Linearized equations for an extended bicycle model

Abstract

The linearized equations of motion for a bicycle of the usual construction travelling straight ahead on a level surface have been the subject of several previous studies [1], [2]. In the simplest models, the pure-rolling conditions of the knife-edge wheels are introduced as non-holonomic constraints and the rider is assumed to be rigidly attached to the rear frame. There are two degrees of freedom for the lateral motion, the lean angle of the rear frame and the steering angle. In the present paper, the model is extended in several ways, while the simplicity of having only two degrees of freedom is retained. The extensions of the model comprise the shape of the tires, which are allowed to have a finite transverse radius of curvature, the effect of a pneumatic trail and a damping term due to normal spin at the tire contact patch, the gradient of the road, the inclusion of driving and braking torques at the wheels and the aerodynamic drag at the rear frame.

Owing to the gradient, the yaw angle of the rear frame is no longer a cyclic coordinate and the kinematic differential equation for its evolution needs to be included. A further consequence is that the stiffness matrix is no longer symmetric, even for zero speed and acceleration. The way of decelerating has a marked influence on the stability characteristics: braking at the rear wheel, braking at the front wheel, aerodynamic drag and riding up an incline influence the lateral dynamics in different ways. The acceleration makes the coefficients of the linearized system time-varying.

A comparison of the derived equations and the results obtained by a multibody dynamic program is made, which shows a complete agreement. The equations can be used for several purposes: firstly, they provide a non-trivial example of a non-holonomic system that can be used to illustrate some of the characteristic properties of systems of this kind; secondly, they can be used as a test problem for the verification of multibody dynamic codes; thirdly, the simple model already yields valuable insight in the effects of several system parameters on the dynamics of a bicycle.