Period-doubling cascade route to chaos in an initially curved microbeam resonator exposed to fringing-field electrostatic actuation

Abstract

Abstract This paper explores the nonlinear dynamics of a piezoelectrically laminated microbeam resonator with an initial curvature, which is subjected to electrostatic actuation caused by fringing fields. The resonator is fully clamped at both of its ends and is coated with two piezoelectric layers, encompassing both the top and bottom surfaces. The fringing field electrostatic force is assessed through finite element modeling, and the resulting data is accurately fitted to a suitable hyperbolic function. The nonlinear motion equation accounts for the geometric nonlinearity and the nonlinear electrostatic force. The motion equation is discretized using Galerkin method and the reduced order system is numerically integrated over the time for the time response. The variation of the first three natural frequencies with respect to the applied electrostatic voltage is determined and the frequency response curve is determined. The bifurcation points have been examined and their types have been clarified based on the loci of the Floquet exponents on the complex plane. The period -doubled branches of the frequency response curves originating from the period doubling (PD) bifurcation points are stablished. It's demonstrated that the succession PD cascades leads to chaotic behavior. The chaotic behavior is identified qualitatively by constructing the corresponding Poincaré section and analyzing the response's associated frequency components. The chaotic response is regularized by applying an appropriate piezoelectric voltage which shifts the frequency response curve along the frequency axis.