Primary resonance of coupled cantilevers subjected to magnetic interaction

Based on a distributed-parameter model, the forced vibration of a cantilever pair excited by a sinusoidal base movement is analyzed. Two cantilevers are coupled at their free ends by a linear spring. A nonlinear concentrated magnetic force acts on the tip of one cantilever, serving at the nonlinear boundary condition of the continuous model. The magnetic force is modeled as a fractional function, strongly dependent on the distance between two magnets. Via the method of multiple scales, the primary resonance is analyzed for all modes. A second-order approximate solution and its stability condition are analytically captured. It is revealed that the frequency–response curves are sensitive to the distance between the two magnets. The curve may exhibit the hardening-type, softening-type or linear behavior due to the existence of the quadratic nonlinearity. The outcomes are supported by the numerical simulations very well.