option window that opens, enter the upper and lower tolerances as shown in the following example:

2.

Click OK to confirm and close the window.

When any derived dimension is analyzed, the default tolerance settings are used for all dimensional contributors. Later you will see that these defaults may be overridden for any individual dimension.


19


Analyzing the Derived Dimension Now that you have a fully constrained sketch model and you have set default tolerances, analyzing the derived dimension is easy.

Tolerance Analysis

20

1.

Select the derived dimension, named CFD. This is the dimension you want to analyze.

2.

From the Analysis menu, choose the Analyze Tolerance command. Results from the analysis are displayed in the Tolerance Analysis Report.


Analyzing the Derived Dimension

The Tolerance Analysis Report The tolerance report shows you information about the analyzed dimension and the dimensions and constraints that contribute to variation in the analyzed dimension.

Analyzed Dimension Information

Description

Analyzed Dimension

The name you assigned to the analyzed item. You can analyze any derived dimension, force, or moment.

Value

The current value of the analyzed dimension.

Failure Rates (ppm)

The number of times per million the design will be out of tolerance.

% in Tolerance

The likelihood that CFD is within tolerance.

Design Limits

The design limits are the goals you set for the analyzed dimension.

Cp / Cpk

Cp and Cpk are process capability indices which represent the ratio of design limits to process capabilities. Both indices assume 3 Sigma manufacturing process capabilities.

Tol Max / Tol Min

The upper and lower tolerance values corresponding to Sigma levels and design goals.

Val Max / Val Min

The value of the dimension plus (+) the TolMax or minus (-) the TolMin: ValMax = Value + TolMax ValMin = Value + TolMin

where the value of TolMin is generally a negative number. Types Included

The c ontributor types i ncluded i n the M onte Carlo a nd Worst Case analyses.

Linear Worst Case

The system estimates the behavior of the model and predicts an outcome based on contributor values being at maximum and minimum tolerance limits.

Non-linear Worst Case

The system sets the values of each dimension to their maximum and minimum tolerance limits and calculates the resulting values for the analyzed dimension. The geometric contributors are then estimated and applied to the analyzed dimension.


21

Tolerance Analysis Quick Tour Contributor Information

Description

Part Name

The dependent assembly (if any) from which the contributor is associated.

Contributor Name

The name of each contributor.

Value

Displays the value and units of dimensional contributors.

Upper or Zone / Lower

The upper and lower tolerance value for the dimensional contributor or the tolerance zone for the constraint contributor.

Type

Contributors are identified by several abbreviations to aid in distinguishing between dimensions and constraints. Symbols are used to supply other relevant information about your model.

Sigma

The Sigma value assigned to the contributor.

MC

The distribution type used by the contributor for the Monte Carlo analysis. Distribution types include Normal (N), Weibull (W), and Uniform (U).

Sensitivity

Sensitivity is a measure of the rate at which small changes in each contributor influence the value being analyzed (CFD). Contributors that have high sensitivities are more tolerance-critical than those with low sensitivities.

Contribution

Taking into account the contributions by all of the contributors, this value shows the percentage accountable to each contributor. That is, a high percentage means that the specific contributor has a major impact on whether or not CFD is within tolerance. Contributors are automatically sorted by their percent contribution. You may also sort by sensitivity.

Upper Contribution Lower Contribution

The system uses the following formulas to calculate the magnitude of change in CFD if the contributor took on its worst case value: Upper Contribution = Upper Tolerance x Sensitivity Lower Contribution = Lower Tolerance x Sensitivity

22


Showing Contributors

Showing Contributors The derived dimension has several contributing dimensions and constraints. After a Tolerance Analysis is performed on the derived dimension, use the Show Contributor command to bring the sketch note forward and locate contributing dimensions or constraints. 1.

Click on row 8 of the Tolerance Analysis Report to select it.

2.

Extend-select row 9. Now both rows are selected.

3.

From the Viewing menu, choose the Show Contributor command. The two perpendicular constraints are highlighted in the sketch.

Interpreting the Results The following are some conclusions that can be made from this analysis: • Because of the way the model was built, certain dimensions and constraints have no effect upon CFD; for example, the width of the linkage base and arm, the diameters of the holes, and the parallels between the edges of the linkage do not contribute to CFD. This information means that, to hold CFD to a designed tolerance range, the tolerances on those contributors are not important. • The probability that CFD falls in the default tolerance range is 56.70%. • The angular dimension contributes the largest percentage towards CFD being out of tolerance (74.8%).


23


When comparing the sensitivity of Angle with the other contributors, the sensitivity of the angular dimension appears low because it has mixed units (in/deg). The sensitivity of Angle is not directly comparable with the other contributors, since the units of these sensitivities are different (in/deg vs. in/in). To compare the sensitivities of Angle with the other contributors, you should normalize the sensitivity value for Angle by converting from in/deg to in/rad. For more information about normalizing contributor sensitivity, see “Comparing Mixed Unit Sensitivities,” page 96.

24


Considering Alternatives

Considering Alternatives The Tolerance Analysis results tell us that CFD is not likely to be in tolerance. The following examples offer alternative design schemes that increase the probability that CFD will be in tolerance. The most effective design schemes use the primary and secondary datums to constrain the geometry: • Alternative 1 demonstrates the effect of tightening the tolerances on the contributors, especially those with large contributions and/or sensitivities. • Alternative 2 demonstrates the positive effect of a dimensioning scheme that reduces the number of datums used to constrain the geometry. • Alternative 3 sets a ± 3 Sigma design goal and demonstrates another way to dimension the model using datums.

Alternative 1 One alternative is to tighten the tolerance on the angular dimension, which is a major contributor to the analyzed dimension. 1.

Select the contributor Angle from the Tolerance Analysis Report.

2.

From the Edit menu, choose the Options command and set an upper and lower tolerance of ± 0.1 degrees.



25

Tolerance Analysis Quick Tour 3.

From the Viewing menu, choose the Reanalyze command.

With Angle held to a tighter tolerance, its percent contribution goes down. Notice that the probability that CFD is within tolerance increased from 56.70% to 86.05%. This is a great improvement, but it may come at a great manufacturing expense. That is, the cost of holding the Angle dimension to a tolerance of ± 0.1 may be very high.

26



Alternative 2 Consider an alternative dimensioning scheme to locate the right hole. Instead of locating it relative to the right edge of the linkage, locate it relative to the primary and secondary datums. 1.

Delete dimensions C and D. The right hole is now underconstrained (that is, under-dimensioned).

2.

Create the locked point-to-line dimensions E and F, as shown below. This locates the hole relative to the model datums.

3.

From the Connections menu, choose the Show Underconstrained Items command to verify that your model is fully constrained.


27

Tolerance Analysis Quick Tour Now analyze CFD for this alternative dimensioning scheme. 4.

Select CFD in the sketch window.

5.

From the Analysis menu,choose the Analyze Tolerance command. The new Tolerance Analysis results are displayed, as shown below.

Notice that the probability that CFD is within tolerance has increased dramatically from 56.70% to 96.61%. Note also that the number of contributors has greatly decreased, and that the angular dimension is no longer a contributor, so it is not necessary to hold it to a tight tolerance. The contributions are spread a little more evenly, with no contributor having an unusually large contribution. Without tightening tolerances, this dimensioning scheme greatly increases the probability that CFD is within tolerance. The machinist locates both holes relative to common model datums.

28



Alternative 3 The following example demonstrates another dimensioning scheme that establishes the left hole as a datum for locating the right hole. This alternative sets a design goal of ± 3 Sigma, which has a 99.73% success rate. If you have a design goal of ± 3 Sigma, then 96.61% is not a high enough probability that CFD is within tolerance. Consider the following dimensioning alternative. 1.

Delete dimension E, which locates the horizontal distance of the right hole from the model datum.

2.

Create the parallel-to-grid dimension G, as shown below. This locks the horizontal distance between the two holes.

3.

Create the parallel-to-grid dimension H, as shown below. This locks the vertical distance between the holes.

4.

From the Connections menu, choose the Show Underconstrained Items command to verify that the model is fully constrained.


29

Tolerance Analysis Quick Tour Now, perform another Tolerance Analysis to see the effect of this new dimensioning scheme. 5.

Select the CFD dimension in the sketch.

6.

From the Analysis menu,choose the Analyze Tolerance command. The results are displayed as shown below.

This dimensioning scheme produces a design that is within ± 3 Sigma. The probability has increased from 96.61% in the previous example, to 99.73%. One reason is that the number of contributors has decreased from 5 to 2. Clearly, if the manufacturer of this linkage can locate the right hole in this manner, you’ll get a better part without incurring the manufacturing expense of tighter tolerances.

30



Non-linear Worst Case Analysis Now that you have a well designed sketch that performs within your stated design limits, you should perform a Non-linear Worst Case analysis. The Non-linear Worst Case analysis will set the tolerance values of each dimension to their maximum and minimum tolerance limits. A Non-linear Worst Case analysis is performed from the Tolerance Analysis Report. 1.

From the Viewing Menu in the Tolerance Analysis Report, select the Set Up Window command. The Tolerance Window Options are displayed.

2.

Choose Yes for Show Non-linear Worst Case. Non-linear Worst Case can be computationally intensive, so it is only calculated when required. Click OK.

3.

An alert is displayed informing you that Non-linear Worst Case calculates the dimensional worst case and uses estimates for geometric contributors. Click OK to dismiss the alert.


31

Tolerance Analysis Quick Tour 4.

The Tolerance Analysis Report is displayed.

When Non-linear Worst Case sets dimension values to their limits, the sensitivities of geometric contributors are recalculated, and the revised sensitivities are used to estimate the geometric contribution. In the example, Non-linear Worst Case values for Tol Max and Tol Min ( ± 0.00613 ) are greater than the ( ± 0.005 ) Tol Max and Tol Min values set for your ± 3 Sigma design goal. The Non-linear Worst Case results are consistent with the Linear Worst Case results. This is an indication that the model is reasonably linear. Non-linear models will produce a noticeable difference between Linear and Non-linear Worst Case.

32



Sigma -0.005 -0.00613

99.73% 3 Sigma Zone

+0.005 +0.00613

Non-linear Worst Case results fall outside of the 3 Sigma Zone.

Refer to the normal distribution curve. The Non-linear Worst Case values are greater than the design limits and fall outside of the ± 3 Sigma zone on the distribution curve. Therefore, in a Non-linear Worst Case situation, when tolerances are stressed to their maximum and minimum values, the analyzed dimension will not be able to handle the variation of the contributing dimensions and constraints. Optional: Spend some time working with the design so Non-linear Worst Case values are ± 0.005 or lower. You can do this by altering the dimensioning scheme, or by tightening up the tolerances on the contributors listed in the Tolerance Analysis Report.


33


Monte Carlo Analysis The Monte Carlo analysis is performed from the Tolerance Analysis Report. Monte Carlo uses statistical random sampling to calculate the mean, standard deviation and probability for the analyzed dimension. With the Monte Carlo analysis, you have the option to investigate the impact of non-normal Weibull and Uniform distributions on the manufacturing process. However, if you do not have adequate information about the manufacturing process for the model, or a good understanding of statistical modeling, it is recommended that you use a Normal distribution. The system uses Normal as the default distribution. For the purposes of this example, all contributors will use the default Normal distribution. The Monte Carlo results for contributors with a Normal distribution should be virtually the same as the results from the standard Tolerance Analysis, which uses RSS probability calculations rather than random sampling. As a result, the Monte Carlo analysis can be used to verify the RSS calculations of the standard Tolerance Analysis. Use the Monte Carlo analysis to analyze CFD. For the example, the sample size is 5000. 1.

From the Edit menu in the Tolerance Analysis Report, select the Monte Carlo Analysis... command.

2.

Enter a sample size of 5000 and choose the Linear method. Note: Monte Carlo Tolerance Analysis results are more accurate with larger sample sizes, since the random sampling has a larger sample for calculating results.

3.

34



Considering Alternatives 4.

The results are displayed in the Tolerance Analysis Report.

The report lists the design limit failure rates for both the standard Tolerance Analysis and Monte Carlo analysis. The Monte Carlo results are listed under the section Linear Monte Carlo Failure Rates. The Percent in Tolerance value for both analyses are virtually the same, with the standard Tolerance Analysis at 99.73% and Monte Carlo calculating the rate at 99.72%.


35


Summary Mechanical Advantage Tolerance Analysis provides a quick and easy-to-use tool to assess the quality of your design schemes. Tolerance Analysis is performed directly from the sketch note, which allows you to re-dimension design schemes based upon the tolerance information generated in the Tolerance Analysis Report. Sketch models should be dimensioned and constrained with properly chosen datums that will provide for a much less tolerance sensitive design. The random creation of dimensions and constraints produces models that are difficult to understand and may produce Tolerance Analysis results that don’t match your expectations. To use the Mechanical Advantage Tolerance Analysis: 1.

Create the sketch model with constraints and dimensions that reflect functional criteria and, if necessary, manufacturing considerations.

2.

Use the Set Default Tolerances command.

3.

Use the Analyze Tolerance command on a derived dimension, force or moment.

4.

Interpret the results and, if necessary, consider changes to tolerances or alternative ways to constrain your model.

5.

Cycle through steps 3 and 4.

The example discussed here is simple; however, you can analyze more complex models just as easily, provided your model is properly created. Finding the optimum dimensioning scheme or tightening tolerances on the major contributors can have a major impact on the function, performance, reliability, and cost of your designs. Properly dimensioned designs can tolerate wide deviations in all contributors.

36


Chapter 4

Using Tolerance Analysis This chapter reviews the terminology used in the Tolerance Analysis Report and provides instructions for performing tasks associated with Tolerance Analysis and the supplementary Monte Carlo and Worst Case analyses. In depth discussion of concepts and strategies are referred to in later chapters.

Understanding Tolerance Analysis Terms To provide you with a more comprehensive understanding of the terms used in discussing Tolerance Analysis, this section details the meaning of each term, how it is derived, and pertinent background information. For more information on interpreting the report, see “Reading the Tolerance Analysis Report,” page 75.


37

Using Tolerance Analysis

Analyzed Dimension, Force or Moment The items that may be analyzed must be derived. The analyzed item must be either a derived dimension, force or moment. Any type of derived dimension may be selected: linear, radial, diameter or angular. A Tolerance Analysis performed on a locked item has no meaning.

Percent in Tolerance This calculated value is the probability that the tolerances applied to the analyzed dimension, force or moment—which can be user-assigned or default—can be attained. It is determined by calculating the area under the normal distribution curve between the limits of the assigned lower to upper tolerances on the analyzed item.

Contributor Mean and Standard Deviation Mechanical Advantage uses the upper and lower tolerances of each contributor with the following equations to calculate the mean and standard deviation of each contributor. upper tolerance + lower tolerance 2

µ

= ---------------------------------------------------------------

σ

upper tolerance – lower tolerance = --------------------------------------------------------------2R

Where:

µ = Mean of that contributor R = Sigma range 3,4,5,6

σ = Sigma value Sensitivity This extremely useful calculation has been used in many different types of analyses. You may recognize it under some of its other names, such as influence coefficient, partial derivative, or lever ratio. Sensitivity shows how a contributor influences the analyzed item when only the effect of that contributor is considered. The units of sensitivity vary depending upon the units of the analyzed item and units of analyzed item units of contributor

the units of the contributors (i.e., ----------------------------------------------------- ). For example, five sets of possible units are: in/in, in/deg, deg/deg, in/lbf or lbf/deg.

38


Understanding Tolerance Analysis Terms

Contribution For each contributor in the Tolerance Analysis Report there are two columns that display contribution, upper and then lower. This is the contribution to the Linear Worst Case of the analyzed item due to the contributor on that row.

Percent Contribution The percentage each contributor contributes to the variance of the analyzed item. This is an important number as it shows how each contributor affects the spread of the normal distribution curve for the analyzed item. The larger the curve spreads, the harder it will be to meet a desired probability in tolerance. If you want to meet a probability goal, you can create the greatest impact by tightening the tolerances on the dimensions with the largest percent contribution. Note: This is not a simple percentage contribution of the contributor to the dimensional worst case total, which is calculated by dividing the upper or lower contribution from a contributor by the dimensional worst case upper and lower calculated values.

Normal Distribution The RSS portion of the Mechanical Advantage Tolerance Analysis assumes a normal distribution of the tolerances about the nominal mean value of the dimension or of the constraint zone. The Greek letter Sigma (σ) is the standard notation used to describe the statistical 2

notion of standard deviation. The variance of the analyzed dimension equals σ . See “Mean and Standard Deviation” on page A-3, for more information about normal distribution in Tolerance Analysis.

Mean For a single contributor the mean, identified by the Greek letter Mu (µ), is defined as the weighted central value in the distribution sample. For a normal distribution the mean occurs at the midpoint of the distribution. It is easy to determine the value of the mean of each contributor, as it is the center of the tolerance range. Some examples are: Dimension

Tolerance

Mean

4.375

+/- .005

0.000

7.125

+.010/–0

0.005

12.6875

+0/–.0016

–0.0008

25.000

+.005/–.010

–0.0075

Note: Mean is figured on the tolerance and not on the dimension. Mechanical Advantage Tolerance Analysis Manual - February 2001

39


Setting Tolerances Tolerance Analysis automatically sets a 0.00 tolerance value for all dimensions, constraints, forces, moments, springs and variables. A tolerance value of 0.00 contributes nothing to the Tolerance Analysis. If you want a tolerance value other than 0.00, you must set the value. To apply tolerance values to dimensions, you can either set a default tolerance or you can assign a specific tolerance to a single dimension that will override the default value in that particular dimension instance. A default tolerance can also be applied to position constraints, which are the only constraint type that can be assigned a default tolerance. For more information about setting default tolerance values, see: • “Default Tolerances by Dimension Type,” page 41. • “Default Tolerances for Dimensions by Precision,” page 42. • “Default Tolerances for Position Constraints or Variables,” page 44. For information about setting tolerances for dimensions that override a default tolerance, see: • “Tolerance Values for Specific Dimensions,” page 45. Tolerance values must be set for specific constraints since default tolerances cannot be set for constraint types. The one exception is for position constraints, which are described above. For more information about setting a tolerance zone for a constraint, see: • “Tolerance Zones for Specific Constraints,” page 48. Tolerance values must be set for specific forces, moments and springs since default tolerances cannot be set for them. For more information about setting a tolerance for a force analysis item, see: • “Tolerance Values for Specific Forces or Moments,” page 49. • “Tolerance Values for Specific Linear or Torsion Springs,” page 50. Tolerance values can be set for several contributors simultaneously as long as the contributor types and units are compatible. For more information about setting tolerance values from the Tolerance Analysis Report, see: • “Setting Tolerance Values from the Tolerance Analysis Report,” page 52.

40


Setting Tolerances

Default Tolerances by Dimension Type Setting a default (preset) tolerance for a type of dimension (e.g. linear dimension) supplies tolerance values for all dimensions of that type. Default tolerance values are set from the Default Tolerance window. To set default tolerances for dimensions: 1.

From the Sketch Note Analysis menu, or the Edit menu in the Tolerance Analysis Report, choose the Set Default Tolerances command. The Default Tolerance window is displayed.

2.

If not already selected, select Type for the Set Dimensions By option.

3.

Enter the tolerance values. Click OK to close the window.

From the Default Tolerances window you can choose a Sigma range and set upper and lower default tolerance values for the following dimensions: • Linear • Radial/Diameter • Angular


41


Default Tolerances for Dimensions by Precision You can set default tolerances based on the Precision setting of the dimension. This This optio option n allo allows ws you you to set set defa defaul ultt tole tolera ranc nces es for for dime dimens nsio ions ns base based d on deci decima mall places. To set the default tolerances based on the Precision setting of the dimension:

42

1.

From From the the Ana Analy lysi siss men menu, u, choo choose se Set Default Tolerances. The Default Tolerances window is displayed.

2.

Click the Precision button for the Set Dimensions By option.

3.

Ente Enterr the the desi desire red d tole tolera ranc ncee for for each each prec precis isio ion n / dime dimens nsio ion n type type comb combin inaation. Click OK or Apply.


Setting Tolerances Usin Using g the the prec precis isio ion n sett settin ing, g, you you can can cont contro roll how how prec precis isel ely y toler toleran ance cess are are appl applie ied d to dimensions. The specified tolerances are applied to dimensions based on the displayed number of decimal places for that dimension. The The foll follow owin ing g exam exampl plee demo demons nstr trat ates es how how preci precisi sion on tole tolera ranc ncee valu values es are are appl applied ied to dimensions in the sketch note.

Based on the values in the example, a dimension that is displayed with three decimal places is applied a tolerance of ± 0.005 . If you use the Option window for that that same same dime dimens nsio ion n and and chan change ge the the numb number er of disp displa laye yed d deci decima mall plac places es to two, two, then a tolerance of ± 0.02 is applied.


43


Default Tolerances for Position Constraints or Variables To set default default tolerances tolerances for position position constraints constraints or variables: variables: 1.

From rom the the Sketc ketch h No Note te An Anal aly ysis sis menu menu,, or the the Ed Edit it menu menu in the the To Tole lera ranc ncee Analysis Report, choose the Set Default Tolerances command. The Default Tolerance window is displayed.

2.

Ente Enterr a tole tolera ranc ncee valu value. e. Clic Click k OK to close the window.

The Position constraint is the only constraint type that can be assigned a default tolerance value.

44


Setting Tolerances

Tolerance for Specific Dimensions and Constraints You can You can spec specif ify y tole tolera ranc ncee valu values es for for spec specif ific ic dime dimens nsio ions ns or cons constr trai aint nts. s. When When a tolerance is applied to a specific dimension, it overrides the default tolerance in that particular instance.

Tolerance Values for Specific Dimensions To speci specify fy a tole tolera ranc ncee for for a spec specif ific ic dime dimens nsio ion n and and over overri ride de the the defa defaul ultt tole tolera ranc nce: e: 1.

Sele Select ct one or more more dime dimens nsio ions ns from rom the the sketc ketch h note. ote. To sele select ct mult multip iple le dimensions, use draw-through select or use your middle mouse button.

2.

From From the the Edi Editt men menu u cho choos osee the the Options command.

3.

Click Click on the the Toler Toleranc ancee drop-d drop-down own list list and and choos choosee either either Plus/Minus, Limit, Symmetric or Basic. See page 46 f or or a description of the different tolerance types.

4.

Enterr an Upper Ente pper and and Lo Lowe werr tole tolera ranc ncee valu valuee in the the dial dialo ogue gue box box and and clic click k OK. If you choose Basic, the tolerances will be zero.

The following is the Angular Dimension Options window. Linear, Arc Length and Radial/Diameter Option windows are quite similar.

A


B

45

Using Tolerance Analysis A

Tolera Tolerance nce dropdrop-dow down n option option list. list. You can can select select from from the five five toleran tolerance ce options:

Assigns the default tolerance value to the dimension.

Overrides the default tolerance value for the dimension with a user specified tolerance value. Requires you to specify Upper and Lower tolerance values for the dimension.

Overrides the default tolerance value for the dimension sion.. Requ Requir ires es you you to defi define ne the the tole tolera ranc ncee by ente enteri ring ng the actual limit values.

46


Setting Tolerances

Assigns the half-width of the tolerance zone, rather than having to enter the plus and minus value individually.

Allows you to define a zero tolerance (as opposed to a default tolerance, which may not be zero). Basic tolerances are denoted with a box surrounding the dimension text.

B

Upper and Lower Tolerance dialog boxes. The values specified in the Tolerance dialog boxes override the default tolerance only for this dimension. The Tolerance dialog boxes are displayed when either the +/- or limit tolerance option is selected. For symmetric , only one tolerance dialog box is displayed.

Note: In accordance with the ASME Y14.5M-1994 standard, default tolerances are not displayed with dimensions on the sketch note.


47


Tolerance Zones for Specific Constraints Assigning a tolerance to a constraint sets the boundaries in which the constrained object must lie. To apply a tolerance zone to a constraint: 1.

Select a constraint from the sketch note.

2.

From the Edit menu choose the Options command. The Constraint Options window is displayed.

3.

For Tolerance type, choose User-defined and enter a value for the tolerance zone.

4.

Click OK to accept the changes and close the options window.

A B

A

Click on User-defined to specify a tolerance. Use the Tolerance pop-up window and click on Default to return the value of the constraint to 0.00.

B

Tolerance Zone. If the constraint is parallel, perpendicular, or at a given angle, the tolerance zone is the width within which the constrained object must lie. If the constraint is a location constraint (concentric, position, fixed point or float pin-in-hole) the tolerance zone is the diameter of the circle within which the center point or fixed point must lie. For more information on Tolerance Zones, see “Perturbing Contributors,” page 97 .

48


Setting Tolerances

Tolerance for Specific Forces, Moments and Springs You can specify tolerance values for specific forces, moments and linear and torsion springs. You cannot set a default tolerance for the force analysis items. The force analysis items are a part of the MA Performance Modeling application. Refer to the Mechanical Advantage Performance Modeling Manual for more information about these items.

