The amplitude decay of a harmonic oscillator damped simultaneously by weak linear and nonlinear damping forces

TL;DR

This paper derives approximate formulas for the amplitude decay of a harmonic oscillator subjected to combined weak damping forces, including constant, linear, and quadratic velocity-dependent forces, validated by numerical comparisons.

Contribution

It introduces a unified analytical approach to model amplitude decay under multiple weak damping forces, enhancing understanding of complex damping scenarios.

Findings

Analytical solutions closely match numerical results for weak damping regimes.

The approach is suitable for undergraduate physics education.

The formulas effectively describe amplitude decay with combined damping forces.

Abstract

We derive approximate expressions for the amplitude decay of harmonic oscillations weakly damped by the simultaneous action of three different damping forces: force of constant magnitude, force linear in velocity, and force quadratic in velocity. Our derivation is based on a basic understanding of the undamped harmonic oscillator and the connection between the energy dissipation rate and the power of the total damping force. 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 damping by combinations of these three forces for an experimentally relevant range of corresponding damping constants. The physical concepts and mathematical techniques we employ are suitable for undergraduate physics teaching.