Nonlinear multi-element interactions in an elastically coupled microcantilever array subject to electrodynamic excitation

In this work, we formulate and investigate a nonlinear initial boundary-value problem for an array of N elastically coupled hybrid microcantilever beams that are subject to electrodynamic excitation. The equations of motion for the individual viscoelastic element consist of two fields: the base component which is common to all cantilevers and the unrestrained component which is excited electrodynamically. The coupling of the elements is obtained via an equivalent linear stiffness that is estimated from experimental measurements of a 5-element array. We employ a Galerkin ansatz to obtain a modal dynamical system that consistently incorporates a quintic nonlinearity due to the combined effects of cubic viscoelasticity and quadratic electrodynamics. We validate the periodic response of a 5-element array with moderate damping and construct numerically a comprehensive bifurcation structure for a 25-element array. The analysis reveals an intricate structure for small damping that includes both quasiperiodic and nonstationary chaotic-like energy transfer between the elements of the array. It is noteworthy that an array with a larger coupling stiffness, corresponding to a smaller distance between adjacent elements, yields a chaotic bifurcation structure for a larger value of viscoelastic damping.