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Nonlinear Dynamics of Longitudinal Vehicle Braking

Nonlinear Dynamics of Longitudinal Vehicle Braking. VSDIA 04, Budapest, Hungary, November 8-10, 2004. Outline. Single-Wheel Braking Model (SWBM) Tractive Properties Choice of Dynamic States Key Features of the EOM Slip Dynamics Two-Wheel Braking Model (2WBM) Closing Remarks.

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Nonlinear Dynamics of Longitudinal Vehicle Braking

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  1. Nonlinear Dynamics of Longitudinal Vehicle Braking VSDIA 04, Budapest, Hungary, November 8-10, 2004

  2. Outline • Single-Wheel Braking Model (SWBM) • Tractive Properties • Choice of Dynamic States • Key Features of the EOM • Slip Dynamics • Two-Wheel Braking Model (2WBM) • Closing Remarks

  3. Single-Wheel Braking Model • Governing Equations

  4. Tractive Properties Friction Law s Longitudinal Wheel Slip m Longitudinal Friction Characteristic

  5. Tractive Properties Wheel Slip • Dimensionless measure of the difference between u and R

  6. Tractive Properties Wheel Slip • Dimensionless measure of the difference between u and R • In the steady-state: • Finite rotation (R 0) with s2(0,1), or • Lockup (R = 0) with s = 1

  7. Tractive Properties Friction Characteristics “Magic Formula” of Pacejka

  8. Tractive Properties Friction Characteristics

  9. Choice of Dynamics States EOM in terms of u and  • u and  are coupled in a complicated way via the slip. where

  10. Choice of Dynamics States EOM in terms of u and s

  11. Equations of Motion Key Features • Slip dynamics are essentially decoupled • Search for steady (constant) slip values s = s* that correspond to steady-state braking

  12. Slip Dynamics • Equilibria • Local Stability

  13. State-Space Description Stable Braking

  14. State-Space Description Impending Possible Lockup

  15. State-Space Description Possible Lockup

  16. State-Space Description Impending Guaranteed Lockup

  17. State-Space Description Guaranteed Lockup

  18. Bifurcation of Slip Dynamics

  19. Lockup Instability • Critical Brake Torque

  20. Brake Torque Error

  21. Brake Torque Error

  22. Brake Torque Error

  23. Two-Wheel Braking Model • Governing Equations

  24. Tractive Properties • Friction Laws • Wheel Slip • Friction Characteristic

  25. Equations of Motion • Same structure as EOM for SWBM

  26. Example State-Space Description

  27. Bifurcation of Slip Dynamics

  28. Bifurcation of Slip Dynamics FRONT WHEEL Stable Braking REAR WHEEL Stable Braking

  29. Bifurcation of Slip Dynamics FRONT WHEEL Stable Braking REAR WHEEL Impending Possible Lockup

  30. Bifurcation of Slip Dynamics FRONT WHEEL Stable Braking REAR WHEEL Possible Lockup

  31. Bifurcation of Slip Dynamics FRONT WHEEL Stable Braking REAR WHEEL Impending Guaranteed Lockup

  32. Bifurcation of Slip Dynamics FRONT WHEEL Stable Braking REAR WHEEL Guaranteed Lockup

  33. Lockup Instability

  34. Lockup Instability

  35. Closing Remarks • Use of wheel slip s provides new insight into vehicle traction; • Entire dynamics (stability and bifurcation) are captured by hb(s) (SWBM) and hbi(s) (2WBM); • New lockup threshold.

  36. Acknowledgments • BUTE Dept. of Applied Mechanics • US-Hungarian Joint Fund for Technological Development • MSU Dept. of Mechanical Engineering • MSU Institute for Global Engineering Education • National Science Foundation

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