Internal Resonances of Nonlinear Double Beams Connected by a Spring and Dashpot

Abstract

This paper investigates the internal resonances of two vibration modes in a nonlinear double-beam system. Two identical beams, simply supported at the both ends, are discretely connected by a set of a spring and dashpot, and one of the beams is subjected to sinusoidal, harmonic excitation. Nonlinearity of the beams comes from a change in tension due to the stretch of the beams. In the theoretical analysis, modal equations of motion for the two, adjacent vibration modes of the system are derived. Then, van der Pol’s method is applied to the modal equations of motion in order to determine the expressions for the frequency response curves. In the numerical calculations, the frequency response curves are shown when the natural frequency of the second vibration mode is close to that of the first vibration mode. The first and second vibration modes of the system are nonlinearly coupled, and internal resonances occur. The time histories of the first and second vibration modes, and the time histories of the displacements of the beams at specific positions are also shown. When the connecting spring stiffness is comparatively small, the first and second vibration modes significantly interact with each other, and the amplitude of the vibration of the beam, which is excited, is larger than that of the other beam at a specific excitation frequency range. The opposite is true at a specific excitation frequency range when the connecting spring stiffness is increased. For a specific value of the connecting spring stiffness, Hopf bifurcation may occur, followed by amplitude modulated motions (AMMs).