Investigation of subcombination internal resonances in cantilever beams

Article type: Research Article

Authors: Arafat, Haider N. | Nayfeh, Ali H.^;

Affiliations: Department of Engineering Science and Mechanics, MC 0219, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA

Note: [] Corresponding author. Tel.: +1 540 231 5453; Fax: +1 540 231 2290; E‐mail: anayfeh@vt.edu.

Abstract: Activation of subcombination internal resonances in transversely excited cantilever beams is investigated. The effect of geometric and inertia nonlinearities, which are cubic in the governing equation of motion, is considered. The method of time‐averaged Lagrangian and virtual work is used to determine six nonlinear ordinary‐differential equations governing the amplitudes and phases of the three interacting modes. Frequency‐ and force‐response curves are generated for the case {\varOmega}\approx\omega_{4}\approx\frac{1}{2}(\omega_{2}+\omega_{5}). There are two possible responses: single‐mode and three‐mode responses. The single‐mode periodic response is found to undergo supercritical and subcritical pitchfork bifurcations, which result in three‐mode interactions. In the case of three‐mode responses, there are conditions where the low‐frequency mode dominates the response, resulting in high‐amplitude quasiperiodic oscillations.

Keywords: beams, subcombination internal resonance, bifurcations

Journal: Shock and Vibration, vol. 5, no. 5-6, pp. 289-296, 1998