Application of the Ritz method to the analysis of non-linear free vibrations of beams

Abstract

This paper presents an analytical solution for geometrically non-linear free vibrations of beams with elastically supported ends in the horizontal direction. The equation of motion is obtained by employing Hamilton's principle and assuming that horizontal inertia forces can be neglected. The Ritz method, with a continuum solution and an iterative procedure, are used for determining the frequencies and non-linear modes of vibrations. The orthogonality conditions for these modes are also discussed. Numerical results for various beam boundary conditions are presented and compared with available results.