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An Advanced Shell Theory Based Tire Model

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Framework for tire simulation analysis and development based on shell theory ... Accuracy of solutions is highly dependent on size of shell grid and CPU ... – PowerPoint PPT presentation

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Title: An Advanced Shell Theory Based Tire Model


1
An Advanced Shell Theory Based Tire Model
The 23rd Annual Tire Society Meeting 2004
  • by
  • D. Bozdog, W. W. Olson

Department of Mechanical, Industrial and
Manufacturing Engineering
2
Outline
  • Objectives
  • Tire Model Formulation
  • Numerical Method
  • Results

3
Motivation
  • Wide variety of tire models
  • Simplistic (spring-damper structures or curve
    fits of experimental data)
  • FEM extremely complex
  • Potential of elasticity based shell theory tire
    model
  • Provide both the benefits
  • Complex analysis
  • Fast computations
  • Assume material properties closed to real values
  • Requires small number of input parameters
  • Can be used for all types of tire design

4
Objectives
  • Short term
  • Determine the deformed shape of tire
  • Internal pressure
  • Vertical loadings
  • Longitudinal and lateral forces
  • Determine the stress-strain distribution in
    structure
  • Long term
  • Framework for tire simulation analysis and
    development based on shell theory
  • Provide a solution for tire modeling for vehicle
    dynamics simulation software

5
Tire model formulation
  • General approach
  • General Linear Thin Shells Theory
  • Mechanics of Laminated Composite Materials
  • Numerical Method

6
Tire model formulation
  • General approach
  • General Linear Thin Shells Theory
  • Mechanics of Laminated Composite Materials
  • Numerical Method

7
Tire model formulation
  • General compatibility equations of plane strain
    thin shells

8
Tire model formulation
  • Shell force and moment resultants

9
Tire model formulation
  • Equilibrium equations for static shell

10
Tire model formulation
  • Applied forces to shell element

11
Tire model formulation
  • General approach
  • General Linear Thin Shells Theory
  • Mechanics of Laminated Composite Materials
  • Numerical Method

12
Tire model formulation
  • Mechanics of Laminated Composite Materials

13
Tire model formulation
  • Mechanics of Laminated Composite Materials
  • Constitutive equations

14
Tire model formulation
  • Mechanics of Laminated Composite Materials
  • Stress and Moment resultants

15
Tire model formulation
  • Mechanics of Laminated Composite Materials

16
Tire model formulation
  • 32x8.8 Type VII aircraft tire

17
System of equations
  • System of equations
  • equilibrium equations
  • constitutive equations
  • compatibility equations
  • boundary conditions
  • 17 equations
  • 12 first order diff. eq.
  • 6 linear eq.
  • 17 unknowns
  • Displacements
  • Strains
  • Force and Moment Resultants

18
Tire model formulation
  • General approach
  • General Linear Thin Shells Theory
  • Mechanics of Laminated Composite Materials
  • Numerical Method

19
Discrete structure
  • Uniform grid

20
Numerical solution
  • Transform system of 17 equations to eight-order
    system of 3 governing partial differential
    equations

21
Numerical solution
  • Apply finite difference method for partial
    derivatives

22
Numerical solution
  • Fit tire profile in f and x direction with
    continous functions

23
Setting parameters
  • Radii of curvature Lamé parameters

24
Iterations
  • Successive iterations for internal and external
    loads
  • pf(f,x), px(f,x) , pz(f,x)
  • Compute after each iteration
  • Radii of curvature R1, R2 and A, B parameters
  • Deformed profile

25
Results
  • Computer code developed using MathematicaTM
    software
  • Strong performance in symbolic computation
  • Advanced mathematical tools
  • Fast execution
  • Iteration performed for pressure load by setting
    pf with 5psi increment for 32x8.8 Type VII
    aircraft tire
  • Results are determined for all 17 variables
  • Displacements
  • Strains Change of Curvature
  • Forces and Moment Resultants

26
Displacements
  • Cross-section tangential displacements 0-95psi
    for 32x8.8 Type VII aircraft tire

27
Displacements
  • Cross-section normal displacements 0-95psi for
    32x8.8 Type VII aircraft tire

28
Displacements
  • Cross-section displacements 0-95psi for 32x8.8
    Type VII aircraft tire

29
Displacements
  • Initial
  • Cross-section displacements for 95psi
  • Previous results by Brewer

30
Conclusion
  • Tire Model
  • Confirm previous results
  • Computer code (still in work)
  • Will have ability to perform complete analysis
    for pressure, longitudinal and lateral external
    forces
  • Theoretical model can incorporate
  • Variable thickness of cross-section
  • Variable stiffness matrix for tread, sidewall and
    bead regions
  • Variable cord path can be incorporated
  • Accuracy of solutions is highly dependent on size
    of shell grid and CPU
  • Code can be customize for specific tires

31
Questions ?
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