Analysis of Hopf Bifurcation of Steering Wheel Shimmy of Automobile

Abstract

It proves that steering wheel shimmy is a vibration of stable limit cycle occurring after Hopf bifurcation, which is elaborated by nonlinear dynamics theory, and the control objectives of shimmy are proposed according to its bifurcation properties. Numerical analysis of bifurcation characteristics has been conducted with a nonlinear shimmy model whose parameters come from a domestic automobile with independent suspension. The results indicate that when the speed reaches 49.98Km/h, supercritical Hopf bifurcation occurs to the system and stable limit cycle appears, i.e. wheels oscillate around the kingpin at the same amplitude; when the speed comes to 76.30 Km/h, Hopf bifurcation occurs again and limit cycle disappears. The bifurcation speed and amplitude of limit cycle match the shimmy speed and amplitude measured from road experiments very well, which confirms the conclusions that shimmy is a vibration of stable limit cycle occurring after Hopf bifurcation at critical speed.