Nonlinear dynamics of a piezoelectrically laminated initially curved microbeam resonator exposed to out-of-plane electrostatic actuation

Abstract

Abstract In this paper the nonlinear dynamics of a piezoelectrically sandwiched initially curved microbeam subjected to out-of-plane electrostatic actuation is investigated. The governing motion equation is derived by minimising the Hamiltonian over the time and discretised to a reduced order model using Galerkin technique. The modelling accounts for nonlinear fringing field and mid plane stretching effect which appears as quadratic and cubic nonlinearities in the motion equation. 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 as well as the piezoelectric excitation on the response of the system are examined and their effect on the bifurcation types are determined. The force response curves assuming the AC voltage as the bifurcation parameter are also introduced; It is illustrated that in contrast with 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.