Study On Stochastic Stability And Stochastic Bifurcation For Railway Vehicle

The hunting stability in railway vehicle system is the inherent nature, and become much more important as the increasing of running speed. So far, most studies of the vehicle stability is mainly focus on the deterministic nonlinear system, however when the vehicle works the random noise disturbance is inevitable, and the deterministic system model is the idealization of actual system, therefore, the stochastic system is much more real and essential, what's more, the requirements of the vehicle model is more and more accurate as the increase of vehicle speed, the influence of random factors cannot be easily neglected. With deeply understanding the movement mechanism of the nonlinear stochastic vibration in vehicle system, we can straighten up its inherent laws, which is meaningfull for the vehicle safety.Based on the above background, the stochastic stability and stochastic bifurcation problem will be researched in this thesis. Starting from the simple wheelset system, and gradually involving the bogie even more complex vehicle systems with the method of theoretical and numerical combination, the specific research content mainly includes the following aspects:(1) Considering the stochastic parametric excitation, the Ito stochastic differential equations of wheelset and bogie Hamilton system are set up respectively. At the same time, the dimension of the lto stochastic differential equations is reduced to one-dimensional energy diffusion process, which comes from the average Ito stochastic differential equation, with the stochastic averaging method. Finally the purpose of transforming the multidimensional space domain to the one dimensional energy domain is achieved, which provides the basis for the study of the the stochastic stability of the wheelset stochastic system.(2) The wheelset system stochastic stability is researched by using the singularity and singular boundary of one-dimensional diffusion process theory, and the global stability conditions for the stochastic wheelset system is obtained.The results show that the singularity and singular boundary have decisive influence on the sample stability of the solution of the diffusion process, which satisfy the stochastic differential equations, as well as the form and the existence of invariant measure.(3) Based on Ito stochastic differential equation, the maximum Lyapunov index analytical formula is deduced theoretically. The D-bifurcation critical condition is obtained due to the study of the D-bifurcation in wheelset stochastic system. And the FPK equation is derived which is dominated by the average Ito stochastic differential equations, the stationary probability density is obtained simultaneously. According to the topology changes, such as the shape and peak, of the probability density, the P-bifurcation condition is obtained. At the same time, the stochastic bifurcation diagram and stochastic limit cycle are defined. Additionally, the difference between D-bifurcation and P - bifurcation is analysised comprehensivly, the comparation of P-bifurcation between the two wheelset systems is also made. At last, the stochastic bifurcation and Hopf bifurcation are compared.(4) Because of the high dimension in bogie system, the stochastic average method has a certain difficulty. The high dimensional FPK equation is derived based on Ito stochastic differential equations of bogie system. The difficulty in constructing the differential scheme is settled with the alternating direction implicit method, and the computational efficiency can be easily improved when combining with the multigrid method, which can effectively solve the problem of higher dimensional FPK equation.Then, the method is applied to the wheelset and the bogie system respectively, and the srochastic bifurcation diagram of bogie system is obtained. At the same time, the numerical results and averaging results are compared, the comparison shows good agreement with each other.(5) The method for calculating the largest Lyapunov exponent from small data sets is reformed when combining with the C-C method, which improves the calculation precision of the largest Lyapunov exponent. This method effectively solves the problem in calculating the largest Lyapunov exponent of high-dimensional system. Then the method is applied to the wheelset and bogie system respectively, and the D-bifurcation critical condition in the stochastic bogie system is obtained, which provides the basis for the research of stochastic stability. At the same time, a numerical method for computing the stochastic critical speed is proposed when combining with the dichotomy method.