Periodic solution and its stability of spring pendulum with horizontal base motion

Abstract

In this paper, the nonlinear dynamic response of spring pendulum with horizontal base motion is studied. The dynamical equations of the system are established by using Lagrange equation. The discrete Fourier transform, harmonic balance method and homotopy continuation method are combined to solve the periodic response of the system, which avoids the limitation of the small amplitude caused by the Taylor expansion in the traditional analytical method. The comparison with the numerical results shows that the proposed method in this paper can not only be used to solve the large amplitude vibration of spring pendulum, but also has a high accuracy. The stability of periodic response is studied by using Floquet theory. The effects of amplitude and frequency of base motion on the periodic response of the system are given, and the bifurcation characteristics of the periodic solution are analyzed. The results show that the influence curve of the base frequency on the periodic response has two peaks, and with the increase of the amplitude of the base motion, the two peaks will shift to the different sides respectively. When the base amplitude is large, the periodic response amplitude changes with the frequency of the foundation motion, and there will be two jumps. The amplitude of the periodic solution increases with the base amplitude. For some base frequencies, the amplitude of the periodic solution will jump with the change of the base amplitude. When the amplitude and frequency of the system are large, the periodic response of the system may be unstable. After the instability, the spring pendulum enters the continuous rotation state, and the amplitude in the breathing direction is great, the system will be destroyed. It is found that Hopf bifurcation may occur in the periodic response of the system corresponding to some base frequencies and amplitudes. The variation of the system response with the base frequency and amplitude after the Hopf bifurcation is studied numerically by the Runge-Kutta method. Complex dynamical behaviors such as periodic motion, almost periodic motion, torus doubling and chaos are found. It is shown that the main path of the system entering chaos is almost periodic torus rupture and paroxysmal. Finally, the influence analysis of the base frequency and amplitude is synthesized, and the transition of the response form on the plane of the basic motion parameters is given. The results of this paper provide a theoretical reference for the analysis and design of spring pendulum in engineering.