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Research Papers

# On the Use of the Subharmonic Resonance as a Method for Filtration

[+] Author and Article Information

Hashemite University, Zarqa, Jordanbkhammad@vt.edu

Ali H. Nayfeh

Virginia Tech, Blacksburg, VA 24061anayfeh@vt.edu

Eihab M. Abdel-Rahman

University of Waterloo, Waterloo, ON, N2L 3G1 Canadaeihab@uwaterloo.ca

J. Comput. Nonlinear Dynam 6(4), 041007 (Apr 12, 2011) (11 pages) doi:10.1115/1.4003031 History: Received April 24, 2010; Revised October 30, 2010; Published April 12, 2011; Online April 12, 2011

## Abstract

We study the feasibility of employing subharmonic resonance of order one-half to create a bandpass filter. A filter made up of two clamped-clamped microbeam resonators coupled by a weak beam is employed as a test design. We discretize the distributed-parameter system using the Galerkin procedure to obtain a reduced-order model composed of two nonlinear coupled Ordinary Differentiation Equations (ODEs). It accounts for geometric and electric nonlinearities as well as the coupling between these two fields. Using the method of multiple scales, we determine four first-order nonlinear ODEs describing the amplitudes and phases of the modes. We use these equations to determine closed-form expressions for the static and dynamic deflections of the structure. The basis functions in the discretization are the linear undamped global mode shapes of the unactuated structure. We found that it is impractical to use the proposed filter structure for subharmonic resonance-based filtering since it cannot produce a single-valued response for small excitation amplitudes. On the other hand, it is feasible to use cascaded uncoupled resonators to build a bandpass filter by operating one in the softening domain and the other in the hardening domain.

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Copyright © 2011 by American Society of Mechanical Engineers
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## Figures

Figure 1

(a) A schematic drawing of two clamped-clamped microbeam resonators coupled by a weak beam and (b) a schematic diagram for one of the resonators

Figure 2

(a) First and (b) second global mode shapes of the filter

Figure 3

Variation of the nonlinear coefficient S3 with the dc voltage

Figure 4

Static (solid line) and Hopf (dashed line) bifurcation curves for the trivial solution for Vdc1=Vdc2=30.0 V. Thin dotted lines represent the excitation amplitudes.

Figure 5

Frequency-response curves for Case I. Stable, unstable, and modulated solutions are represented by solid, long-dashed, and short-dashed lines, respectively. SpPF and SbPF stand for supercritical and subcritical pitchfork bifurcations, respectively.

Figure 6

Frequency-response curves for Case II. Stable, unstable, and modulated solutions are denoted by solid, long-dashed, and short-dashed lines, respectively. SpPF and SbPF represent supercritical and subcritical pitchfork bifurcations, respectively.

Figure 7

Frequency-response curves for Case III. Stable and unstable solutions are denoted by solid and dashed lines, respectively. SpPF and SbPF are supercritical and subcritical pitchfork bifurcations, respectively.

Figure 8

Frequency-response curves for Case IV. Stable and unstable solutions are denoted by solid and dashed lines, respectively. SpPF and SbPF stand for supercritical and subcritical pitchfork bifurcations, respectively.

Figure 9

Variation of the threshold Vac,th with the dc voltage. The solid line is for Q=1150, and the dashed line is for Q=2000.

Figure 10

Bifurcation diagrams for Q=1150 (solid lines), Q=750 (dashed lines), and Q=330 (dotted lines) for Vdc1=Vdc2=30.0 V

Figure 11

Bifurcation diagrams for the trivial solution for Vdc1=30.0 V and Vdc2= (a) 30.1 V, (b) 30.3 V, (c) 30.5 V, (d) 29.9 V, (e) 29.7 V, and (f) 29.5 V

Figure 12

Frequency-response curves for Vdc1=30.0 V and Vdc2=30.5 V. Stable and unstable solutions are denoted by solid and dashed lines, respectively. SpPF and SbPF stand for supercritical and subcritical pitchfork bifurcations, respectively.

Figure 13

Frequency-response curves for Vdc1=30.0 V and Vdc2=29.5 V. Stable and unstable solutions are denoted by solid and dashed lines, respectively. SpPF and SbPF are supercritical and subcritical pitchfork bifurcations, respectively.

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