Abstract: The nonlinear relationship between slip angle and lateral force has been considered with Magic Formula of tire to construct a shimmy model of three degree of freedom, parameters of which came from a domestic motor vehicle with independent suspension. It proves that steering shimmy is a vibration phenomenon of stable limit cycle occurring after Hopf bifurcation, which is elaborated by theories of ordinary differential equation and nonlinear dynamics. First, the bifurcation speed and existence of limit cycle are determined with Hurwitz determinant. Then stability of limit cycle is determined by the first coefficient of Lyapunov. Finally the amplitude of limit cycle is computed by simulation with Matlab. The results indicate that when the speed reaches 46.14 km/h, supercritical Hopf bifurcation occurs to the system and stable limit cycle appears, I.e. wheels oscillates around the kingpin at the equal amplitude, which can maximize 2.982°; when the speed comes to 70.50 km/h, supercritical Hopf bifurcation occurs again and limit cycle disappears. The bifurcation speed and amplitude of limit cycle are equal to the shimmy speed and amplitude from road experiments of the vehicle, which confirms the conclusion.

Key words: Dependent suspension, Hopf bifurcation, Limit cycle, Steering wheel shimmy