Nonlinear dynamics of a piezoelectrically laminated initially curved microbeam resonator exposed to fringing-field electrostatic actuation

In this paper, the nonlinear dynamics of a piezoelectrically sandwiched initially curved microbeam subjected to fringing-field electrostatic actuation is investigated. The governing motion equation is derived by minimizing the Hamiltonian over the time and discretized to a reduced-order model using the Galerkin technique. The modelling accounts for nonlinearities due to the fringing-field electrostatic force, initial curvature and mid-plane stretching. The electrostatic force is numerically computed using finite element simulation. The nonlinear dynamics of the microbeam in the vicinity of primary resonance is investigated, and the bifurcation types are determined by investigating the location of the Floquet exponents and their configuration with respect to the unit circle on the complex plane. The branches on the frequency-response curves, which originate from the period-doubling bifurcation points, are introduced, and the transition from period-1 to period-2 response is demonstrated by slight sweep of the excitation frequency over the time. The effect of DC and AC electrostatic excitation and the piezoelectric excitation on the response of the system are examined, and their effect on the bifurcation types is determined. The force response curves assuming the AC voltage as the bifurcation parameter are also introduced; it is illustrated that in contrast to in-plane electrostatic excitation, in fringing field-based resonators the resonator is not limited by pull-in instability, which is substantially confining the amplitude of the motion in in-plane resonators.