Exact mathematical solution for nonlinear free transverse vibrations of beams

TL;DR

This paper presents the first exact mathematical solution for nonlinear free transverse vibrations of beams, transforming the governing equations into coupled forms and analyzing mode shapes and frequencies.

Contribution

It introduces an exact solution approach for nonlinear beam vibrations, linking mode shapes with geometrical and material properties, and compares results with Galerkin method for validation.

Findings

Exact solutions match Galerkin method results

Mode 2 in-plane vibration develops at the nth transverse mode

Amplitude to length ratio indicates when nonlinear effects dominate

Abstract

In the present paper, an exact mathematical solution has been obtained for nonlinear free transverse vibration of beams, for the first time. The nonlinear governing partial differential equation in un-deformed coordinates system has been converted in two coupled partial differential equations in deformed coordinates system. A mathematical explanation is obtained for nonlinear mode shapes as well as natural frequencies versus geometrical and material properties of beam. It is shown that as the th mode of transverse vibration excited, the mode 2 th of in-plane vibration will be developed. The result of present work is compared with those obtained from Galerkin method and the observed agreement confirms the exact mathematical solution. It is shown that the governing equation is linear in the time domain. As a parameter, the amplitude to length ratio has been proposed to show when the nonlinear terms become dominant in the behavior of structure.