Modeling the amplitude and energy decay of a weakly damped harmonic oscillator using the energy dissipation rate and a simple trick

Abstract

We demonstrate how to derive the exponential decrease of amplitude and an excellent approximation of the energy decay of a weakly damped harmonic oscillator without solving the associated equation of motion and without insight into the analytical form of its solution. This is achieved using a basic understanding of the undamped harmonic oscillator and the connection between the damping force's power and the energy dissipation rate. The trick is adding the energy dissipation rates corresponding to two specific pairs of initial conditions with the same energy. In this way, we obtain a first-order differential equation from which we get the time-dependent amplitude and the energies corresponding to each pair of considered initial conditions. Comparing the results of our model to the exact solutions and energies yielded an excellent agreement. The physical concepts and mathematical techniques we employ are well-known to first-year undergraduates.