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 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 harmonic oscillator 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 harmonic oscillator, 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 for undergraduate-level or advanced high school physics instruction.

Figures (7)

• Figure 1: Coulomb damping, kinetic to total mechanical energy ratio, comparison of the exact result (dashed black line) and approximation (full red line) representing Eq. \ref{['eq:EkEm_ratio']} for damping strengths $\gamma_0 = 0.050$ (left pane) and $\gamma_0 = 0.080$ (right pane).
• Figure 2: Coulomb damping, mechanical energy decay, exact (circles) and approximation (full red line) shown here for damping strengths $\gamma_0 = 0.050$ (left) and $\gamma_0 = 0.080$ (right).
• Figure 3: Coulomb damping, approximate envelope $\mathcal{\tilde{E}}_{m}$ (solid black line, Eq. \ref{['eq:approxEm_Coulomb']}) of both, exact kinetic (dot dashed line) and potential (dashed line) energy. Comparison is given for two damping strengths $\gamma_0 = 0.050$ (left) and $\gamma_0 = 0.080$ (right).
• Figure 4: Coulomb damping, exact displacement (dashed black line) against approximate displacement $x(\tau)$ (solid red line) given in Eq. \ref{['eq:Coulomb_x_app']}. Comparison is presented for two damping strengths: $\gamma_0 = 0.050$ (left pane) and $\gamma_0 = 0.080$ (right pane). The dot-dashed vertical line serves to indicate a pre-established instant (see Eq. \ref{['eq:coulomb_e_full']}) at which motion ceases.
• Figure 5: Newton damping, kinetic to total mechanical energy ratio, comparison of the exact (numerical) result (dashed black line) and approximation (solid red line) representing Eq. \ref{['eq:EkEm_ratio']} for damping strengths $\gamma_2 = 0.20$ (left pane) $\gamma_2 = 0.60$ (right pane).