Motorcycle local stability analysis under acceleration and braking by model linearization and eigenproblem solution

A study on the dynamic stability of a motorcycle under severe acceleration and braking conditions is presented. The purpose is to analyze the experimentally observed oscillatory behavior, such as wheel bounce or chatter and rear suspension pitch. This phenomenon greatly affects the handling performance and the safety of sports motorcycles. A plane motion lumped parameter model of the vehicle is defined by the variational virtual work approach. Tire longitudinal forces are modeled using Pacejka's nonlinear equations for quasi steady states; the kinematical model of the chain drive system is also included. The obtained nonlinear system of second order ordinary differential equations is linearized in symbolic form and transformed into the state space. The same equations of motion are numerically integrated in a commercial multibody code to compute the time histories simulating different maneuvers at quasi constant values of vehicle longitudinal acceleration. Local stability is evaluated via the model eigenvalues that are computed with respect to instantaneous equilibrium points evaluated by numerical integration. Root locus plots are traced to evaluate the graphical representation of damping and frequency associated with previously obtained eigenvalues, under different operating conditions. Coupling of standard key vehicle modes (front/rear wheel hop and suspension pitch) are also visualized by eigenvector animations. The influence of tire characteristics on the damping values of these key motorcycle modes is pointed out. The sensitivity of structural parameters on the stability of the lightly damped and critical modes is then discussed in detail.