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

ADCATS

A Closed Form Solution for Nonlinear Tolerance Analysis

Geoff Carlson

Brigham Young University

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ADCATS

Outline• Statistical Tolerance Analysis

Overview

• Assembly Functions

• Tolerance Analysis Methods

• Acceleration Analysis Method

• Skewness Approximation

• Summary

ADCATS

Statistical Tolerance Analysis

Component Variation:xi ± dxi

QualityFraction

GivenGiven FindFind

AssemblyTolerance

AssemblyFunction

LL UL

Assembly Variation:ui ± dui

ADCATS

Statistical Tolerance Analysis:Two and Three-dimensional assemblies

• u, represents the dependant assembly dimension

• xi, represents the component dimensions in the assembly

• , represents the contribution of each component dimension

2

ii

dxx

udu

ix

u

ADCATS

Statistical Tolerance Analysis: Two-dimensional Example

u

y

tan

yu

22

dyy

ud

udu

22

2 tan

1

sin

dydy

du

ADCATS

u

y

Statistical Tolerance Analysis: Example: Nonlinear Assembly Function

• Limits of the input variable, , are symmetrically distributed

• Distribution of the output variable, u, is skewed

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

n

iix

n 11

1

+s-s

x

n

iix

n 1

21

22 )(

1

n

iix

n 1

313 )(

1

n

iix

n 1

414 )(

1

ADCATS

Nonlinear Assemblies:Assembly Function Representation

• Explicit: xi = set of input variables

uj = set of output variables

• Implicit:xi = set of input variables

uj = set of output variables

)(ij

xfu

0),( ji uxh

ADCATS


• Linearized:

dh = change in the assembly function

dxi = small changes in the assembly dimensions, xi

duj = the corresponding kinematic changes, uj

0),(

jj

xi

i

xjix du

u

hdx

x

huxdh

0),( jj

yi

i

yjiy du

u

hdx

x

huxdh

0),(

jj

ii

ji duu

hdx

x

huxdh

ADCATS


• Linearized:

A = assembly dimension sensitivities

B = kinematic sensitivitiesdX = column vector of assembly dimension variations

dU = column vector of kinematic variations B-1A = tolerance sensitivity matrix

0 dUBdXAdh

dXABdU 1

ADCATS

Example: Nonlinear AssembliesOne-way Clutch Assembly Function

• Explicit:

• Implicit:

• Linearized:

1

a

b

c2

c1

f

2

cf

ca11 cos

0)270cos(2

)90cos( 11 fcbhx

0)270sin(2

)90sin(2 11 f

ca

hy

df

dc

da

d

dbdU

2063.04142.02078.0

1841.8306.161227.8

1

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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

ADCATS

Solutions to Nonlinear Systems:MCS

10,000 Sets of Parts

Assembly Histogram

Count the Rejects

LL UL

Random No. Generator

Assembly Function

ADCATS

Solutions to Nonlinear Systems:MSM

Component Input Moments

Assembly Output Moment

Taylor Series Expansion--Second Order

1

1

))((2

1

2)(2

2

1

)(

n

i

n

ij

joxjxioxixjxix

hn

i

ioxix

ix

hn

i

ioxixix

hoRRy

ADCATS


• The first four raw moments of the output distribution can be found by applying the expected value operator to y:

n

iiibyE

112)(

1

1 122

2

14

232

22 22)(n

i

n

ijjiijijii

n

iiiiiiiiii bbbbbbbyE

...)(1

333

n

iiibyE

...6)( 222

1

1

1 1

24

44

iij

n

i

n

i

n

ijiii bbbyE

ADCATS


• The first four raw moments can be centralized using the following equations:

• where, i is the ith central moment of R

oRyER )()(1

222 )()()( yEyER

3233 )(2)()(3)()( yEyEyEyER

422344 )(3)()(6)()(4)()( yEyEyEyEyEyER

ADCATS

Solutions to Nonlinear Systems:SOTA

• First and second order sensitivities are found using finite difference formulas:

i

iiii

i x

uxxuuxxu

x

u

2

,,

22

2 ),(),(2),(

i

iiiii

i x

uxxuuxuuxxu

x

u

j

i

jjiijjii

i

jjiijjii

ji x

x

uxxxxuuxxxxu

x

uxxxxuuxxxxu

xx

u

2

2

),,(),,(

2

),,(),,(

2

ADCATS

Solutions to Nonlinear Systems:TAKS

Variation Model Kinematic Model

a

a

b

c2

c1

f

b

c1

c2

f

0

][

)()90()90(

1

1

2

2

21

f

fc

cc

ba

i

ii

c

ii

ii

ef

feec

eec

ebea

0

)

(

)(

)(

)90(

)90(

)()(

)(

21

21

21

211

fccba

fccba

ccba

ccbacba

baa

i

i

i

ii

ii

edf

ef

ced

eedc

edbeda

Velocity equation:Small displacement equation:

ADCATS

Solutions to Nonlinear Systems:Method Comparison

• Relative Effort

One-way Clutch

MCS 100k 340,000

MCS 30k 102,300

SOTA 41

Linear 1

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?