Tolerance Values for Specific Forces or Moments To specify a tolerance for a force or moment: 1.

Select either a force or a moment from the sketch note.

2.

From the Edit menu, choose Options. An Options window is displayed. The Force and Moments Options windows are very similar.

A

A

Tolerance drop-down option list.

Four of the five tolerance options are available for forces and moments: Default, Plus/Minus, Limit and Symmetric. The Basic option, which sets the tolerance to a value of zero, is not available. For information about each option in the list, see “Tolerance Values for Specific Dimensions,” page 45.


49


Tolerance Values for Specific Linear or Torsion Springs To specify a tolerance for either a linear or torsion spring: 1.

Select any part of the spring.

2.

From the Edit menu, choose Options to open the Spring Options window. The Linear and Torsion Spring Options windows are similar.

A

B

A

Drop-down list to set the tolerance for the Spring Rate. Spring Rate is a characteristic of the spring that specifies how much force per unit length the spring exerts.

B

Drop-down list to set the tolerance value for the Free Length (linear spring) or Free Angle (torsion spring). Free Length/Free Angle is the size/angle of the spring in the un-deflected state.

Four of the five tolerance options are available for the springs: Default, Plus/Minus, Limit and Symmetric. The Basic option, which sets the tolerance to a value of zero, is not available. For information about each option in the list, see “Tolerance Values for Specific Dimensions,” page 45.

50


Setting Tolerances

Auto Tolerance for Float Pin-in-Hole Constraints Unlike other constraints, the float pin-in-hole automatically calculates a tolerance for the constraint. The following formula is used to automatically calculate the tolerance zone: Float = maximum hole diameter - minimum pin diameter

The calculated tolerance can be overridden by setting a user defined tolerance, in which case the constraint behaves like a concentricity constraint. The auto tolerance is overridden in the same way as the other tolerances. For more information about overriding tolerance values, see “Tolerance Zones for Specific Constraints,” page 48 and for additional information about the auto tolerance calculations for the float constraint, see“Float Pin-in-Hole Constraint,” page 164. Note: The manual method requires you to update the constraint tolerance whenever changes occur in the pin or hole nominal values or tolerances.

Constraints that do not have Tolerance Zones You cannot assign a tolerance value to a constraint that does not have a tolerance zone. The following constraints do not have tolerance zones. • Coincident point. • Point-at-midpoint. • Sum to. • Sum to Zero. • Product. • Scale.


51


Constraints that use Tolerance Zones with Profile Tolerances The following constraints only have tolerance zones when used on a region with a profile tolerance. • Tangent. • Tangent-to-region. • Point-on-object. • Point-on-region. Tolerance values are not directly assigned to these constraints; instead, you must apply the tolerance to a region which defines the profile. Use the Option window of the region with the profile tolerance to assign a tolerance zone to the constraint.

Setting Tolerance Values from the Tolerance Analysis Report From the Tolerance Analysis Report, you can assign tolerance values for the analyzed dimension as well as contributing dimensions, constraints, forces, moments or variables. Contributors are listed under the heading Contributor Information.

Tolerance Values for the Analyzed Item You can assign or edit the tolerance value of the analyzed dimension, force or moment from the Tolerance Analysis report.

52

1.

From the Edit menu in the Tolerance Analysis report, select Set Design Limits . The Analyzed Dimension Options window is displayed.

2.

From the Tolerance drop-down list you can specify the tolerance value.


Setting Tolerances

Tolerance Values for Constraints, Dimensions, Forces, Moments and Variables You can edit tolerance values for constraints, dimensions, forces, moments and variables from the Tolerance Analysis Report. Even though Springs are not included as contributors to the Tolerance Report, the spring characteristics—such as the spring rate variable and the free length—are included. To assign a tolerance value to specific contributors: 1.

Select one or more contributors from the Tolerance Report by clicking on and highlighting the contributor row. To select multiple contributors, use the draw-through select or extendselect method. To set tolerance values for multiple contributors, the contributor types and units must be compatible. For example, constraints and dimensions should not be selected together, and contributors with different units, such as inches and degrees, should not be selected together.

2.

From the Edit menu, choose the Options command. The Tolerance Contributor Options window is displayed.

3.

Enter in the tolerance values and click OK or Apply.

When you edit the value of a contributor, the system re-analyzes the design and updates the report and the model to reflect the changes to the contributor. If the contributor you are changing is part of a dependent assembly, the system updates the master assembly.


53


Tolerance Analysis You do not need to set individual or default tolerances to perform a Tolerance Analysis. If you don’t, the system supplies a tolerance of 0.00, and proceeds with the analysis. Constraints and dimensions with a tolerance value of 0.00 contribute nothing to the Tolerance Analysis results, but may still contribute to the sensitivity analysis, which is a function of the model and the way it is dimensioned and constrained, not of its tolerances. For more detailed information, see “The Importance of Sensitivity in Tolerance Analysis,” page 94.

Performing the Analysis Tolerance Analysis can only be performed on a derived dimension, force or moment. An item is derived when it is fully constrained by existing constraints and dimensions. To analyze the tolerance on a derived dimension, force or moment:

54

1.

Select a derived dimension, force or moment.

2.

Choose Analyze Tolerance in the Analysis menu.


Tolerance Analysis 3.

The system displays the Tolerance Analysis Report.


55


Performing Multiple Tolerance Analysis You can select multiple derived dimensions, forces or moments and perform a tolerance analysis on each of the selected items. Performing an analysis on multiple items is very similar to performing an analysis on a single derived dimension, force or moment. The only difference is that you must select more than one item to be analyzed. The following methods can be used to select multiple derived dimensions, forces or moments: • Extend-select. • Draw-through select. Other geometry can be included in the selection since MA identifies the derived items. • Select a Group that contains derived dimensions, forces or moments. For example, consider the following sketch which contains two derived dimensions, named Top Dim and Bottom Dim.

To analyze both dimensions simultaneously:

56

1.

Extend-select the Top Dim and Bottom Dim.

2.

From the Analysis menu, choose Analyze Tolerance. MA verifies that both dimensions are derived and performs an analysis for each dimension.


Tolerance Analysis 3.

The resulting Tolerance Analysis displays one report at a time. In the example, the report for Bottom Dim is displayed.

4.

To view the results for the other derived dimension, choose Choose Analyzed from the Viewing menu. The sub-menu that opens lists both of the analyzed dimensions. Choosing Top Dim will display the other report results.


57


Reanalyzing the Analyzed Dimension, Force or Moment If you make changes to contributor tolerance values using the Tolerance Analysis Report Options command, the Report will automatically update. Changes made to the sketch model will cause the results in the Report to become invalid. If the Report requires updating, the system displays the message “Reanalyze to enable select” or “Results may be invalid.” If you get one of these messages: • From the Viewing menu of the Tolerance Analysis Report, choose the Reanalyze command.

Showing the Analyzed Dimension, Force or Moment After a Tolerance Analysis is performed on a derived dimension, force or moment, use the Show Analyzed Dimension command to bring the sketch note forward and locate the analyzed item: • From the Tolerance Report Viewing menu, choose the Show Analyzed Dimension command. You can only highlight the analyzed item if the Tolerance Report is up-to-date. See “Reanalyzing the Analyzed Dimension, Force or Moment,” page 58 for more information about updating the Tolerance Analysis.

Showing Contributors to the Analyzed Dimension, Force or Moment Derived dimensions, forces or moments can have several contributing dimensions and constraints. After a Tolerance Analysis is performed, use the Show Contributor command to bring the sketch note forward and locate contributing dimensions, constraints, forces, moments or variables. To locate contributors to the analyzed item: 1.

From the Tolerance Analysis Report, click anywhere on the contributor row to select that contributor. You can extend-select other contributors or you can select every contributor in the report by choosing the Select All command from the Edit menu.

2.

From the Viewing menu of the Tolerance Analysis Report, choose the Show Contributor command.

You can only select contributors if the Tolerance Analysis Report is up-to-date. For more information about updating the Tolerance Analysis, see “Reanalyzing the Analyzed Dimension, Force or Moment,” page 58.

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Tolerance Analysis

Showing Contributors to the Dependent Assembly If any dependent assemblies appear in the Tolerance Analysis, the following considerations apply: • For any contributor that is part of a dependent assembly, there will be one row in the Tolerance Analysis Report for each copy of the dependent assembly. For example, if one dependent assembly contains three contributors and the assembly appears twice in the Tolerance Analysis, there may be six rows in the contributor table relating to that assembly. The tolerances will always be the same as you set them in the master assembly. • When you select a row representing a contributor in a dependent assembly and choose Show Contributor, the dependent assembly highlights, and so does the individual contributor in the master assembly. • Although the tolerance is the same for each assembly, the percent contribution and the sensitivity may be quite different since each assembly may relate differently to the analyzed dimension. For additional information about dependent assemblies in the Tolerance Analysis Report, see “Using Dependent Assemblies for Tolerance Analysis,” page 174.

Sorting the Tolerance Analysis Report You can sort the Tolerance Analysis Report either by contribution (highest percentage first), by magnitude of sensitivity (highest number first—the sign is ignored), or by Part Name. To sort the Tolerance Analysis Report by contribution or sensitivity: • From the Viewing menu, choose the Sort By Contribution command, or the Sort By Sensitivity command. The system sorts the window as you specify. The sensitivity is a ratio of the change in the analyzed dimension, force or moment units of analyzed item units of contributor

to a change in the contributor (i.e., ----------------------------------------------------- ). The absolute value of the sensitivity will vary greatly depending on the units involved. For example, deg/in will generally produce very large sensitivities, and conversely, in/deg will generally produce very small sensitivities, both relative to in/in or deg/deg sensitivities. When MA sorts by sensitivity, it considers the numerical values of the sensitivities and takes no account of the units involved. Thus, a comparison of sensitivities is only meaningful where the units are the same. For more information about contributor units, see “Comparing Mixed Unit Sensitivities,” page 96. To sort the Tolerance Analysis Report by Part Name: • From the Viewing menu in the Tolerance Analysis Report, choose Sort By Part Name. The contributors are sorted by the Part Name that is listed in the Part Name column of the report.


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Tolerance Distribution Graph The Tolerance Distribution Graph helps you visualize the tolerance distribution of an analyzed dimension, force or moment. To view the graph: • From the Viewing menu of the Tolerance Report window, choose the Plot Distribution Graph command. The following example is a visual representation of the tolerance information in the Tolerance Analysis Report.

A

B D

C

A

60

Target. The target curve is calculated and displayed only if tolerance design limits are specified for the analyzed item. The target curve is a normal curve with a standard deviation that is calculated from the tolerance zone of the design limits and its desired sigma value.


Tolerance Analysis B

Current. The current curve graphs the actual statistical distribution for the analyzed item and displays the failure rate in parts-per-million and the sigma value.

C

Previous curve. The previous curve is a “current” curve that was saved from the previous analysis. The previous curve can be compared with the current curve of an updated analysis.

D

Tolerance design limits. The tolerance design limits for the analyzed item are displayed on the graph as green vertical bars.

The bottom X-axis of the graph is the value of the resultant. The top X-axis shows the mean value of the current distribution and the number of sigmas out from the mean.

Preserving the Current Graph You can save the “current” curve and compare it with the results of an updated tolerance analysis. When you preserve a current graph, the “current” curve is saved and denoted on the graph as the “previous” curve. To preserve the current curve: • From the Edit menu of the Distribution Graph Window, choose the Preserve Current Graph command. This feature allows you to update contributor tolerance values and compare the results with your previous analysis.

Resetting the Graph Resetting the graph, removes the distribution information for the “previous” curve from the distribution window. To reset the graph: • From the Edit menu in the Distribution Graph window, choose the Reset Graph command.

Printing the Graph To print the graph: • From the File menu in Distribution Graph window, choose the Print command. For additional information about printing with Mechanical Advantage, see the Mechanical Advantage User Basics and Set Up Guide. I


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Duplicate Contributors When modeling with dependent assemblies, a single contributor may show up two or more times in the Tolerance Analysis Report when you expect it to only show up once. Combining duplicate contributors causes the Tolerance Analysis results to change. In some cases it is necessary to combine contributors to get the correct results, while in other cases the contributors must remain separate to produce correct results. For more information about when to combine contributors and the impact on the Tolerance Analysis Report, see“Combine Duplicate Contributors,” page 175.

Combining Duplicate Contributors Duplicate contributors are easily identified from the Tolerance Analysis Report since they are grouped together in the report as a result of a shared name.

To combine duplicate contributors:

62

1.

Select one or more contributor rows from the Tolerance Analysis Report.

2.

From the Edit menu, choose the Combine Duplicates command. You may choose as many rows as you like, only the duplicates are combined.


Tolerance Analysis When a contributor is combined into a single entry, an Ampersand “&” symbol is displayed in the Type column for the contributor row.

When a reanalysis is performed, the system will take into account the merged status of these contributors. The merged status of a contributor is remembered, even if the note is closed.

Separating Duplicate Contributors You can undo the effect of the Combine Duplicates command. To separate combined contributors in the Tolerance Analysis Report: 1.

Select a combined contributor from the Tolerance Analysis Report.

2.

From the Edit menu, choose the Separate Duplicates command.

Combined contributors are always displayed in the Tolerance Analysis Report, even when the sensitivity and percent contribution values are zero.


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Writing to an ASCII file You can convert a Tolerance Analysis Report to a standard format, ASCII (American Standard Code for Information Interchange), and save it in a file for use in other applications. To convert a Tolerance Analysis Report to ASCII format: 1.

From the Convert menu, choose the Write to ASCII File command. The system displays the Write to ASCII option window.

2.

Type in a directory (optional) and the file name; or, use Browse to specify the directory or file. You can use upper and lower case letters, numbers, and the following other characters: , @ . - # + =.

3.

Select either the ASCII-formatted or Comma-delimited format. ASCII-formatted is used for ASCII-based printouts (not postscript). Comma-delimited is used for exporting to other applications, especially spreadsheets and databases.

4.

Select to either Include or Exclude Zero Contributors. This option specifies if contributors with a zero upper and lower contribution are written to the ASCII file. Contributors with zero contribution are included in the Report if they have a sensitivity value, if they are questionable (e.g., due to poorly directioned datums), or if they are a centerline constraint.

5.

Click OK. The system puts all the text into an ASCII file, with the name and in the directory you specify. If you don’t specify a directory, Mechanical Advantage puts the file in your home directory.

If the filename you specify already exists, you can either replace the file, or append the new output to the existing file.

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Worst Case Analysis

Worst Case Analysis Linear Worst Case and Non-linear Worst Case are extensions of Tolerance Analysis. For Linear Worst Case, the effects of each contributor are computed by multiplying the tolerance limits by the sensitivity at the nominal value. In comparison, Non-linear Worst Case will actually perturb the dimension, force, variable and moment contributors and eliminate errors due to sensitivities that vary as these contributors are perturbed. Refer to “Sensitivity Analysis,” page 93, for information about sensitivities and perturbing contributors. Linear Worst Case analysis is performed every time you analyze a derived quantity. In Linear Worst Case analysis, the system uses nominal sensitivity values to provide a linear estimate of the impact of each contributor at maximum and minimum tolerance limits. The results are an estimate of the worst case and are usually a close approximation of Non-linear Worst Case. When the worst case is critical, you should perform a Non-linear Worst Case analysis to obtain results that more accurately predict the behavior of the model. In Non-linear Worst Case, the system sets the values of each dimension, force, variable and moment to their maximum and minimum tolerance limits and estimates the values for geometric contributors. When Non-linear Worst Case sets each of these values to their limit, the sensitivities of geometric contributors are recalculated, and the revised sensitivities are used to estimate the geometric contribution. This method, when used to revise the contribution estimate for geometric contributors, will reduce the effect of higher order terms but will not eliminate the effect of non-linear geometric contributors.


65


Performing a Linear Worst Case Analysis The Linear Worst Case analysis is calculated every time you perform a Tolerance Analysis. 1.

Select a derived dimension, force or moment from the sketch note.

2.

From the Analysis menu choose the Analyze Tolerance command. The results are displayed in the Tolerance Analysis Report.

A

A

66

Linear Worst Case results are displayed in the Tolerance Analysis Report.


Worst Case Analysis

Performing a Non-linear Worst Case Analysis To use Non-linear Worst Case, you must first perform a Tolerance Analysis on a derived dimension, force or moment: 1.

Select a derived dimension, force or moment from the sketch note.

2.

From the Analysis menu choose the Analyze Tolerance command. The results are displayed in the Tolerance Analysis Report.

3.

From the Viewing menu of the Tolerance Analysis Report, choose the Set Up Window command. The Tolerance Option window is displayed.

4.

Choose Yes for Show Non-linear Worst Case and click OK to close the window.

5.

An alert is displayed informing you that Non-linear Worst Case calculates the worst case for dimensions, forces, variables and moments, and uses estimates for geometric contributors. Click OK to dismiss the alert.


67

Using Tolerance Analysis 6.

The results are displayed in the Tolerance Analysis Report. For more information on reading the report, see “Worst Case Analyses,” page 82.

A

A

68

Non-linear Worst Case results are displayed in the Tolerance Analysis Report.


Monte Carlo Analysis

Monte Carlo Analysis Monte Carlo analysis is an extension of Tolerance Analysis capabilities that uses random sampling to predict tolerance results. For information about random sampling in Monte Carlo, see “Monte Carlo Analysis,” page 84. Another feature of Monte Carlo allows you to investigate the impact of nonnormal distribution types for contributors. Using non-normal distributions allows you to design your model to more accurately reflect the manufacturing process. However, if you do not have adequate information about the manufacturing process, or a strong background in statistical modeling, it is recommended that you use a Normal distribution.

Monte Carlo Distribution Types Normal is the default distribution type and is recommended for most Monte Carlo analyses. You can set the distribution type for any contributor, choosing from: • Normal • Uniform • Weibull Uniform and Weibull distribution types are only used in the Monte Carlo analysis. For more information about Monte Carlo distribution types see “Distribution Types,” page 119. To change the default setting for Monte Carlo distribution types: 1.

From the Tolerance Analysis Report, select a contributor by clicking anywhere on the contributor row.

2.

From the Edit menu choose the Options command. The Tolerance Contributor Option window is displayed.

3.

Select a Monte Carlo distribution type and click OK to close the window.

Note: If the Monte Carlo distribution types are not displayed in the Options window, then the contributor is not set for inclusion in Monte Carlo analysis. For information on how to include the contributor in Monte Carlo analysis, see “Contributor Inclusion in Worst Case and Monte Carlo Analyses,” page 72 .


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Performing a Monte Carlo Analysis Before performing the Monte Carlo analysis, the system must have up-to-date sensitivity values. If the Tolerance Analysis is not up-to-date, the system attempts to re-analyze before proceeding with Monte Carlo. Non-linear Monte Carlo analysis perturbs all dimensions, variables, moments and forces in the contributor list to determine if additional contributors are to be added to the contributor list as a result of non-linear effects. Banner messages may be displayed during this re-analysis to inform you of updates. 1.

Select a derived dimension, force or moment and perform a Tolerance Analysis.

2.

From the Edit menu of the Tolerance Analysis Report, choose the Monte Carlo Analysis… command. The Monte Carlo option window is displayed.

3.

Enter a sample size. Note: The upper limit on sample size is 999,999.

4.

Choose either the linear or non-linear method. The linear method is quick but is based on linear sensitivities only. The non-linear method is generally more accurate but takes a considerable amount of time. For more information see “Linear and Non-Linear Methods,” page 125. Note: Since the non-linear method is computationally intensive, Mechanical Advantage provides status information about the progress of the analysis. The analysis may be interrupted with the Stop key.

70


Monte Carlo Analysis 5.

Click OK. The results are displayed in the Tolerance Analysis Report.

A

A

Statistical Failure Rates for Monte Carlo based on your design limits.


71


Contributor Inclusion in Worst Case and Monte Carlo Analyses You have complete control over which contributors are included in Worst Case and Monte Carlo analyses. Contributors that are included by default are always included unless you specify otherwise, and contributors that are not included by default can be set for inclusion. A plus (+) sign appears in the Type column of the Tolerance Analysis Report to show that a contributor will be included in Monte Carlo and Worst Case analyses.

Contributors Included by Default The following contributor types are included by default in Non-linear Worst Case and Monte Carlo analysis: • Dimensions. • Position constraints. • Concentricity constraints. • Float pin-in-hole constraints. • Profile tolerances. • Variables. • Forces. • Moments. You can exclude a default contributor from inclusion in Worst Case and Monte Carlo:

72

1.

From the Tolerance Analysis Report, select a default contributor by clicking anywhere on the contributor row.

2.

From the Edit menu choose the Options command. The Tolerance Contributor Options window is displayed.

3.

Choose No for the Use in Worst Case and Monte Carlo option.

4.

Click OK to exclude the contributor from Worst Case or Monte Carlo.


Contributor Inclusion in Worst Case and Monte Carlo Analyses

A

B

A

Contributor type. The contributor is a linear dimension, which is included by default in Monte Carlo and Worst Case analyses.

B

Selecting No overrides the default inclusion in Monte Carlo and Worst Case analyses for this contributor.

Specifying Contributor Inclusion The following contributors must be set for inclusion in Worst Case and Monte Carlo: • Parallel. • Perpendicular. • Collinear. • Fixed length/radius. • Equal length/dimension. • Fixed point. • Line at angle.


73

Using Tolerance Analysis To set a contributor for inclusion in Worst Case and Monte Carlo: 1.

From the Tolerance Analysis Report, select one or more contributors by clicking anywhere on the contributor row.

2.

From the Edit menu choose the Options command. The Tolerance Contributor Options window is displayed.

3.

Select Yes for Use in Worst Case and Monte Carlo. The Monte Carlo Distribution Options are displayed. See “Monte Carlo Distribution Types,” page 69 f or more information.

4.

Click OK to include the contributor in Worst Case and Monte Carlo.

A

B C

A

Contributor type. The contributor is a perpendicular constraint, and is not included by default in Worst Case or Monte Carlo.

B

If the contributor is not included by default in Worst Case and Monte Carlo analyses, choose Yes for Use in Worst Case and Monte Carlo. Selecting Yes overrides the default setting that excludes the contributor from inclusion in Worst Case and Monte Carlo and displays the distribution option toggle for Monte Carlo.

C

74

Monte Carlo Distribution Type options. The system defaults to a normal distribution. Click to select a different distribution type. For detailed information about Monte Carlo distribution types, see “Advanced Topics for Monte Carlo,” page 119. Mechanical Advantage Tolerance Analysis Manual - February 2001

Chapter 5

Reading the Tolerance Analysis Report This chapter provides instructions for reading and interpreting the results in the Tolerance Analysis Report.

Report Header Statistical Failure Rates

Worst Case Monte Carlo

Contributor Information


75

Reading the Tolerance Analysis Report

Report Header Analyzed Dimension, Force or Moment The name you assigned to the derived dimension, force or moment being analyzed.

Value The current value of the analyzed dimension, force or moment.

Statistical Failure Rates The The syst system em calcu calcula late tess Stat Statis isti tica call Failu Failure re Rates Rates at four four diff differ eren entt Sigm Sigmaa valu values es and and at your design limits. The default Sigma values are ± 3 , ± 4 , ± 5 , and ± 6 Sigma. You Yo u may may speci specify fy alter alterna nate te valu values es by choo choosi sing ng the the Set Set Up Wind Window ow comm comman and d from from the Analysis menu of the Tolerance Analysis Report.

Sigma Values The Tolerance Analysis Report displays the failure rate, in parts per million (ppm), at a specific Sigma value. Sigma is a statistical parameter that is used to describe the distribution about the mean of a process or procedure. A ± 3 Sigma igma desig esign n goalwil goalwilll produ roduce ce 2700fai 2700failu lure ress per per mill millio ion n. If youare striv trivin ing g for 0 defects, a ± 6 Sigma design goal has a failure rate of less than 1 per million. Design Designing ing for ± 6 Sigma Sigma allo allows ws for for some some variat variation ion in mean mean but approa approache chess 0 defects.

A B

76

±4 Sigma is always 63 ppm.

A

The failure rate @

B

The failure failure rate rate @ Design Limits (i.e. based on your your current current design) design) is 432,981 ppm.


Statistical Failure Rates

Design Limits The Design Limits are the goals you set for the analyzed dimension, force or moment.

A B

C

D

A

The The Desi DesignLim gnLimit itss for for the the anal analyz yzed ed dime dimens nsio ion, n, forc forcee or mome moment nt are are disp displa laye yed d in this row.

B

The numb number er of times times per per million million the the analyz analyzed ed dimens dimension ion,, force force or moment moment will exceed the set tolerance range.

C

The percentage percentage the analyz analyzed ed dimensio dimension, n, force force or moment moment will will be within the set tolerance range.

