Internal resonance in forced vibration of coupled cantilevers subjected to magnetic interaction

Abstract

Forced vibration is investigated for two elastically connected cantilevers, under harmonic base excitation. One of the cantilevers is with a tip magnet repelled by a magnet fixed on the base. The cantilevers are uniform viscoelastic beams constituted by the Kelvin model. The system is formulated as a set of two linear partial differential equations with nonlinear boundary conditions. The method of multiple scales is developed to analyze the effects of internal resonances on the steady-state responses to external excitations in the nonlinear boundary problem of the partial differential equations. In the presence of 2:1 internal resonance, both the first and the second primary resonances are examined in detail. The analytical frequency–amplitude response relationships are derived from the solvability conditions. It is found that the frequency–amplitude response curves reveal typical nonlinear phenomena such as jumping and hysteresis in both primary resonances as well as saturation in the second primary resonance. The frequency–amplitude response curves may be converted from hardening-type single-jumping to double-jumpings, and further to softening-type single-jumping by adjusting the distance between two magnets. It is also found that the unstable parts of the frequency–amplitude response curves correspond to quasi-periodic motions. The finite difference scheme is proposed to discretize both the temporal and the spatial variables, and thus the numerical solutions can be calculated. The analytical results are supported by the numerical solutions.