Damped harmonic oscillator revisited: the fastest route to equilibrium

TL;DR

This paper investigates the convergence rates of damped harmonic oscillator solutions, demonstrating that optimal underdamped solutions can reach near-equilibrium faster than critical damping, with experimental confirmation and specific exceptions.

Contribution

It introduces a method to identify optimal underdamped solutions that reach equilibrium faster than other damping types, challenging traditional assumptions.

Findings

Optimal underdamped solutions can reach the energy threshold sooner.

Critical damping is not always the fastest to equilibrium.

Experimental validation supports theoretical results.

Abstract

Theoretically, solutions of the damped harmonic oscillator asymptotically approach equilibrium, i.e., the zero energy state, without ever reaching it exactly, and the critically damped solution approaches equilibrium faster than the underdamped or the overdamped solution. Experimentally, the systems described with this model reach equilibrium when the system's energy has dropped below some threshold corresponding to the energy resolution of the measuring apparatus. We show that one can (almost) always find an optimal underdamped solution that will reach this energy threshold sooner than all other underdamped solutions, as well as the critically damped solution, no matter how small this threshold is. We also comment on one exception to this for a particular type of initial conditions, when a specific overdamped solution reaches the equilibrium state sooner than all other solutions. We confirm some of our findings experimentally.