Nonlinear Frequency Veering in a Beam Resting on an Elastic Foundation

Abstract

In this work we present an investigation into nonlinear frequency veering of an elastic Euler— Bernoulli beam resting on a Winkler elastic foundation subjected to a static lateral load and hinged—hinged, with a torsional spring at one end. The beam is assumed to have an initial quarter-sine shape rise owing to a constant differential edge settlement. The beam static deflection is obtained by using a combined numerical—analytical procedure that accounts for the induced nonlinear axial force to mid-plane stretching. The harmonic balance method is used to solve for the nonlinear natural frequency of the associated nonlinear temporal problem obtained using the assumed mode method. The results are presented in the form of characteristic curves which show variation of the nonlinear natural frequency of the three modes of a selected range of system parameters. These results indicate that, depending on the system parameters, the nonlinear natural frequency of various modes can exhibit complicated behaviors not found using linear analysis.