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 WHEELPossible Lockup

FRONT WHEELStable Braking

REAR WHEELImpending Guaranteed Lockup

Bifurcation of Slip Dynamics

Bifurcation of Slip Dynamics

FRONT WHEELStable Braking

REAR WHEELGuaranteed 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