Hybrid Analytical Technique for Nonlinear Vibration Analysis of Thin‐Walled Beams

Abstract

A two‐step hybrid analytical technique is presented for the nonlinear vibration analysis of thin‐walled beams. The first step involves the generation of various‐order perturbation functions using the Linstedt‐Poincaré perturbation technique. The second step consists of using the perturbation functions as coordinate (or approximation) functions and then computing both the amplitudes of these functions and the nonlinear frequency of vibration via a direct variational procedure. The analytical formulation is based on a form of the geometrically nonlinear beam theory with the effects of in‐plane inertia, rotatory inertia, and transverse shear deformation included. The effectiveness of the proposed technique is demonstrated by means of a numerical example of thin‐walled beam with a doubly symmetric I‐section. The solutions obtained using a single‐spatial mode were compared with those obtained using multiple‐spatial modes. The standard of comparison was taken to be the frequencies obtained by the direct integration/fast Fourier transform (FFT) technique. The nonlinear frequencies obtained by the hybrid technique were shown to converge to the corresponding ones obtained by the direct integration/fast Fourier transform (FFT) technique well beyond the range of applicability of the perturbation technique. The frequencies and total strain energy of the beam were overestimated by using a single‐spatial mode.