On the Application of the Multiple Scales Method on Electrostatically Actuated Resonators

Abstract

We investigate modeling the dynamics of an electrostatically actuated resonator using the perturbation method of multiple time scales (MTS). First, we discuss two approaches to treat the nonlinear parallel-plate electrostatic force in the equation of motion and their impact on the application of MTS: expanding the force in Taylor series and multiplying both sides of the equation with the denominator of the forcing term. Considering a spring–mass–damper system excited electrostatically near primary resonance, it is concluded that, with consistent truncation of higher-order terms, both techniques yield same modulation equations. Then, we consider the problem of an electrostatically actuated resonator under simultaneous superharmonic and primary resonance excitation and derive a comprehensive analytical solution using MTS. The results of the analytical solution are compared against the numerical results obtained by long-time integration of the equation of motion. It is demonstrated that along with the direct excitation components at the excitation frequency and twice of that, higher-order parametric terms should also be included. Finally, the contributions of primary and superharmonic resonance toward the overall response of the resonator are examined.

Linear and nonlinear dynamics of micro and nano-resonators: Review of recent advances

“…This signifies the importance of the expansion in revealing the effective nonlinearities due to the electrostatic term. In addition, it was revealed in [303] that the Taylor expansion of the electrostatic term yields the same dynamic response to that of the cross-multiplied one up to the same order using perturbations. To further analyze the dynamic response, we analyze Eq.…”

“…Following the details of the multiple time scales perturbations method and using the method of reconstitution, we end up with two modulation equations that govern the amplitude a and phase  of the response [304].…”

“…Following the multiple time scales method and using the method of reconstituting, we ended up with two modulation equations[304]. The final dynamic solution is…”