Periodic Solutions to Nonlinear Euler-Bernoulli Beam Equations

TL;DR

This paper investigates the existence of time-periodic solutions in nonlinear Euler-Bernoulli beam equations with variable coefficients, employing advanced mathematical techniques to handle parameter-dependent solutions in a measure-theoretic framework.

Contribution

It introduces a novel application of Lyapunov-Schmidt reduction and Nash-Moser iteration to establish periodic solutions for nonlinear beam equations with variable coefficients.

Findings

Existence of families of periodic solutions for a broad set of parameters.

Solutions exist on a Cantor set with asymptotically full measure as perturbation parameter approaches zero.

The methods extend the understanding of nonlinear vibrations in variable coefficient beams.

Abstract

Bending vibrations of thin beams and plates may be described by nonlinear Euler-Bernoulli beam equations with $x$-dependent coefficients. In this paper we investigate existence of families of time-periodic solutions to such a model using Lyapunov-Schmidt reduction and a differentiable Nash-Moser iteration scheme. The results hold for all parameters $(\epsilon,\omega)$ in a Cantor set with asymptotically full measure as $\epsilon\rightarrow0$.