Resonance and Periodic Solutions for Harmonic Oscillators with General Forcing

TL;DR

This paper characterizes resonance and periodic solutions for forced harmonic oscillators with general forcing functions, extending classical results beyond sinusoidal inputs and providing accessible proofs and examples.

Contribution

It offers a new characterization of resonant solutions for general forcing functions, including discontinuous cases, with elementary proofs suitable for students.

Findings

Resonance depends on the relationship between forcing and natural frequencies.

Conditions on Fourier modes determine the existence of resonant solutions.

Examples illustrate various constructions of resonant solutions.

Abstract

We discuss the notion of resonance, as well as the existence and uniqueness of periodic solutions for a forced simple harmonic oscillator. While this topic is elementary, and well-studied for sinusoidal forcing, this does not seem to be the case when the forcing function is general (perhaps discontinuous). Clear statements of theorems and proofs do not readily appear in standard textbooks or online. For that reason, we provide a characterization of resonant solutions, written in terms of the relationship between the forcing and natural frequencies, as well as a condition on a particular Fourier mode. While our discussions involve some notions from $L^2$-spaces, our proofs are elementary, using this the variation of parameters formula; the main theorem and its proof should be readable by students who have completed a differential equations course and have some experience with analysis. We provide several examples, and give various constructions of resonant solutions. Additionally, we connect our discussion to notions of resonance in systems of partial differential equations, including fluid-structure interactions and partially damped systems.