Nonlinear vibration of two coupled cantilevers under magnetic action

The investigation is motivated by the application potential in energy harvesting. Two identical cantilevers are fixed at a same vibrating structure and connected by a linear spring at the free ends. A permanent magnet is attached to the free end of the lower cantilever, another permanent magnet is fixed at the vibrating structure, and the two magnets are repulsive and vertically facing to each other. The base excitation is a harmonic movement. 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 presence of 2:1 internal resonance, both the first and the second primary resonances are examined in details. 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. Especially, the frequencyamplitude response curves may be converted from hardening-type single-jumping to doublejumpings, 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.