轮对非线性动力学系统蛇行运动的解析解

在对铁路车辆系统的极限环幅值和非线性临界速度进行分析时通常采用数值方法, 不便于研究其随系统参数的变化规律. 轮对系统保留了影响车辆系统动力学性能的几个关键要素: 如轮轨几何非线性约束、轮轨接触蠕滑关系和悬挂系统等, 可以反映铁路车辆系统蛇行运动的本质特性. 轮对系统自由度少、参数少, 可以采用解析方法进行分析. 本文选取合适的特征量把轮对非线性动力学方程无量纲化, 得到了带有小参数的两自由度微分方程; 采用多尺度方法对该方程进行了解析求解; 给出了轮对系统极限环幅值的解析表达式并对其稳定性进行了判定; 给出了轮对系统的分岔速度解析表达式, 并进而获得系统的非线性临界速度的解析表达式. 在对得到的解析解用数值结果进行验证后, 用得到的解析解进行了系统参数影响分析. 传统的分岔图计算方法(如降速法、路径跟踪法等)需对微分方程进行大量数值积分计算方可求解系统的非线性临界速度值, 而通过本文获得的解析表达式可直接给出系统的非线性临界速度值和极限环幅值, 便于研究轮对系统动力学特性随参数的变化规律,进行快速方案比对和筛选, 为转向架结构优化设计提供参考. 

ANALYTICAL SOLUTION OF THE HUNTING MOTION OF A WHEELSET NONLINEAR DYNAMICAL SYSTEM

Numerical methods are usually used to analyze the amplitude of limit cycle and nonlinear critical speed of railway vehicle system, which is inconvenient to study the rule of variation with vehicle system parameters. The wheelset system retains several critical elements that affect the dynamic performance of the railway vehicle system, such as the geometric nonlinear constraints of the wheel-rail, the wheel-rail contact creep relationship, and the suspension system, which can reflect the essential characteristics of the hunting motion of the railway vehicle system. The wheelset system has fewer degrees of freedom and parameters, which can be analyzed by the analytical method. In this paper, the nonlinear dynamics equations are nondimensionalized by choosing appropriate characteristic parameters, and the two-degree-of-freedom nonlinear differential equations with small parameters are obtained. The method of multiple scales is used to solve the equations analytically. The analytical expressions of the amplitude of the limit cycle of the wheelset system are given and its stability is judged. The analytical expressions of the bifurcation speed of the wheelset system are given, and then the analytical expressions of the nonlinear critical speed of the wheelset system are obtained. After the analytical solutions are verified by the numerical results, the influence of wheelset system parameters is analyzed by using the analytical solutions. The traditional calculation methods of bifurcation diagram (such as speed reduction method, path-following method, etc.) require a large number of numerical integration calculations on the differential equations to solve the nonlinear critical speed of the system. However, the analytical expressions obtained in this paper can directly give the nonlinear critical speed and amplitude of the limit cycle of the wheelset system, which is convenient for studying the rule of variation of the dynamic performance of the wheelset system with parameters and for quick comparison and screening of schemes, and provide a reference for the optimization design of bogie structure.