非线性转子-轴承系统的周期解及近似解析表达式

通过对普通打靶方法进行改造提出一种确定非线性系统周期轨道及周期的新型打靶算法。首先通过改变系统的时间尺度，将非线性系统周期轨道的周期显式地出现在非线性系统的系统方程中，然后对传统打靶法进行改造，将周期也作为一个参数一起参与打靶法的迭代过程，迭代过程包含对周期轨道和周期的求解，迭代过程中的增量通过优化方法选择，从而能迅速确定出系统的周期轨道及其周期。应用所求的结果结合谐波平衡方法求得了非线性系统的周期轨道的近似解析表达式，理论上通过增加谐波的阶数任何精度的周期解都可以得到。最后将该方法应用于非线性转子轴承系统，求出了在某些参数下转子的周期解及其近似解析表达式，通过与四阶Runge-Kutta数值积分结果比较，验证了方法的有效性，计算结果对于转子系统运动的定量控制有重要理论指导意义。

Periodic solution and its approximate analytic expressions of the nonlinear rotor-bearing system

In this paper, a generalized shooting method and a harmonic balancing method to determine the periodic orbit, its period and the approximate analytic expression of the nonlinear bearing-rotor system are presented. At first, by changing the time scale, the period of the periodic orbit of the nonlinear system is drawn into the governing equation of the system explicitly. Then, the generalized shooting procedure is recompiled; the period takes part into the iteration procedure of the shooting method as a parameter. The iteration procedure includes the solving of the periodic orbit and its period. The increment value changed in the iteration procedure is selected by using the optimization method, and then the periodic orbit and its period of the system are determined rapidly. The approximate analytic expression of the nonlinear bearing-rotor system is obtained by using the results solved combined with the harmonic method. The periodic solutions of any precision can be obtained by adding the number of the order of the harmonics theoretically. At last, applying this method into the nonlinear rotor-bearing system, the periodic solution and its approximate analytic expression of the nonlinear bearing-rotor system are obtained under the condition of some parameters. The validity of the method has been verified by comparing with the results of the 4-order Runge-Kutta numerical integral method. The results make important theoretic sense in the controlling of the motion of the rotor-bearing system.