Non-linear response of buckled beams to 1:1 and 3:1 internal resonances

Abstract

The non-linear response of a buckled beam to a primary resonance of its first vibration mode in the presence of internal resonances is investigated. We consider a one-to-one internal resonance between the first and second vibration modes and a three-to-one internal resonance between the first and third vibration modes. The method of multiple scales is used to directly attack the governing integral–partial–differential equation and associated boundary conditions and obtain four first-order ordinary-differential equations (ODEs) governing modulation of the amplitudes and phase angles of the interacting modes involved via internal resonance. The modulation equations show that the interacting modes are non-linearly coupled. An approximate second-order solution for the response is obtained. The equilibrium solutions of the modulation equations are obtained and their stability is investigated. Frequency–response curves are presented when one of the interacting modes is directly excited by a primary excitation. To investigate the global dynamics of the system, we use the Galerkin procedure and develop a multi-mode reduced-order model that consists of temporal non-linearly coupled ODEs. The reduced-order model is then numerically integrated using long-time integration and a shooting method. Time history, fast Fourier transforms (FFT), and Poincare sections are presented. We show period doubling bifurcations leading to chaos and a chaotically amplitude-modulated response.