Hopf bifurcation character of an interactive vehicle–road shimmy system under bisectional road conditions

The adhesion coefficients of a bisectional road have significant coupled influences on the shimmy characteristics of the front wheels of a vehicle. A four-degree-of-freedom model for a representative sport utility vehicle was established. This model considered the adhesion coefficients of a bisectional road and the friction of the steering system of the suspension. The existence and stability of the system’s limit cycles were qualitatively determined using the Hopf bifurcation theorem and the centre manifold theory based on the model. The influences of the adhesion coefficients on the Hopf bifurcation characteristics of the system were calculated using a numerical method. The results showed that the road adhesion coefficient μ₁ of the left front wheel and the road coefficient μ₂ of the right front wheel significantly affected the vehicle shimmy and coupling relationship when different. Keeping μ₁ at a certain value, the swing angles and the angular velocities of the two front wheels consistently decreased when μ₂ decreased. The phenomenon repeatedly occurred when the difference Δμ between the adhesion coefficients increased. Moreover, the discrepancy between the amplitude of the left front wheel and the amplitude of the right front wheel is much more apparent when both the adhesion coefficients are larger. Good agreement between the shimmy characteristics of the two wheels was also found when comparing the results using the centre manifold reduced-dimensions method with the numerical method. Furthermore, a higher reduced order caused the reduced system to be closer to the original system.