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

Single-Wheel Braking Model Governing Equations

Tractive Properties s Longitudinal Wheel Slip   Longitudinal Friction Characteristic Friction Law

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

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 

Tractive Properties Friction Characteristics “Magic Formula” of Pacejka

Tractive Properties Friction Characteristics

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

Choice of Dynamics States EOM in terms of u and s

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

Slip Dynamics Equilibria Local Stability

State-Space Description Stable Braking

State-Space Description Impending Possible Lockup

State-Space Description Possible Lockup

State-Space Description Impending Guaranteed Lockup

State-Space Description Guaranteed Lockup

Bifurcation of Slip Dynamics

Lockup Instability Critical Brake Torque

Brake Torque Error

Two-Wheel Braking Model Governing Equations

Tractive Properties Friction Laws Wheel Slip Friction Characteristic

Equations of Motion Same structure as EOM for SWBM

Example State-Space Description

Bifurcation of Slip Dynamics

FRONT WHEEL Stable Braking REAR WHEEL Stable Braking

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

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

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

FRONT WHEEL Stable Braking REAR WHEEL Guaranteed Lockup

Lockup Instability

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.

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