Response Surface Methodologyand

Its application to automotivesuspension designs

Tatsuyuki AmagoOffspring of candidate for former

general (SHOGUN)

Toyota Central R&D Labs., Inc 2

OutlineI. Introduction & Basis of RSM

1. History of RSM2. What’s RSM3. Why is RSM4. Least square method5. Design Of Experiment (DOE)

II. Its application to automotive suspension designs

1. Size optimization for beam stiffens2. Size optimization for section beam property

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I. Introduction and Basis of Response surface Methodology(RSM)

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History of RSM1951 Box & WilsonBox & WilsonContributed RSM of Quadratic Polynomials

1988 ~ 1990 Design Of Experiments (DOE)TaguchiTaguchi DOE (Taguchi Method)MyersMyers & Montgomery& Montgomery DOE RSM

1992 ~ 1994 Quality engineeringThe Process of a semiconductor … etc

1995 Optimization for Numerical AnalysisHaftkaHaftka Composite Wing Structural OptimizationShiratoriShiratori Design Optimization of automotive Seat frame

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What’s RSMApproximation OptimizationApproximation Function = Response surfaceResponse surfaceLeast square method & Design of Experiments

Design Variable 1 Design

Vari

able

2

State Variable

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Why is R.S.M. Conventional OptimizationOptimization Problem setup

Design Variable,Objective Function,Constraint

Sensitivity Calculation　　　　

OptimizationPart

Optimum ValueA solution is not obtaineda nonlinear large problem.

Analysis (FEM etc)　　

Huge calculation Time & Resources

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Why is R.S.M.

Optimization problem setupDesign Variable,Objective Function, Constraint

Optimization calculateUsing the Response Surface

Optimum valueCalculation is

very early

Response surface creationResponse surface creationA function is approximated.A function is approximated.

Design ValiableDesign ValiableDesign ValiableDesign Valiable

Obje

cti

ve F

uncti

on

Obje

cti

ve F

uncti

on

Obje

cti

ve F

uncti

on

Obje

cti

ve F

uncti

on

ResuponseResuponseResuponseResuponse Surface Surface Surface Surface

Analysis ResultAnalysis ResultAnalysis ResultAnalysis Result

Optimum ValueOptimum ValueOptimum ValueOptimum Value

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Response surface creation

Design of Experiments

Parametric Design

Analysis (FEM etc)

Response surface creationApproximation for

Objective function & Constraint Design ValiableDesign ValiableDesign ValiableDesign Valiable

Obj

ecti

ve F

unct

ion

Obj

ecti

ve F

unct

ion

Obj

ecti

ve F

unct

ion

Obj

ecti

ve F

unct

ion ResuponseResuponseResuponseResuponse

Surface Surface Surface Surface

Analysis ResultAnalysis ResultAnalysis ResultAnalysis Result

Optimum ValueOptimum ValueOptimum ValueOptimum Value

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Response surface

Design Variables : ( )nixi 1=

( ) ε+= nxxfy 1

Least Square Method PolynomialsExponentialLogarithm … etc

Neural NetworkSpline interpolationLagrange interpolation

State Variable :

Linearized

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Least square method (1)

Quadratic polynomials∑∑∑<==

+++=n

jijiij

n

iiii

n

iii xxxxy ββββ

0

2

00

Example of two variables for simplification

Linearizedxxxxxxxxxxxxxy

→===

+++++=

5214223

21

215224

21322110

,,ββββββ

Used Linear functionCoefficient of functionEvaluation of function Easily obtained by Statistics

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Least square method (2)

Number of Experiments : nNumber of variable : k

=

=

=

=

+=

nnnknn

k

k

n xxx

xxxxxx

y

yy

ε

εε

β

ββ

2

1

2

1

21

22221

11211

2

1

1

11

εεεεββββ

εεεεββββ

Xy

Xy

Error Sum of Square : L=ε T ε MinimizeLeast square estimations of β : b

( ) yXXXb TT 1−=

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Least square method (3)

Sum Square of Error :

Sum Square of Regression :

Total Sum of Square :

Coefficient of multiple determination : R2

yXbyy TTT −=SSE

nTSSR 2−= yXb TT

Total Sum : ∑=

=n

iiyT

1

nTSyy 2−= yyT

SyySSESyySSRR −== 12

Adjusted Coefficient of multiple determination : Rad2

)1()1(12

−−−−=

nSyyknSSERad

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Design Of Experiments (DOE)Parameter design for efficient experiment