D

The maxi maximu mum m and mini minimu mum m tole tolerrance ance rang ange you you set for for the the anal analyz yzed ed dimen imen-sion, force or moment.

In the above example, the tolerance range set in the design limits (labeled in the repor eportt as as Tol Tol Max and and To Tol Min Min)) is is ± 0.005 . Bas Based ed on the the cu curren rrentt des desig ign n the the par part will be manufactured within acceptable specifications only 56.70% of the time. If you you are are desig designin ning g to achi achieve eve a ± 3 Sigma Sigma succ success ess rate, rate, the the values values listed listed for for the T o oll M ax ax a nd nd T ol ol Min c ol olumn ( ± 0.01913 ) m us ust b e br brought within ±3 Sigma i n th the stated stated design design limits. limits. In other other words, words, the value value list listed ed for for ± 3 Sigma Sigma Tol Max and Tol Min must be lower than, or equal to, the Tol Max and Tol Min valu alues state tated d for for the the des desig ign n lim limit itss—in —in th this exam exampl ple, e, low lower than than ± 0.005 .

Percent in Tolerance The percent in tolerance is the likelihood that the design will be manufactured within the set tolerance range.

Failure Rate The failure rate is the number of times per million the design will be out of tolerance.


77


Tol Max and Tol Min The Tol Max and Tol Min are the upper and lower tolerance values required to meet meet Sigm Sigmaa goal goalss and and desi design gn goal goalss. When When the the To Toll Max and and To Toll Min valu values es for a Sigm Sigmaa rang rangee are are less less than than or equa equall to the the valu values es stat stated ed for for desi design gn limit limits, s, the the desi design gn will achieve the success rate corresponding to that Sigma range.

A

B

78

A

A ± 3 Sigma success rate is achieved when the Tol Max and Tol Min tolerance values for ± 3 Sigma are less than or equal to the Tol Max and Tol Min values lues required for the design limits.

B

The Desi Design gn Limits Limits for Tol Tol Max Max and Tol Tol Min Min remain remain cons constan tant. t.



Cp and Cpk Indices Cp and Cpk are process capability indices which represent the ratio of design limits limits to to proce process ss capa capabil bilitie ities. s. Both Both indices indices assume assume ± 3 Sigma Sigma manuf manufactu acturin ring g process capabilities. Cp and Cpk numbers lower than 1.0 indicate the manufacturing process cannot relia reliabl bly y meet meet the the desi design gn goal goals. s. Cpk Cpk is more more cons conser erva vati tive ve than than Cp beca becaus usee it take takess into account the possibility that the process mean may be shifted from the mean of the design limit range. When When calc calcul ulat atin ing g Cp and and Cpk, Cpk, Mech Mechan anic ical al Ad Adva vant ntag agee trea treats ts all all cont contri ribu buto tors rs as if they they were were at their their proc proces esss limit limits. s. The The cont contri ribu buto tors rs are are in turn turn used used to calc calcul ulat atee the the resu result ltan antt Sigm Sigmaa rang rangee for for the the stat statis isti tica call analy analysi sis, s, and and data data rang rangee for for Mo Mont ntee Carl Carlo. o. As a result, it is important to verify that the tolerances on all contributors are greater than or equal to the process limits. When When the the proc proces esss for for any any indi indivi vidu dual al cont contri ribu buto torr can can be cont contro roll lled ed more more prec precis isel ely y than than indic indicated ated by the the stand standard ard ± 3 Sigma Sigma rang range, e, you you can tighte tighten n the contri contribut butor or Sigma range accordingly. ly. In othe ther words, if the process can be held to ± 6 Sigma, the Sigma range for the contributor can be adjusted, and the impact of the better control will be reflected in the Cp and Cpk calculations.

Cp Cp compares the total design limit range against the process capabilities. The formula for Cp is: For statistical analysis: Design Limit Range ±3 σ

----------------------------------------------------------------

For Monte Carlo analysis: Design Limit Range --------------------------------------99.73% of Data Range

Note Note that that ± 3 Sigma Sigma repr represe esents nts a total total spre spread ad of of 6 Sigm Sigmaa and and is the same same as 99.73%.


79


Cpk Cpk considers the affect of a shifted mean. In the following calculations µd represents the mean for the design limits of the analyzed dimension. The following formula is used to calculate µ d for Monte Carlo analysis:

µ d = Mean

of the data samples

See “Mean and Standard Deviation” on page A-3, for information about calculating mean for a statistical analysis. Tolerance Analysis performs the following calculations to determine Cpk. Cpk is the lesser of the following: For statistical analysis:

µd – Lower

Design Limit

---------------------------------------------------------------------------3σ

Upper Design Limit – µ d

---------------------------------------------------------------------------3σ

For Monte Carlo analysis:

µd – Lower Design Limit -------------------------------------------------------------------------------------------------------------------------------------µd – Data Limit at 0.135% of Data Range Upper Design Limit – µ d -----------------------------------------------------------------------------------------------------------------------------------------Data Limit at 99.865% of Data Range – µ d

Note that the denominator in all Cpk calculations is essentially 49.865% of the process range.

80



Val Max and Val Min The Value of the analyzed dimension + the Tol Max or Tol Min. Val Max = Value + Tol Max Val Min = Value + Tol Min

In the following example, the Val Max for CFD @ ± 3 Sigma is calculated: Val Max = 5.9426 + 0.01913 Val Max = 5.9618

Types Included The contributor types included in the analysis. For more information about contributor types, see “Type,” page 86.


81


Worst Case Analyses In wo wors rstt case case analy analyse ses, s, the the syst system em anal analyz yzes es the the beha behavi vior or of the the mode modell and and pred predic icts ts an outcome based on contributor values being at maximum and minimum tolerance limits. In the following example, the tolerance ranges for both Linear and Non-linear Worst Case are outside of the design limits.

A B C A

Design Limits Tolerance Tolerance Values. Values. The allowable allowable amount amount of variation variation for the the analyzed dimension.

B

Line Linear ar Wors Worstt Case Case Tole Tolera ranc ncee Valu Values es.. The The syst system em esti estima mate tess the the mini minimu mum m and and maximum amount of variation in the analyzed dimension. Compare these values to the design limits to determine if the dimension will be in tolerance during a worst case scenario.

C

Non-linear Non-linear Worst Worst Case Case Toleranc Tolerancee Values. Values. The system system calculates calculates the minimum and maximum amount of variation in the analyzed dimension and esti estima mates tes the the valu values es for for geom geomet etri ricc cont contri ribu buto tors rs.. Comp Compar aree thes thesee valu values es to the the desi design gn limit limitss to dete determ rmin inee if the the dime dimens nsio ion n will will be in tole tolera ranc ncee duri during ng a wors worst t case scenario.

If you you are are desi design gnin ing g for for wo wors rstt case case and and the the wo wors rstt case case tole tolera ranc ncee valu values es are are larg larger er than the values stated in the design limits, you will need to improve your design by re-allocating constraints or dimensions, or by tightening tolerances on sensitive contributors.

82


Worst Case Analyses In the the foll follow owin ing g exam exampl ple, e, the the desi design gn was was impr improv oved ed and and the the tole tolera ranc ncee resu results lts indi indi-cate cate the analyz analyzed ed dimen dimensio sion n has has a ± 3 Sigma Sigma succ success ess rate. rate. Howe However ver,, the Linear Linear and Non-linear Worst Case tolerance values are still larger than the tolerance range set in the design limits.

A

B C

D

A

±3 Sigm Sigma a Tol Toler eran ance ce Val Value ues. s. The The maxi maximu mum m and and mini minimu mum m tole tolera ranc ncee valu values es requ requir ired ed for for the the des desig ign n to have have ± 3 Sigm Sigma a succ succes esss rate rates. s.

B

Line Linear ar Wors Worstt Case Case Tol Toler eran ance ce Val Value ues. s.

C

NonNon-li line near ar Wors Worstt Case Case Toler Toleran ance ce Valu Values es..

D

Desi Design gn Limit Limitss for for toler toleran ance ce..

If you you are are desi desig gning ning for worst orst cas case, the the wors orst cas case and and ± 3 Sigma igma tole tolera ranc ncee valu values es should be less than the tolerance range set in the design limits. See “M “Model odel Building Essentials for Tolerance Analysis,” page 131 for for more information abou aboutt prop proper er dime dimens nsio ioni ning ng tech techni niqu ques es or see see “Se “Sensi nsitiv tivity ity Ana Analys lysis, is,”” pag pagee 93 for more information about improving tolerance results.

Linear Worst Case Linear Worst Case is performed every time you run a Tolerance Analysis. The system uses sensitivity values to provide a linear estimate of the outcome based on contributor values being at maximum and minimum tolerance limits. When designing for worst case, the worst case values listed on the report for Tol Max and Tol Min should be less than the Tol Max and Tol Min values set for your design limits.

Non-linear Worst Case Non-lin Nonlinear ear Wors Worstt Case Case prov provid ides es tole tolera ranc ncee resu result ltss for for a situ situat atio ion n wh when en all all cont contri ribbutor utorss are are simu simult ltan aneo eous usly ly at the the same same extr extrem eme. e. The The syst system em sets sets the the valu values es of each each dimension to their maximum and minimum tolerance limits and calculates the resulting values for the analyzed item. The geometric contributors are then estimated and applied to the analyzed item.


83


Monte Carlo Analysis The The Mo Mont ntee Carl Carlo o anal analys ysis is is a stat statis isti tica call type type of Tole Tolera ranc ncee An Anal alys ysis is that that is base based d on a random sampling. In Monte Carlo analysis, the system randomly considers up to one one mill millio ion n poss possib ible le toler toleran ance ce valu values es for for each each cont contri ribu buto torr and and calcu calculat lates es the the mean, standard deviation and probability for the analyzed dimension, force or moment in each occurrence. A Monte Carlo analysis that is performed on a derived item whose contributors have have a norm normal al dist distri ribu buti tion on can can be used used to veri verify fy the the resu result ltss of the the RSS RSS Tole Tolera ranc ncee Analysis.

A

C

B

A

The The per percent cent in tole tolerrance ance for for the the anal analyz yzed ed item item bas based on stand tanda ard RSS RSS Tole Tolerrance Analysis calculations.

B

The The perc percen entt in tole tolera ranc ncee for for the the anal analyz yzed ed item item base based d on Mont Montee Carl Carlo o rand random om sampling techniques with a sample size of 5000.

C

The number number of failures failures per million million based on the random random sampling sampling of of 5000 5000 possible outcomes.

Note: Statistical probability rates are less accurate with small sample sizes.

Monte Carlo allows you to set the manufacturing distribution method to investigate gate the impact impact of non non-no -norma rmall distri distribut bution ions. s. For additio additional nal inform informatio ation n on Mon Monte te Carlo distribution types, see “D “Distribution istribution Types,” page 119. 119.

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Contributor Contributor Information The sectio section n labeled labeled Contri Contribut butor or Inform Informati ation on on the tolera tolerance nce report report lists lists inform informaation about the dimensions, constraints, variables, forces and moments that contribute to the variation of the design. Items listed in the Tolerance Analysis Report are contributors to the analyzed quantity. By defaul default, t, contrib contributo utors rs are listed listed by percen percentt contri contribut bution ion;; the contri contribut butor or with with the highest percentage is listed first. The Tolerance Analysis Report can also be sorted by magnitude of sensitivity or by part name. See “Sorting the Tolerance Analysis Report,” page 59 f or or more information.

Part Name A Part Part Name Name colu column mn allo allows ws you you to inst instan antl tly y reco recogn gniz izee if a cont contri ribu buto torr is from from an asse assemb mbly ly part part.. If the the cont contri ribu buto torr is asso associ ciate ated d with with a name named d depe depend nden entt asse assemb mbly ly,, then that name is listed. If the dependent assembly is not named, then the report will list the name of the master assembly. For example, the following sketch design is made up of several dependent assembly parts: LINK, FOLLOWER, BASE and CRANK.


85

Reading the Tolerance Analysis Report The tolerance report uses the Part Name column to identify which dependent assembly the contributor is associated with.

Contributor Name The name of each contributor. Contributors are named using the Options comm comman and d from from the the Edit Edit menu menu.. Un Unna name med d cont contrib ribut utor orss are are assi assign gned ed a defa defaul ultt name name consisting of the contributor type and a numeric string.

Value For contributors that are dimensions, forces, variables or moments, this column show showss thei theirr valu valuee and and unit units. s. Cons Constr train aintt cont contri ribu buto tors rs have have no entr entry y in this this colu column mn..

Upper or Lower Zone These two columns display the upper and lower tolerance value for the contributor.

Type The Type column lists the following contributor information: • Cont Contri ribu buto torr type. type. • Contri Contribut butor or symbol symbol.. • Contributor inclusion in Monte Carlo or Non-linear Worst Worst Case analysis.

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Contributor Types Geometric Tolerancing. Mechanical Advantage uses the following encoding system to indicate the geometric tolerancing contributor types. Symbol Type of Tolerance Characteristic

MA Constraint/Tolerance

P

Profile

Profile of a surface

Profile

O

Orientation

Angularity Perpendicularity Parallelism

Line-at-Angle Perpendicular Parallel

L

Location

Position concentricity

Concentricity

D

Dimension

Dimensional constraint

Linear, radial, diameter or angular dimension, fixedlength, fixed-radius, equallength or equal-radius constraint

Performance Modeling Tolerancing. Forces, moments and variables can be listed as contributors in the Tolerance Analysis Report. However, only the constraints and dimensions that do not involve any external links will be included in the results of the analysis.

The following encoding system is used to indicate the Performance Modeling contributor types. Symbol

Type of Tolerance

F

Force

M

Moment

V

Variable


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Contributor Symbols The following symbols, when displayed in the Type column, indicate additional information about contributors. Symbol

Description

?

A red question mark indicates the contributor has a poorly chosen datum or that the geometry is under-constrained. This symbol is displayed following a type symbol in the type column. For example: D? indicates the following potential problems with the dimensional contributor: • It is a line-to-line dimension and its parallel constraint has a poorly chosen datum. • The geometry is under-constrained. • Displays in Critical Parameter Analysis if a contributor encountered a solution error during a Perturb attempt.

+

A blue plus sign indicates the contributor is included in Non-linear Worst Case or Monte Carlo analysis. This symbol is displayed before a type symbol in the Type column.

&

Indicates a combined contributor. Duplicate contributors are a result of multiple dependent assemblies contributing to the Tolerance Analysis results. For example, a parallel contributor included in Non-linear Worst Case and Monte Carlo analysis with a duplicate contributor that has been combined will have the following representation in the type column: +O&

cl

Indicates a potential problem because the centerline was at the nondatum end of the contributor. A centerline has no constrainable endpoints so it cannot provide a contribution to the tolerance. A regular line should be used if the contributor needs to be part of the analysis. Indicates Least Material Condition (LMC). An LMC symbol in the Type column indicates a bonus contributor. An LMC symbol in the Upper or Zone column indicates the contributor that the LMC is set for. Indicates Max Material Condition (MMC). An MMC symbol in the Type column indicates a bonus contributor. An MMC symbol in the Upper or Zone column indicates the contributor that the MMC is set for.

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Sigma The Sigma column displays the Sigma value that is assigned to the contributor.

MC The MC column of the Tolerance Analysis Report displays the Monte Carlo distribution type assigned to that contributor. A contributor can be assigned either Normal (N), Weibull (W), or Uniform (U) distribution. For more information on distribution types, see “Distribution Types,” page 119.


89


Contributor Results The contributor results section lists the sensitivity, contribution and upper and lower worst case contribution for each contributor.

Sensitivity Sensitivity is the rate of change of the analyzed dimension, force or moment with respect to a contributor. Mathematically, it is the first partial derivative of the analyzed item with respect to a contributor. It is sometimes described as a measure of the leverage of a contributor. For example, if the sensitivity is 1.0, a change of 1 unit in the contributor accounts for a change of 1 unit in the analyzed item. If the sensitivity is 2.0, a 1 unit change in the contributor accounts for a 2 unit change in the analyzed item. Contributors that have high sensitivities are more tolerance-critical than those with low sensitivities. Some sensitivities may have minus signs. The minus sign indicates that an increase in the value of a contributor is reflected as a decrease in the value of the analyzed item. The system takes the sign into account when it calculates the Worst Case for the analyzed item. That is, when the system calculates the Worst Case, some contributors take on their positive tolerance limits while others take on their negative.

Contribution Taking into account contributions by all of the contributors, this value shows the percentage accountable to each contributor. A high percentage indicates the specific contributor has a major impact on whether or not the analyzed dimension, force or moment is within tolerance. By default, contributors are automatically sorted by their percent contribution. You may also sort by sensitivity, see “Sorting the Tolerance Analysis Report,” page 59.

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Contributor Results

Upper Contribution / Lower Contribution Given the sensitivity and the tolerance you specified for each contributor, the system calculates the magnitude of the change in the analyzed item if the contributor took on its worst case value. The following formula is used to calculate contribution. Upper Contribution = Upper Tolerance Zone Lower Contribution = Lower Tolerance Zone

× Sensitivity

× Sensitivity

The following example demonstrates how the system calculates the Upper Contribution for contributor # 1. Upper Contribution = 0.01

× 1.00

Upper Contribution = 0.01


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Chapter 6

Sensitivity Analysis This chapter demonstrates the relationship between sensitivity and the behavior of your design.

Sensitivity in the Design Model Sensitivity ( f i ) shows the ratio of change imposed on a derived dimension, force or moment by a small change in a contributing dimension, constraint, variable, force or moment. This type of analysis provides you with critical insight into the behavior of your model. The following concept provides you with a great deal of information about your designs: Resulting change in the analyzed dimension f i = ------------------------------------------------------------------------------------Small change in the contributor

Understanding the relationship between a change in a contributor and the resulting change in the analyzed item will provide you with the following types of information: • Significant insight into the behavior of your model. • Information on what to change, by how much, and in what direction in order to meet design goals. • A guide for tolerancing your design while meeting functional and manufacturing requirements. The system performs sensitivity analysis without taking any tolerance information into account. Sensitivity is purely a function of the constraints, dimensions, variables, forces and moments in your model. Sensitivity is used in conjunction with tolerance information to calculate percent contribution and the probability that the analyzed quantity is within the specified tolerance. When the system computes the sensitivity of a contributor, it assumes that all other constraints are perfectly maintained. This can lead to results where unexpected contributors have a surprising effect upon the analyzed item.


93

Sensitivity Analysis

The Importance of Sensitivity in Tolerance Analysis The following example is a fully constrained sketch model which is used throughout the chapter to demonstrate how a change in a contributor influences the analyzed dimension.

You can create this sketch model to investigate sensitivity analysis for yourself. The sketch is set up to use inches and all linear dimensions are point-to-point. The parallel and perpendicular constraints are constrained to the lines dimensioned by A and B, which are the implicit datums for the model. Note: Datums are a point of reference from which you should constrain your model. Constraining dimensions and constraints to a common datum will reduce the amount of sensitivity in the design. For more information on datums and model building techniques, see “ Datums,” page 131.

94


The Importance of Sensitivity in Tolerance Analysis

Sensitivity Magnitude When the derived dimension labeled E is analyzed, the Tolerance Analysis results are displayed.

There are seven contributors to dimension E. Each contributor has a sensitivity magnitude that is listed in the Tolerance Analysis Report. The sensitivity magnitude is the ratio of change a contributor has on the analyzed dimension. When calculating the sensitivity for each contributor, the system considers the effect of perturbing each contributor, one at a time, while maintaining all other constraints. See “Perturbing Contributors,” page 97 for more information. A high sensitivity magnitude indicates that a small change in the contributor will produce a large change in the analyzed dimension. A one inch change in a contributor with a sensitivity magnitude of 1.0 inches/inches produces a 1.0 inch change in the analyzed dimension; whereas the same change in a contributor with a sensitivity magnitude of 5.0 inches/inches produces a 5.0 inches change in the analyzed dimension. High sensitivities are less desirable than low sensitivities. A well-designed model is not overly sensitive to changes in contributors. Mechanical Advantage Tolerance Analysis Manual - February 2001

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Comparing Mixed Unit Sensitivities In the following example, the sensitivity of Para1 (1.00 inches/inches) cannot be directly compared to the sensitivity of the angular dimension (–0.070 inches/ degree) due to the differing sensitivity units.

The units of sensitivity are determined by dividing the contributor units into the units of the analyzed dimension as follows: Analyzed dimension units Sensitivity units = ------------------------------------------------Contributor units

If the units of the analyzed dimension are inches and a contributor has units of degrees, the sensitivity units will be in/deg. Mixed unit sensitivities cannot be compared directly with non-mixed sensitivities unless they are first normalized. Mixed unit sensitivities including length and angle can be normalized by converting the degrees to radians. To convert from length/degree, multiply the sensitivity as follows: length 180 degree length Sensitivity  ------------ × ----------  ------------  = Sensitivity  ------------  degree π  radian   radian 

To convert from degrees/length, use the following equation:

π radian degree radian Sensitivity  ------------  × ----------  ------------ = Sensitivity  ------------   length  180  degree  length 

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Perturbing Contributors

Perturbing Contributors Perturbing contributors demonstrates the impact of contributor sensitivity on the design and is helpful in determining why a constraint scheme is not optimal. Use the Perturb command to see the impact of contributors on the analyzed dimension. When you perturb parallel and perpendicular constraints, the system makes the lines slightly out of parallel or perpendicular to simulate the effects on the analyzed quantity. Note: Derived dimensions, forces and moments may not be perturbed.

To Perturb a contributor: 1.

Select a contributor from the sketch note.

2.

From the Analysis menu, choose the Perturb command. The system displays the Perturb of Sketch Note window.

3.

Enter a value for Perturb by. Click OK or Apply.

Contributors can also be perturbed from the tolerance report window: 1.

Select a contributor from the tolerance report.

2.

From the Viewing menu in the tolerance report, choose Perturb Contributor. The sketch note is brought forward and the selected contributor is highlighted.

3.

From the Perturb of Sketch Note window that is displayed, enter a value for Perturb by.

4.

Click OK or Apply.

In the following examples each contributor listed in the Tolerance Analysis Report is perturbed, one at a time, to demonstrate how changes to the contributor effect the analyzed dimension.


97


Perturbing the Angular Dimension Perturbing the angular dimension so that it is slightly more and less than 45 ° has an impact on the sketch model.

The system must maintain all other constraints when perturbing the angular dimension, causing the analyzed dimension to change when the angle changes. When the sensitivity of the angular dimension is a negative value, a decrease in the angular dimension will increase the analyzed dimension, and an increase in the angular dimension will decrease the analyzed dimension. Since the analyzed dimension is linear, and the contributor is an angle, the units of sensitivity are inches/degree. The sensitivity magnitude of –0.070 inches/degree means that changing the angle + 1 ° changes the analyzed dimension by –0.070 inches. Even though the angular dimension has a small sensitivity magnitude, it has the single greatest impact on the analyzed dimension. When interpreting sensitivity results, make careful note of the units of the sensitivity. For example, angular dimensions have sensitivities that are in units of inches/degrees, which tend to be smaller than linear dimensions, which are in units of inches/inches. Even though angular sensitivities appear to be small, they can have a large impact on the analyzed dimension. When the angular dimension is locked at a 45 ° angle, it contributes 90% to the analyzed dimension being in tolerance.

98



Perturbing Parallel Constraint 2 The following figure demonstrates how the sketch adjusts when the second contributor, Para2, is perturbed. Tolerance Zone

Since all other constraints are maintained, perturbing the lines so they are more or less parallel causes the analyzed dimension to adjust. Parallel constraints, as described in ASME Y14.5M-1994, use a tolerance zone (as shown above). The Tolerance Analysis Report shows Para2 has a sensitivity of 1.00. For geometric constraints, the negative or positive sign of the sensitivity has no real meaning; only the magnitude has relevance. Since parallel uses a tolerance zone, the units are inches/inches. Therefore, a sensitivity of 1.00 inches/inches means that a tolerance zone of 0.001 inches causes the analyzed dimension to vary by 0.001 inches. The sensitivity of Para2 (1.00 inches/inches) cannot be directly compared to the sensitivity of the angular dimension (–0.070 inches/degree) due to the differing units. Even though the sensitivity of Angle is smaller than the sensitivity of Para2, the percent contribution to the tolerance of the analyzed dimension is much greater. This is a result of the technique used to constrain the sketch model.


99


Perturbing Dimension C The system maintains all the constraints in the sketch and the model adjusts when dimension C is perturbed.

The Tolerance Analysis Report shows dimension C has a sensitivity of –1.00 inches/inches. The negative sensitivity means that decreasing dimension C by 0.001 inches increases the analyzed dimension by 0.001 inches.

