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

Exact mathematical solution for nonlinear free transverse vibrations of beams