Large-amplitude free vibrations of beams-a discussion on various formulations and assumptions

Abstract

Various analytical formulations of the problem of large-amplitude free vibrations of simply supported beams with immovable ends based on the Rayleigh-Ritz method with one-term approximations for axial and transverse displacements are presented in this paper. Many controversial points, raised by different investigators at different times, such as whether axial displacements should be included in the strain energy or not, and whether or not the strain-displacement relationship can be linearized, are discussed herein. It is observed that the formulation wherein the axial displacement is neglected and, further, the quadratic term in the strain displacement relation is linearized leads to an equation of motion, which when solved, based on the simple harmonic oscillations assumption, yields exactly the same non-linear frequency as do the perturbation method, the Ritz-Galerkin method and the elliptical integral solution with axial displacement included, no linearization of non-linear terms and without the harmonic oscillations assumption.