Hopf Bifurcation Study Of Hunting Behavior Of Vehicle Systems

Railway vehicle is a strongly nonlinear system with nonlinear wheel-rail contact relation and suspensions.According to the requirements of dynamic performance,suspension parameters can be optimized in order to obtain good vibration reduction and reduce wheel-rail contact forces.Here,dampers,as energy disspipation elements,strongly affect the train operating performance like hunting stability.Firstly,the influence of suspension damping on the hunting stability,especially the carbody primary hunting under different modal vibrations,is studied.And the effects of wheel-rail contact geometry and yaw damper nolinear parameters on the system Hopf bifurcation behaviour are analyzed.Besides,due to excessive wear of wheel profile,the nonlinear critical speed is greatly reduced,that may cause wheel flange-rail collision even at a normal speed.Thus,the complex dynamic phenomena from the collision are further investigated.The main research contents in the thesis are as following:(1)Summarizes the theoretical research results on lateral motion stability,bifurcation,chaos and asymmetry of vehicle hunting motion from different angles at home and abroad.On the basis of those research results,the main research work in this thesis is introduced.(2)Explains the hunting motions phenomena of vehicle system through the long-term tracking dynamic test and roller rig test.Hunting stability differences under suspension parameter variation and wheel wear condition in a complete operation cycle are compared.(3)Establishs a 7-DOF vehicle lateral dynamic model by taking a bogie with larger positioning stiffness as a rigid bogie.The modes of vehicle hunting motion are calculated with different speeds and suspension parameters.In order to suppress the carbody hunting motion,effects of different damping on carbody sway and yaw mode stability are studied.Finally,the influence of different damping parameters on the critical speeds of carbody primary and bogie secondary hunting motions is disccussed.The results show that when the lateral damping is small,carbody lower sway transfers from unstable to stable state with the vertical damping increasing,and then return to instability.Meanwhile,the stability of the carbody yaw motion is guaranteed at a larger longitudinal damping.As the vertical damping is small,the upper sway of the car body is instability,stability and instability successively with the lateral damping increasing.Moreover,When the longitudinal damping is small,the increaseing lateral damping,contributes to improving the yaw motion stability of the car body.(4)Based on the bogie linear critical speed calculation by using Lienard-Chipart stability criterion,Central Manifold Theorem is adopted to reduce the system dimension.Subsequently,a symbol discriminant formula of Hopf bifurcation type is deduced by the use of Normal Form theory considering the wheel rolling radius and contact angle.The results show that bogie linear critical speed with wheel high effective conicity is slightly lower than that with low effective conicity.If the second or third order coefficient of wheel rolling radius increases or the third order coefficient of contact angle decreases,the bogie system could transfer from subcritical Hopf bifurcation to supercritical Hopf bifurcation.(5)Comprehensively analyzes the effect of yaw damping and its series stiffness on the bifurcation type of a bogie system.After calculating the linear critical speed,the system is reduced to the plane system by Central Manifold theorem,and the symbolic bifurcation expression is derived by Normal Form theory.The results show that reasonable matches of first-order damping parameter and series stiffness coefficient of yaw damper could increase the system critical speed.The maximum linear critical speed could be obtained by adjusing the fisrt order coefficient of the series stiffness,which could acquire entirely different characteristics of Hopf bifurcation feature.From the point of view of Hopf bifurcation,the third order damping coefficient of yaw damper has strong impact on the bogie system with high tread conicity.When the first-order damping coefficient is constant,it is found that the bogie system with high tread conicity enters the subcritical Hopf bifurcation region firstly,as the third-order damping coefficient decreases.(6)Analyzes the influence of different tread conicity on vehicle critical speed.The differences between bogies and wheelsets lateral displacements from the front and rear sides are comparied.The asymmetric phenomenon of vehicle system is studied based on the maximum lateral displacements in the left and right sides of the car body.And the dynamic characteristic of the system is recognized by using Poincare map and Lyapunov exponent.The result shows that linear and nonlinear critical speeds decrease with the increase of tread conicity.Comparied with the wheelset,the bogie frame lateral motion is easily affacted by the tread conicity.Besides,the multiple periodic bifurcation and chaos of the vehicle hunting motion may appear when wheel and flange collision occurs,especially for the asymmetrical motions.When the tread conicity increases to a certain value,it is found that the vehicle motion leans to one side in a small speed interval as vehicle speed decreases,and this particular asymmetrical motion phenomenon may disappear as the tread conicity continuously increases.