Research Papers

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

[+] Author and Article Information
Saad Ilyas

Physical Sciences and Engineering (PSE),
King Abdullah University of Science
and Technology,
Thuwal 23955-6900, Saudi Arabia
e-mail: saad.ilyas@kaust.edu.sa

Feras K. Alfosail

Physical Sciences and Engineering (PSE),
King Abdullah University of Science
and Technology,
Thuwal 23955-6900, Saudi Arabia
e-mail: feras.alfosail@kaust.edu.sa

Mohammad I. Younis

Physical Sciences and Engineering (PSE),
King Abdullah University of Science
and Technology,
Thuwal 23955-6900, Saudi Arabia
e-mail: mohammad.younis@kaust.edu.sa

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received June 30, 2018; final manuscript received January 29, 2019; published online February 15, 2019. Assoc. Editor: Eihab Abdel-Rahman.

J. Comput. Nonlinear Dynam 14(4), 041006 (Feb 15, 2019) (8 pages) Paper No: CND-18-1291; doi: 10.1115/1.4042694 History: Received June 30, 2018; Revised January 29, 2019

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.

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Grahic Jump Location
Fig. 1

Schematic of an SDOF spring–mass–damper system representing an MEMS microbeam resonator under parallel-plate electrostatic actuation

Grahic Jump Location
Fig. 2

Analytical and numerical frequency–response curves of the resonator against the frequency detuning parameter σ for (a) VDC = 10 V, VAC = 10 V, and μ = 0.01; (b) VDC = 15 V, VAC = 5 V, and μ = 0.01; and (c) VDC = 1 V, VAC = 15 V, and μ = 0.01. The detuning parameter here is defined as Ω=(1/2)ωn+σ. Note here that the frequency response equations obtained from MTS give only the dynamic solution, and hence the static solution is superimposed afterward to compare with the LTI results.

Grahic Jump Location
Fig. 3

Analytical frequency–response curves of the resonator against the frequency detuning parameter σ for (a) VDC = 10 V, VAC = 10 V, and μ = 0.01; (b) VDC = 15 V, VAC = 5 V and μ = 0.01; and (c) VDC = 1 V, VAC = 15 V μ = 0.01, showing the contribution from primary and superharmonic components to the overall resonators response. The detuning parameter here is defined as σ=Ω−(1/2)ωn. The response shown here comprises of the dynamic solution obtained visa MTS superimposed onto the static solution.



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