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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 PropertiesFriction Characteristics
“Magic Formula” of Pacejka
Tractive PropertiesFriction 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 StatesEOM in terms of u and s
Equations of MotionKey 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
Brake Torque Error
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
Bifurcation of Slip Dynamics
FRONT WHEELStable Braking
REAR WHEELStable Braking
Bifurcation of Slip Dynamics
FRONT WHEELStable Braking
REAR WHEELImpending Possible Lockup
Bifurcation of Slip Dynamics
FRONT WHEELStable Braking
REAR WHEEL
Possible Lockup
FRONT WHEELStable Braking
REAR WHEEL
Impending Guaranteed Lockup
Bifurcation of Slip Dynamics
Bifurcation of Slip Dynamics
FRONT WHEELStable Braking
REAR WHEEL
Guaranteed Lockup
Lockup Instability
Lockup Instability
Closing Remarks
• Use of wheel slip s provides new insight into vehicle traction;
• Entire dynamics (stability and bifurcation) are captured by hbs (SWBM) and hbis (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