100



Perturbing Parallel Constraint 1 The following example demonstrates how the sketch adjusts when the parallel constraint, Para1, is perturbed within the tolerance zone.

e n o Z e c n a r e l o T

Para1 has a sensitivity of 1.00 inches/inches. The magnitude of the sensitivity indicates that if the two lines constrained by the parallel constraint, Para1, are allowed to go out of parallel within a tolerance zone of 0.001 inches, the analyzed dimension will vary by 0.001 inches. For constraint contributors the positive or negative sign of the sensitivity has no relevance.


101


Perturbing Dimension B The following example demonstrates how the sketch adjusts when dimension B is perturbed.

The sensitivity of dimension B is +1.00 inches/inches. A positive sensitivity of 1.00 indicates that an increase of 0.001 inches in dimension B causes the analyzed dimension to increase by 0.001 inches.

102



Perturbing Dimension A The following example demonstrates the effect of perturbing dimension A.

The sensitivity of dimension A is +1.00 inches/inches. A positive sensitivity of 1.00 indicates that an increase of 0.001 inches in dimension A causes the analyzed dimension to increase by 0.001 inches. The effect is the same as dimension B.


103


Perturbing the Perpendicular Constraint The example demonstrates how the sketch adjusts when the perpendicular constraint is perturbed. Tolerance Zone

Perpendicular constraints, like parallel constraints, use a tolerance zone. Perturbing Perp causes dimension B to move within the tolerance zone of the perpendicular constraint. The movement of the line dimensioned by B causes the line dimensioned by C to move since the parallel constraint, Para2, is maintained. Para2 causes line C to rotate, keeping it parallel to line B. Since the parallel constraint is maintained, the sensitivity of the perpendicular constraint is less than 1.00. The sensitivity, as shown in the Tolerance Analysis Report, is 0.667 inches/ inches. As with all geometric constraint contributors, a positive or negative sign for sensitivity is meaningless. The 0.001 inches tolerance zone for the perpendicular constraint causes a deviation in the analyzed dimension of 0.000667 inches.

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The Effect of Angular Dimensions on Sensitivity

The Effect of Angular Dimensions on Sensitivity Angular dimensions increase the amount of sensitivity in the design, resulting in a decrease in the probability that the dimension will be in tolerance. The following examples demonstrate how angular dimensions influence the behavior of the design and Tolerance Analysis results. Alternative dimensioning schemes are suggested for reducing contributor sensitivities.

Tightening the Lock on the Angular Dimension Locking sharp angles usually amplifies sensitivities. The increased sensitivity results in a decrease in the probability that the analyzed dimension is in tolerance. Using the following sketch model, the example demonstrates how sensitivities change when the angular dimension is changed from 45 ° to 30 ° .


105

Sensitivity Analysis In the following Tolerance Report, the angular dimension is locked at 30 ° .

Compare the above Tolerance Analysis results with the following Tolerance Report, where the angular dimension is locked at 45 ° .

Several of the sensitivities change. For example, the sensitivity of the angular dimension doubles, from -0.70 inches/degree to -0.140 inches/degree. The increase indicates the change in Angle causes the analyzed dimension to be twice as sensitive to variations in the angular dimension. For angles that are less than 30 ° , the sensitivity is even greater. The increased sensitivities cause the percent in tolerance rate to decrease from 31.6% to 16.5%.

106



Re-Dimensioning the Angular Dimension Using the same sketch model, the following example demonstrates the impact of re-dimensioning the angular dimension so the 30 ° angle is deleted and replaced with the 45 ° angular dimension.


107

Sensitivity Analysis The derived dimension is analyzed and the results displayed below.

The Tolerance Analysis Report indicates the sensitivity of Para2 increased from 1 to 5. As a result, Para2 now contributes 32% to the analyzed dimensions tolerance results.

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Perturbing Parallel Constraint 2 Perturbing Para2 demonstrates the impact of maintaining the angular dimension, Angle2. Since all constraints are maintained, the angular constraint forces the right side of the model to pivot around the lower right corner, thereby amplifying deviations from the parallelism. High sensitivities like this are very undesirable and should be avoided.


109


Replacing the Angular Dimension with a Linear Dimension Using the same sketch model, the following example demonstrates the effect of replacing the angular dimension, Angle2, with the linear dimension F.

110


The Effect of Angular Dimensions on Sensitivity The derived dimension is analyzed and the Tolerance Analysis Report is displayed.

In this example the number of contributors is greatly decreased, from seven to three. The sensitivity of Para2 is still high (3.00), but due to the reduction in contributors, the probability that the analyzed dimension is in tolerance has increased to 63.4%.


111


The Effect of Analyzing Linear Dimension Types If the analyzed dimension is linear, it is important to know what type of linear dimension it is. The same contributor might produce different sensitivities for a point-to-point dimension and a point-to-line dimension. For example, creating a point-to-line dimension between two lines that are parallel gives you the functional equivalent of a line-to-line dimension. The dimension type should reflect the way the part will be manufactured. For example, measuring the distance of one corner from a face implies a point-to-line dimension whereas a point-to-point dimension measures corner to corner. The sensitivity to changes in contributors is dependent upon the type of analyzed dimension. To get valid Tolerance Analysis results, create dimensions that reflect the way measurements are made when the model is manufactured. If you expect to measure the top line of a part relative to the left surface, then create a line-topoint dimension. If you want to measure from corner to corner, then use a pointto-point dimension. When you are creating locked dimensions, the dimension type may be unimportant. However, when you analyze derived dimensions, the dimension types are quite different. The following examples demonstrate the effects of sensitivity on derived linear dimensions.

112


The Effect of Analyzing Linear Dimension Types

Perturbing Parallel Constraint 1 The following example demonstrates how the four types of derived linear dimensions react when the contributor, Para1, is perturbed. The values of the derived dimensions are not relevant; what matters is whether the values are the same or different from each other. Line_to_Line: 6.332 Point_to_Line: 6.332

Analyzed Dimensions

Line_to_Point: 6.332 Poi n t_ to_ Poi n t:6.341


113


Perturbing Parallel Constraint 2 This example demonstrates how the four types of derived linear dimensions react when the contributor, Para2, is perturbed. 3 2 L i n e: 6 .3 to_ L i n e_

81 3 L i n e: 5. _ o t t_ n i o P

Analyzed Dimensions

Line_to_Point: 5.870 Point_to_Point: 5.870

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The Effect of Analyzing Linear Dimension Types

Perturbing the Perpendicular Constraint The example demonstrates how the four types of derived linear dimensions react when the perpendicular contributor, Perp, is perturbed. L in e _t o _ L in

e : 5 .6 99

P oi n t _t o _ L in

e : 5 .6 99

L ine _t o _ P o int : 5 .6 99

Analyzed Dimensions

Point_to_Point: 5.741


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Perturbing the Angular Dimension This example demonstrates the impact when the angular dimension, Angle, is perturbed. Line_to_Line: 6.301 Point_to_Line: 6.301 Line_to_Point: 6.301

Analyzed Dimensions

Point_to_Point: 6.301

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Summary

Summary The results of a sensitivity analysis correspond to the constraints, dimensions, variables, forces and moments used to create a design. The system performs a sensitivity analysis on each contributor listed in the Tolerance Analysis Report. A well-designed model is not overly sensitive to deviations in contributors. When the system computes the sensitivity of a contributor, it assumes that all other constraints are perfectly maintained. This can lead to results where unexpected contributors have a surprising effect upon the analyzed dimension. Since sensitivities are computed without regard for the tolerance information assigned to the contributors, the sensitivity of a contributor is a result of the dimensioning scheme used to create the design. The sensitivity magnitude listed in the Tolerance Analysis Report indicates the ratio of change between the contributor and the analyzed dimension. A high sensitivity magnitude indicates that a small change in a contributor will elicit a large change in the analyzed dimension. When possible, high sensitivities should be avoided. • Linear sensitivities, which are in units of inches/inches, should be kept near or below 1.0. Linear sensitivities higher than 2.0 probably indicate a poorly dimensioned part. • In contrast, it is common for assemblies to have sensitivities greater than 1.0. Note: It is important to know what type of linear dimension is being analyzed.

Always consider percent contribution when comparing the sensitivities of angular and linear dimensions. Even though sensitivities for angular dimensions, which are in units of inches/degree, are usually much smaller than those of linear dimensions, which are in units of inches/inches, the angular contributors might have a larger impact upon the analyzed dimension then the sensitivity indicates. As a result, you should look at both the sensitivity and the percent contribution for the contributor. Angular dimensions can lead to high sensitivities and unexpected contributors. This is particularly true for angles close to 0 ° and180 °. In cases such as these, you may get much more tolerance-insensitive designs by replacing the angular dimension or dimensions with combinations of linear dimensions. When interpreting sensitivity magnitude, the positive or negative sign of the sensitivity indicates the direction the analyzed dimension will change for a given change in the contributor. The sign of non-dimensional contributors, however, has no real meaning since the contributor is perturbed within a symmetric zone. When confronted with a Tolerance Analysis that produces counter-intuitive results, you can perturb questionable contributors to see their effect on the analyzed dimension. Perturbing your model in this way can be very helpful in determining why your constraint scheme is not optimal.


117


118


Chapter 7

Advanced Topics for Monte Carlo This chapter is intended for an advanced user with a strong statistical background who wants to investigate the impact of non-normal statistical distributions. The Monte Carlo analysis is a statistical type of Tolerance Analysis that is based on a random sampling. In Monte Carlo analysis, the system randomly considers up to 1 million (999,999) possible tolerance values for each contributor and calculates the mean, standard deviation and probability for the analyzed dimension, force or moment. The random sampling of Monte Carlo differs from the standard Tolerance Analysis, which uses RSS mathematical equations for calculating tolerance results. As such, Monte Carlo provides an alternate method for calculating probability, mean and standard deviation for the analyzed item and can be used to verify the results of the RSS Tolerance Analysis. Monte Carlo analysis is available as part of the Tolerance Analysis.

Distribution Types Monte Carlo analysis offers two non-normal distributions: Weibull and Uniform. The distribution type you choose for a contributor should be based on the manufacturing process used to construct that particular contributor. If you do not have adequate information about the manufacturing process for the model, or a good understanding of statistical modeling, it is recommended that you use a Normal distribution. Normal is the default distribution. For information about setting distribution types, see “Monte Carlo Distribution Types,” page 69.


119

Advanced Topics for Monte Carlo

Normal The system defaults to a Normal distribution for all analyses. A Normal distribution has two parameters: the mean ( µ ) , and the standard deviation ( σ ) . The mean and standard deviation for each contributor are computed from its tolerance values. The mean and standard deviation for the contributor are automatically calculated and displayed in the Contributor Options window. To view the parameter values: 1.

Select a contributor row from the Tolerance Analysis Report.

2.

From the Edit menu, choose the Options command.

3.

The Tolerance Contributor Options window is displayed.

B

A

120

A

The Mean value for the contributor.

B

The Standard Deviation for the contributor.


Distribution Types

Weibull Use the Weibull distribution to model distributions that are skewed about the mean. Even though Weibull distributions are most often used to model the failure rate or life expectancy of a part or system, it has manufacturing applications since it can also be used to model an exponential distribution. The Weibull distribution has three parameters: location, scale and shape. Note: A two-parameter Weibull is a special case in which the location parameter is 0.0. Exponential distribution is a special case of the Weibull distribution where the location parameter is 0.0 and the shape parameter is 1.0.

You can edit parameter values from the Tolerance Contributor Options window. 1.

Select a contributor row from the Tolerance Analysis Report.

2.

From the Edit menu in the Tolerance Analysis Report, choose the Options command.

3.

The Tolerance Contributor Option window is displayed.

A

C

B A

Location parameter - minimum value.

B

Scale parameter - characteristic value.

C

Shape parameter - slope.


121

Advanced Topics for Monte Carlo Estimates for default values for the location and scale parameters are dependent upon the contributor type. You may need to edit these values and supply the proper shape parameter value. When shape parameter value ≥ 1.0 , the distribution about the mean decreases: Location

0

Scale

0

Shape

3.5000

Dimensional Contributors The following are default location and scale parameter values for a contributor with upper and lower tolerances and a nominal value. Location:

location = L i

where L i is the Lower Tolerance Value for the contributor. Scale:

scale = 0.0

Edit the parameter values to generate random deviations (or offsets) from the nominal value of the contributor. You should use offsets, rather than actual values corresponding to a nominal value, to avoid the problems that occur if you change the nominal value for a contributor, but neglect to change the corresponding location and scale parameters. Edit the parameter values so that location is in the following range: L i ≤ location

< scale

where L i is the Lower Tolerance Value for the contributor. • When location < L i , the system alerts you with a beep and continues to perform the analysis. • When location ≥ scale , the system displays the following alert message. Click OK to dismiss the alert. The system requires you to correct the situation before performing the analysis.

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Distribution Types

Constraint Contributors The following are the estimated default values for location and scale parameter when the contributor has a tolerance zone or a true position tolerance value. Location:

location = 0.0

When working with constraint contributors, you cannot edit the location value. Scale:

T scale = 2

Where T is a true position tolerance value. You can edit the scale value, which should most often be in the following range: 0 < scale

≤T

When the scale parameter value is outside of this range, the system displays an alert message. Shape:

For additional error checking, the system requires the following for shape parameters: shape

> 0.0

The system will display an alert when shape ≤ 0.0 , and require you to correct the value.


123


Uniform A Uniform distribution has simple minimum and maximum values which you control through the upper and lower tolerance values or the zone or true position values.

A

A

124

Over the Range. The computed minimum and maximum values are shown as information.


Linear and Non-Linear Methods

Linear and Non-Linear Methods When you perform a Monte Carlo analysis, the system lets you choose between a linear or a non-linear analysis. The method choices of linear and non-linear are significantly different. A linear analysis is extremely fast, but assumes constant sensitivities. This method is useful for a quick analysis that is based on distributions other than normal. A non-linear analysis is more accurate since it accounts for varying sensitivities for dimensional contributors. A non-linear analysis performs the following steps: 1.

The system sets all contributors to their maximum values and performs a Tolerance Analysis. This allows the system to find previously undetected contributors which have sensitivities of zero when dimensions and constraints are at nominal values.

2.

The system updates the Tolerance Analysis and perturbs the contributors that are set for inclusion in Monte Carlo. This is identical to your changing the values of all the dimensions, but much faster. The values are based on deviates from the distributions you specify. When the design is linked to external notes, non-linear Monte Carlo incorporates the effects of inter-note propagation into the analysis. The contributors are perturbed the number of times you specify in the sample size.

Since the non-linear method is computationally intensive, Mechanical Advantage provides status information about the progress of the analysis. The analysis may be interrupted by pressing the Crtl-C keys.


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Model Building for Non-Linear Monte Carlo Non-linear Monte Carlo analysis is an advanced Tolerance Analysis capability which is useful in analyzing designs with significant non-linearity. Use the following model building guidelines to obtain valid results for non-linear Monte Carlo analysis.

Contributors from Multiple Dependent Assemblies In non-linear Monte Carlo the system updates the model to calculate each sample. When working with dependent assemblies, the system assumes the contributors in each dependent assembly are identical. When identical contributors are present, the model updates the parent master assembly and all identical contributors, and their associated dependent assemblies will have the same value. However, in the manufacturing environment, identical components are not exactly alike due to manufacturing variances. Non-linear Monte Carlo analysis is unable to model these variances because it treats identical components as if they were exactly alike. This is especially problematic when identical contributors from identical components have different, even opposite, sensitivities. This possibility is quite plausible because identical components are used in different locations of the design model. For proper tolerance results in non-linear Monte Carlo, it is important that only one dependent assembly is created from each master assembly. When multiple uses of a component are needed, copy the master and all of its dimensions and constraints, and then make a new dependent assembly from the new master.

Equal-Length Constraints In non-linear Monte Carlo, equal-length constraints, including equal-radius and equal-dimension, should be deleted for best results. Because non-linear Monte Carlo actually updates the model for each sample, the equal-length constraints cause equal variances to be propagated for equal-dimensions. To prevent this, these constraints should be deleted so that equaldimensions can have independent variations.

Linked-In Dimensions Similarly, links that drive contributor dimensions should be removed to allow the system to vary the dimension. In situations where the link is not removed, the linked note should be open so that propagation will be enabled, especially if the link propagates variations that are important to the analysis. Even though Mechanical Advantage provides random number functions which can be used to generate variances and drive linked dimensions, good results should be obtained by deleting links to dimensions which allows the non-linear Monte Carlo analysis to generate the variances for the dimensions.

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User Control of Monte Carlo Solution Failure Limit

User Control of Monte Carlo Solution Failure Limit Monte Carlo simulations will automatically stop after a defined number of solution failures. Solution failures generally occur because the sketch model cannot be solved with the values calculated for the given sample. For example, solution failures will occur in the following instances: • A line which was constrained point-on-circle no longer intersects the circle. • A mechanism reaches a lock-up state. The default number of solution failures is one percent of the number of samples (e.g., a 5000 run simulation can have up to 50 solution errors before stopping). The environment variable MAMCMAXFAIL allows you to override the maximum number of failures. For example, if your environment is using the “c-shell”: setenv MAMCMAXFAIL 100

For UNIX, this variable must be set in the window where the “ma” command is executed to start the Mechanical Advantage. For Windows NT, use the following steps to set the environment variable:

→ Control Panel.

1.

From the Start menu, choose Settings

2.

From the control panel, choose the System icon. The System Properties window is displayed.

3.

From the System Properties window, choose the Environment Tab.

4.

Create the variable MAMCMAXFAIL as a system or user variable and enter the maximum solution failure rate.

Note: You must set the environment variable before starting Mechanical Advantage. You do not need to reboot your machine as long as the variable was set before Mechanical Advantage was started.


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Plotting Monte Carlo results in a DCAP Note The results of a Monte Carlo computation can be plotted in a Mechanical Advantage DCAP note. A Plot Monte Carlo DCAP Note is distributed with Mechanical Advantage and the Note can be located from the MA Desktop File menu, using the New command. To export the Monte Carlo results and import them into a DCAP Note: 1.

From the Convert menu in the Tolerance Analysis Report, choose the Export Monte Carlo Results... command.

The results are written to a file which can then be imported into a Plot Monte Carlo DCAP note. 2.

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From the Convert menu in the Data Table for DCAP Note window, choose the Read From ASCII File command. The DCAP note displays a histogram of the analyzed dimension values calculated during the Monte Carlo analysis.


Monte Carlo TOLLOG

Monte Carlo TOLLOG Once you have performed a Monte Carlo analysis, the system reports a Monte Carlo Percent in Tolerance value in the Monte Carlo Failure Rates section of the Tolerance Analysis Report. The result is displayed to two or more places after the decimal. If you want to record the results of each sample in a file, you can set the environment variable TOLLOG to specify the file name in which you want the results recorded. The following Monte Carlo information is recorded from the Tolerance Report into the TOLLOG: • Geometric deviations. • Flag indicating whether in or out of tolerance. For example: Sample 1 Mir from 1 to 1.00185962546018 Sample 1 ldim368 from .4E-01 to .39434522713638E-01 Sample 1 Lcr from .149999999999999 to .148769179703715 Sample 1 Cgw from .25E-01 to .216205818821324E-01 Sample 1 Cir from .7 to .701200190990842 Geometric deviation 1 perp113 by -.112622828827704E-03 Results 1 .912136727507377E-01 deviation: .62136727507386E-02 OUTTOL Sample 2 Mir from 1 to 1.00007393283054 Sample 2 ldim368 from .4E-01 to .39574096357804E-01 Sample 2 Lcr from .149999999999999 to .149382593754479 Sample 2 Cgw from .25E-01 to .23191430368929E-01 Sample 2 Cir from .7 to .699393355469761 Geometric deviation 2 perp113 by .525327474714113E-04 Results 2 .883558955965564E-01 deviation: .335589559655737E-02 INTOL

These results include one (1) geometric contributor. INTOL means the resultant sample was in tolerance, and OUTTOL means it was out of tolerance.


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Chapter 8

Model Building Essentials for Tolerance Analysis This chapter describes model building techniques that are essential to properly dimension and constrain your model. Because all Tolerance Analyses are performed on the geometry in your sketch note, the construction of the design significantly impacts your tolerance results.

Datums The following sections are designed to improve your model-building techniques by improving your ability to implicitly define datums as you constrain your geometry. A datum is the piece of geometry that provides the control reference for your model. A good dimensioning scheme uses a common point of reference to constrain the geometry in a design. Doing this will reduce the tolerance deviations that accumulate when constraints are randomly applied. The application of constraints and dimensions are an important part of the process of constructing your sketch model. The way in which these constraints and dimensions are applied to the model can significantly impact your tolerance results. For the best results, it is important that you have an implicit notion of datums and that you identify common points of reference for constraining and dimensioning your design. These datums may also be the same geometry that you define with the datum tool. In Mechanical Advantage, the following eight constraints (geometric tolerances) use a datum relationship. • Parallel. • Perpendicular. • Collinear. • Equal-length. • Equal-radius. • Concentric to. • Position. • Float pin-in-hole. Mechanical Advantage Tolerance Analysis Manual - February 2001

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Model Building Essentials for Tolerance Analysis An implicit datum is defined when a constraint is applied to the geometry. In the figure below, line A was made parallel to line B. The constraint symbol has an arrow at one end, showing that line B is the implicit datum. Derived angular dimension

A C

Implicit datum or the parallel constraint

B

D

When you perform a Tolerance Analysis, line A is adjusted a small amount to consider its effect on the derived angular dimension. The datum line is held fixed when the system considers the contribution of the parallel constraint.

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Datums Not all constraints reference datum sketch objects. As shown in the table below, the only constraints that reference datums in Tolerance Analysis are parallel, perpendicular, collinear, equal-length, equal-radius, concentric, position and float pin-in-hole. Constraint

Has tolerance?

References a datum during Tolerance Analysis?

parallel

yes

yes

perpendicular

yes

yes

collinear

yes

yes

tangent

no

—

equal-length

yes

yes

equal-radius

yes

yes

equal-dimension

no

—

concentric

yes

yes

position

yes

yes

float pin-in-hole

yes

yes

bias pin-in-hole

no

—

coincident point

no

—

point-on-object

no

—

point-at-midpoint

no

—

product

no

no

sum-to or sum-to-zero

no

no

scale

no

no

line-at-angle

yes

no

fixed-point

yes

no

fixed-length

yes

paper units

fixed-radius

yes

paper units

dimension

yes

paper units

Notice that the equal-dimension constraint cannot be given a tolerance and therefore never shows up as a contributor to Tolerance Analysis. This is true for all constraints for which you cannot specify a tolerance.


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Model Building Essentials for Tolerance Analysis

Constraints that Reference Datums The datums for parallel, perpendicular, collinear, equal-length, equal-radius, float pin-in-hole and concentric constraints are defined by how you create the constraint. The piece of geometry that is picked as the destination for the constraint (that is, the geometry you click on after choosing the appropriate Constrain command) is the datum. To reverse the datum and non-datum geometry, choose Swap Datum from the Tolerance Analysis Report header.

Each contributor is analyzed separately. The Tolerance Analysis Report shows you the effect that a small change in each contributor has on the analyzed dimension. When the contributor is a constraint that references a datum, the small change should be made on the non-datum sketch object. You guarantee that the small change is made on the non-datum sketch object by making sure that other constraints and dimensions anchor the datum appropriately.

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Datums

Parallel Constraints Parallel constraints lock the orientation (that is, the angle) of one line relative to another. The datum line, therefore, must have its orientation fixed by other constraints or dimensions when you do the Tolerance Analysis. The figures below show parallel constraints whose datums are properly chosen. Notice that in each case the orientation of the datum is fixed due to other constraints. If any datum is reversed, Tolerance Analysis results for that contributor might be invalid. That is, while the Tolerance Analysis results are always mathematically correct (reflecting the way you created your model), they might not represent what you expect from an engineering standpoint. The following examples demonstrate parallel constraints with properly chosen reference datums.

Orientation fixed by angular dimension relative to the grid.


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Perpendicular Constraints Like parallel constraints, perpendicular constraints lock the orientation of one line relative to another. The datum line, therefore, must have its orientation fixed by other constraints or dimensions when you perform the Tolerance Analysis. The figures below shows two examples of perpendicular constraints with properly chosen datums. Reversing the datum on any perpendicular constraint might produce invalid Tolerance Analysis results for that contributor.

Orientation of datum fixed by line-at-angle constraint.

136

Orientation of datum fixed by two dimensions relative to the grid.


Datums

Equal-Length and Equal-Radius Constraints Equal-length and equal-radius constraints are similar to one another. Equal-length equates the length of one line to another, and equal-radius equates the radius of one arc or circle to another. For these constraints, the datum object must have its length or radius fully determined by other constraints. The following examples demonstrate equal-length and equal-radius constraints with properly chosen reference datums.


