Periodic Solutions to Nonlinear Euler-Bernoulli Beam Equations

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 $\epsilon\rightarrow0$.