On Nonlinear Response Near-Half Natural Frequency of Electrostatically Actuated Microresonators

Abstract

This paper deals with the nonlinear response of electrostatically actuated cantilever beam microresonators near-half natural frequency. A¯rst-order fringe correction of the electrostatic force, viscous damping, and Casimir e®ect are included in the model. Both forces, electrostatic and Casimir, are nonlinear. The dynamics of the resonator is investigated using the method of multiple scales (MMS) in a direct approach of the problem. The reduced order model (ROM) method, based on Galerkin procedure, is used as well. Steady-state motions are found. Numerical simulations are conducted for uniform microresonators. The in°uences of damping, actuation, and fringe e®ect on the resonator response are found. NomenclatureConstants " 0 ¼ 8:854 Â 10 À12 C 2 N À1 m À2 Permittivity of free space } ¼ 1:055 Â 10 À34 J Á s Dirac constant c ¼ 2:998 Â 10 8 m Á s À1 Speed of light ¼ 1:85 Â 10 À5 kg Á m À1 s À1

Bifurcation Type Change of AC Electrostatically Actuated MEMS Resonators due to DC Bias

“…MMS is a perturbation method; therefore "the first few steps reveal the important features of the solution and the remaining ones give small corrections"; that is, "one may calculate just few terms in a perturbation series" [33]. Throughout this paper we only discuss a first-order uniform expansion of the solution [18][19][20][21][34][35][36][37][38][39][40][41]. Thus, an approximate solution to this weakly nonlinear system in the form of a first-order uniformly valid expansion is sought:…”

“…A soft voltage having both components of DC and AC is applied between the cantilever and the ground plate. The partial differential equation of motion of the cantilever is as follows [20,21]:…”

“…The pseudo-arclength continuation technique has been employed to examine the nonlinear static and dynamic behavior. Frequency and voltage responses [18][19][20][21] have been reported for EA-MEMS-CRs for soft AC near half natural frequency. This resulted into primary resonance of the system.…”

“…Forces taken into account are due to electrostatic actuation, fringe effect, damping, and beam stiffness. MMS and ROM [18][19][20][21] are used to investigate the resulting nonlinear parametric resonance. It is showed that the subcritical Hopf bifurcation, due to only soft AC, becomes a cyclic fold bifurcation due to the DC bias.…”

Bifurcation and pull-in voltages of primary resonance of electrostatically actuated SWCNT cantilevers to include van der Waals effect

“…They reported both frequency and voltage responses. Both methods MMS and ROM were used to investigate primary and parametric resonance of electrostatically actuated MEMS resonators [4][5][6][7][8][9][10].…”