Asymptotic analysis and limiting phase trajectories in the dynamics of spring pendulum

Abstract

Spring pendulum is a widely discussed two degree-of-freedom (DOF) mechanical systems in numerous references. In this paper the asymptotic approach and limiting phase trajectories (LPT) have been applied to analyze the two DOF mathematical model of a spring pendulum. The LPT and multiple timescale (MTS) methods are effective tools of the investigation of non-linear systems. Some interesting and important aspects of dynamics of the system are discussed. The main attention is focused on the non-steady-state vibrations when the energy is intensively exchanged. Then with increasing values of the selected parameters, a sudden change in the character of vibrations is observed. These phenomena are very well described by the LPT. The method allows to determine the critical values of the parameters responsible for the mentioned transitions. Our analytical studies are verified by numerical calculations.