The dynamic motion of a parametrically excited microbeam-string affected by nonlinear damping is considered asymptotically and numerically. It is assumed that the geometrically nonlinear beam-string, subject to only modulated alternating current voltage, is closer to one of the electrodes, thus resulting in an asymmetric dual gap configuration. A consequence of these novel assumptions is a combined parametric and hard excitation in the derived continuum-based model that incorporates both linear viscous and nonlinear viscoelastic damping terms. To understand how these assumptions influence the beam's performance, the conditions that lead to both principal parametric resonance and a three-to-one internal resonance are investigated. Such conditions are derived analytically from a reduced-order nonlinear model for the first three modes of the microbeam-string using the asymptotic multiple-scales method which requires reconstitution of the slow-scale evolution equations to deduce an approximate spatio-temporal solution. The response is investigated analytically and numerically and reveals a bifurcation structure that includes coexisting in-phase and out-of-phase solutions, Hopf bifurcations, and conditions for the loss of orbital stability culminating with nonstationary quasi-periodic solutions and chaotic strange attractors.

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