ADCATS

Acceleration Analysis MethodExample: One-way Clutch

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ADCATS


• Acceleration Equation:

0

2902702

180180

270901

ii

iiii

iiiii

fece

feceefec

efeecebea cba

ADCATS


• Resolving the acceleration equation into real and imaginary parts and organizing into matrix form yields:

• where,

f

c

a

DB

f

c

a

CB

f

c

a

ABb 1211 2

f

c

a

ABb

1

ADCATS


Closed Form Sensitivity Kinematic Acceleration Coefficients

2

2

a

3

2

2

)1(coscos)()1(cossin2

d

cf

d

2

2

c

2

2

f

fa 2

ca 2

fc 2

3

cos)(

d

cf

3

3

2

cos)(sincos2

d

cf

d

32

)1(coscos)(sin

d

cf

d

3

2

2

cos)(sin

d

cf

d

3

2

2

cos)1)(cos()1(cossin2

d

cf

d

3

2

2

cos)1)(cos()1(cossin22

d

cf

d

fc

2a

2c

2f

ca

fa

3

cos)(

d

cf

3

2

2

)1(coscos)()1(cossin2

d

cf

d

3

3

2

cos)(sincos2

d

cf

d

32

)1(coscos)(sin2

d

cf

d

3

2

2

cos)(sin2

d

cf

d

ADCATS

Acceleration Analysis Method:Skewness

• Variance from TAKS method

• Skewness from Acceleration Method?

df

dc

da

d

db

2063.04142.02078.0

1841.8306.161227.8

2222

2 )2063.0()4142.0()2078.0( dfdcda

f

c

a

DB

f

c

a

CB

f

c

a

ABb 1211 2

?3

ADCATS

Skewness Approximation:Acceleration Analysis

• Solving the acceleration equation for :

• Simplifying:

fcd

cf

dfa

d

cf

d

cad

cf

df

d

cf

d

cd

cf

da

d

cf

fd

cd

ad

3

2

23

2

2

32

2

3

3

2

2

3

2

2

2

3

cos)1)(cos(sin)1cos2(2

cos)(sin2

)1(coscos)(sin2

)()(cossincos2

cos)1)(cos()1(cossin2cos)(

cos1cos1

fcbfabcabfbcbabfbcbab cfafacffccaafca 222222

ADCATS


Acceleration Equation Groups

Operation Transformed Acceleration Equation Groups

Related Raw Moment

none

ib

iib

ijb fcbfabcab cfafac

222 2

)2(21

yxbij

222222222 fcbfabcab cfafac

222 fbcbab ffccaa )(yE

)( 3yE

222 fbcbab ffccaa

fbcbab fca 3)( xbi

333333 fbcbab fca

)( 2yE

n

iiiibyE

12)(

1

1 122

22 )(n

i

n

ijjiijbyE

n

iiibyE

13

33)(

3233 )(2)()(3)( yEyEyEyE

Raw moments with terms directly from the acceleration equation

ADCATS


Standardized Skewness Inputs

Standardized Skewness of

a3 c3 f3

MSM Kinematic Approx.

Error (%)

0.0 0.0 0.0 -0.093555 -0.457E-6 99.999

0.1 0.1 0.1 -0.170698 -0.077737 54.459

0.5 0.5 0.5 -0.473756 -0.388685 17.957

1.4 1.4 1.4 -1.124679 -1.088316 3.233

Number of Terms 80 9 -

ADCATS

Skewness Approximation:MSM Raw Moments

First three raw moments:

n

iiiibyE

12)(

1

1 122

2

14

232

22 22)(n

i

n

ijjiijijii

n

iiiiiiiiii bbbbbbbyE

222222

2

1

1

1 1

242

232

23

222

32242

1 1

3322

1

133

3

52

42

61

333

]}[366{

]336

363[

]66[

]33[)(

kjjiijkkikjjjkiijkikijkkijii

n

i

n

ij

n

jk

jiijiijiijijijjiii

jijiijiijjjijjjiii

n

i

n

ijj

jiijjjiijiijji

n

i

n

ijjiij

iiiiiiiiiii

n

iii

bbbbbbbbbbbb

bbbbbbb

bbbbbbb

bbbbbbb

bbbbbbyE

ADCATS

Skewness Approximation:Truncated MSM Raw Moments

1

1 122

2

14

232

2

22

21

n

i

n

ijjiijijii

n

iiiiiiiiii

bbbblock

bbbbblock

Second raw moment blocks

E(y2) Blocks

Neglected Terms

block 1 i4

block 2 entire block

Truncated second raw moment:

n

iiiiiii bbbyE

132

22 2)(

ADCATS


Third raw moment blocks

222

2222

1

1

1 1

24

2

23

2

23

22

2

3224

2

1 1

33221

33

3

5

2

4

2

61

3

3

]}[366{4

]336

363[3

]66[2

]33[1

kjjiijkkikjjjkiijkikijkkijii

n

i

n

ij

n

jk

jiijiijiijijijjiii

jijiijiijjjijjjiii

n

i

n

ijj

jiijjjiijiijjii ij

jiij

iiiiiiiiiii

n

iii

bbbbbbbbbbbbblock

bbbbbbb

bbbbbbbblock

bbbbbbbblock

bbbbbbblock

ADCATS


E(y3) Blocks

Neglected Terms

block 1 i5, i6

block 2 i3j3

block 3 i2j4

block 4 entire block

Truncated third raw moment:

]36

36[

6]3[)(

23

2

23

22

2

321 1

1

1224

2

13

33

jiijijijjiii

jijiijiijjji

n

i

n

ijj

n

i

n

ijjiijjiiiii

n

iii

bbbbb

bbbbb

bbbbbbyE

ADCATS


Standardized Skewness

i3 MSM Truncation Approx.

Error (%)

Kinematic Approx.

Error (%)

0.0 -0.93555 -0.093600 0.049 -0.457E-6 99.999

0.1 -0.170698 -0.170800 0.067 -0.077737 54.459

0.5 -0.473756 -0.474075 0.068 -0.388685 17.957

1.4 -1.124679 -1.125448 0.068 -1.088316 3.233

Number of terms

80 42 - 9

ADCATS

Acceleration Analysis MethodSummary

• 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

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

Documents

adcats solutions

adcats nonlinear assemblies

adcats example

order slide

nonlinear systems

assembly dimensions

skewed slide

assembly function dx