nonlinear dynamics of longitudinal vehicle braking vsdia 04, budapest, hungary, november 8-10, 2004

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


Impending Possible Lockup


Possible Lockup


Impending Guaranteed Lockup


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


FRONT WHEELStable Braking

REAR WHEELStable Braking



REAR WHEELImpending Possible Lockup



REAR WHEEL

Possible Lockup


REAR WHEEL

Impending Guaranteed Lockup




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 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

nonlinear dynamics of longitudinal vehicle braking vsdia 04, budapest, hungary, november 8-10, 2004

Documents