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

hub

roller

spring

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

hub

roller

spring

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

hub

roller

spring

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

Documents

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