New approach to approximate analytical solutions of a harmonic oscillator with weak to moderate nonlinear damping: Part I

TL;DR

This paper presents a new analytical method for approximating solutions of a damped harmonic oscillator with nonlinear damping, validated against numerical solutions for various damping types, and demonstrating high accuracy.

Contribution

The paper introduces a novel ansatz-based approach for deriving approximate solutions for nonlinear damping in harmonic oscillators, extending analytical techniques to quadratic and Coulomb damping cases.

Findings

Excellent agreement with numerical solutions for quadratic damping.

Good approximation for Coulomb damping until near the end of motion.

Improved variants of solutions enhance accuracy for Coulomb and combined damping.

Abstract

We introduce a new approach to deriving approximate analytical solutions of a harmonic oscillator damped by purely nonlinear, or combinations of linear and nonlinear damping forces. Our approach is based on choosing a suitable trial solution, i.e. an ansatz, which is the product of the time-dependent amplitude and the oscillatory (trigonometric) function that has the same frequency but different initial phase, compared to the undamped case. We derive the equation for the amplitude decay using the connection of the energy dissipation rate with the power of the total damping force and the approximation that the amplitude changes slowly over time compared to the oscillating part of the ansatz. By matching our ansatz to the initial conditions, we obtain the equations for the corresponding initial amplitude and initial phase. Here we demonstrate the validity of our approach in the case of damping quadratic in velocity, Coulomb damping, and a combination of the two, i.e. in this paper we consider purely nonlinear damping, while the dynamics with combinations of damping linear in velocity and nonlinear damping will be analyzed in a follow-up paper. In the case of damping quadratic in velocity, by comparing our approximate analytical solutions with the corresponding numerical solutions, we find that our solutions excellently describe the dynamics of the oscillator in the regime of weak to moderately strong quadratic damping. In the case of Coulomb damping, as well as in the case of a combined Coulomb and quadratic damping, our approximate analytical solutions agree well with the corresponding numerical solutions until the last few half-periods of the motion. Therefore, for these two cases, we introduce improved variants of our approximate solutions which describe the dynamics well until the very end.