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Research Papers

On the Dynamics of Two Mutually-Coupled, Electromagnetically-Actuated Microbeam Oscillators

[+] Author and Article Information
Andrew B. Sabater

School of Mechanical Engineering, Birck Nanotechnology Center, and Ray W. Herrick Laboratories,  Purdue University, West Lafayette, IN 47907

Jeffrey F. Rhoads1

School of Mechanical Engineering, Birck Nanotechnology Center, and Ray W. Herrick Laboratories,  Purdue University, West Lafayette, IN 47907jfrhoads@purdue.edu

1

Address all correspondence to this author.

J. Comput. Nonlinear Dynam 7(3), 031011 (Apr 04, 2012) (9 pages) doi:10.1115/1.4005999 History: Received June 06, 2011; Revised January 05, 2012; Online April 04, 2012; Published April 05, 2012

This work describes the modeling, analysis, predictive design, and control of self-excited oscillators, and associated arrays, founded upon electromagnetically-actuated microbeams. The study specifically focuses on the characterization of nonlinear behaviors arising in isolated oscillators and small arrays of nearly-identical, mutually-coupled oscillators. The work provides a framework for the exploration of larger oscillator arrays with different forms of coupling and feedback, which can be exploited in practical applications ranging from signal processing to micromechanical neurocomputing.

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Copyright © 2012 by American Society of Mechanical Engineers
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Figures

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Figure 1

Model of the beam in three dimensions. As shown in the inset figure, the magnetic field B→ is oriented at an angle α with respect to the vertical reference.

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Figure 2

Schematic diagram of the beam element with a description of the variables used for modeling. Note that u, v, and ψ are the longitudinal, transverse, and angular displacements, respectively, and s is the arc length.

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Figure 3

Schematic diagram of an array of electromagnetically-actuated microcantilever oscillators. Each microbeam is assumed to be spaced such that it is mechanically isolated. Each microbeam has two current loops: one for actuation and one for sensing.

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Figure 4

Bifurcation diagram of the steady-state amplitude for the parameters in Table 3, α = π/4 and ɛ −2 K = 5 × 103 . The rest state always exists, however it is only stable for ɛ −2 G greater than −234.4. For ɛ −2 G less than this value, the system has a stable limit cycle.

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Figure 5

Bifurcation diagram of the steady-state amplitude for the parameters in Table 3, ɛ −2 G =  − 1000 and ɛ −2 K = 5 × 103 . The rest state always exists, however it is only stable for |α|<0.349. For α outside this domain, the system has a stable limit cycle.

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Figure 6

Bifurcation diagram for the two oscillator case when c¯>γ1. The parameters for the system are ɛ −2 G =  −200, ɛ −2 K = 1 × 104 , α = π/4 and δ = 0.05. The rest state is stable when |Gc|<14.653. When Gc exceeds this critical value, the symmetric phase locked solution is stable and the oscillators have limit cycles of identical frequency, with a phase difference related to the frequency mistuning.

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Figure 7

Bifurcation diagram for the two oscillator case, with ɛ −2 G =  −2 × 103 , ɛ −2 K = 5 × 103 , α = π/4 and δ = 0. The solutions in this case are qualitatively the same as in the case when there is a frequency mistuning, with two major differences: the symmetric phase drift solution does not exist and the symmetric phase locked solution is always stable for a nonzero value of Gc .

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Figure 8

Bifurcation diagram for the two oscillator case, with ɛ −2 G =  −2 × 103 , ɛ −2 K = 5 × 103 , α = π/4 and δ = 0.05. The symmetric phase drift solution (solid blue) is the only stable solution when |Gc|<1.465. For larger Gc , the only stable solution is the larger symmetric phase locked solution (solid red). The asymmetric (black dashed) and smaller phase locked solutions (red dashed) exist for certain values of Gc , but these solutions are not stable.

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Figure 9

Bifurcation diagram for the two oscillator case, with ɛ −2 G =  −2 × 103 , ɛ −2 K = 5 × 103 , α = π/4 and Gc  = 1. The larger symmetric phase locked solution is the only stable solution for |δ|<0.034 and outside this range the symmetric phase drift solution is the only stable solution. Sensing the amplitude of the phase locked solution could be exploited to implement a mass sensor.

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Figure 10

Response of two mutually-coupled oscillators, where the displacement of oscillator 1 is shown in red and the displacement of oscillator 2 is shown in blue. With ɛ −2 G =  −2 × 103 , ɛ −2 K = 5 × 103 , α = π/4 and Gc  = 1, a 4 ng mass is added to the tip of oscillator 2 at 0.120 μs. This results in a change in δ and causes the system to transition from a stable symmetric phase locked state to a symmetric phase drift state, as was predicted by the bifurcation diagram in Fig. 9. This mass deposition event can be detected by monitoring either oscillator’s amplitude or the relative phase between the oscillators.

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