= for obtained better regression formulation

Response surface by least square method

Variance covariance matrix ( V(b)=cov(bi,bj) )by least square estimations b

( ) 12)( −= XXb TV σσ2 : Error Variance of state variable y Unknown

Minimize coefficient variance

Minimize for diagonal of (XT X)-1

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Orthogonal designMainly used for linear polynomialOrthogonal arraysLinear 2-Level factorial design [ L8(27), L16(215) …]Quadratic 3-Level factorial design [ L9(34), L27(313) …]

Orthogonal polynomialsChebyshev orthogonal polynomials

Efficient for low order & no interactionsLarge design number for highest-orderBut, very easy

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Central Composite Design (CCD)

Mainly used for quadratic polynomialParametric design2-level full factorial design nF = 2k

Center point n0 > 1Two axial point on axis of each design Variable

at distance of design origin. nR = 2kTotal Number of design n = 2k + 2k + n0

No direction dependability = Rotatable design Better design for quadratic polynomial

2 variablemodel

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Computer support design A-optimalityMoment matrix :Sum of diagonal value for M inverse : trace(M-1)

Minimize[ trace(M-1) ] Found XConsideration for diagonal value

Therefore, Not rotatable design

D-optimalityMaximize[ M ] Found X

Consideration for Full value by Mtherefore, rotatable design

This method best parametric design. But, difficult

nXXM

T=

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Improvement of RSUsed high order PolynomialZooming methodDomain decomposition methodKriging model ….. etc

First domain

First Optimal value

Second domain

Decomposed domainUsed low order polynomial

Zooming Method Domain decomposition Method

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RSM Program on ExcelProgram for Orthogonal design approach

3 factors and3 levels Design

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II. Its application to automotive suspension designs

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Trailing Twist Axle SuspensionMixture of structures and mechanismsGood for FF automobilesAdvantages

Simple structures & low costNo suspension frameHigh stiffness

Twisting CrossbeamCritical Design Part

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Examples

1) Size optimization for beam stiffens2) Size optimization for section beam property

y

z

Shear center

Center of gravity

Torsion

Bending y

z

Shear center

Center of gravity

Iy

Iz

J

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1) Size Optimization for stiffnessDesign variableThickness : t and Forming length : L

Objective functionMinimize total mass

ConstraintTorsional and Bending stiffness

t

L

2r

Section of beam

Torsion

Bending

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Optimization techniqueParametric Studies Based on DOE

Analysis for FEM

Optimization used RSMOptimum Design variables

Initial design variable for next step

Optimization used FEM

Torsion Bending

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Parametric Studies Based on DOETwo factors, three levels in DOE

L/r = 0.0,0.5,1.0 where r=50(mm)t=1.0,2.5,4.0 (mm)

Evaluation functionTotal volume : VTorsional stiffness : GT

Bending stiffness : GS

4192.51811740.56

2.50.5510.54403

2.502101tL/r

L9(34) Orthogonal arrays

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Response surface

t

Total volumeL/r

tL/r

tL/r

Bending stiffness

V

Torsion stiffness

GT GB

Feasibleregion

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Optimization Using FEA(Ansys)

Initial value (Central value) L/r=0.5, t=2.5(mm)

Not Converged

More than 80 iteration

Case 2

Initial value ( Using RSM) L/r=0.9, t=2.5(mm)

Optimal Solution L/r=0.987, t= 1.94(mm)

13 iteration

Case 1

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Optimal design

Optimal design L/r = 0.987, t = 1.94(mm)

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2) Optimize for Cross Section Properties

Design variableThickness : t and Forming length : L

Objective functionMinimize Polar moment of Inertia J

Constrainty position of shear center ey≧10(mm)Second moment of Inertia Iy≧900,000 (mm4)Second moment of Inertia Iz≧100,000 (mm4)

t

L

2r

Section of beam

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Parametric Studies Based on DOETwo factors, three levels in DOE

L/r = 0.0,0.5,1.0 where r=50(mm)t=1.0,2.5,4.0 (mm)

Evaluation functiony position of shear center eySecond moment of Inertia Iy,IzPolar moment of Inertia J y

z

Shear center

Center of gravity

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Response surface

00.2

0.40.6

0.811

2

3

4

0

10

20

30

00.2

0.40.6

0.81

00.2

0.40.6

0.811

2

3

4

01́ 1062́ 1063́ 106

00.2

0.40.6

0.81

00.2

0.40.6

0.811

2

3

4

5000007500001́ 106

1.25́106

1.5́106

00.2

0.40.6

0.81 1

00.2

0.40.6

0.811

2

3

4

0

500000

1́ 1061.5́106

00.2

0.40.6

0.8

L/r L/rt

J

tL/r

Iy

L/rt

t

ey

Iz

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Optimal design

Optimal Solution L/r=1.0, t=2.47(mm)

Response Surface Methodologyand

Its application to automotivesuspension designs

Thank you very much !