A method to determine the periodic solution of the non-linear dynamics system

In this paper, the generalized shooting method and the harmonic balancing method to determine the periodic orbit, its period and the approximate analytic expression of the non-linear bearing–rotor system are presented. At first, by changing the time scale, the period of the periodic orbit of the non-linear system is drawn into the governing equation of the system explicitly. Then, the generalized shooting procedure is sought out. The increment value changed in the iteration procedure is selected by using the optimization method. The procedure involves determining the periodic orbit and its period of the system, and the stability of the periodic solution is determined by using Floquet stability theory. The validity of such method is verified by determining the periodic orbit and period of the forced van der Pol equation. Secondly, the periodic solution of the non-linear rotor–bearing system is expanded into Fourier series according to the character of the solution obtained by using the generalized shooting method. Then the approximate analytic expression of the periodic solution of the system is obtained by using the harmonic balancing method. Theoretically, the solution with any precision can be obtained by adding the number of the harmonics. At last, the periodic orbit, period and approximate analytic expressions of the periodic solution of the non-linear rotor–bearing system are provided.