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Collinear Constraints Collinear constraints relate both the orientation and the position of one line relative to another. Consequently, a properly chosen datum for a collinear constraint must have both its orientation and its position fully determined by other constraints. In the example below, the collinear constraint has a properly chosen datum. Its orientation is fixed, since it is parallel to a line that has a line-at-angle constraint, and its position is fixed due to the locked dimension from the fixedpoint constraint. The following example shows a collinear constraint with a properly chosen reference datum.

138


Datums

Concentric and Float Pin-in-Hole Constraints Concentric and float pin-in-hole constraints, in effect, constrain the center of one circle or arc to another. The datum circle or arc must have a center whose location is fixed by other constraints or dimensions. In the following example the concentric constraint has a properly chosen datum. The center of the datum is constrained to the fixed point.


139


Position Constraints Position constraints use datums to constrain the location of a circle by referencing either the gridded centerpoint of another circle, or two lines that are not parallel to each other. When the datum is a circle or an arc, the location of the centerpoint must be fixed by other constraints or dimensions. In the following example the position constraint has a properly chosen datum. The datum circle is properly constrained by dimension A and dimension B.

In the following example the position constraint has two properly chosen datums, the two perpendicular lines, labeled Datum 1 and Datum 2.

140


Datums

Choosing Datums for Constraints Here are the important points to remember when choosing datums for constraints: • If you do a Tolerance Analysis when a contributing constraint has an inappropriate datum, the Tolerance Analysis calculations for that contributor might be invalid. • Choose datum objects that are appropriately constrained by other constraints or dimensions. • Appropriately constrained datums have the following characteristics: Constraint

Characteristics of a good datum

Parallel or Perpendicular

Line whose orientation (angle) is fully determined.

Equal-length

Line whose length is fully determined.

Equal-radius

Circle or arc whose radius is fully determined.

Collinear

Line whose orientation and position are fully determined.

Concentric or float pin-in-hole

Circle or arc whose center is fully constrained.

Position

Two lines that are not close to parallel to each other, or a circle or arc whose center is fully constrained.

Note: No other geometric constraints reference datum objects.


141


Tolerance Analysis Results and Bad Datums If any contributor has a poorly chosen datum, the Tolerance Analysis Report shows a red “?” (question mark) in the Type column. This indicates that the sensitivity calculated for that contributor probably has no physical meaning. Therefore, the contribution values for all contributors may be invalid. The example below shows a sketch where the base line (constrained line-at-angle) is made perpendicular to dimension B. When dimension A is analyzed, a red “?” is displayed in the Type column of the Tolerance Analysis Report, indicating that the perpendicular constraint has a poorly chosen datum.

A

B

142

A

The Perp1 constraint has a poorly chosen datum.

B

Perp1 shows a red “?” in the Tolerance Analysis Report, indicating a poorly chosen datum.


Datums When this happens, use the Show Contributor command in the Tolerance Analysis Report Viewing menu to find the constraint with the bad datum. Swap the datum for this constraint with the Swap Datum command, located on the Tolerance Analysis Report header, then Reanalyze. The revised contributor results are shown below.

Consider the following when working with constraint datums: • Deleting the “grounding” constraints in your model (that is, fixed-point and line-at-angle) and recreating them elsewhere can make previously good datums inappropriate. When creating line-at-angle and fixed-point constraints, take care that existing constraints still reference appropriate datums. • If a constraint references an inappropriate datum, swap the datum.


143


Constraint Schemes The constraint scheme used in your model should match the way the design will be manufactured, as well as reflect critical functional constraints. For example, consider the following simple staircase model. The critical functional dimension is labeled CFD, which is a derived dimension due to constraints that are not shown.

There are many ways to constrain this model so that CFD is derived. The examples will consider the following constraint schemes: • Chained constraints: constraining each piece of geometry to the next. • Baseline constraints: constraining all geometry to a common datum.

144


Constraint Schemes

Chained Constraints In the following figure, the stairs are constrained by means of a chain of perpendiculars. The datum for each is the next lower step. Notice that all datums “flow” in the same direction: toward the fully constrained base. This is much better than a random flow of datums.


145


Analyzing CFD produces the tolerance results shown above. The perpendicular and fixed-length constraints are listed as contributors: the steps are constrained in a string, or chain, of perpendiculars and are labeled perp1–perp8 in the report; the fixed-length constraints on the vertical steps are labeled flen1 – flen4. If the staircase is manufactured by measuring each step surface perpendicular to the previous one, you would expect the errors in each measurement to compound and CFD would quite likely be out of tolerance. The Tolerance Analysis results reflect this, showing many contributors. The perpendiculars at the bottom of the steps (perp1 and perp2) have larger sensitivities due to their distance from CFD. Aside from considering better ways to constrain (and manufacture) this staircase, you can reduce the tolerance on the contributors with high sensitivities. A reduction in the tolerances on perpendiculars perp1 and perp2, which have a sensitivity of 4.0, will produce sixteen (that is, four squared) times the benefit of a reduction in the tolerance on the perpendiculars whose sensitivity is 1.0. For more information about sensitivity, see “Sensitivity Magnitude,” page 95. 146


Constraint Schemes

Baseline Constraints When dimensioning a staircase it is better to relate all of the step surfaces to a common baseline, or datum. In the following example, the horizontal step lines are all constrained to a common datum, the baseline. Also, all vertical step lines are constrained perpendicular to the baseline. These perpendiculars are not shown since they do not contribute to CFD. The Tolerance Analysis results show a marked improvement over the chained constraints in the previous example.


147


Using a baseline constraint scheme reduces the number of contributors from twelve to eight. Notice also that no contributor has a sensitivity greater than 1. The probability that CFD is in tolerance is improved, from 29.23% to 71.12%. Clearly, relating as many constraints as possible to the common datum produces a much better model. In general, you should strive for models that have small numbers of contributors, with all contributors having balanced sensitivities. Take care when comparing sensitivities, however. You cannot directly compare sensitivities involving angles with those that are purely linear. For example, when you analyze a linear dimension, the sensitivity of a parallel constraint is in units of inches/inches (or the paper units you specify). However, the units of an angular dimension contributor are inches per degree. For more information, see “Comparing Mixed Unit Sensitivities,” page 96.

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Choosing Appropriate Dimension Schemes

Choosing Appropriate Dimension Schemes The previous examples demonstrated why chained constraint schemes produce models that have large numbers of contributors and are less likely to be within the desired tolerance. Constraining multiple objects to a common datum produces models that are less tolerance-critical and therefore more desirable. Now we illustrate the same point when considering various ways to dimension your model. Consider the following simple part. The critical functional dimension is labeled CFD, and this part is to be turned on a lathe.

The following examples consider three different ways to dimension the part: chained dimensions, baseline dimensions, and direct dimensions. We simplify the following discussion by considering only the contributions made by the dimensions. Contributions by geometric constraints are disregarded. Also, for clarity, most constraints are not shown.


149


Chained Dimensions The figure below illustrates chained dimensions. CFD is a derived dimension because dimensions B, C, and D are locked. Dimension A has no effect upon CFD. (In ASME Y14.5M-1994, CFD is called a reference dimension.)

150



Analyzing CFD produces the Tolerance Analysis results shown above. There are six contributors, three of which are the dimensions B, C, and D. The remaining three contributors are the shown parallel constraints. They have 0% contribution to the change in CFD, since we set up the tolerances for all geometric constraints to be 0. Thus, we consider only the effects due to the dimensional tolerances. The probability that CFD is within tolerance is 91.67%. Models that are dimensioned with chains tend to have large numbers of contributors and decrease the likelihood that the analyzed dimension is within the specified tolerance. Chained dimensions compound the errors of individual contributors and can lead to large accumulated errors for the overall dimension.


151


Baseline Dimensions The figure below illustrates baseline dimensions. Note that CFD is derived, since dimensions A and D are locked. Dimensions B and C have no effect upon CFD.

152



Analyzing CFD produces the Tolerance Analysis results shown above. There are only four contributors, two of which are the dimensions A and D. In comparison with the chained dimension example, using a baseline reduces the number of contributors from seven to four and the probability that CFD is in tolerance increases from 91.7% to 96.6%. Use a baseline dimensioning scheme to minimize the number of contributors and to reduce the accumulated error of each individual contributor on the analyzed dimension. Reducing tolerance deviation to the analyzed dimension improves Tolerance Analysis results. Note: Baseline dimensioning often reflects better manufacturing practice as well. For example, it is easier and more reliable for the lathe worker to measure from a stable baseline than by chaining measurements.


153


Direct Dimensions With direct dimensioning, CFD is locked. That is, it is not a derived (or reference) dimension. This produces the greatest likelihood that CFD is in tolerance. Since there are no contributors to CFD, other than the tolerance on the dimension itself, the probability that CFD is in tolerance corresponds with the Sigma rate assigned to the dimension. For example, if CFD is manufactured at ± 3 Sigma, than the likelihood that the dimension will be in tolerance is 99.73%.

Clearly, direct dimensioning is desirable for critical dimensions. However, for those cases where it is impractical (for example, for manufacturing reasons), baseline dimensioning is almost always more desirable than chain dimensioning.

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Summary of Model Building Essentials

Summary of Model Building Essentials The following summary of model building guidelines will help you to create models that produce valid Tolerance Analysis results. When dimensioning and constraining your model, choose appropriate baselines and centerlines. Ground them with fixed point and line-at-angle constraints early in the model-building process. A well dimensioned model relates as many objects as possible to the baselines or centerlines. Chained constraints and dimensions produce Tolerance Analysis results with large numbers of contributors. When modeling with chained constraints and dimensions, analyzed dimensions are less likely to be within tolerance. Refrain from modeling your design with random constraint schemes. While baseline constraint schemes are generally better than chained constraint schemes, both methods produce valid Tolerance Analysis results. Randomly created constraints will almost certainly produce invalid Tolerance Analysis results and are difficult to understand and modify. Use the Tolerance Analysis Report to identify contributors that have high sensitivities. High sensitivities often require unnecessarily tight tolerances and imply that your constraint scheme is not optimum. Poorly chosen datums and chained constraints and dimensions can contribute to high sensitivities.


155


156


Chapter 9

Advanced Model Building for Tolerance Analysis This chapter describes advanced model building techniques that influence your Tolerance Analysis results. Complex sketch models often have design schemes that produce unexpected results for Tolerance Analysis. Using the model building techniques described in this chapter will help to ensure that all contributors to the analyzed item are accounted for in the Tolerance Analysis Report.

Profile Tolerances Profile tolerances can be applied to grouped geometry, such as groups, regions, and assemblies. Profile tolerances applied to a group or region contained within a master assembly are propagated to any dependent assemblies. Changes to the profile tolerance are propagated to all dependents. The tolerances are updated whenever the profiles on a master group or region are updated. In general, the best practice is to apply Profile tolerances to groups and regions, but not to assemblies. This practice will help to reduce problems associated with duplicate geometry and nested assemblies. Profile tolerances are included by default in Monte Carlo and Worst Case analyses.


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Advanced Model Building for Tolerance Analysis

Material Modifiers In the ASME Y14.5M-1994 specification, material modifiers allow a geometric tolerance on a feature to be dependent on the size of a feature or on the material condition of the datums. Mechanical Advantage supports the application of material modifiers for the position of a circular feature. The modifier is only applicable to feature size (pin or hole diameter), and is not applicable to the datum material conditions. Using material modifiers, you can specify that the allowable Position of a hole is dependent on the size of the hole. For example, by using a maximum material condition (MMC) modifier, you can specify that as a hole gets larger it does not require as tight of a position control to still function as desired (e.g., for assembly clearance). For additional information about using material modifiers in RSS analysis, see Appendix B - Incorporation of Material Condition Modifiers into RSS Statistical Analyses.

Setting the Modifier Type Material modifiers can be specified in the constraint option window for either Position or Concentricity constraints: 1.

Select either a Position or Concentricity constraint from the sketch note.

2.

From the Edit menu, choose Options. The Constraint Options window is displayed.

3.

Select the Modifier type, and if applicable select a Feature type. You can optionally enter a Bonus Factor. Click OK.

B

A C

A

Modifier type. Specifies the tolerance as RFS, MMC, or LMC.

RFS Regardless of Feature Size Maximum Material Condition MMC LMC Least Material Condition 158


Material Modifiers B

Feature. Available for MMC and LMC. Specifies if the feature is internal (hole), external (pin) or currently is undefined.

C

Bonus Factor. When a material modifier is specified, a bonus contributor is added to the Tolerance Analysis contributor list. The bonus factor is the amount of the bonus tolerance that should be applied in RSS calculations. The Bonus Factor has a default value of zero, which produces a zero percent contribution in Tolerance Analysis.

Worst Case Application of Material Modifiers Material modifiers are useful for a worst case analysis, such as insuring a minimum assembly clearance. A bonus contributor is added to the Tolerance Analysis contributor list when a material modifier is specified. The bonus contributor is always placed under the associated constraint that it modifies. The contribution of the bonus tolerance depends on the modifier type. For MMC on a hole, the bonus contributes fully to the worst case when the hole is at LMC, and has no contribution when the hole is at MMC; for example:

Material modifiers can lead to unexpected results in certain circumstances, as illustrated in the following contributor list. If the Position constraint has more than double the sensitivity of the diameter dimension, then both the upper and lower worst case results are obtained by placing the hole at LMC and applying the full bonus tolerance. This is due to the fact that the bonus tolerance has a higher impact than the size tolerance (as indicated by the sensitivity).


159


Statistical Applications of Material Modifiers One of the underlying assumptions of RSS analysis is that each contributor is independent. However, the application of a material modifier allows you to create a dependency between size and position. In many real processes, there is unlikely to be a real dependence between size and position. For example, when drilling a hole, the size is determined by the drill bit size and the location is determined by how accurately the work piece is located relative to the drill bit. As such, the bonus tolerance associated with the material modifier need not be included in the statistical analysis. However, if there is a dependence, the Bonus Factor can be used to include the modifier in the RSS analysis (i.e., by increasing the Position tolerances).

A

A

Monte Carlo Dependence. Specify if the constraint tolerance and size should be treated as Dependent or Independent. Allows you to model a more precise notion of dependence for Monte Carlo analysis. • Independent (Indep.): the Bonus Factor is utilized and the analysis is done in the same fashion as RSS. • Dependent (Depend.): the feature size is calculated first for each sample, and the applicable bonus tolerance range is calculated based on this feature size. As such, the bonus range is adjusted from sample to sample to reflect the feature size variation. The location of the feature is then randomly determined based on this bonus range plus the unmodified position tolerance.

160


Position Constraint

Position Constraint Position constraints are used to constrain the location of a circle by referencing either the gridded centerpoint of another circle, or two datum lines that are not parallel to each other. The position constraint eliminates the need for a “construction” circle (i.e. a circle constrained with a concentric constraint) to add a position tolerance. The position constraint constrains selected circles, circular arcs, or circular segments. By selecting multiple circles, circular arcs and segments, you can apply the position constraint to several pieces of geometry at the same time. To set a position constraint: The position constraint works on one or more selected circles, circular arcs, or circular segments. 1.

Select (or extend select) any desired circles, arcs, or circular segments.

2.

From the Constrain menu, choose Position .

3.

In response to the “Pick Datum 1” prompt that is displayed in the bottom left corner of your screen, pick a datum. The datum can be a line or a circle. If you pick a circle, you will not need to pick a second datum. Special considerations apply for datum circles, for information see “Using a Circle as a Datum,” page 162.

4.

Ifyou pickeda lineforthefirst datum,youare prompted topick a second datum. In response to the “Pick Datum 2” prompt, displayed in the bottom left corner of your screen, pick a second line that is not parallel to the first datum line.

Dimensions and constraints are created to position the selected geometry.


161


Using a Circle as a Datum Circles may be used as valid datum objects for position constraints. When a position constraint references a datum circle, the constraint uses the gridded dimension to the centerpoint of the datum circle to constrain the centerpoint location of the new circle. A dependent assembly should be created if you plan to rotate the sketch. For example, in the following sketch a position constraint is used to constrain the centerpoint position of circle A in relation to the datum circle centerpoint.

circle A datum circle

When the sketch is rotated 45˚, the geometry rotates while the circles maintain their position in reference to the grid.

A dependent assembly is recommended since it maintains the alignment of the circles within the geometry when the dependent is rotated.

162


Position Constraint

Editing a Position Constraint Each position constraint uses geometric dimensions as points of reference to constrain the basic distance between the datums and the circle center. To manage the dimensions, refer to the following summary of the editing behavior of the position constraint: • Deleting a position constraint automatically removes the associated basic dimensions. • If a basic dimension associated with a position constraint is selected for deletion, an Alert window is displayed telling you that constraints will also be deleted. If Delete And Continue is selected, the position constraint will also be removed (and the other basic dimension). The alert is not raised if the position constraint is being deleted simultaneously, or if the dimension is being deleted because the datum it references is selected for deletion. • Copying all geometry for a position constraint (both datum lines and the circle) causes the position constraint and associated dimensions to be copied. • Moving the datum lines or the circle does not break the position constraint.

Tolerance for a Position Constraint To assign a tolerance to the position constraint: 1.

Select the constraint.

2.

From the Sketch Note Edit menu, choose Options . The Constraint Options window is displayed.

3.

Using the Tolerance drop-down list, you can either use the default tolerance or specify a user-defined tolerance.


163


Float Pin-in-Hole Constraint The float pin-in-hole constraint models an assembly situation where a part with a pin is inserted into a part with a hole and the pin is constrained to float freely within the hole.

Automatic Tolerance Calculation The following formula is used to automatically calculate the tolerance for the float pin-in-hole constraint: Float = maximum hole diameter - minimum pin diameter

where: maximum hole diameter = hole diameter + upper hole tolerance

and minimum pin diameter = pin diameter + lower pin tolerance

The calculated tolerance corresponds to the maximum clearance between the pin and the hole (i.e. the smallest pin in the largest hole). The tolerance calculations are likely to be conservative for most assembly modeling situations.

164


Float Pin-in-Hole Constraint

Special Considerations for Float Pin-in-Hole The float pin-in-hole constraint is not appropriate in all circumstances since the constraint was designed to model an assembly situation where a pin floats within a hole. In particular, the constraint was not designed for modeling and analyzing a fit clearance. For example, consider the following situation:

In the example, the analyzed dimension is the “fit clearance” between the two circles. If the smaller circle is constrained to the larger circle with a float pin-inhole constraint, the automatic tolerance is calculated as follows: Float = 1.5 - 1.0 + 0.1 + 0.1 = 0.7 units


165

Advanced Model Building for Tolerance Analysis When a Tolerance Analysis is performed for the “fit clearance” dimension, the following contributor list is generated:

Summing up the upper and lower contributions and adding them to the nominal radial clearance (fit clearance) produces the following results: Max Fit Clearance = 0.7 units Min Fit Clearance = -0.2 units

As the negative value for Min Fit Clearance indicates, the tolerance for “fit clearance” is not calculated properly since the inner circle will never extend beyond the outer circle. The discrepancy is a result of an incorrect application of the float pin-in-hole constraint to model a fit clearance. The float constraint has an automatic tolerance based on the maximum diametrical clearance. As such, if the constraint is used in an analysis of the clearance itself, the possible motion of the pin relative to the hole is over-estimated. In general, measurements that include either the pin or hole size dimension and the float constraint in the same analysis can produce erroneous results. Note: You can use the float constraint to model a fit clearance if the constraint and dimensions are associated with different dependent assemblies that have the same master.

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Pass-Through and Locked Constraints

Pass-Through and Locked Constraints This section describes how the tolerance effect of equal-to, scaled-from, sum-to and product constraints affect contributor inclusion in the Tolerance Analysis Report. When working with these constraints, it is important to understand the notion of upstream and downstream contributors. Upstream contributors for a derived entity are the items that drive the derived value. In the equation C = A × B , the items A and B are upstream of C, since their values drive C. The downstream items for an entity, on the other hand, are (derived) items whose values are driven by the entity. In the equation, C is downstream of A and B. Using the equal-to and scaled-from constraints can result in a situation where a derived quantity (i.e., a downstream contributor) is listed as a contributor to the analyzed item. In this situation, however, it is sometimes preferable to have the upstream contributors (i.e., the items that drive the derived quantity) listed in the report, rather than the derived quantity. If you do not want the derived item listed as a contributor to the analyzed item, you have the option to set the tolerance effect for equal-to and scaled-from constraints as either locked, or pass-thru. • When the constraint is set to pass-thru, the contributors that drive the derived item are passed-through to the tolerance report. • When the setting is set to locked, the derived item is listed as a contributor to the analyzed dimension. For information about setting the tolerance effect, see “Setting the Tolerance Effect,” page 168. For a detailed example of how the equal-to and scaled-from constraints can result in a derived dimension being a contributor to the analyzed dimension, see “EqualTo and Scaled-From Constraints,” page 169. For information about when to use a locked or pass-thru tolerance effect for the equal-to and scaled-from constraints, see “Choosing the Tolerance Effect,” page 168. For the product, sum-to and sum-to-zero constraints, the contributors that drive the derived item are automatically passed through to the report. For more information about how these constraints affect contributors to the tolerance report, see “Product, Sum to and Sum to Zero Constraints,” page 173.


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Setting the Tolerance Effect For the equal-to and scaled-from constraints, you can set the tolerance effect for the contributor to be either locked, or pass-thru. To set the tolerance effect: 1.

Select an equal-to or scaled-from constraint.

2.

From the Edit menu, chose the Options command. The Constraint Options Window is displayed.

3.

Choose Pass-Thru for Tolerance Effect.

4.

Click OK.

Choosing the Tolerance Effect The locked option should be chosen when the features constrained by the equalto or scaled-from constraint are the same size but independent from each other. For example, an equal-length-dimension constraint is applied as a convenience to enable multiple dimensions to be changed simultaneously. Typically this is done for multiple occurrences of a similar feature, such as multiple holes in a hole pattern, or multiple fillets. The locked status of the constraint allows each of the constrained features to be treated as an independent feature for tolerance analysis purposes. A pass-thru tolerance effect is appropriate when using the equal-to or scaled-from constraints in connection with functional modeling. For example, you should choose the pass-thru status when creating equal and opposite forces. In most cases, if a dimension is constrained to be equal to a derived dimension, the status of the constraint should be set to pass-thru. This is because equal dimension constraints that are applied to derived dimensions are nearly always done in connection with functional modeling. In these cases, the pass-thru status enables the contributors to the derived dimension to appear in the tolerance analysis. A pass-thru tolerance effect is also appropriate when the features at each end of the constraint are not independent from each other. In this case, the equal length constraint is not so much a convenience, but rather, it’s a modeling necessity. 168



Equal-To and Scaled-From Constraints When applied to dimensions or lines, the equal-to and scaled-from constraints are given a locked status by default. When these constraints are used to constrain dimensions or lines to derived dimensions, the locked status of the constraint precludes the identification of upstream contributors for Tolerance Analysis. Note: When applied to forces and moments the constraint is pass-thru by default.

The following example demonstrates the impact of locked dimensions on the Tolerance Analysis Report and provides instructions to include upstream dimensions and constraints using the pass-thru option. There are three derived dimensions in the following example: C, D, and F. • Dimensions A and B drive the value of dimension C. • Dimension D is constrained Dimension Equal To dimension C. Therefore, dimensions A and B drive the value of dimension D. • Dimensions E and D drive the value of dimension F. Since dimensions A and B drive dimension D, then it is also true that dimensions A, B and E drive the value of dimension F.


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Advanced Model Building for Tolerance Analysis When dimension F is analyzed, the system lists dimensions D and E as contributors but does not list dimensions A or B, even though A and B drive the value of D. The Dimension Equal To constraint treats D as a locked dimension and thus prevents the system from reading dimensions A and B as upstream contributors.

Contributors that are upstream of the Dimension Equal To constraint are included in the Tolerance Analysis when you override the ‘locked’ setting of the equal dimension constraint.

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Pass-Through and Locked Constraints Use the pass-thru option to override the default ‘locked’ setting of the Dimension Equal To constraint. 1.

Select and highlight the constraint.

2.

From the Edit menu, choose the Options command.

3.

Choose Pass-Thru for Tolerance Effect and click OK.

When treated as pass-thru constraints, the analysis identifies contributors that are upstream of the Dimension Equal To constraint which drive the analyzed dimension.


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When dimension D is driven by a pass-thru constraint, Tolerance Analysis can identify dimensions A and B as upstream contributors that drive the analyzed dimension. As a result, dimension D is no longer listed as a contributor.

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Product, Sum to and Sum to Zero Constraints For the product, sum-to and sum-to-zero constraints, the tolerance effect is always passed through. The tolerance contributors will vary depending on whether or not the upstream contributors are locked or derived values. Consider the following example of a product constraint: C = A×B

The value for C is derived due to the product constraint. When a tolerance analysis is performed on C, the following factors influence the contributor list: • If A or B is a locked value, then the locked value is listed as a contributor in the report. • If A or B is a derived value, then the contributors to A or B are automatically listed as contributors to C in the report.


