A Nonlinear Model for Vehicle Braking

The purpose of this study is to investigate and understand the dynamics of motor vehicles un-der straight ahead braking and accelerating condi-tions. Two vehicle braking models are considered— a quarter-car, or single wheel model, and a half-car, or two wheel model—and a nonlinear analysis is un-dertaken for each. A brief survey of the main char-acteristics and features of a single wheel acceleration model is also conducted. The equations of motion describing the single wheel model shown in Figure 1 are presented and the need to quantify the available friction force for braking is specified. This motivates an investigation of the tire/road interface and leads to the introduc-tion of force coefficient characteristics µ as a function of longitudinal wheel slip s. The equations of motion are hence cast into a framework which is convenient for a nonlinear dynamic analysis. The forward vehi-cle speed u is customarily chosen as an independent variable for such an investigation. Whereas the an-gular velocity of the wheel ω is typically chosen as the second independent variable for braking anal-ysis [1], this paper utilizes wheel slip in its stead. This results in a single dimensionless function h(s) of longitudinal wheel slip that completely character-izes the nonlinear dynamic behavior of the system. It will be shown that considerable insight is gained from such a description. Criteria for the local sta-bility of finite slip in braking and wheel lockup are outlined and a qualitative vector field description of the braking model is presented and discussed. The main results of the investigation include domains of stable and unstable braking, local stability criteria, and a description of the transition to wheel lock-up as the brake torque is varied. The single wheel model equations of motion can be cast in the form u ̇ = −µ(s)g s ̇ = g u h(s) where g is the acceleration due to gravity and µ(s) is an experimentally derived friction coefficient charac-teristic. It is defined by the ratio of the longitudinal braking force X to the reaction force Z of the singl