An Advanced Shell Theory Based Tire Model

An Advanced Shell Theory An Advanced Shell Theory Based Tire Model Based Tire Model

by by

D. Bozdog, W. W. OlsonD. Bozdog, W. W. Olson

Department of Mechanical, Industrial and Manufacturing Engineering

The 23rd Annual Tire Society Meeting 2004

September 20, 2004 2Tire Society Meeting 2004

OutlineOutline

ObjectivesObjectives

Tire Model FormulationTire Model Formulation

Numerical MethodNumerical Method

ResultsResults


MotivationMotivation

Wide variety of tire modelsWide variety of tire models Simplistic (spring-damper structures or curve fits of Simplistic (spring-damper structures or curve fits of

experimental data)experimental data) FEM extremely complexFEM extremely complex

Potential of elasticity based shell theory tire Potential of elasticity based shell theory tire modelmodel Provide both the benefitsProvide both the benefits

Complex analysisComplex analysis Fast computationsFast computations

Assume material properties closed to real valuesAssume material properties closed to real values Requires small number of input parametersRequires small number of input parameters Can be used for all types of tire designCan be used for all types of tire design


ObjectivesObjectives Short termShort term

Determine the deformed Determine the deformed shape of tireshape of tire Internal pressureInternal pressure Vertical loadingsVertical loadings Longitudinal and Longitudinal and

lateral forceslateral forces

Determine the stress-Determine the stress-strain distribution in strain distribution in structure structure

Long termLong term

Framework for tire Framework for tire simulation analysis and simulation analysis and development based on development based on shell theory shell theory

Provide a solution for Provide a solution for tire modeling for vehicle tire modeling for vehicle dynamics simulation dynamics simulation softwaresoftware


Tire model formulationTire model formulation General approachGeneral approach

General Linear Thin Shells TheoryGeneral Linear Thin Shells Theory

Mechanics of Laminated Composite MaterialsMechanics of Laminated Composite Materials








Tire model formulationTire model formulation General compatibility equations of plane General compatibility equations of plane

strain thin shells:strain thin shells:

0

111

1111

1111

11

11

21

212

21

2

1

zzz

B

v

AR

B

A

u

BR

AwwB

B

wA

AABK

w

AR

uB

AB

w

BR

v

BK

w

BR

vA

AB

w

AR

u

AK

A

u

B

A

B

v

A

B

R

wu

B

AB

v

B

R

wv

A

AB

u

A


Tire model formulationTire model formulation Shell force and moment resultantsShell force and moment resultants


Tire model formulationTire model formulation Equilibrium equations for static shellEquilibrium equations for static shell

0

0)()(

0)()(

0)()(

0)()(

0

21

21

2

1

R

M

R

MNN

ABQA

MB

MBMAM

ABQB

MA

MAMBM

ABpR

ABN

R

ABNAQBQ

ABpR

ABQ

AN

BNBNAN

ABpR

ABQ

BN

ANANBN

z


Tire model formulationTire model formulation Applied forces to shell elementApplied forces to shell element

,sin,

tansin,)(

]cos[tan,)(cos,

2

1

12

liz

l

l

Frhpp

u

vFvsignp

u

vFusignrhp







Tire model formulationTire model formulation Mechanics of Laminated Composite Mechanics of Laminated Composite

MaterialsMaterials

22

22

22

12

2

1

sincoscossincossin

cossin2cossin

cossin2sincos

662616

262212

161211

QQQ

QQQ

QQQ



MaterialsMaterials

Constitutive equations:Constitutive equations:

zK

zK

zK

0

0

0

kkkK

K

K

QQQ

QQQ

QQQ

z

QQQ

QQQ

QQQ

662616

262212

161211

0

0

0

662616

262212

161211

][



MaterialsMaterials

Stress and Moment resultantsStress and Moment resultants

2/

2/

2/

2/;

h

h

kk

h

h

kk

zdz

M

M

M

dz

N

N

N

n

k

z

z

n

k

z

z

k

k

k

k

zdz

M

M

M

dz

N

N

N

11 11

;



MaterialsMaterials

K

K

K

DDDBBB

DDDBBB

DDDBBB

BBBAAA

BBBAAA

BBBAAA

M

M

M

N

N

N

662616662616

262212262212

161211161211

662616662616

262212262212

161211161211

12

~)(

3

1

~)(

2

1

)(

3][2

][][1

][

31

3

1

][][1

][

21

2

1

][1

][11

kkk

n

kkijkk

k

n

kij

kk

n

kkijkk

k

n

kij

k

n

kkijkk

k

n

kij

thtQhhQD

htQhhQB

tQhhQA


K

K

K

M

M

M

N

N

N

38.2100091.12448.52

087.5139.2291.12400

039.2212.1148.5200

091.12448.5221.385400

91.12400085.93506.4035

48.520006.403595.2003

Tire model formulationTire model formulation

32x8.8 Type VII aircraft tire32x8.8 Type VII aircraft tire


System of equationsSystem of equations System of equations:System of equations:

equilibrium equationsequilibrium equations constitutive equationsconstitutive equations compatibility compatibility

equationsequations boundary conditions boundary conditions

17 equations17 equations 12 first order diff. eq.12 first order diff. eq. 6 linear eq.6 linear eq.

