Exact mathematical solution for nonlinear free transverse vibrations of beams

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 s th mode of transverse vibration excited, the mode 2s 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 governing equation is linear in time domain. As a parameter, the amplitude to length ratio (Λ⁄l) has been proposed to show when the nonlinear terms become dominant in the behavior of structure