Damped harmonic oscillator revisited: a new approach to energy decay in the case of Coulomb, Stokes, and Newton damping

Abstract Approximate formulas are derived to describe energy loss in a harmonic oscillator (HO) that experiences three distinct damping mechanisms: constant-magnitude (Coulomb), velocity-proportional (Stokes), and velocity-squared (Newton), using fundamental mathematical methods and physical insight. Our methodology leverages an understanding of the free HO and the inherent link between energy dissipation rates and the power exerted by damping forces. We establish a direct analytical framework for assessing the energy of a damped HO, obviating the need for amplitude-based equations. The simplicity of our findings is accompanied by their remarkable accuracy when validated against exact or computational simulations. In addition to an excellent approximate description of the energy decay, we also show how to derive an exact solution in the case of Stokes damping without relying on the standard procedure for solving second-order differential equations. The theoretical underpinnings and mathematical strategies employed are well-suited to undergraduate-level or advanced high school physics instruction. In the appendix, we propose a MATLAB-based computer experiment for the Newton damping case, illustrating the comparison between the approximate energy-based solution and a numerical reference solution.