Damped harmonic oscillator revisited: the fastest route to equilibrium

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...

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Bibliographic Details
Published inarXiv.org
Main Authors Lelas, Karlo, Poljak, Nikola, Jukić, Dario
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 09.07.2023
Subjects
Damping
Energy resolution
Equilibrium
Harmonic oscillators
Initial conditions
Measuring instruments
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Summary: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.
ISSN:2331-8422