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Research Papers

Parametrically Excited Electrostatic MEMS Cantilever Beam With Flexible Support

[+] Author and Article Information
Mark Pallay

Department of Mechanical Engineering,
Binghamton University,
4400 Vestal Parkway East,
Binghamton, NY 13902

Shahrzad Towfighian

Department of Mechanical Engineering,
Binghamton University,
4400 Vestal Parkway East,
Binghamton, NY 13902
e-mail: stowfigh@binghamton.edu

1Corresponding author.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received April 21, 2016; final manuscript received September 19, 2016; published online December 7, 2016. Assoc. Editor: Izhak Bucher.

J. Vib. Acoust 139(2), 021002 (Dec 07, 2016) (8 pages) Paper No: VIB-16-1229; doi: 10.1115/1.4034954 History: Received April 21, 2016; Revised September 19, 2016

Parametric resonators that show large amplitude of vibration are highly desired for sensing applications. In this paper, a microelectromechanical system (MEMS) parametric resonator with a flexible support that uses electrostatic fringe fields to achieve resonance is introduced. The resonator shows a 50% increase in amplitude and a 50% decrease in threshold voltage compared with a fixed support cantilever model. The use of electrostatic fringe fields eliminates the risk of pull-in and allows for high amplitudes of vibration. We studied the effect of decreasing boundary stiffness on steady-state amplitude and found that below a threshold chaotic behavior can occur, which was verified by the information dimension of 0.59 and Poincaré maps. Hence, to achieve a large amplitude parametric resonator, the boundary stiffness should be decreased but should not go below a threshold when the chaotic response will appear. The resonator described in this paper uses a crab-leg spring attached to a cantilever beam to allow for both translation and rotation at the support. The presented study is useful in the design of mass sensors using parametric resonance (PR) to achieve large amplitude and signal-to-noise ratio.

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Figures

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Fig. 1

Layout of the resonator with cross section of the beam and top down view of electrode and beam tip

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Fig. 2

Fixed versus flexible support

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Fig. 3

The convergence of nondimensional mass (m), damping (c), and stiffness (k) values to the fixed boundary case (dashed lines) as Rr and Rt increase. Good convergence occurs at approximately 104 and above.

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Fig. 4

Case I: Rr frequency response (Rt=10,000) at 100 V AC. Numbers next to lines indicate the value of Rr. Solid lines indicate downsweep and dashed lines indicate upsweep. Frequency step of 20 Hz.

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Fig. 5

Case II: Rt frequency response (Rr=10,000) at 100 V AC. Numbers next to lines indicate the value of Rt. Solid lines indicate downsweep and dashed lines indicate upsweep. Frequency step of 20 Hz.

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Fig. 6

Case III: Rr = Rt frequency response at 100 V AC. Numbers next to lines indicate the value of Rr and Rt. Solid lines indicate downsweep and dashed lines indicate upsweep. Frequency step of 20 Hz.

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Fig. 7

Rr=Rt=10 time response in steady-state regime with an approximately 50 μm amplitude. Driven at 10,880 Hz at a voltage of 100 V. An initial displacement of 40 μm was used to reach the higher energy state in the backsweep region.

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Fig. 8

Frequency response of Rr=Rt=10 showing higher‐order parametric resonances. Frequency step of 20 Hz.

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Fig. 9

Bifurcation diagram for Rt = 10,000 and Rr = 2 showing peak amplitude against frequency for the last 50 cycles in the steady-state regime. Backsweep only. Includes both primary and fundamental parametric resonances. Frequency step of 10Hz.

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Fig. 10

Bifurcation diagram for Rt = 10,000 and Rr = 25 showing peak amplitude against frequency for the last 50 cycles in the steady-state regime. Backsweep only. Frequency step of 10Hz.

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Fig. 11

Poincaré section of the chaotic attractor for Rt = 10,000, Rr = 2, and f = 12 kHz. A total of 1500 points were sampled. The simulation was run for 2000 cycles, with sample points starting after cycle 500 to ensure the system was on the chaotic attractor and had negligible transient effects.

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Fig. 12

Information dimension for Rr = 2 and Rt = 10,000 at f = 12 kHz. Initial conditions are nondimensional.

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Fig. 13

Zoomed in portion of the Rr = 2 bifurcation diagram showing intermittent chaos and period 5 behavior. A 0.1 Hz frequency step.

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Fig. 14

Time response of Rr = 2 in period 5 regime at 12.44 kHz

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Fig. 15

Backsweep frequency response of Rr=Rt=10 for low voltages. Numbers next to lines indicate voltage level. Frequency step of 5 Hz.

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Fig. 16

Frequency response of support design 3 as listed in Table 2: Rr = 5.33 and Rt = 2203.26, at 100 V AC. Solid lines indicate downsweep and dashed lines indicate upsweep. Frequency step size was 20 Hz.

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Fig. 17

Backsweep frequency response of design 3, with Rr = 5.33 and Rt = 2203.26. Numbers next to lines indicate voltage level. Frequency step of 5 Hz.

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