Study On Mechanism And Control Strategies Of Vehicle Steering Wheel Shimmy With Bifurcation Theories

Bifurcation theories of nonlinear dynamics and stability theories of ordinary differential equation (ODE) have been considered to elaborate the bifurcation character of steering wheel self-excited shimmy of motor vehicle. It proves theoretically that self-excited shimmy is a vibration phenomenon of stable limit cycle (LC) occurring after Hopf bifurcation. Some control objectives to attenuating shimmy have been proposed on the basis of its bifurcation property, and preliminary exploration has been made on research methods of shimmy control strategies. A simple shimmy model of three degree of freedom (DOF), parameters of which come from a domestic motor vehicle with independent suspension, has been constructed as an example to conduct numerical analysis of bifurcation property, road test and numerical simulation. The results confirm the conclusions. This dissertation consists of six chapters. Chapter one first introduces the concept of shimmy and summarizes its five characteristics: 1) shimmy is not a phenomenon occurring to all vehicles, and it happens more likely to used vehicles than new ones; 2) occurrence of shimmy is related to intensity of excitement, such as road perturbations, etc. and shimmy occurs when the excitement is intense enough; 3) shimmy does appear between certain limits of speed, and its amplitude increases with speed in neighborhood of critical shimmy point; 4) shimmy most frequently appears in the range of 40 Km/h to 70 Km/h, and it has larger amplitude and lower frequency than that appearing above 100 Km/h; 5) the shimmy frequency, amplitude and critical speed of a vehicle remain almost unchanged under different road conditions. Then a systematical overview of the achievements in shimmy documented in the references is presented. Based on the above overview, recommendations of the most critically needed improvements in the study of steering wheel shimmy are given. Bifurcation analysis of shimmy must be conducted in order to make further exploration into this phenomenon and seek some strategies to attenuate shimmy. The focus and significance of the research studied in the paper are elaborated in the final. Chapter two first presents some fundamental concepts related to bifurcation of nonlinear dynamics, such as phase space,singular point, LC and bifurcation, etc. It demonstrates that the nonlinear system with either stable LC or unstable LC is unstable. Then the study on the topological normal form for fold and Hopf which are only two kinds of generic equilibria bifurcation occurring in one-parameter continuous-time system have been elaborated by centre manifold theorem and Poincaré-Birkhoff normal form. Finally, it proves theoretically that self-excited shimmy is a vibration phenomenon of stable LC occurring after Hopf bifurcation. A simple shimmy model of 3 DOFs, parameters of which come from a motor vehicle with independent suspension, has been constructed in chapter three. The nonlinear relationship between slip angle and lateral force has been considered with Magic Formula of tire, and the non-holonomic rolling constrain between tire and ground is deduced by string model equations. Detailed numerical analysis has been made on bifurcation property of shimmy system with smooth nonlinearity. This chapter also introduces the algorithms to determine whether equilibria bifurcation will happen in a system when the speed varies, and the algorithms to determine the type and the speed of bifurcation, stability, period and amplitude of limit cycle occurring after bifurcation. The computation results indicate that when the speed reaches 46.14 Km/h, supercritical Hopf bifurcation occurs in the system and stable limit cycle appears, i.e. wheels oscillates around the kingpin at the equal amplitude which increases as the speed goes up; when the speed comes to 56.34 Km/h, the amplitude of limit cycle decreases as the speed goes up; supercritical Hopf bifurcation occurs again and limit cycle disappears when the speed comes to 70.50 Km/h. Numerical simulation with Simulink, a toolbox of Matlab, has been made on the shimmy system and the results confirm the validity of the calculation results. The calculated bifurcation speed and amplitude of limit cycle also matchthe shimmy speed and amplitude measured in road experiments. It confirms the conclusions that the critical bifurcation speed is the shimmy speed of actual vehicles and self-excited shimmy is vibration of stable limit cycle occurring after Hopf bifurcation at critical speed. Chapter four first discusses the description of clearance and dry friction nonlinearities of steering wheel shimmy model in time domain and changes those non-smooth nonlinearities into linear functions about amplitude and frequency of shimmy by sinusoidal-input describing function (SIDF) analysis methods in frequency domain. Then the influence of clearance and dry friction nonlinearities on steering wheel shimmy has been analyzed by eigenvalue method. The validity of linear shimmy model constructed by SIDF has been confirmed by simulation with Simulink. The results show that the unstable LC happens in the shimmy system due to the existence of dry friction, which best explains why occurrence of shimmy relates to intensity of excitement and why the shimmy occurs when the excitement is intense enough. The clearance factor influences the shimmy mainly in two aspects: one is that it will intensify and deteriorate the shimmy phenomenon and amplify the amplitude of the existent limit cycle; the other is that the larger clearance will make the speed limit of shimmy even larger. It explains well why shimmy occurs more likely to used vehicles than new ones. On the basis of the bifurcation character of steering wheel shimmy, Chapter 5 presents the shimmy control objectives, i.e. if the stable limit cycle appears, minimize the amplitude so as to make it disappear; if the unstable limit cycle occurs, maximize the amplitude to improve the system resistance to shimmy. It expounds the concept of bifurcation parameter sensitivity and analyzes the parameter sensitivity of the shimmy system with smooth nonlinear factors constructed in chapter 3. It explains the variability of factors contributing to shimmy and why the measures applicable to weaken shimmy in sample vehicle A have no apparent effect on sample vehicle B, and sometimes are even useless. It provides the theoretical foundation and the effective algorithmic program that can analyze the cause of shimmy for the specific vehicle. On the basis of the concept of bifurcation parameter percentage sensitivity, an optimized method to weaken and even eliminate shimmy has been put forward. The idea has been proposed by the author and is not subject to the textual research in other references. With numerous...