Hybrid analytical technique for evaluating the sensitivity of the nonlinear vibrational response of beams

Abstract

A three-step hybrid analytical technique is presented for evaluating the nonlinear vibrational response as well as the first-order sensitivity coefficients of thin-walled beams (derivatives of the nonlinear frequency with respect to material and geometric parameters of the beam). The first step involves the generation of various-order perturbation functions, and their derivatives with respect to the material and geometric parameters of the beam, using the Linstedt-Poincaré perturbation technique. The second step consists of using the perturbation functions as coordinate (or approximation) functions and then computing the amplitudes of these functions and the nonlinear frequency via a direct variational procedure. The third step consists of using the perturbation functions, and their derivatives, as coordinate functions and computing the sensitivity coefficients of the nonlinear frequency via a second application of the 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, transverse shear deformation, and cross-sectional warping 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.