An Advanced Shell Theory Based Tire Model

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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
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Outline

• Objectives
• Tire Model Formulation
• Numerical Method
• Results

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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

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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

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Tire model formulation

• General approach
• General Linear Thin Shells Theory
• Mechanics of Laminated Composite Materials
• Numerical Method

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Tire model formulation

• General approach
• General Linear Thin Shells Theory
• Mechanics of Laminated Composite Materials
• Numerical Method

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Tire model formulation

• General compatibility equations of plane strain
  thin shells

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Tire model formulation

• Shell force and moment resultants

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Tire model formulation

• Equilibrium equations for static shell

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Tire model formulation

• Applied forces to shell element

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Tire model formulation

• General approach
• General Linear Thin Shells Theory
• Mechanics of Laminated Composite Materials
• Numerical Method

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Tire model formulation

• Mechanics of Laminated Composite Materials

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Tire model formulation

• Mechanics of Laminated Composite Materials
• Constitutive equations

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Tire model formulation

• Mechanics of Laminated Composite Materials
• Stress and Moment resultants

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Tire model formulation

• Mechanics of Laminated Composite Materials

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Tire model formulation

• 32x8.8 Type VII aircraft tire

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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

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Tire model formulation

• General approach
• General Linear Thin Shells Theory
• Mechanics of Laminated Composite Materials
• Numerical Method

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Discrete structure

• Uniform grid

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Numerical solution

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

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Numerical solution

• Apply finite difference method for partial
  derivatives

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Numerical solution

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

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Setting parameters

• Radii of curvature Lamé parameters

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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

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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

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Displacements

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

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Displacements

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

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Displacements

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

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Displacements

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

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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

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Questions ?