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Using Dependent Assemblies for Tolerance Analysis The following sections detail the advantages of using dependent assemblies for Tolerance Analysis, and describes the common scenario of duplicate contributors that occurs when modeling with dependent assemblies.

Advantages of Using Dependent Assemblies Dependent assemblies are recommended for Tolerance Analysis since they produce the most reliable and accurate results. Master and Dependent assemblies must be located in the same note for Tolerance Analysis to provide accurate results. Part management. Component definition and management is improved by using a master assembly for each part. It’s an efficient way to organize the work.

Industry practice is to create separate drawings for each part, and to then create assembly drawings that reference the individual parts. With MA you can create master assemblies for each component part. All the master assemblies are required to be in the sketch note in which the components are assembled for tolerance analysis. Model stability. Model stability and performance is greatly enhanced by using dependent assemblies. MA creates systems of equations to model and analyze geometric problems. These sets of equations can become quite complex and timeconsuming to solve. By using dependent assemblies, MA is able to break the equation set into more easily solved subsets. Easier to build complex assemblies of parts. Dependent assemblies act like rigid bodies since they always have exactly 3 degrees of freedom. This makes it much easier to build complex assemblies of parts.

Using dependent assemblies also makes it easier to modify the assembly scenario. Multiple assembly scenarios may be explored quickly and easily when dependent assemblies are used. Update parts. Master assemblies may be modified without changing the assembly of parts. This makes it much easier to make changes to individual parts. Reduce unexpected results. Certain constraint schemes that are valid and useful for individual parts (master assemblies) will encounter problems in the assembly of parts. For example, gridded dimensions are useful in master assemblies, but may cause unexpected results in kinematic assemblies of parts.

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Using Dependent Assemblies for Tolerance Analysis

Combine Duplicate Contributors When modeling with dependent assemblies, it is common for a single piece of geometry to show up two or more times in the Tolerance Analysis Report. The following example demonstrates how a single contributor shows up multiple times in the Tolerance Analysis Report when you want it to only show up once. The sketch model is designed to simulate the vertical movement of an Arm Assembly as it pivots between a 30 ° and 80 ° angle. Even though the sketch will simulate the movement of one assembly, a proper dimensioning scheme uses two dependent assemblies to gather tolerance information at both the 30 ° and 80 ° angle.


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Advanced Model Building for Tolerance Analysis In the example, the two dependent assemblies gather tolerance information at both the 30 ° and 80 ° angles and analyze the movement of the model between the angles. For more information about constructing assemblies in Sketch Note, see the Mechanical Advantage Sketch Note Manual.

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Using Dependent Assemblies for Tolerance Analysis The dependent assemblies provide tolerance information for the analyzed dimension when the Arm is at both a 30 ° and 80 ° angle. As a result, the same piece of geometry is listed twice in the report and the sensitivity and percent contribution varies depending on the angular alignment. Duplicate contributors are easily identified from the Tolerance Analysis Report because they are grouped together as a result of a shared name.

Proper dimensioning enables Tolerance Analysis to gather the appropriate tolerance information for the moving part. However, because the duplicate contributors do not accurately reflect that the model simulates a single moving part the contributors should be combined in the Tolerance Analysis Report.


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Advanced Model Building for Tolerance Analysis When the contributors are combined, the range of values for sensitivity and percent contribution are calculated and displayed. For information about how to combine contributors, see “Combining Duplicate Contributors,” page 62.

Note: Combining the duplicate contributors changes the RSS Tolerance Analysis results for sensitivity. This is because, when compounding variance, the sensitivity of combined contributors are squared after summing the sensitivities, rather than squaring the constituent sensitivities and then summing their squares. For more information about the calculation of variance, see “ Mean and Standard Deviation,” page A-3.

For contributors with one degree of freedom, the combined sensitivity is the sum of the individual sensitivities: f

c

= f

where f

1

c

+f

2

is the combined sensitivity.

For concentricity and position contributors, the combined sensitivity is calculated by a vector sum, which accounts for the directions of maximum sensitivity: c = f 1 + f 2

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Using Dependent Assemblies for Tolerance Analysis

When not to Combine Duplicate Contributors The following example demonstrates how a single contributor can accurately show up multiple times in the Tolerance Analysis Report. The following assembly models a camera aperture that is dimensioned using several dependent camera leaf assemblies.

Each dependent assembly represents a physical part of the camera aperture.


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Advanced Model Building for Tolerance Analysis When a Tolerance Analysis is performed on the derived dimension, the report lists several duplicate contributors. When dependent assemblies are used to dimension this type of model, contributors should not be combined.

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Model Building with Projection Lines

Model Building with Projection Lines Projection lines that are properly dimensioned are a powerful modeling tool for 3D Tolerance Analysis. Projection lines constrain the orientation of different views in your sketch model so that the dimensions in one view drive the corresponding features in another view. Using projection lines ensures that each view is incorporated into the Tolerance Analysis. The unintended side effects of using projection lines in an improperly designed model are sensitivity multiplication and contributor duplication. The following examples describe proper model building techniques for projection lines.

Side View Orientation


Top View Orientation

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Model Building to Avoid Sensitivity Multiplication When using projection lines, proper design is required to prevent an increase in the ratio of contributor sensitivity. An increase in sensitivity reduces the ability of the design to withstand variation and increases the need to apply tolerances to control variation. Use the following model building guidelines to avoid sensitivity multiplication in your design.

Parallel Projection Line Orientation The orientation for parallel projection lines MUST be controlled by a common datum. When working with projection lines that are parallel, you have two constraint scheme options: • Chain projection lines back to a common datum. The orientation of the first projection line can be constrained to a piece of geometry, and then all subsequent parallel projection lines can be constrained to the first projection line. For more information, see“Chained Constraints,” page 145. • Always use the same datum to constrain projection line orientation. In the following example, the orientation for each projection line is constrained parallel to the center line, which is the common datum for the entire sketch. Projection Lines

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Model Building with Projection Lines

Constraining Projection Lines Use either perpendicular and parallel or line-at-angle orientation constraints to constrain the orientation of a projection line. You should never mix line-at-angle constraints with parallel and perpendicular constraints to constrain projection line orientation since this will cause the projection lines to effectively have different datums. • Parallel and perpendicular constraints act as line-at-angle constraints when they are used to constrain the orientation of projection lines. • Constraining orientation with parallel and/or perpendicular constraints will give your model more flexibility, whereas using all line-at-angle constraints may reduce the flexibility of your model. For example, if you change the orientation of a feature in the sketch model, and you want the orientation of the projection lines to update in response to that change, a line-at-angle constraint will prevent the orientation of the projection line from updating. You should not use tangent or point-on-object constraints to constrain the orientation of projection lines. Tangent or point-on-object constraints cause the projection line to have a different datum than the other projection lines in the sketch. Tangent or point-on-object may be used to constrain the starting location, but never the orientation.


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Model Building to Avoid Duplicate Contributors When you use orientation constraints to control the orientation of a surface, the corresponding surfaces in the auxiliary views should have their orientation controlled by the projection line that connects the surfaces. In other words, you should not use orientation constraints in auxiliary views to control surfaces whose orientation is constrained in the primary view. For example, consider the top surface of a block. In the front view the orientation may be controlled by a parallel constraint to the bottom of the block. Top Surface of Front View

A

B

Front View

Right Side View

A

Projection line emanating from top surface of the front view.

B

Projection line emanating from the bottom of the front view.

In the right side view, the projection lines are used to control the height of the block. It is important that the points at the bottom of the right side view are constrained point-on-object to the projection line emanating from the bottom of the front view. Note: Failure to observe this model building technique can result in extra orientation contributors and sensitivity magnification.

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Chapter 10

Critical Parameter Analysis This chapter describes concepts and provides step-by-step instructions for performing a Critical Parameter Analysis. Critical Parameter Analysis (CPA) is a powerful extension of Mechanical Advantage Tolerance Analysis capabilities. CPA applies the power of Tolerance Analysis to non-geometric parameters, enabling users to rapidly analyze and gain insight to the factors affecting performance and reliability in their designs.

Performance Model Parameters Critical Parameter Analysis extends Tolerance Analysis capabilities beyond geometric factors to analyze performance model parameters. Performance models are mathematical representations of design function, consisting of user written equations, geometrically derived equations, and empirical data such as physical properties. Examples of performance parameters include response time, torque, current, force, stress, weight, velocity, inertia and temperature. Critical Parameter Analysis calculates the sensitivity and percent contribution of contributors that cause variation in performance parameters. Taken together, the sensitivity and percent contribution provide valuable information about a design. A Critical Parameter Analysis is always performed from a sketch note, on derived dimensions, variables, forces, or moments that are driven by parameters within the sketch note. CPA considers inter-note propagation in the analysis of parameters. When variables are linked to Math, DCAP, Program, Solid, and other Sketch Notes, they are included in the analysis and recognized as contributors if they affect the parameter being analyzed.


185

Critical Parameter Analysis

Special Considerations for Critical Parameter Analysis Point-to-Point Dimensions Point-to-point dimensions contain the mathematical notion of absolute value. As a result, when a negative tolerance is applied to a point-to-point dimension at or near zero, it has the same effect as applying a positive tolerance. Because this would produce erroneous results, CPA prevents you from running an analysis that includes a point-to-point dimension at or near zero. Tolerance Analysis does not have this problem because it focuses on a manufacturing oriented analysis rather than a functional analysis. This allows different mathematical methods to be used which handles these dimensions properly. When using Critical Parameter Analysis, all point-to-point dimensions at or near zero should be replaced with point-to-line, gridded, or line-to-line dimensions.

Geometric Contributors CPA includes profile and concentricity contributors, such as position and float pin-in-hole constraints, in the analysis for both dimensions and variables. As in Tolerance Analysis, the sensitivity for a concentricity contributor is in the direction of maximum impact. The distribution assumptions for the concentricity contributor are the same as in Tolerance Analysis. The process of maximizing concentricity sensitivities increases the computational requirements and some additional time should be expected when conducting a CPA. Important Consideration. If multiple copies of the same dependent assembly are used in an analysis and contain concentricity contributors, the maximization process finds the single direction for all assemblies that maximizes the analyzed object. As such, no duplicate contributors are reported. This differs from Tolerance Analysis, which independently maximizes each concentricity.

Variable Contributors Since CPA originates from sketch note, locked variables are only included as contributors to CPA when they are linked out from the originating sketch note to other notes, such as a math note. For example, a locked variable in a math note is only included as a contributor to CPA when it is linked out from the originating sketch note to the math note. For more information on variables, see the Mechanical Advantage Sketch Note Manual .

186


Performing Critical Parameter Analysis

Performing Critical Parameter Analysis These steps outline the general process for performing a Critical Parameter Analysis. See “Example 1: Critical Parameter Analysis,” page 191 for further illustration. 1.

From the File menu, choose the Set Update Scope command. Because CPA considers inter-note propagation when performing the analysis, the Update Scope must be set to everything and immediately . Click OK to confirm and close the option window.

2.

Tolerance ranges may be specified for contributors and for the analyzed object. From the Analysis menu, choose the Set Default Tolerances... command. Or, to set tolerances for specific items, choose Options from the Edit menu.

3.

In order for potential contributors to be recognized, they must be connected to the sketch note from which the analysis is being performed. Create the variables in the sketch note, and then link out to all corresponding variables in other notes that are potential contributors.

4.

Select the dimension, variable, force or moment to be analyzed. Any derived object that is driven by a parameter in another note may be selected for analysis. If the variable you wish to analyze is not linked in to the sketch note, create the variable and then link in to it from the corresponding parameter in the other note.

5.

From the Analysis menu, choose the Analyze Parameters command.

6.

If there are unopened notes that are directly or indirectly connected to the sketch note in which the analysis is being performed, the system opens the notes automatically. In this case, you must re-choose the Analyze Parameters command after all connected notes have been opened. If CPA contributors already exist for the object being analyzed, you will be asked if you want to reuse the previous CPA results. Select which results, if any, you would like to reuse and choose OK to begin the analysis. For more information about reusing CPA results, see “Saving CPA Contributor Results,” page 188.

7.

CPA results are reported in the same manner as a Tolerance Analysis Report. Similarly, the CPA results can be further analyzed using Monte Carlo analysis.

8.

To reduce the computational time required to perform additional analysis on the analyzed object, you can save the CPA contributor information. From the Viewing menu of the CPA report, choose Set Up Window. To save the results, select Yes in the Save CPA Contributors toggle.


187


Saving CPA Contributor Results When a CPA is first performed, the Mechanical Advantage conducts a computationally intensive analysis to identify which parameters have non-zero sensitivities. These parameters and their sensitivity values are listed as contributors in the CPA report. You can save the contributor list and sensitivity information to reduce the time required to perform a reanalysis of the same dimension or variable. To save CPA results: 1.

Perform a CPA on a dimension, variable, force or moment.

2.

From the Viewing menu in Report window, choose Set Up Window. The Tolerance Window Options window is displayed.

3.

To save the contributor information, select Yes in the Save CPA Contributors toggle.

The following considerations should be reviewed before using the Save CPA Contributors option: • If you already have saved CPA results and you set the control to No, any saved results on the currently analyzed dimension are deleted. • If you have multiple dimensions with saved CPA results, then selecting No deletes any saved results for the currently analyzed dimension, but does not delete the CPA results for the other dimensions. • The Save CPA Results toggle retains its setting for future analyses. Therefore, if the toggle was set to No in a previous analysis, then a new analysis also has this setting. If the new analysis is performed on a dimension that already has saved CPA results, then these results will not be updated. • To update saved CPA results, set the toggle to Yes before you perform a new analysis.

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Saving CPA Contributor Results Using saved contributor information: 1.

If the Analyze Parameters or Reanalyze command is used on an object that already has saved results, the CPA Calculation Control window is displayed:

The CPA Calculation Control window lets you decide if the previously saved contributor information should be reused. Choose to reuse: • Nothing. This option will re-perform the complete analysis, including determining the possible contributors. • Contributor List. Only re-uses the contributor list. By selecting this option, you are prompted to specify if the analysis should re-use the existing directions for the position constraints. • Contributor List & Sensitivities. Re-uses the contributor list and saved sensitivities. 2.

If a contributor was deleted since the last analysis, an alert is displayed to warn you that the results may no longer be valid.

3.

The analysis is performed based on the options you selected. If saved information is used, the “Contributor Information” will be called “Saved Contributor List and Sensitivities” or “Saved Contributor List.”

Note: If uncertainty exists as to whether model changes will impact the contributors to the CPA, it is advisable to not reuse saved information.


189


Specifying Contributor Inclusion Critical Parameter Analysis includes all parameters with non-zero sensitivities in the CPA report. During a reanalysis, you can exclude the effect of a particular contributor from inclusion in the report. Since the sensitivity and contribution information for the excluded contributor are not calculated, this information can not be saved with other CPA results information.

Excluding Contributors The Exclude Contributor command removes the effect of the contributor from the CPA results. 1.

Select and highlight a contributor row in the CPA report.

2.

From the Viewing menuof the CPA results window, choose the Exclude Contributor command. The contributor row is changed to a light blue color. The contribution for the contributor is listed as N/A and the contributor will be excluded from future analyses.

You can remove the exclude contributor setting at any time: 1.

Select and highlight the excluded contributor.

2.

From the Viewing menu of the CPAresults window, choose the Include Contributor command. The Include Contributor command removes the exclusion setting and the contributions are included in the analysis results. A reanalysis occurs to update the sensitivities.

Tolerance Distribution Graph The Tolerance Distribution Graph helps you visualize the tolerance distribution of the analyzed dimension or variable. To view the graph: • From the Viewing menu of the CPA Report window, choose the Plot Distribution Graph command. For more information about the Tolerance Distribution Graph, see “Tolerance Distribution Graph,” page 60.

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Example 1: Critical Parameter Analysis

Example 1: Critical Parameter Analysis The following example provides detailed instructions for performing a Critical Parameter Analysis on parameters that affect the performance of a trapezoidal beam.

Analyzing Performance Parameters The geometry is created in a sketch note and the preliminary steps of setting the Update Scope and Default Tolerances have already been taken. The dimensions and physical properties of the sketch note are linked out to equations in the math note. The first parameter affecting the performance of the trapezoidal beam that we want to analyze is Stress, defined in the math note by the equation Stress = M*C/Ix. To perform an analysis on this parameter, we must first link it into the sketch note. 1.

In the sketch note, use the equation tool to create a corresponding variable, Stress. Click on a spot in the sketch note where the text is to be displayed, and type in the variable name: Stress.


191

Critical Parameter Analysis 2.

In the math note, select the variable Stress so that only Stress is highlighted. From the Connections menu choose the Link command and click on the corresponding variable in the sketch note.

Note: When you select Stress from the math note, double click and highlight Stress and be sure the entire equation is NOT selected.

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Example 1: Critical Parameter Analysis Potential contributors affecting Stress include Ix, Force, and Length. Ix is the moment of inertia about the X axis, a geometrically derived variable which is automatically included in the analysis. Force and Length, however, are non-geometric parameters; in order for them to be included as potential contributors in the analysis, they must be connected with the sketch note. 3.

In the sketch note, use the equation tool to create the corresponding variables, Force and Length. Be sure to specify the values already assigned in the math note.


193


194

4.

From the sketch note, select and highlight the variable Force by clicking on it.

5.

From the sketch note Connections menu, choose the Link command and click on the corresponding variable in the math note. This step is repeated to link the Length variable out from the sketch note to the math note.


Example 1: Critical Parameter Analysis 6.

From the sketch note, select and highlight dimension C. From the Connections menu in the sketch note, choose the Link command and link the dimension to the corresponding variable in the math note. Dimension C is the distance from the base of the beam to the centroid.

Dimension C


195

Critical Parameter Analysis The variable Ix is the moment of inertia about the X axis. Ix is a geometrically derived variable which is automatically included in the analysis. The next step is to Link Ix from the sketch note into the math note. 7.

From the sketch note, select and highlight the entire assembly by clicking on the assembly symbol.

8.

From the Group menu in the sketch note, choose the Show Physical Properties command. The Physical Properties window is displayed. From the Physical Properties window, select the row for Ix properties.

9.

From the Connections menu in the Physical Properties window, choose the Link command and link the property Ix to the corresponding variable in the math note.

Now you are ready to perform the Critical Parameter Analysis.

196


Example 1: Critical Parameter Analysis 10. In the sketch note, select the Stress parameter. From the Analysis menu, choose the Analyze Parameters command. In a matter of seconds, the system displays the results of the analysis.

The dimensional contributors and variables are automatically identified and displayed. The sensitivity and percent contribution are shown for each contributor. The probability of Stress being in tolerance is also calculated and displayed. A linear or non-linear Monte Carlo analysis can also be performed on the CPA results. The Critical Parameter Analysis Report displays information in the same format as the Tolerance Analysis Report. For more information on reading the report, see “Analyzed Dimension, Force or Moment,” page 76.


197


3D Property Analysis Critical Parameter Analysis can also be used to analyze the properties of a three dimensional assembly. Using Mechanical Advantage Solid Note, which is a three dimensional solid modeling tool, a Critical Parameter Analysis is performed on the surface area of the trapezoidal beam assembly. The assembly must first be linked from the sketch note into the solid note.

198

1.

From the sketch note, select and highlight the trapezoidal beam assembly by clicking on the assembly symbol.

2.

From the Connections menu in the sketch note, choose the Link command and link the assembly into the solid note by clicking in the solid note window. The assembly is displayed in solid note.

3.

From the Model menu in solid note, choose the Extrude command. Enter the value for length, Length = 10, which defines the depth of the assembly for solid note.


Example 1: Critical Parameter Analysis The solid note assembly is displayed:

4.

From the Viewing menu in solid note, choose the Open Variable Window command. The Solid Note Variable window is displayed.

5.

Select the Length variable in the sketch note. From the Connections menu, choose the Link command and link the variable Length from the sketch note out to the variable Length in the Solid Note Variable window.


199

Critical Parameter Analysis The next step is to create the corresponding Surface Area variable in the sketch note and link it from the solid note. 6.

200

From the sketch note, use the equation tool to create the variable Surface Area.


Example 1: Critical Parameter Analysis 7.

Sele Selectthebea ctthebeam m inthesoli inthesolid d noteby noteby clic clicki kingon ngon it. it. FromtheMo FromtheMode dell menu menu,, Show Physic Physical al Proper Propertie tiess comman choose choose the Show command. d. The Physic Physical al PropProperties window is displayed.

8.

From From the Physic Physical al Prop Propert erties ies window window,, selec selectt the Surface Area row row by clic clicki king ng on it. it. With With the the row row stil stilll sele select cted ed,, clic click k the the Link button, and then click on the Surface Area variable in the sketch note.


201


9.

202

Perf Perfor orm m the the anal analys ysis is by sele select ctin ing g the the targ target eted ed para parame mete ter— r—Surface Area—in the sketch note and from the Analysis menu choose the Analyze Parameters command.


Example 1: Critical Parameter Analysis

Critica Criticall Parame Parameter ter Analys Analysis is displa displays ys inform informatio ation n in the same same format format as Tolera Tolerance nce Anal An alys ysis is.. For For more more info inform rmati ation on on read readin ing g the the repo report rt,, see see “Anal “Analyzed yzed Dimen Dimension, sion, Force or Moment,” page 76. 76. Crit Critica icall Para Parame meter ter An Anal alys ysis is can can also also be used used to perf perfor orm m othe otherr 3D Tole Tolera ranc ncee An Analalyses yses.. 3D dime dimens nsio ions ns can can be anal analyz yzed ed by usin using g prof profil iles es link linked ed from from a sket sketch ch note note to crea create te soli solids ds in a soli solid d note note.. Resu Result ltss from from a 3D refe refere renc ncee dime dimens nsio ion n (mea (measu sure re)) can then be linked back into a sketch note variable and analyzed. In some cases, it may be necessary to create other links to drive solid note parameters that may be contributors.


203


Example 2: Testing Deflection The The foll follow owin ing g is a conc concep eptu tual al exam exampl plee that that demo demons nstr trat ates es the the abil ability ity of CPA CPA to test test the rate of deflection when force is applied to a pipe.

The above sketch is a detail view of a pipe. The sketch visually represents what happens to the pipe when force is applied at a specific point. In the sketch, the force vector displays the maximum amount of force that is applied to a specific point on the pipe.

204


Example 2: Testing Deflection The following variables have been created in the sketch: Variable sy symbol

Definition

Defl

The amount of deflection when force is applied to the pipe.

E

Elasticity of steel.

Force

Amount of force (in lbs.).

Iy

The moment of inertia.

Area

The area of the pipe.

The variables are linked to a math note and used in the following equations.


205

Critical Parameter Analysis In this example, Critical Parameter Analysis analyzes the design to determine the rate of deflection. The design limits for the pipe state the pipe cannot deflect more or less than .879 inches with a tolerance range of +/- .125.

The Critical Parameter Analysis displays information in the same format as Tolerance Analysis. For more information on reading the report, see “Analyzed Dimension, Force or Moment,” page 76.

206


Appendix A

Computing Tolerance Analysis with RSS Probability Calculations This appendix describes the RSS calculations that are used to derive Tolerance Analysis results.

Derived Dimensions, Forces, and Moments In Mechanical Advantage, a derived object (dimension, force or moment) is one where the constraints completely determine the value of the object. While the system does not actually build a representation of this function, a derived object A can be expressed as a function of all the constraints and locked dimensions, forces, moments and variables in the design: A = f (v , v , ..., v ) 1 2 n where each argument of this function is a potential contributor. For Tolerance Analysis, this function is linearized: dA = f dv + f dv + ... + f dv + HOT 1 1 2 2 n n where dA is the change in the derived object caused by changes dv in each varii able v , f is the partial derivative of the function f with respect to the variable v , i i i and HOT are higher-order terms which are ignored. (There are situations where these higher-order terms should not be ignored because they have a significant effect on the derived object A.)


A-1

Computing Tolerance Analysis with RSS Probability Calculations

Sensitivity Coefficients The first and most time-consuming step in Tolerance Analysis is the computation of the partial derivatives f , f , ..., and f . The numeric values for these partial 1 2 n derivatives are called sensitivity coefficients. Other names commonly used are influence coefficients, marginal costs, and lever arm ratios. Essentially, the number f is the ratio of a small change in the analyzed dimension A due to a small i change in the variable v , f :=: delta (A) / delta (v ). i i i Once the sensitivity coefficients have been computed, the remaining Tolerance Analysis computations are simple and fast. They all rely on the fact that those non-linear higher-order terms are ignored, and that the analyzed dimension is essentially a linear combination of the contributors: dA = f dv + f dv + ... + f dv 1 1 2 2 n n Once the sensitivity coefficients have been computed, the Tolerance Analysis program: 1. 2.

Computes the mean µ and standard deviation σ for each of the contribi i utors v using its tolerance specification. i Computes the mean µ and standard deviation σ for the analyzed A A object using the linearized formulas.

