Underdamped harmonic oscillator driven by a train of short pulses: Analytical analysis

TL;DR

This paper develops analytical solutions for an underdamped harmonic oscillator driven by periodic short pulses, enabling parameter estimation through nonlinear curve fitting for various pulse trains including Dirac, square, and Gaussian pulses.

Contribution

It introduces two analytical solution forms for classical UHO driven by different pulse trains, including approximate harmonic solutions up to second order for Gaussian and Dirac comb pulses.

Findings

Analytical solutions for time-periodic and harmonic responses are derived.

Solutions for Gaussian and square pulses approach Dirac comb solutions as pulse width decreases.

Harmonic solutions enable experimental parameter determination via curve fitting.

Abstract

A theoretical model of an underdamped harmonic oscillator (UHO) driven by periodic short pulses may find plenty of applications in classical, semiclassical, and quantum physics. We present here two different forms of analytical solutions: {\it time-periodic solutions} and {\it harmonic solutions} for one-dimensional classical UHO driven by three different trains of short pulses. They are a Dirac comb, a train of square pulses, and a train of Gaussian pulses with the same pulse-to-pulse time interval $T$ and pulse width $2\tau$. Two solutions for square and Gaussian pulses approach to that of the Dirac comb when the pulse width $2\tau \rightarrow 0$ as expected. In particular, the harmonic solutions for Dirac comb and Gaussian pulses could be expressed approximately with harmonic terms of the repetition frequency $\omega_{\rm R} = 2\pi/T$ up to the second order. The presented analytical solutions would provide a practical way to determine experimentally the system parameters such as the underdamped oscillation frequency $\omega = \sqrt{\omega_0^2-\gamma^2}$, the natural frequency $\omega_0$, and the damping rate $\gamma$, by nonlinear curve fitting procedures for different driving force parameters of $T$ and $2\tau$.