  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.
