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ADCATS A Closed Form Solution for Nonlinear Tolerance Analysis Geoff Carlson Brigham Young University.

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Presentation on theme: "ADCATS A Closed Form Solution for Nonlinear Tolerance Analysis Geoff Carlson Brigham Young University."— Presentation transcript:

1 ADCATS A Closed Form Solution for Nonlinear Tolerance Analysis Geoff Carlson Brigham Young University

2 ADCATS Outline Statistical Tolerance Analysis Overview Assembly Functions Tolerance Analysis Methods Acceleration Analysis Method Skewness Approximation Summary

3 ADCATS Statistical Tolerance Analysis Component Variation: x i ± dx i Quality Fraction Given Find Assembly Tolerance Assembly Function LLUL Assembly Variation: u i ± du i

4 ADCATS Statistical Tolerance Analysis: Two and Three-dimensional assemblies u, represents the dependant assembly dimension x i, represents the component dimensions in the assembly, represents the contribution of each component dimension

5 ADCATS Statistical Tolerance Analysis: Two-dimensional Example

6 ADCATS Statistical Tolerance Analysis: Example: Nonlinear Assembly Function Limits of the input variable, , are symmetrically distributed Distribution of the output variable, u, is skewed

7 ADCATS Four Moments of a Distribution First Moment: –mean - measure of location Second Moment: –standard deviation - measure of spread Third Moment: –skewness - measure of symmetry Fourth Moment: –kurtosis - measure of peakedness +s -s x

8 ADCATS Nonlinear Assemblies: Assembly Function Representation Explicit: x i = set of input variables u j = set of output variables Implicit: x i = set of input variables u j = set of output variables

9 ADCATS Nonlinear Assemblies: Assembly Function Representation Linearized: dh = change in the assembly function dx i = small changes in the assembly dimensions, x i du j = the corresponding kinematic changes, u j

10 ADCATS Nonlinear Assemblies: Assembly Function Representation Linearized: A = assembly dimension sensitivities B = kinematic sensitivities dX = column vector of assembly dimension variations dU = column vector of kinematic variations B -1 A = tolerance sensitivity matrix

11 ADCATS Example: Nonlinear Assemblies One-way Clutch Assembly Function Explicit: Implicit: Linearized:

12 ADCATS Solutions to Nonlinear Assemblies Monte Carlo Simulation (MCS) Method of System Moments (MSM) Second Order Tolerance Analysis (SOTA) Tolerance Analysis Using Kinematic Sensitivities (TAKS) Acceleration Analysis Method

13 ADCATS Solutions to Nonlinear Systems: MCS 10,000 Sets of Parts Assembly Histogram Count the Rejects LL UL Random No. Generator Assembly Function

14 ADCATS Solutions to Nonlinear Systems: MSM Component Input Moments Assembly Output Moment Taylor Series Expansion--Second Order

15 ADCATS Solutions to Nonlinear Systems: MSM The first four raw moments of the output distribution can be found by applying the expected value operator to y:

16 ADCATS Solutions to Nonlinear Systems: MSM The first four raw moments can be centralized using the following equations: where,  i is the i th central moment of R

17 ADCATS Solutions to Nonlinear Systems: SOTA First and second order sensitivities are found using finite difference formulas:

18 ADCATS Solutions to Nonlinear Systems: TAKS Variation ModelKinematic Model Velocity equation:Small displacement equation:

19 ADCATS Solutions to Nonlinear Systems: Method Comparison Relative Effort

20 ADCATS Acceleration Analysis Method: Can we extend the kinematic velocity analogy? Can second order sensitivities be obtained from an acceleration analysis of a kinematic model? Can skewness be approximated from the acceleration analysis?

21 ADCATS Acceleration Analysis Method Example: One-way Clutch

22 ADCATS Acceleration Analysis Method Example: One-way Clutch Acceleration Equation:

23 ADCATS Acceleration Analysis Method Example: One-way Clutch Resolving the acceleration equation into real and imaginary parts and organizing into matrix form yields: where,

24 ADCATS Acceleration Analysis Method Example: One-way Clutch Closed Form Sensitivity Kinematic Acceleration Coefficients

25 ADCATS Acceleration Analysis Method: Skewness Variance from TAKS method Skewness from Acceleration Method?

26 ADCATS Skewness Approximation: Acceleration Analysis Solving the acceleration equation for  : Simplifying:

27 ADCATS Skewness Approximation: Acceleration Analysis Acceleration Equation Groups OperationTransformed Acceleration Equation Groups Related Raw Moment none Raw moments with terms directly from the acceleration equation

28 ADCATS Skewness Approximation: Acceleration Analysis Standardized Skewness Inputs Standardized Skewness of   a3  c3  f3 MSM Kinematic Approx. Error (%) 0.0 -0.093555-0.457E-699.999 0.1 -0.170698-0.07773754.459 0.5 -0.473756-0.38868517.957 1.4 -1.124679-1.0883163.233 Number of Terms 809-

29 ADCATS Skewness Approximation: MSM Raw Moments First three raw moments:

30 ADCATS Skewness Approximation: Truncated MSM Raw Moments Second raw moment blocks E(y 2 ) Blocks Neglected Terms block 1  i4 block 2entire block Truncated second raw moment:

31 ADCATS Skewness Approximation: Truncated MSM Raw Moments Third raw moment blocks

32 ADCATS Skewness Approximation: Truncated MSM Raw Moments E(y 3 ) Blocks Neglected Terms block 1  i5,  i6 block 2  i3  j3 block 3  i2  j4 block 4entire block Truncated third raw moment:

33 ADCATS Skewness Approximation: Truncated MSM Raw Moments Standardized Skewness  i3 MSMTruncation Approx. Error (%) Kinematic Approx. Error (%) 0.0-0.93555-0.0936000.049-0.457E-699.999 0.1-0.170698-0.1708000.067-0.07773754.459 0.5-0.473756-0.4740750.068-0.38868517.957 1.4-1.124679-1.1254480.068-1.0883163.233 Number of terms 8042-9

34 ADCATS Acceleration Analysis Method Summary Second order sensitivities can be obtained directly from acceleration analysis Sensitivities can be used with MSM –Increased efficiency –No iteration required Truncated MSM equation provide a good estimate of output skewness

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