On the approximate solutions to a damped harmonic oscillator with higher-order nonlinearities on a rough plane

Abstract

The harmonic oscillator is an important physical model in both classical mechanics and quantum mechanics which describes the dynamics of an object displaced from its equilibrium position. The damped harmonic oscillator has already been well studied on a smooth plane. However, it is hard to avoid energy dissipation caused by a plane in real cases. Hence, in this paper, the approximate solutions to a damped harmonic oscillator with higher-order nonlinearities on a rough plane is explored mainly by the Cardano-Tartaglia formula. In approximate solutions, an approximate solution is provided under specific prerequisite and a case is studied given the initial conditions. Also, the system error of the approximate solution in approximate solutions is analyzed in error analysis under the restrictions of the initial conditions. In this paper, only a short time span of the damped harmonic oscillator dynamics is analyzed because of the existence of two terms which are mentioned in error analysis.