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An Advanced Shell Theory Based Tire Model

The 23 rd Annual Tire Society Meeting 2004. An Advanced Shell Theory Based Tire Model. by D. Bozdog, W. W. Olson. Department of Mechanical, Industrial and Manufacturing Engineering. Outline. Objectives Tire Model Formulation Numerical Method Results. Motivation.

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An Advanced Shell Theory Based Tire Model

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  1. The 23rd Annual Tire Society Meeting 2004 An Advanced Shell Theory Based Tire Model by D. Bozdog, W. W. Olson Department of Mechanical, Industrial and Manufacturing Engineering

  2. Outline • Objectives • Tire Model Formulation • Numerical Method • Results Tire Society Meeting 2004

  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 Tire Society Meeting 2004

  4. 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 Objectives Tire Society Meeting 2004

  5. Tire model formulation • General approach • General Linear Thin Shells Theory • Mechanics of Laminated Composite Materials • Numerical Method Tire Society Meeting 2004

  6. Tire model formulation • General approach • General Linear Thin Shells Theory • Mechanics of Laminated Composite Materials • Numerical Method Tire Society Meeting 2004

  7. Tire model formulation • General compatibility equations of plane strain thin shells: Tire Society Meeting 2004

  8. Tire model formulation • Shell force and moment resultants Tire Society Meeting 2004

  9. Tire model formulation • Equilibrium equations for static shell Tire Society Meeting 2004

  10. Tire model formulation • Applied forces to shell element Tire Society Meeting 2004

  11. Tire model formulation • General approach • General Linear Thin Shells Theory • Mechanics of Laminated Composite Materials • Numerical Method Tire Society Meeting 2004

  12. Tire model formulation • Mechanics of Laminated Composite Materials Tire Society Meeting 2004

  13. Tire model formulation • Mechanics of Laminated Composite Materials • Constitutive equations: Tire Society Meeting 2004

  14. Tire model formulation • Mechanics of Laminated Composite Materials • Stress and Moment resultants Tire Society Meeting 2004

  15. Tire model formulation • Mechanics of Laminated Composite Materials Tire Society Meeting 2004

  16. Tire model formulation • 32x8.8 Type VII aircraft tire Tire Society Meeting 2004

  17. 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 System of equations Tire Society Meeting 2004

  18. Tire model formulation • General approach • General Linear Thin Shells Theory • Mechanics of Laminated Composite Materials • Numerical Method Tire Society Meeting 2004

  19. Discrete structure • Uniform grid Tire Society Meeting 2004

  20. Numerical solution • Transform system of 17 equations to eight-order system of 3 governing partial differential equations Tire Society Meeting 2004

  21. Numerical solution • Apply finite difference method for partial derivatives Tire Society Meeting 2004

  22. Numerical solution • Fit tire profile in f and x direction with continous functions Tire Society Meeting 2004

  23. Setting parameters • Radii of curvature & Lamé parameters Tire Society Meeting 2004

  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 Tire Society Meeting 2004

  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 Tire Society Meeting 2004

  26. Displacements • Cross-section tangential displacements 0-95psi for 32x8.8 Type VII aircraft tire Tire Society Meeting 2004

  27. Displacements • Cross-section normal displacements 0-95psi for 32x8.8 Type VII aircraft tire Tire Society Meeting 2004

  28. Displacements • Cross-section displacements 0-95psi for 32x8.8 Type VII aircraft tire Tire Society Meeting 2004

  29. Displacements • Initial • Cross-section displacements for 95psi • Previous results by Brewer Tire Society Meeting 2004

  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 Tire Society Meeting 2004

  31. Questions ? Tire Society Meeting 2004

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