3.

Calculates the tolerance and the value of the analyzed object at plus and minus various standard deviation multiples.

4.

Computes as a percent the amount that each variable contributes to the variation in A, then sorts the variables on this contribution.

5.

Estimates the probability that A will satisfy the tolerance limits you placed on it through the object Option window or by setting the default tolerance for that dimension type.

6.

Computes the upper and lower worst-case values for dA by moving each dv to its upper or lower tolerance limit. i

The following paragraphs explain how these calculations are performed.

A-2


Mean and Standard Deviation

Mean and Standard Deviation The Mechanical Advantage uses the tolerance specification for each variable to determine the mean µ and standard deviation σ for that variable. i i

Dimension, Force, Moment and Variable Contributors The following formula calculates the mean ( µ i ) for a dimension, force, moment or variable contributor with a normal distribution:

µi

Ui + Li = ----------------2

where U i is the upper tolerance value and L i is the lower tolerance value. The following formula calculates the standard deviation (σ i ) for a dimension, force, moment or variable contributor with a normal distribution:

σi

Ui – Li = ----------------2×R

where U i is the upper tolerance value and L i is the lower tolerance value and R is the Sigma range (3,4,5,6).

Contributors with Tolerance Zones Contributors with a tolerance zone (T), have a single tolerance value (e.g., parallel, line at angle) and a sigma range (R). With tolerance zones, it is assumed that µ i = 0.0 The following formula calculates the standard deviation for all tolerance zone contributors, except profile tolerances, with a normal distribution:

σi

T = ---R

For profile tolerances, the following formula calculates the standard deviation for a profile tolerance zone contributor with a normal distribution:

σi

T = ------------2×R

Note: Contributors that constrain two degrees of freedom (fixed points, concentric circles, collinears) are treated slightly differently. For information about contributors with two degrees of freedom, see “Concentric Constraints,” page A-21.


A-3


Analyzed Object The calculation of the mean µ

and standard deviation σ relies on a theorem A A from basic statistics and the use of the linearized formula for dA. The theorem states that, if each dv is a random variable with mean µ , then dA is a random i i variable with mean µ : A

µA = f 1µ1 + f 2µ2 + ... + f nµn 2 The theorem goes on to state that if each dv has a variance σ , then dA has i i 2 variance σ : A

σA2 = f 12 σ12 + f 22 σ22 + ... + f n2σn2 In each case, the variance is the square of the standard deviation. These two formulas relate the mean and standard deviation of each contributor v to the mean i and standard deviation of the analyzed object A. The contribution of each dv is the fraction that it contributes to the variance of i dA: 2 2 2 contribution (dv ) = 100 f σ / σ i i i A This formula represents the percentage impact of each contributor on the variation in the analyzed object. A variable with a large contribution will have the greatest impact on the probability that the analyzed object will be within its tolerance (see “Probability in Tolerance,” below). The sum of all the contributions is 100%. Notice that this contribution is the ratio of the squares of the standard deviations. If dv has a small contribution such that doubling its tolerance range would not i make a large change to the variance of the analyzed object, this doubling would 2 double the standard deviation σ but quadruple (2 = 4 times) the value of the varii 2 ance σ , and hence would approximately quadruple the contribution of v . Or, to i i take another example, suppose you reduce the tolerance on a contributor from .005 in. to .003 in. Although the new tolerance has the ratio of .003/.005 = .6 to 2 the old, the change in the contribution is .6 , or 36%. This means that you can shrink a large contributor by less than you might think in order to increase to an acceptable level the probability of achieving your specified tolerance. A reasonable design goal is to have several of the largest contributors have approximately the same contribution, but because of this second-order relationship, do not try to get the contributions exactly equal. Other design considerations can have far greater significance than having equal contributions. A-4


Probability in Tolerance

Probability in Tolerance Mechanical Advantage uses the central limit theorem of statistics to compute the probability in tolerance. This theorem states that as the number of variables dv i increases, the analyzed dimension dA approaches a Gaussian (or normal) distribution. Given that dA has (approximately) a Gaussian distribution with mean and standard deviation, then the probability that dA is within its tolerance specification is the area under the Gaussian bell-shaped curve:

1 Prob = -------------σA π 2

UA

∫

LA

e

1 x – µA – --- --------------σ 2

2

dx

where the limits of the integration, L and U , are the lower and upper tolerance A A specification, respectively, for the analyzed object A, µ is the computed mean, A and σ is the computed standard deviation. A ±1

σ Area = probability in tolerance

lower

d

upper


A-5


Worst-Case Values The upper and lower worst-case values are computed by respectively maximizing and minimizing the value of dA using the linearized formula: dA = f dv + f dv + ... + f dv 1 1 2 2 n n where each of the variables dv , dv , ..., dv is constrained by its tolerance spec1 2 n ification. (For the meaning of the variables associated with dimensions and constraints, see “Contributors and their Variables,” page A-7.) These are two simple optimization problems, one for the upper worst case and one for the lower worst case, whose solutions can be quickly computed. For one-dimensional contributors that have no relationship with any other contributor, the tolerance specification becomes the inequality L ≤ dv ≤ U i i i and the only contribution to the analyzed object is the term contribution value = f dv i i When f is positive, this contribution reaches its maximum when dv is at its upper i i limit and its minimum when dv is at its lower limit. Conversely, when f is negai i tive, this contribution reaches its maximum when dv is at its lower limit and its i minimum when dv is at its upper limit. When f is zero, this contributor has no i i effect on the linearized formula for the analyzed object. For each contributor, Mechanical Advantage computes these worst-case contributions and displays them in the columns labeled “Upper Contribution” and “Lower Contribution” in the Tolerance Analysis Report. Note: Contributors that constrain two degrees of freedom (concentric circles, position, float pin-in-hole, fixed points, collinears) are treated slightly differently. For information about contributors with two degrees of freedom, see “Concentric Constraints,” page A-21.

This simple formula for computing the worst-case results fails when there are relationships between contributors. For example, ASME Y14.5M-1994 carefully describes how a linear dimension and a parallelism specification should interact. Because the Mechanical Advantage database does not record which contributors interact with which other contributors, this system is unable to handle these interactions properly. For this reason contributors marked with a “+” in the “Type” column of the results window are included in the worst-case analysis. The contributors with no “+” in the “Type” column are not included in the worst-case analysis, although the terms that would have been added to the worst-case values are displayed.

A-6


Contributors and their Variables

Contributors and their Variables Each contributor to an analyzed object has a variable associated with it. When the contributor is a dimension, force, moment, or equation, this variable is reasonably obvious, but you need to understand what the variable associated with each constraint means. The analyzed object itself is one of these variables. You can replace the variable dv with the variable dA in any of the following definitions i of dv to get the formula for the analyzed object. The following discusses the net i effect of each variable, not necessarily exactly how the system handles it. Dimensions are discussed first, followed by constraints.

Linear dimensions The Mechanical Advantage has several different linear dimensions, all of which are represented in the data base by the formula:

dim =

2

( P 1x − P 2x ) + ( P 1y − P 2y )

2

where the first point, (P

, P ), is a certain point on the first object and the 1x 1y second point, (P , P ), is a certain point on the second object. These two 2x 2y objects are the two geometric objects that appear in highlight color when the dimension is selected. Since the tolerance specification is how much this formula can vary from the ideal value of zero, the variable is the difference between this th formula and zero. If this dimension is the i variable v , then: i

dv i= dim ñ

2

( P 1x − P 2x ) + ( P 1y − P 2y )

2

Unfortunately, the exact choice of the points on each object that are used for the dimension can have an effect on the sensitivity coefficients of other constraints, so this choice is given here for each type of dimension.


A-7


Point-to-Point Dimension If the two objects are both points, then the Euclidean distance formula is used.

dim P2 (P 2x , P2y )

P1 (P1x , P1y )

dim =

A-8

(P1x - P2x ) 2 + (P1y - P2y )2



Point-to-Line Dimension When one object is a point and the other is a line, then the distance is between the given point and the point on the extended (infinite) line that is closest to the given point. Here, (P , P ) is the given point and (P , P ) is the point on the line 1x 1y 2x 2y that is closest to the given point, and the line from P1 to P2 is perpendicular to the given line. When you draw a line, Mechanical Advantage really creates a line of infinite length projected from the end points you specify. So, the point on the line closest to the point to which you are drawing the dimension may not be on the line segment, but will be on the infinite line that passes through the ends of that line segment.

P1 (P 1x , P 1y )

dim

P2 (P2x , P2y )

dim =

(P1x - P2x ) 2 + (P1y - P2y ) 2


A-9


Line-to-Line Dimension When both objects are lines, then the line toward which you drew the dimension line when you created the line-to-line dimension is the datum line, and the distance between the two lines is the distance from the midpoint of the non-datum line to the datum line. If you set the parallel constraint before you draw the lineto-line dimension, the datum of the dimension will be the datum already established for the parallel constraint.

P1 (P1x , P1y )

dim

P2 (P 2x , P2y )

dim =

A-10

(P1x - P2x ) 2 + (P1y - P2y )2



Gridded Point-to-Point Dimension A dimension that is point-to-point relative to the grid (one that displays the little grid symbol) is treated as a point-to-line dimension with the imaginary line being perpendicular to the orientation and through one of the points.

P1 (P1x , P1y ) P2 (P2x , P2y )

P 2x – P1x

Radial Dimensions The Mechanical Advantage data base uses the radius of each circle and arc to calculate the formula for a radial dimension. The variable for this dimension is: dv = dim – radius (circle) i Similarly, the variable for a diameter dimension is: dv = dim – 2 radius (circle) i

Fixed-Length Constraints A fixed-length constraint is a point-to-point distance between the end points of the line segment, and is treated as such. Therefore, fixed-length constraints are listed as type “D” in the Tolerance Analysis Report.

Fixed-Radius Constraints A fixed radius is a radial dimension on the circle or arc, and is treated as such. Therefore, fixed-radius constraints are listed as type “D” in the Tolerance Analysis Report.


A-11


Angular Dimensions The angular dimension from a line to horizontal is given by: dim - argument (P

2y

–P ,P –P )=0 1y 2x 1x

where the line segment goes from the point (P ,P ) to the point (P ,P ) and 1x 1y 2x 2y argument is the angle that fixes the direction of a complex number. (See Webster’s Ninth New Collegiate Dictionary , s.v. “argument,” definition 6b.)

P2 (P2x , P2y )

argument (P2y – P1y , P2x – P1x )

P1 (P1x , P1y )

A-12


Contributors and their Variables The angular dimension between a line and some direction other than horizontal (for example, vertical, to the grid), has an offset; hence, its variable is: dv = dim – argument (P – P , P – P ) – offsetAngle i 2y 1y 2x 1x For example, the offset angle for an angular dimension to vertical is 90 degrees (or 270 degrees, depending upon the initial orientation of the line).

P2 (P2x , P2y )

dim

P1 (P1x , P1y )

dim = arctan (P2y - P1y , P2x - P1x ) + 90


A-13

Computing Tolerance Analysis with RSS Probability Calculations The angular dimension between two lines is the difference between their angles with the horizontal: dim

= argument (P – P , P – P ) 2y 1y 2x 1x – argument (P – P , P – P ) 4y 3y 4x 3x

where (P ,P ) and (P ,P ) are the end points of the first line and (P ,P ) 1x 1y 2x 2y 3x 3y and (P ,P ) are the end points of the second line. The variable for such a dimen4x 4y sion is: dv = dim – argument (P – P , P – P ) i 2y 1y 2x 1x + argument (P – P , P – P ) 4y 3y 4x 3x

P4 (P4x , P4y ) P2 (P2x , P2y )

α−β

P3 (P3x , P3y )

P1 (P1x , P1y )

β

α

Notice that the variables are changes to the angle between the two lines. This means that the distance that the end points will move because of this variable depends upon the length of the lines. Thus, longer lines will result in larger sensitivity coefficients for angular dimensions.

A-14


Contributors and their Variables Since all angular dimensions are expressed in degrees, there is a conversion factor applied to their sensitivities that involves the length of the lines and the conversion from radians to degrees. This conversion factor is approximately 57.3 degrees per radian (0.0175 radians per degree). When a linear dimension is being analyzed, the sensitivity coefficients for angular dimensions are multiplied by 0.0175, making for comparatively small coefficients. Conversely, when the system analyzes an angular dimension, it multiplies the sensitivity coefficients on linear dimensions by 57.3, making for comparatively large coefficients. The major impact of this feature is that when the Tolerance Analysis results are sorted by sensitivity the angular dimensions tend to be at the bottom of the list, even when their relative contributions are larger than those of linear dimensions.


A-15


Line-at-Angle Constraints ASME Y14.5M-1994 specifies that an angularity tolerance be different from a tolerance on an angular dimension, and Mechanical Advantage makes exactly the same distinction. According to ASME Y14.5M-1994, the tolerance on a line-atangle constraint is a zone that describes how much the surface represented by this line can differ from an ideal surface at the specified angle. In Mechanical Advantage, a line constrained at angle α has the variable: dv = –sin α (P – P ) + cos α (P – P ) i 2x 1x 2y 1y This is the distance between the two end points of the line measured in the direction perpendicular to the ideal line:

“true” end point dv i

“true” line

ideal line

α “true” start point

Notice the distinction between this variable (dv ) and the change in the angle dα. i If you replace this line-at-angle constraint with an angular dimension α, then you have changed variables from the above dv to dα. The change to the affected i sensitivity coefficients will have the ratio π ⁄ 180 times the length of the line because the linearized formula for dv in terms of dα is: i dv i = d α [ degrees ]

A-16

( π ⁄ 180 ) [ radians ⁄ degree] length ( line )



Parallel Constraints According to ASME Y14.5M-1994, a parallelism tolerance specification has a datum surface. The tolerance specification describes a zone in which the other, non-datum surface represented by this line can differ from an ideal surface that is exactly parallel to the datum. In the Mechanical Advantage, a parallel constraint is treated exactly like a line-at-angle constraint, except that the angle α is determined by the datum line. The variable for a parallel constraint is the distance between the two end points of the non-datum line measured in the direction perpendicular to the datum line.

dvi ideal line “true” line

α

datum


A-17


Perpendicular Constraints According to ASME Y14.5M-1994, a perpendicularity tolerance specification has a datum surface. The tolerance specification describes a zone in which the other, non-datum surface represented by this line can differ from an ideal surface that is exactly perpendicular to the datum. In the Mechanical Advantage, a perpendicularity constraint is treated exactly as a line-at-angle constraint, except that the angle α is determined by the datum line. The variable for a perpendicular constraint is the distance between the two end points of the non-datum line measured in the direction parallel to the datum line.

dv i “true” line

ideal line

datum

α

A-18



Equal-Dimension Constraints The concept of equal dimensions is not covered by ASME Y14.5M-1994. Since there are two dimensions connected by each equal-dimension constraint, and each has its own variable and contributes independently to the analysis, these equaldimension constraints are removed from the analysis. Each of these dimensions has its own sensitivity and contribution to the Tolerance Analysis, and any sensitivities this dimension may have from other contributors will not be propagated through the equality constraint to the other dimension. This is as if you had deleted the equal-dimension constraint and locked the non-datum dimension.

datum

Note: Equal dimension constraints have a pass-thru option to pass the contribution through to the Tolerance Analysis Report. Note: Product and summation constraints behave in manner that is similar to equal dimension constraints that have the pass-thru option set.

Equal-Length Constraints The concept of equal lengths is not covered by ASME Y14.5M-1994. In Mechanical Advantage, each equal-length constraint is represented by two hidden pointto-point dimensions, one for the length of each line, and by an equal-dimension constraint between these two dimensions. One of the lines, the one you selected after choosing the Length Equal To command, is the datum line. The equallength constraint is treated as if the length dimension on the non-datum line were the contributing dimension. This is as if you had deleted the equal-length constraint and replaced it with a locked dimension on the length of the non-datum line. There is no contributor associated with the datum end of an equal-length constraint.


A-19


Equal-Radius Constraints The concept of equal radii is not covered by ASME Y14.5M-1994. In Mechanical Advantage, each equal-radius constraint is represented by two hidden radial dimensions, one for the radius of each arc, and by an equal-dimension constraint between these two radial dimensions. One of the arcs, the arc you selected after choosing the Radius Equal To command, is the datum arc. This equal-radius constraint is treated as if the radial dimension on the non-datum arc were the contributing dimension. This is as if you had deleted the equal-radius constraint and replaced it with a locked dimension on the radius of the non-datum arc. There is no contributor associated with the datum end of an equal radius constraint.

datum

A-20



Concentric Constraints Concentric constraints are the most common of the two-degree-of-freedom constraints. These constraints have two variables instead of one and result in two sensitivity coefficients instead of one. These two variables and two sensitivities are merged into one in the tolerance report. The following paragraphs explain how this merging is done. The variables for a concentric constraint are: dv = C - C x 2x 1x dv = C - C y 2y 1y where the first (datum) circle has center at (C

, C ) and the second (non1x 1y ). In the figure on the next page, the distance

datum) circle has center at (C , C 2x 2y between these two points is labeled “error.” The tolerance specified for a concentric constraint is a true position tolerance; this means that the true center must lie in a circle with its center at the other center’s specified location and its diameter equal to the tolerance specification. If the true position tolerance specification is Tol, then this means that:

dv

2 x +

2

dv y

≤

1 2

Tol

(Note: Tol is a diameter, not a radius.)

Position Constraint Position constraints are handled the same as concentric constraints. For information about concentric constraints, see “Concentric Constraints,” page A-21.


A-21


min.

C2

error dvx

dvy

C1

sensitivity

fx fy

max.

Tol When the system performs the sensitivity analysis, it computes one sensitivity coefficient for each of the two variables associated with a concentric constraint. In the picture above, the vector of the two coefficients is labeled “sensitivity.” These two sensitivity coefficients are merged into one in the tolerance window. The following paragraphs explain this process in more detail. The two terms in the linearized formula for dA are: dA = ... + f dv + f dv + ... x x y y Since the tolerance zone is symmetric, the mean values for these two variables are both zero and contribute nothing to the formula for the mean of the analyzed object.

A-22


Contributors and their Variables Since the tolerance zone is circular, the standard deviations (hence also the variances) for these two variables are assumed to be identical:

σx = σy = σ and the contribution to the variance of the analyzed dimension becomes:

σA2 = ... + f x2σx2+ f y2σy2 + ... The sensitivity coefficient for a concentric constraint is reported in the Tolerance Analysis Report as the one number:

sensitivity =

2

f x

+

2

f y

so that the contribution to σ is A

σA2 = ... + (sensitivity)2 σ2 + ... just as for the contributors that constrain only one degree of freedom. This always reports a positive value for the sensitivity coefficient, so any sense of the direction of maximum sensitivity is lost.

True position type contributors (e.g., concentric constraints, which have two degrees of freedom), have a single tolerance value (T) and a Sigma range (R). To calculate the standard deviation for true position type contributors, the system calculates a special factor (SF) which is a value based on the Sigma range. SF values are based on two-dimensional random distribution formulas. See Abramowitz and Stegun, Handbook of Mathematical Functions (1964:936) for information about how SF values are derived. For true position type contributors, it is assumed that µ i = 0.0 . The following formula calculates the standard deviation for a true position contributor with a normal distribution:

σi

= T × SF R = 3, SF = 0.14537612 R = 4, SF = 0.11371301 R = 5, SF = 0.09326077 R = 6, SF = 0.078970878


A-23

Computing Tolerance Analysis with RSS Probability Calculations These factors relate the Rayleigh distribution to a normal distribution. For the worst case analysis, the problem is to find values for the two variables dvx and dvy that satisfy the tolerance specification: 2 x +

dv

2

dv y

≤

1 2

Tol

and that maximize (or minimize) the two terms: contribution = f x dvx + f y dvy The proof of the Cauchy-Schwartz inequality (see any basic text on vector analysis or linear algebra, such as Apostol 1961, 260) includes the proof that this sum reaches its maximum and minimum when the vector (dvx, dvy) is parallel to the vector (f x, f y). If L is the (signed) ratio of the lengths of these two vectors, the maximum occurs when: dvx = L f x dvy = L f y The tolerance specification becomes: 2

2

(f x + f y )L

2

≤

1 2

Tol

which translates into: – Limit ≤ L≤ + Limit where

2 2 1 Limit = --- Tol ⁄ f + f 2 x y

A-24


Contributors and their Variables The maximum contribution occurs when L = + Limit, and the minimum contribution occurs when L = – Limit. At L = + Limit, the maximum contribution becomes: max = f x (L f x) + f y (L f y ) = L (f x2 + f y2)

1 = Tol 2

2

f x

+

2

f y

= half the tolerance zone times the sensitivity and the minimum contribution is just the negative of this number.

Fixed-Point Constraints If the fixed point is specified to be at (Xspec, Yspec), then the two variables for this constraint are: dvx = Px - Xspec dvy = Py - Yspec Other than the fact that (Xspec, Yspec) cannot vary, each fixed-point constraint is handled like a concentric constraint. All the discussions about concentric constraints also apply to fixed-point constraints.


A-25


Collinear Constraints While a collinear constraint limits two degrees of freedom (position and orientation), its tolerance specification is not a true position specification. The two variables are the distances from the start and end points of the non-datum line to the datum line. The variables for collinear constraint are dv S and dvE: dvS = –sin (α)(PSx – P1x) + cos (α)(PSy – P1y) dvE = –sin (α)(PEx – P1x) + cos (α)(PEy – P1y) where P1 = (P1x,P1y) is a point on the datum line, PS = (PSx,PSy) is the start point, and PE = (PEx,PEy) is the end point of the non-datum line. The angle α is determined by the datum line. The tolerance specification for a collinear constraint, according to ASME Y14.5M-1994, is a zone that limits how far these two points can move off the datum line. This tolerance translates into the inequalities: – Zone ≤ dv s ≤ Zone

and – Zone ≤ dv E ≤ Zone

where Zone is the tolerance zone specification.

A-26



PS

dv S

d a t u m l i n e

“ t r u e ” l i n e

i d e a l l i n e

+ Zone

dv E

α

PE - Zone

Tol

When performing the sensitivity analysis, Mechanical Advantage computes one sensitivity coefficient for each of the two variables associated with a collinear constraint. The two terms in the linearized formula for dA are dA = ... + f SdvS + f EdvE + ... Since the tolerance zone is symmetric, the mean values for these two variables are both zero and contribute nothing to the formula for the mean of the analyzed dimension. The tolerance zone is the same for both variables, so the standard deviations (and therefore the variances) for these two variables are assumed to be identical:

σS = σE = σ and the contribution to the variance of the analyzed dimension becomes

σA2

= ... + f S2σS2 + f E2σE2 + ... = ... + (f S2 + f E2)σ2 + ...


A-27

Computing Tolerance Analysis with RSS Probability Calculations The sensitivity coefficient for a collinear constraint could be reported in the Tolerance Analysis Report as the one number:

sensitivity =

2

f x

+

2

f y

so that the contribution to σA is sensitivity squared times standard deviation squared, just as for contributors that restrict one degree of freedom. For the worst case analysis, the problem is to find values for the two variables dvS and dvE that satisfy the tolerance specification: – Zone ≤ dv s ≤ Zone

and – Zone ≤ dv E ≤ Zone

and that maximize (or minimize) the two terms: contribution = f SdvS + f EdvE The maximum occurs when dvS is either +Zone or –Zone, whichever has the same sign as f S. A similar statement applies to dvE, with the maximum contribution of: contribution

= |f S| Zone + |f E| Zone = ( |f S| + |f E| ) Zone

The minimum contribution is the negative of this number. The sensitivity coefficient for a collinear constraint appears in the Tolerance Analysis Report as the one number: sensitivity = |f S| + |f E| so that the maximum contribution to dA is sensitivity times tolerance specification, just as for the contributors that restrict one degree of freedom. The sensitivity was chosen because it is always larger than the root mean square sensitivity that would be correct for computing σA. The Mechanical Advantage sensitivity for collinear constraints, then, is a conservative approximation, always greater than or equal to the actual sensitivity.

A-28



Forces Forces are represented by the following formula: force =

± K ( P1x – P 2x )

2

+ (P

1y

–P

) 2y

2

where K is the conversion factor between force and length units (in force units per length unit). P and P are the tip and tail points of the force. The choice 1 2 of sign is such that forces are initially positive and invert when the force tip passes through the force tail point. If the force is the i

dv = force i

th

variable v , then: i

± ( P1x – P2x )

2

+ (P

1y

–P

) 2y

2

Moments Moment values are direct variables in the system, since the values for a locked moment are not associated with any geometry. Moments are represented as: moment = m where m

nominal

nominal

If this moment is the i

is the nominal moment value. th


dv = moment – m i nominal


A-29


Equations (Variables) As their name implies, equations directly represent their variables in the system. They are given by: variable = v where v

nominal

nominal

If the variable is the i

is the assigned value for the variable.

th


dv = variable – v i nominal

A-30


Appendix B

Incorporation of Material Condition Modifiers into RSS Statistical Analyses

Introduction Position tolerance depends on feature size when Material Condition Modifiers— maximum material condition (MMC) and least material condition (LMC)—are applied. This position-size dependency presents a challenge to RSS tolerance analysis because RSS analyses require that variables (contributors) be independent. This appendix will examine this apparent conflict and propose a method whereby MMC and LMC, under certain circumstances, may be included in RSS analyses.

