Research On The Nonlinear Stability And Hopf Bifurcation Transformation Mechanism Of High-speed Railway Vehicles

As the operation and rapid development of high-speed railway trains,the dynamic environment of high-speed vehicle systems becomes more complex,which deteriorates vehicle system dynamic behaviors.Therefore,how to improve the three major dynamic indexes of high-speed vehicle systems,namely,running stability,ride comfort and curve passing performance,becomes particularly critical.As one of the most important problems related to the running safety of railway vehicles,the lateral stability of high-speed railway vehicles has a direct influence on the critical running speed on the track.Once hunting motions occur for running high-speed vehicles,the operation quality of railway vehicle systems will deteriorate rapidly,which causes a rapid decrease of the ride comfort and the running safety,an increase of the dynamic interaction between the wheel and the rail at the same time,and a damage to the track.Therefore,a derailment may happen to railway vehicles under this condition.In conclusion,more research on the mechanism of the lateral running stability of high-speed railway vehicles and mastering its inherent laws have important theoretical meaning and guiding value to the running safety of high-speed railway vehicles.According to different research purposes,a lateral dynamic model of a single wheel-set system,a bogie system and a high-speed railway vehicle are established in this thesis,respectively.Since the model of a single wheel and a bogie are mainly used for mechanism research on hunting motions and Hopf bifurcation transformation,they are simplified during the modeling process to facilitate the theoretical analysis.However,the lateral stability analysis of the high-speed vehicle model is mainly based on numerical methods.In order that the lateral dynamic model of high-speed railway vehicles can describe the actual model more closely,a realistic wheel/rail contact relation,yaw dampers and secondary lateral dampers represented by Maxwell models of a stiffness and a damping in series,anti-roll bar device and bump stops with clearances,etc.are fully considered in the modelling process.Based on the theoretical analysis of simple systems and the numerical simulation of complex systems,the mechanism of hunting motions of high-speed railway vehicles is systematically studied in this thesis,and the influences of primary system parameters on the lateral dynamics of a certain type vehicle are obtained.In this thesis,the main research work includes:(1)Based on Kalker's linear creep theory,Wagner's flange force model represented by a fifth-order polynomial function with respect to the lateral displacement of the wheel-set,gravitational stiffness and yaw gravitational stiffness,etc.,the differential equations of lateral motions of a single wheel-set model are established,where only the lateral and yaw motions are considered.According to the Center Manifold Theorem and Normal Form Theory from nonlinear dynamics,the first-order Lyapunov coefficient expression of the corresponding normal forms on the center manifolds of the single wheel-set model is derived at the Hopf bifurcation point.Thus,according to the sign of the first Lyapunov coefficient,the Hopf bifurcation type of the wheel-set model can be determined.When the first Lyapunov coefficient is positive,a subcritical Hopf bifurcation happens to the wheel-set system.When the first Lyapunov coefficient is negative,a supercritical Hopf bifurcation happens to the wheel-set system.When the first Lyapunov coefficient is zero,which is a degenerate case,the bifurcation analysis becomes more complicated.At this time,an additional control parameter needs to be introduced to complete the Bautin bifurcation analysis of the single wheel-set model.The analysis result shows that the single wheel-set model transfers between these two kinds of Hopf bifurcation through a Bautin bifurcation.(2)Based on the single wheel-set model,the differential equations of lateral motions of a bogie model are established considering the secondary suspension system of the railway vehicles.Only the lateral and yaw motion of the front and back wheel-set and the bogie frame are considered in this model.Compared with the model of a single wheel-set,the dimension of the bogie system increases greatly.If the Center Manifold Theorem and the Normal Form Theory are applied in the bogie model,the theoretical derivation process would be very cumbersome.However,introducing the multiple-linear vector functions makes it possible to conduct the dimensionality reduction and regularization process of high-dimension systems simultaneously.The first-order Lyapunov coefficient expression of the corresponding normal forms on the center manifolds of the bogie model is derived at the Hopf bifurcation point.According to the sign of the first-order Lyapunov coefficient,the Hopf bifurcation type of the bogie system can be determined.When the first Lyapunov coefficient is zero,a Bautin bifurcation analysis of the bogie system can be done by introducing an additional control parameter.Similarly,the bogie model system transfers between these two kinds of Hopf bifurcation through a Bautin bifurcation.(3)A lateral dynamic model of a high-speed railway vehicle is established considering the actual wheel and rail profile,yaw dampers and secondary lateral dampers represented by Maxwell models of the stiffness and damping in series,anti-roll bar device and bump stops with clearances.A wheel/rail contact table is generated by wheel/rail contact calculation program RSGEO.A numerical routine for calculating the Hopf bifurcation point is compiled using MATLAB.Based on the routine mentioned above and multiple-linear vector functions,a routine for calculating the first Lyapunov coefficient of the corresponding normal forms on the center manifolds of a high-dimension system is also implemented using MATLAB,which makes it possible to determine the Hopf bifurcation type without drawing the bifurcation diagram,when values of the system parameters are given.Because of the nonsmoothness of the high-speed railway vehicle system,an event-driven strategy is introduced to detect the switching surfaces of different movement states during the numerical integration.The lateral stability of a certain type vehicle is studied by numerical analysis,and a supercritical Hopf bifurcation is found in this model with new wheel profiles.But it shows subcritical bifurcation features at a realistic operation speed.The bifurcation diagram of a certain type vehicle is drawn by increasing and decreasing the running speed gradually with the original system parameter values.Based on the results of the event-driven strategy in the decreasing process of the running speed,it is found that a certain type vehicle jumps from a periodic motion of a high amplitude to an equilibrium through a grazing bifurcation.(4)The influences of primary system parameters such as the longitudinal and lateral stiffness of the axle box,the longitudinal and lateral stiffness of the air spring,the stiffness and damping of the secondary lateral damper,the stiffness and damping of the yaw damper and the wheel/rail contact relation,etc.,on the lateral stability of a certain type vehicle are investigated by numerical analysis.It is shown that,nearly all the key parameters within normal conditions do not have any influence on the Hopf bifurcation type of a certain type vehicle except for the longitudinal stiffness of the axle box.The nonlinear critical speed of a cerntain type vehicle can be increased by lowering the longitudinal stiffness of the axle box,the stiffness of the secondary lateral damper,and the stiffness of the yaw damper,or by increasing the longitudinal and lateral stiffness of the air spring and the first-stage damping of the yaw damper.It should be noticed that a too high or too low damping of the secondary lateral damper can deteriorate the nonlinear critical speed of a certain type vehicle.The stiffness and the unloading force of the yaw damper determine if the running safety monitoring of a certain type vehicle is possible.With respect to four different standard wheel profiles,a supercritical Hopf bifurcation happens to a certain type vehicle in all cases,and the linear critical speed is far higher than the nonlinear critical speed.However,with the increase of the running mileage,which leads to wheel wear,the linear critical speed of a certain type vehicle decreases gradually,and eventually collides with the nonlinear critical speed.When the running mileage reaches 90,000 km,the instability motion of a certain type vehicle becomes more complicated.