Nonlinear oscillations of strings and beams

TL;DR

This paper explores time-periodic solutions to nonlinear wave and beam equations, revealing complex structures and stability properties that enhance understanding of nonlinear oscillations in strings and beams.

Contribution

It introduces a new class of large-energy solutions with complex mode structures and provides a systematic framework for their analysis and stability assessment.

Findings

Identification of a new class of large-energy solutions

Discovery of stable solutions within this class

Fractal-like structure of solutions

Abstract

We investigate the time-periodic solutions to the nonlinear wave and beam equations and uncover their intricate, fractal-like structure. In particular, we identify a new class of large-energy solutions with complex mode compositions and propose a systematic framework for their analysis. A Floquet stability study reveals that this class contains solutions that are linearly stable, suggesting that they may play a significant role in the nonlinear dynamics of the systems.