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 s Longitudinal Wheel Slip   Longitudinal Friction Characteristic Friction Law

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

6. Tractive Properties Dimensionless measure of the difference between u and  R Wheel Slip In the steady-state: 1.Finite rotation   R    with s 2 , or 2.Lockup   R  with s 

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

8. Tractive Properties Friction Characteristics

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

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

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

32. FRONT WHEEL Stable Braking REAR WHEEL Guaranteed Lockup

33. Lockup Instability

35. Closing Remarks Use of wheel slip s provides new insight into vehicle traction; Entire dynamics (stability and bifurcation) are captured by h b  s  (SWBM) and h bi  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