17 unknowns17 unknowns DisplacementsDisplacements

StrainsStrains

Force and Moment Force and Moment ResultantsResultants

KKK ,,,,,

wvu ,,

QQNNN ,,,,

MMM ,,







Discrete structureDiscrete structure Uniform gridUniform grid


Numerical solutionNumerical solution Transform system of 17 equations to eight-Transform system of 17 equations to eight-

order system of 3 governing partial order system of 3 governing partial differential equationsdifferential equations

),(),,,,,(

),(),,,,,(

),(),,,,,(

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

zph

pg

pf


Numerical solutionNumerical solution Apply finite difference method for partial Apply finite difference method for partial

derivativesderivatives

4

)1,1()1,1()1,1()1,1(

)1,(),(2)1,(

),1(),(2),1(

2

)1,()1,(

2

),1(),1(

2

22

2

22

2

jifjifjifjiff

jifjifjiff

jifjifjiff

jifjiff

jifjiff


Numerical solutionNumerical solution Fit tire profile in Fit tire profile in and and direction with direction with

continous functionscontinous functions

directioninkaat

directioninkaas

m

kk

n

kk

max

max

10

10

]cos[)(

]cos[)(


Setting parametersSetting parameters Radii of curvature & Lamé parametersRadii of curvature & Lamé parameters

)()()]([2)(

))]([)((),(

22

2/322

1

ssss

ssR

cossin

cossin)(arctan

ss

ss


IterationsIterations

Successive iterations for internal and Successive iterations for internal and external loadsexternal loads pp, p, p p pzz

Compute after each iterationCompute after each iteration Radii of curvature R1, R2 and A, B parametersRadii of curvature R1, R2 and A, B parameters Deformed profileDeformed profile


ResultsResults Computer code developed using MathematicaComputer code developed using MathematicaTMTM

softwaresoftware Strong performance in symbolic computationStrong performance in symbolic computation Advanced mathematical toolsAdvanced mathematical tools Fast executionFast execution

Iteration performed for pressure load by setting pIteration performed for pressure load by setting p with 5psi increment for 32x8.8 Type VII aircraft tirewith 5psi increment for 32x8.8 Type VII aircraft tire

Results are determined for all 17 variablesResults are determined for all 17 variables DisplacementsDisplacements Strains & Change of CurvatureStrains & Change of Curvature Forces and Moment ResultantsForces and Moment Resultants


DisplacementsDisplacements Cross-section tangential displacements 0-Cross-section tangential displacements 0-

95psi for 32x8.8 Type VII aircraft tire95psi for 32x8.8 Type VII aircraft tire


DisplacementsDisplacements Cross-section normal displacements 0-95psi Cross-section normal displacements 0-95psi

for 32x8.8 Type VII aircraft tirefor 32x8.8 Type VII aircraft tire


DisplacementsDisplacements Cross-section displacements 0-95psi for Cross-section displacements 0-95psi for

32x8.8 Type VII aircraft tire32x8.8 Type VII aircraft tire


DisplacementsDisplacements InitialInitial Cross-section displacements for 95psiCross-section displacements for 95psi Previous results by BrewerPrevious results by Brewer


ConclusionConclusion Tire ModelTire Model

Confirm previous resultsConfirm previous results Computer code (still in work)Computer code (still in work)

Will have ability to perform complete analysis Will have ability to perform complete analysis for pressure, longitudinal and lateral external for pressure, longitudinal and lateral external forcesforces

Theoretical model can incorporateTheoretical model can incorporate Variable thickness of cross-sectionVariable thickness of cross-section Variable stiffness matrix for tread, sidewall and Variable stiffness matrix for tread, sidewall and

bead regionsbead regions Variable cord path can be incorporatedVariable cord path can be incorporated

Accuracy of solutions is highly dependent on size Accuracy of solutions is highly dependent on size of shell grid and CPU of shell grid and CPU

Code can be customize for specific tiresCode can be customize for specific tires


Questions ?Questions ?

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An Advanced Shell Theory Based Tire Model