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Research Papers

Theoretical and Experimental Investigation of Two-to-One Internal Resonance in MEMS Arch Resonators

[+] Author and Article Information
Feras K. Alfosail

Physical Sciences and Engineering Division,
King Abdullah University of Science and
Technology,
Thuwal 23955-9600, Saudi Arabia

Amal Z. Hajjaj

Physical Sciences and Engineering Division,
King Abdullah University of Science
and Technology,
Thuwal 23955-9600, Saudi Arabia

Mohammad I. Younis

Physical Sciences and Engineering Division,
King Abdullah University of Science and
Technology,
Thuwal 23955-9600, 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 May 22, 2018; final manuscript received October 7, 2018; published online November 19, 2018. Assoc. Editor: Eihab Abdel-Rahman.

J. Comput. Nonlinear Dynam 14(1), 011001 (Nov 19, 2018) (15 pages) Paper No: CND-18-1222; doi: 10.1115/1.4041771 History: Received May 22, 2018; Revised October 07, 2018

We investigate theoretically and experimentally the two-to-one internal resonance in micromachined arch beams, which are electrothermally tuned and electrostatically driven. By applying an electrothermal voltage across the arch, the ratio between its first two symmetric modes is tuned to two. We model the nonlinear response of the arch beam during the two-to-one internal resonance using the multiple scales perturbation method. The perturbation solution is expanded up to three orders considering the influence of the quadratic nonlinearities, cubic nonlinearities, and the two simultaneous excitations at higher AC voltages. The perturbation solutions are compared to those obtained from a multimode Galerkin procedure and to experimental data based on deliberately fabricated Silicon arch beam. Good agreement is found among the results. Results indicate that the system exhibits different types of bifurcations, such as saddle node and Hopf bifurcations, which can lead to quasi-periodic and potentially chaotic motions.

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Figures

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Fig. 1

A schematic of the clamped–clamped arch beam electrothermally tuned and electrostatically actuated

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Fig. 2

A scanning electron microscope image of the studied silicon clamped–clamped arch beam

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Fig. 3

(a) Natural frequency variation against the applied electrothermal voltage VTh and (b) variation of the frequency ratio with VTh

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Fig. 4

Frequency response curves showing the transverse deflection of the midpoint of the arch as a function of the detuning parameter σ1: (a) VDC=45V, VAC=0.5V, σ2=−1.8. () MTS solution from Eq. (36) neglecting the influence of cos(2Ωt) by setting F2VAC=0 and F3VAC=0, (▶) 4-modes Galerkin solution of Eq. (11) and (b)VDC=15V, VAC=20V, σ2=−1.8 () MTS solution from Eq. (36) neglecting the influence of cos(2Ωt) by setting F2VAC=0 and F3VAC=0, (◆) experimental data.

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Fig. 5

Frequency response curves showing the amplitudes of the first and third modes as a function of the detuning parameter σ1 for VDC=45V, VAC=1V, σ2=−7.42, μ1=1.35, μ2=0.65, F2VDC=3.17, F3VDC=0.01785. (—) Stable solution, (– –) unstable solution. The influence of cos(2Ωt) is neglected by setting F2VAC=0 and F3VAC=0.

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Fig. 6

Time history, corresponding phase space, and Fourier frequency spectrum for VDC=45V, VAC=1V, σ2=−7.42, μ1=1.35,μ2=0.65, F2VDC=3.17, F3VDC=0.01785, (a) σ1=−1.6, (b) σ1=−0.73. The influence of cos(2Ωt) is neglected by setting F2VAC=0 and F3VAC=0.

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Fig. 7

Frequency response curves showing the amplitudes of the first and third modes as a function of the detuning parameter σ1 for VDC=45V, VAC=2V, σ2=−7.42, μ1=1.35, μ2=0.65, F2VDC=6.34, F3VDC=0.035. (—) Stable solution, (– –) unstable solution. H denotes Hopf bifurcation. The influence of cos(2Ωt) is neglected by setting F2VAC=0 and F3VAC=0.

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Fig. 8

Time history, Poincare section, and fast Fourier transform spectrum for VDC=45V, VAC=4V, σ2=−7.42, μ1=1.35,μ2=0.65, F2VDC=12.68, F3VDC=0.074σ1=−3.074

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Fig. 9

Force response curves showing the first and third mode amplitudes as a function of the applied VAC voltage for VDC=45V, μ1=1.35,μ2=0.65, σ2=−7.42: (a)σ1=−0.5 and (b)σ1=3. (—) Stable solution, (– –) unstable solution. H denotes Hopf bifurcation.

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Fig. 10

Frequency response curves showing the transverse deflection of the midpoint of the arch as a function of the detuning parameter σ1 for VDC=15V, VAC=30V, σ2=0.826, μ1=3.85,μ2=2: (a) experimental data, (b) the inset figure shows the full range of the frequency response curve. MTS solution using single excitation term, cos(Ωt), by setting F2VAC=0 and F3VAC=0, and (c) MTS solution using simultaneous excitation. (—) Stable solution, (– –) unstable solution. (–·–) Unstable solution due to Hopf bifurcation. H denotes Hopf bifurcation.

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Fig. 11

Frequency response curves showing the amplitudes of the first and third modes as afunction of the detuning parameter σ1 for VDC=15V, VAC=30V, σ2=0.826, μ1=3.85,μ2=2. (—) Stable solution, (– –) unstable solution. (–·–) Unstable fixed solution due to Hopf bifurcation. H denotes Hopf bifurcation.

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Fig. 12

Fourier frequency coefficient variation from Eqs. (A3a) and (A3b) as a function of the detuning parameter for VDC=15V, VAC=30V, σ2=0.826, μ1=3.85,μ2=2. (—) Stable solution, (– –) unstable solution.

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