Exact mathematical solution for nonlinear free transverse vibrations of beams

In the present paper, an exact mathematical solution is obtained for the nonlinear free transverse vibration of beams for the first time. The governing nonlinear partial differential equation in un-deformed coordinates system is converted in two coupled partial differential equations in deformed coordinates system. Then, a mathematical explanation is obtained for the nonlinear mode shapes as well as natural frequencies versus geometrical and material properties of the beam. It is shown that as the sth mode of transverse vibration is excited, the 2sth mode of the in-plane vibration will be developed. The results of the present work is compared with those obtained by the Calerkin method and the observed agreement will confirm the exact mathematical solution. It is shown that the governing equation is linear in the time domain. As a parameter, amplitude to length ratio (A/l) is proposed to show when the nonlinear terms become dominant in the behavior of structure. (C) 2020 Sharif University of Technology. All rights reserved.