The vertical load of the tire has a significant influence on the lateral force, so the influence of the dynamic load on vehicle shimmy should be taken into account. Based on the dynamic model of a quarter vehicle, a three-degrees-of-freedom dynamic model of the shimmy system with consideration of the road roughness excitation is established by applying the second Lagrange equation. The response characteristic of the system is investigated by the numerical simulations. Moreover, the complexification-averaging method is used to obtain the analytical expression of the shimmy angle of the front wheel, and then, the stability of periodic solutions of the system is evaluated based on the bifurcation theory. Finally, the saddle-node bifurcation and Hopf bifurcation of the shimmy system are studied. The influence of the system parameters on the bifurcation characteristic of the system is also investigated, and the results obtained by using the complexification-averaging method are compared with the numerical examples.

To analyze the influence of the road roughness excitation on vehicle shimmy, a 12 degrees-of-freedom dynamic model of vehicle–road system is developed. The Hopf bifurcation theory is used to study the system stability. On this basis, the natural frequency and modal properties of the vehicle system are elaborated. It can be found that the roll mode plays a crucial role in the vehicle stability. Then, the dynamic shimmy behavior exposed to the sinusoidal and random road roughness excitations is investigated with the help of the modal analysis and the largest Lyapunov exponent. Furthermore, the numerical results are verified through the measurement results, and the influence of the front wheel track on vehicle shimmy is also examined. The results show that the decrease of the front wheel track is an effective way to attenuate vehicle shimmy for different road roughness excitations.

In this paper, the variation of tire longitudinal force and vehicle motions is studied when vehicle passing through the boundary of two road surfaces with different friction coefficient. The critical longitudinal slip of total sliding on the front and back roads is obtained by theoretical analysis. Then the variation of longitudinal force in different range of longitudinal slip is analyzed and a mathematical model is established to describe the transient force characteristics of tire. The longitudinal force, the longitudinal slip, the forward speed, and the angular velocity of the wheel on μ-step road are tested experimentally for different initial longitudinal slips, compared the tested results with those calculated by mathematical model. The results show that the initial longitudinal slip before passing through the road boundary is directly related to the change of longitudinal force and vehicle motions. The larger the initial longitudinal slip is, the more obvious the transient characteristics of the tire will be. For the vehicle driving from high-adhesion-road to low-adhesion-road. When the initial longitudinal slip is small, the longitudinal force decreases rapidly at first, and then quickly returns to the original value and remains constant on the low-adhesion-road. When the initial longitudinal slip exceeds the maximum recoverable longitudinal slip, the reduced longitudinal force cannot be recovered or can only be partially recovered on the low-adhesion-road. For the case of low-adhesion-road to high-adhesion-road, the longitudinal force will continue to increase as the tire passes through the boundary, and then gradually decrease to the original value when contact area passes through the boundary completely. In generally, the increased initial slip rate will not cause the longitudinal force irrecoverable. The relevant conclusions can provide theoretical references for vehicle dynamics and ABS control.

Since the vehicle is a complex mechanical system with many subsystems, the influence of the dynamic coupling between the subsystems of the vehicle on shimmy should be taken seriously. Therefore, a 12 degrees-of-freedom dynamic model of vehicle shimmy system with consideration of the dynamic coupling between the vertical motion and the lateral motion of the vehicle is established. In particular, the influence of the vertical load of the tire on the nonlinear cornering force is also considered. Then, the dynamic stability of the shimmy system is discussed with the help of the system eigenvalues, and the influence of the damping of the front suspension on the dynamic stability is also examined from the viewpoint of the energy transfer. On this basis, the damping of the front suspension is selected as a bifurcation parameter, and the two-dimensional center manifold of the high dimensional shimmy system is obtained by means of the center manifold theory. Finally, the Hopf bifurcation characteristic of the shimmy system is analyzed, and the analytical solutions of the shimmy system are derived for different damping of the front suspension. The results show that the increase of the damping of the front suspension is beneficial to attenuate vehicle shimmy and improve the dynamic stability of the vehicle.