Position-Size Dependence versus Independence in Manufacturing Although MMC and LMC modifiers imply a position-size dependency, for many manufacturing processes this dependency applies only to measurement and inspection. The manufacturing reality is that position and size are often independent. For example, the position tolerance of a drilled hole generally does not depend on its diameter, even if an MMC or LMC modifier is called out. Therefore, for parts manufactured by processes in which size and position are independent, a reasonable basis exists for treating size and position as independent variables in RSS analyses, even if MMC or LMC modifiers are specified on the drawing. If position and size are to be treated as independent variables, why is there a need to consider MMC and LMC modifiers in RSS analyses? The reason is that when parts are inspected, they are inspected against drawings. The inspection process pits the position-size dependency tolerance standard against the position-size independence of the manufacturing process. RSS analyses can be useful in predicting the success rate.


B-1


Tolerance Modeling Proceeding on the basis that position and size are independent for some manufacturing processes, how should the MMC and LMC tolerances be modeled to reflect both engineering intent as embodied in the drawings, and manufacturing reality? This paper will present two key points: • How to apply the notion of a bonus position tolerance to RSS calculations? • How to determine the percentage of the bonus tolerance that should be applied to the RSS analysis?

RSS Calculations with Bonus Tolerances The term bonus tolerance is often used to describe the additional position tolerance that is added to the basic position tolerance, depending on feature size. The maximum bonus tolerance is equal to the total size range of the feature. For example, for a 1.000 ± 0.005 hole with a position tolerance of 0.003@MMC, the base position tolerance is 0.003, and the maximum bonus tolerance is 0.010. For the purpose of this discussion, the term bonus tolerance is generally not the maximum bonus tolerance. It can be thought of as the bonus tolerance available for a given sample. If the sample is at the maximum departure allowed from the condition specified by the MMC or LMC modifier, then the bonus tolerance will equal the maximum bonus tolerance. For statistical analysis purposes, the bonus tolerance can also be thought of as the difference between the manufacturing process tolerance and the base position tolerance. This assumes that the base tolerance is smaller than the manufacturing process limits. A separate section of this paper discusses how to determine the bonus tolerance for statistical analysis. The discussion that follows assumes that an appropriate bonus tolerance has been determined. In order to apply the bonus tolerance to an RSS analysis, recall the basic RSS formula for calculating variance:

σ

2

= (s

σ ) 1 1

2

+ (s

σ ) 2 2

2

+ ... ( s

σ ) n n

2

EQ 1

where:

σ s

2

n

σn

B-2

= Variance of the analyzed dimension = Sensitivity of variable (contributor) n = Standard deviation of variable n


Tolerance Modeling To calculate the contribution of position and size you can write the following expression:

σ

2

= ... + ( s

σ ) p p

2

+ (s

σ ) s s

2

+ ... ( s

σ ) n n

2

EQ 2

where: s

p

σp

= Sensitivity of the position variable = Standard deviation of the position variable

s = Sensitivity of the size variable s

σs

= Standard deviation of the size variable

Because position as modeled in Mechanical Advantage is a 2 degree-of-freedom variable, Mechanical Advantage treats it as having a Rayleigh distribution and, as described on page A-23 of the Tolerance Analysis Manual, calculates σ as p follows:

σp

= T

p

× SF

EQ 3

where: T

p

= Total position tolerance

SF = Special factor based on the sigma range of the manufacturing process. The default sigma range is assumed to be ± 3 σ .


B-3

Incorporation of Material Condition Modifiers into RSS Statistical Analyses However, for position tolerances with MMC or LMC modifiers, the total position tolerance, T , can be represented as the base position tolerance plus the bonus p tolerance: T

p

= T

+T

base

EQ 4

bonus

Therefore:

σp

= (T

σbase

= T

σbonus σp

=

base

+T

base

= T

bonus

) × SF

EQ 5

× SF

bonus

EQ 6

× SF

EQ 7

σ base × σ bonus

EQ 8

Referring back to EQ 2, you can substitute the right hand side of EQ 8 for

σ

2

= ... + ( s

(σ +σ )) p base bonus

2

+ (s

σ ) s s

2

+ ... ( s

σ ) n n

2

σp : EQ 9

Expanding the calculations pertaining to the position contribution you obtain:

( s p ( σbase + σ bonus ) )

2

= s

p

2

 σ base

2

+ 2(σ

σ

base bonus

2 ) + σ bonus  

Recognizing that the contribution of the base position tolerance is s

2

σ base

2

, p you can separate the contribution of the base tolerance from the bonus tolerance: B ase contribution = s

2 p

Bonus contribution = s

σ base

p

2

2

 σbonus

EQ 10 2

+ 2( σ



σ ) base bonus 

EQ 11

This calculation is the basis for reporting the bonus contribution separate from the contribution of the base position tolerance in the Mechanical Advantage tolerance report.

B-4


Tolerance Modeling By representing T

as a ratio of T you can show a further interesting bonus base relationship between the Base contribution and the Bonus contribution. r = T

⁄ Tbase

bonus

σ bonus

= r×σ

EQ 12

base

Substituting this relationship back into EQ 11 and simplifying, you end up with the following expression for the bonus contribution: Bonus contribution = s

2 p

2 2

σbase ( r

+ 2r )

Or, in other words: 2 Bonus contribution = Base contribution × ( r + 2r )


EQ 13

B-5

Incorporation of Material Condition Modifiers into RSS Statistical Analyses 2 Referring to the factor ( r + 2r ) as a contribution multiplier, you can easily create a table as a function or r: r

r 2+2r

0

0

0.1

0.21

0.2

0.44

0.3

0.69

0.4

0.96

0.5

1.25

0.6

1.56

0.7

1.89

0.8

2.24

0.9

2.61

1.0

3.00

1.1

3.41

1.2

3.84

1.3

4.29

1.4

4.76

1.5

5.25

As can be seen, the effect of the bonus tolerance grows exponentially. The bonus contribution equals the base contribution when the ratio r is approximately 0.414.

B-6


Tolerance Modeling

Determining the Appropriate Bonus Tolerance for RSS Analysis The bonus tolerance is a variable that depends on feature size. The value of the bonus tolerance ranges from 0 to T , where T is the total tolerance range max max of the size callout. T

max

is the maximum bonus tolerance.

As stated earlier, RSS analysis requires that variables be independent. In order to satisfy this requirement, you must choose a specific value for the bonus tolerance that can be used for statistical modeling. Obviously T

should be used for worst case analysis. But it is overly consermax vative for statistical analyses. In choosing the bonus value to use for statistical analyses, you will use the following assumptions: • Manufacturing process limits are specified at ± 3 σ (this is the baseline used in C

p

and C

pk

calculations).

• Size tolerances of a feature are normally distributed. The tolerance limits represent the ± 3 σ limits of the distribution. In other words, feature size will lie within the tolerance limits 99.73% of the time. • For a given callout, process capabilities are required such that the combination of feature size and position meet the specifications 99.46% of the time. This last assumption comes from elementary statistics. Recall that for independent variables, the joint probability of A and B occurring equals the probability of A times the probability of B. P

AB

= P

A

× PB

EQ 14

If you assume that the processes for controlling size and position each have ± 3 σ limits, and apply EQ 12, you get the following: P P P

size pos

s&p

= 0.9973 = 0.9973 = 0.9973

2

= 0.9946


EQ 15

B-7

Incorporation of Material Condition Modifiers into RSS Statistical Analyses If the range of size values is normally distributed between the tolerance limits, then the range of bonus tolerances is also normally distributed between 0 and T . In fact, because of the direct relationship between the bonus tolerance and max the size tolerance, whatever the distribution of the size range, the bonus tolerance distribution will exactly match. Similarly, the distribution of total position tolerance limits ( T ) will also follow the distribution of the size tolerances. p It is important to realize that there are two distinctly different distributions that are important to this analysis. The first is a distribution of tolerance limits. As indicated in the previous paragraph, this distribution directly corresponds to the size distribution. The second distribution is the position value distribution, which is the manufacturing process distribution. It is quite different from the tolerance limit distribution. For example, it has a mean of zero and its spread (standard deviation) may be quite different. Your goal is to find the manufacturing distribution that will satisfy a tolerance limit distribution. Or to put it another way, you must answer the question, “What is the probability that position will lie within the manufacturing distribution, and also lie within the tolerance limit?” This is a joint probability question. You begin by defining the two independent probabilities: P P

A B

= probability that a position range lies within the manufacturing limits. = probability that the position range is within the tolerance limit.

Let ± p equal the limitsof a range of position values. If you let the range bounded by p grow to the ± 3 σ limits of the manufacturing process, then P = 0.9973 . A Recognizing that p is a radial value and that the position tolerance T

p

is a diame-

tral value, you can establish the relationship: T

p

≥ 2p

EQ 16 essentially states that the tolerance T

EQ 16

p

must exceed the position value in

order to meet the tolerance callout.

B-8


Tolerance Modeling You also know that the following limits exist for T : p T

base

≤ Tp ≤ ( T base + Tmax )

With this in mind you can rewrite P P

B

= probability that 2p ≤ T

p

EQ 17

B

as follows:

≤ ( Tbase + T max )

Turning your attention back to the joint probability of P

and P , recall from A B EQ 15 that the joint probability of meeting the position and size requirements is 2 0.9973 = 0.9946 . A case can be made that the joint probability of P

and P is identical because A B the probability distribution for position tolerances directly parallels the distribution of size values. If P

A

× PB

= 0.9946 , and P

A

The probability distribution of T

= 0.9973 , then P

p

B

= 0.9973 .

is such that between the lower limit = T

base

and upper limit = ( T

+T ) the probability = 0.9973. But from the defibase ma x nition of P , the lower limit = 2p . This means that 2p = T . In other words, B base the manufacturing process limits must be less than or equal to the base position tolerance in order for the parts to be acceptable. To the extent that the manufacturing process limit ( 2p ) is larger than T will be reduced, and therefore the joint probability P


A

× PB

base

,P

B

will also be reduced.

B-9


Summary For best results in tolerance analysis, manufacturing process limits should be used whenever possible. In the case of position tolerances with MMC or LMC modifiers, the process capability limits should be less than or equal to the base position tolerance. Otherwise the parts will fail to meet specifications. If the manufacturing process tolerance exceeds the position base tolerance, the difference may be entered into the MA as a bonus tolerance. The effect of the bonus tolerance will be shown separately from the base tolerance in accordance with the calculations presented earlier in this appendix. If you wish to adopt a conservative approach, you may apply a bonus tolerance of your own choosing. Bonus tolerances are entered into Mechanical Advantage by specifying a percentage of the feature size range that will be used as the bonus tolerance. For example, if the size tolerance is ± 0.003 , and a percentage of 40% is entered, the bonus tolerance will be set at 0.4 × 0.006 = 0.0024 . If the tolerance range is changed the bonus tolerance will update automatically. The contribution of a bonus tolerance will equal that of the base tolerance when the ratio of the bonus tolerance to the base tolerance is approximately .41. When the ratio rises to 1, the effect of the bonus tolerance will be 3x greater than that of the base tolerance.

B-10


Index Commands are displayed in bold text.

Numerics 3D Property Analysis See Critical Parameter Analysis 3D Tolerance Analysis See Projection lines

A Analyze Tolerance 54 Analyzed dimension contributors to A-7 Analyzed dimension, showing 58 Angular dimension perturbing 98 sensitivity 105–111 ASCII file converting a tolerance report to 64 exclude zero contributors 64 include zero contributors 64 Assembly contributor multiplication See Duplicate contributors dependent, in tolerance analysis 59 working with duplicate contributors See Duplicate contributors Auto tolerance float pin-in-hole 51

B Basic tolerance 47 Bonus factor 158 material modifier

159

C Choose Analyzed 57 Choose analyzed multiple tolerance analysis

Collinear constraint datums for 138 tolerance specification for A-26 Combining duplicate contributors See Duplicate contributors Command Analyze Parameters 187, 197 Analyze Tolerance 54 Choose Analyzed 57 Combine Duplicates 62 Exclude Contributor 190 Export Monte Carlo Results 128 Extrude 198 Include Contributor 190 Link 192 Monte Carlo Analysis 70 Open Variable Window 199 Pass-Thru 168, 169 Perturb 97 Plot Distribution Graph 60, 190 Position 161 Read From ASCII File 128 Reanalyze 58 Reset Graph 61 Save CPA Contributor Info 188 Separate Duplicates 63 Set Default Tolerance 41, 44 Set Default Tolerances 42 Set Up Window 67 Show Analyzed Dimension 58 Show Contributor 58, 59 Show Physical Properties 196 Sort By Contribution 59 Sort by Part Name 59 Sort By Sensitivity 59 Stopkey 125 Swap Datum 134 Write To ASCII File 64

Constraint collinear tolerance specification and variables for A-26 concentric tolerance specification and variables for A-21 equal to 169 Pass-Thru 167 equal-length, and tolerance proper datums for 137 float pin in hole 164 auto tolerance calculation 164 fit clearance 165 special consideration 165 line-at-angle tolerance specification and variables for A-16 parallel proper datums for 135 perpendicular proper datums for 136 position 161 circle datums 162 datums 161 editing 163 proper datums for 140 tolerance 163 product 173 scaled from 169 Pass-Thru 167 sum to 173 sum to zero 173 that have no tolerance 51 Constraint schemes baseline 147 chained 145 for model building 144 Contribution 39 Contributor derived 167

57


I-1

Index Contributor information contributor results 90 contributor symbols 88 contributor types geometric 87 performance modeling 87 name 86 reading the report 85–89, 90 type 86 upper and lower contribution 91 upper or lower zone 86 value 86 Contributor mean 38 Contributor part name 85 Contributor symbols & 88 + 88 ? 88 cl 88 L 88 M 88 Contributor type geometric type symbols 87 performance modeling type symbols 87 reading the Type column 86 Contributors to tolerance analysis calculated A-2 selecting 58 setting tolerance values 52 showing 58 updating the master assembly 53 Converting tolerance analysis report to ASCII format 64 Converting degrees to radian 96 Cp and Cpk indices 79–80 CPA See Critical Parameter Analysis Critical Parameter Analysis 3D property analysis 198 absolute value 186 Analyze Parameters command 187, 197 contributor inclusion 190 creating variables 191 Extrude command 198 inclusion of geometric contributors 186 Link command 192 linking physical properties 196 linking to Solid Note 198, 199 linking variables 192

I-2

overview for performing the analysis 187 performance parameters 185 physical properties 196 point-to-point dimensions 186 reading the report 197 save CPA contributor info 188 Show Physical Properties command 196 step by step example 191 testing deflection example 204 tolerance distribution graph 190 variable contributors 186 variable definitions 205 variable symbols 205

D Datums 131 choosing datums for constraints 141 collinear constraint datum 138 concentric constraint datum 139 equal-length constraint datum 137 equal-radius constraint datum 137 model building 14–18, 25–30 primary datum 14 secondary datum 15 parallel constraint datum 135 perpendicular constraint datum 136 position constraint datums 140 Default tolerance 46 default tolerance window 41, 44 overriding 45 setting default tolerances by dimension type 41 by precision setting 42 for position constraint 44 for variables 44 Dependent assemblies See Duplicate contributors Dependent assembly assembly of parts 174 in tolerance analysis 59, 174 advantages 174 model stability 174 part management 174 update parts 174 Derived contributor 167

Derived items dimension 38 force 38 moment 38 Dimension angular calculated A-12 linear 112 calculated A-7 Dimensioning schemes 149 baseline 152 chained 150 direct 154 Direct dimensioning of critical dimension 154 Distribution types See Monte Carlo Downstream contributor derived contributor 167 Duplicate contributors avoiding 184 combining 62 dependent assemblies 177 projection lines 184 separating 63 when to combine 175 working with duplicates

E Editing contributor in tolerance analysis 58 Enter. See Return key Equal-dimension constraint, and tolerance A-19 Equal-length constraint, and tolerance A-19 Equal-length constraints as pass-through constraints 169 Equal-radius constraint, and tolerance A-20 datums for 137 Exporting tolerance analysis table to ASCII file 64

F Feature LMC and MMC 159 Feature type material modifier 158


Index Fixed length constraint A-11 Fixed point constraint, variables for A-25 Fixed radius constraint, defined A11 Float pin-in-hole auto tolerance 51, 164 special considerations 165 Force analyze tolerance 54 setting tolerance 49, 53

G Gaussian distribution See Normal distribution

I Influence coefficient INTOL See TOLLOG

A-2

L Least Material Condition 88 Lever arm ratio A-2 Limit tolerance 46 Linear dimension calculated A-6 perturbing 100, 102, 103 sensitivity 100, 102, 103 types 112–116 sensitivity 112–116 Linear Monte Carlo See Monte Carlo Linear spring setting tolerance 50 Linear worst case analysis contributor inclusion 72–74 performing 66 Linearized function, derived dimension defined as A-1 LMC 88 feature 159 modifier type 158

M MAMCMAXFAIL 127 environment variable UNIX 127 Windows NT 127 See Monte Carlo Marginal cost A-2 Material Modifier worst case 159 Material modifier 158 bonus factor 158, 159 feature type 158 LMC 158 MMC 158 modifier type 158 monte carlo dependence 160 RFS 158 set modifier 158 statistical application 160 Max Material Condition 88 feature 159 Mean 39 Mean for tolerance contributor calculated A-2 MMC 88 feature 159 modifier type 158 Model building 131 angular dimension 107 baseline constraints 147 chained constraints 145 constraint schemes 144 datums 131 Swap Datum 134 dimension schemes 149 duplicate contributors in assemblies 175 equal-length constraints See Equal-length constraints for tolerance analysis datums 131 geometric contraints that reference datums 134 Monte Carlo See Monte Carlo profile tolerances 157 projection lines 181 See Projection lines replacing angles with lines 110 sensitivity multiplication 182


Model building to avoid sensitivity 182 datums 182 Orientation 183 parallel and perpendicular constraints 183 point-on-object constraint 183 Tangent constraint 183 Modifier type LMC 158 material modifier 158 MMC 158 RFS 158 Moment analyze tolerance 54 setting tolerance 49, 53 Monte Carlo advanced topics 119 contributor inclusion 72–74 distribution types 119 calculating standard deviation A-23 linear method 125 MAMCMAXFAIL 127 material modifiers 160 non-linear method 125 non-linear model building 126 equal-length constraints 126 linked-in dimensions 126 multiple dependent assemblies 126 normal 120 calculating mean 120 calculating standard deviation 120 parameter values 120 See Normal distribution performing 70 plotting in DCAP 128 Export Monte Carlo Results 128 Read From ASCII File 128 reading the MC column distribution types 89 reading the report 84 sample size limit 70 solution failure limit set environment variable 127 Stopkey 125 tolerance zone calculating standard deviation A-23

I-3

Index TOLLOG See TOLLOG uniform 124 over the range 124 weibull constraint contributors 123 dimensional contributors 122 generate random deviations 122 generating offsets 122 parameters 121 See also Weibull distribution Multiple tolerance analysis 56 Choose Analyzed 57

N Non-linear Monte Carlo See Monte Carlo Non-linear worst case analysis contributor inclusion 72–74 performing 67 Normal distribution 39 3 Sigma 33 calculating mean 120 calculating standard deviation 120, A-23 default distribution See Monte Carlo worst case analysis 33

O OUTTOL See TOLLOG Over the range See Monte Carlo

P Parallel constraint datums for 135 perturbing 99, 101, 109 sensitivity 99, 101 tolerance applied to A-17 part name contributor 85 Pass-Thru 168 Percent contribution 39 Percent in tolerance 38

I-4

Perpendicular constraint datums for 136 perturbing 104 sensitivity 104 tolerance applied to A-18 Perturbing contributors 97 angular dimensions 98 for sensitivity 97–104 parallel constraints 101 perpendicular constraints 104 tolerance zone 99 Pin and Hole float 51 Plus/Minus tolerance 46 Position 161 Position constraint circle datum 162 datum 161 editing 163 tolerance value 163 true positioning 161 Primary datum See Datums Probability in tolerance calculated A-5 Product constraint tolerance effect 173 Profile tolerance tolerance zone 52 Profile tolerances 157 groups and regions 157 Projection lines 3D Tolerance Analysis 181 avoiding duplicate contributors 184 avoiding sensitivity multiplication 182 chained 182 constraining orientation 181 duplicate contributors 181 parallel 182 proper model building 181 sensitivity multiplication 181 controlling orientation 182 to constrain orientation 181

Q Quick tour Monte Carlo 34–35 non-linear worst case 31–33 step by step instructions 13–36 tolerance analysis 20 tolerance analysis report 21–22

R Radial dimension calculated A-11 Radians converting See Sensitivity analysis Reanalyze 58 Return key 3 RFS modifier type 158 Root Sum Squared RSS See also Root Sum Squared

S Secondary datum See Datums Selecting contributors to tolerance analysis 58 Sensitivity 38 equation to determine sensitivity 93 Sensitivity analysis 90 angular dimensions 105 calculated A-2 comparing units 96 converting to radians 96 datums 94 linear dimensions 112 mixed unit sensitivity 96 negative sensitivity 100 parallel constraints 101 perpendicular constraints 104 perturbing contributors 97 sensitivity magnitude 95, 98 summary 117 Sensitivity coefficients A-2 Sensitivity magnitude See Sensitivity analysis Sensitivity multiplication 182 chained projection lines 182 parallel constraints 183 perpendicular constraints 183 See Projection lines Set Default Tolerances 42 Setting tolerance values See Tolerance values Show Analyzed Dimension 58 Show Contributor 58, 59


Index Showing analyzed dimension in tolerance analysis 58 Solid Note variable window 199 Sort By Contribution 59 Sort by Part Name 59 Sort By Sensitivity 59 Sorting tolerance analysis window 59 Springs linear 50 setting tolerance 50 torsion 50 Standard deviation 38 Sum-to constraint tolerance effect 173 Symmetric tolerance 47

T Tolerance analysis and dependent assemblies 59 critical parameter analysis See also Critical Parameter Analysis default tolerances for dimensions 40 model building See Model building 131 performing multiple analysis 56 performing the analysis 54 sensitivity 59, 90 setting tolerances 40 showing analyzed dimension 58 showing contributors 58 tolerance distribution graph 60 Tolerance analysis report analyzed dimension, force, moment 76 Combine Duplicates 62 contribution 90 contributor information 85–89 contributor results 90 Cp and Cpk indices 79–80 design limits 77 failure rate 77 linear worst case 83 MC 89 Monte Carlo 84 non-linear worst case 83 percent in tolerance 77 Reanalyze 58 report header 76

Separate Duplicates 63 setting tolerances 52 Show Analyzed Dimension 58 Show Contributor 58 sigma values 76 Sort By Contribution 59 Sort By Sensitivity 59 statistical failure rates 76 tol max and tol min 78 types included 81 upper and lower contribution 91 val max and val min 81 value 76 worst case analyses 82 Write to ASCII File 64 Tolerance analysis terms analyzed dimension dimension 38 force 38 moment 38 contribution 39 contributor mean 38 mean 39 normal distribution 39 percent contribution 39 percent in tolerance 38 sensitivity 38 standard deviation 38 Tolerance distribution graph 60 preserve current graph 61 printing 61 reset the graph 61 Tolerance type plus/minus 46 Tolerance types basic 46 default 46 limit 46 symmetric 46 Tolerance value auto 51 constraints that cannot have tolerance 51 setting for a constraint 48 setting for a dimension 45 setting for force 49 setting for moment 49 setting for multiple contributors 53 constraints 53 dimensions 53 forces 53 moments 53 variables 53


setting for multiple dimensions 45 setting for springs 49 setting from the report 52 Tolerance zone 99 calculating standard deviation A-23 constraints that cannot have tolerance 51 parallel constraints 99, 101 perpendicular constraints 104 profile tolerance 52 TOLLOG environment variable INTOL OUTTOL See Monte Carlo Torsion spring setting tolerance 50

U Uniform distribution over the range See Monte Carlo Upstream contributor derived contributors

167

V Variable setting tolerance 53 Variables creating 191 in critical parameter analysis 191 linking 192 See Critical Parameter Analysis

W Weibull distribution constraint contributors default parameter values 123 editing parameter values 123 dimensional contributors default parameter values 122 editing parameter values 122 editing parameter values 121 location parameter 121 scale parameter 121 shape parameter 121 two-parameter distribution 121

I-5

Tolerance

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