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Research Papers

Analysis of a Chaotic Electrostatic Micro-Oscillator

[+] Author and Article Information
S. Towfighian

Department of Mechanical Engineering, University of Waterloo, Waterloo, ON, N2L 3G1, Canadastowfigh@engmail.uwaterloo.ca

G. R. Heppler

Department of Systems Design Engineering, University of Waterloo, Waterloo, ON, N2L 3G1, Canadaheppler@uwaterloo.ca

E. M. Abdel-Rahman1

Department of Systems Design Engineering, University of Waterloo, Waterloo, ON, N2L 3G1, Canadaeihab@uwaterloo.ca

1

Corresponding author.

J. Comput. Nonlinear Dynam 6(1), 011001 (Sep 27, 2010) (10 pages) doi:10.1115/1.4002086 History: Received May 08, 2009; Revised March 04, 2010; Published September 27, 2010; Online September 27, 2010

The closed-loop dynamics of a chaotic electrostatic microbeam actuator are presented. The actuator was found to be an asymmetric two-well potential system with two distinct chaotic attractors: one of which occurs predominantly in the lower well and a second that visits a lower-well orbit and a two-well orbit. Bifurcation diagrams obtained by sweeping the ac voltage amplitudes and frequency are presented. Period doubling, reverse period doubling, and the one-well chaos through period doubling are observed in amplitude sweep. In frequency sweep, period doubling, one-well, and two-well chaos, superharmonic resonances and on and off chaotic oscillations are found.

Copyright © 2011 by American Society of Mechanical Engineers
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References

Figures

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Figure 1

Schematic of microbeam oscillator. The arrows indicate the electrostatic field.

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Figure 2

Nondimensional deflection of the beam tip versus voltage Vdc

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Figure 3

Closed loop system schematic

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Figure 4

Nondimensional deflection of the beam tip versus voltage, Vref, for the closed-loop system

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Figure 5

Roots of the equilibrium equation q1 in complex plane

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Figure 6

Effect of the controller gain G on the static deflection versus voltage, Vref, curve

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Figure 7

Stability diagram in the parameter space of controller gain, G, and voltage, Vref

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Figure 8

Effect of amplification factor, Ψ, on static pull-in graph for G=0.8

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Figure 9

Nondimensional circular frequency versus input voltage Vref

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Figure 10

The nondimensional phase portrait and beam deflection, Vs, for system parameters of G=4, Ψ=1.5274 V, r=100, μ=0.73, Vac=0 V, Vdc=26.2 V, and initial condition of (q1=0.3, q̇1=0, Vs=0.009)

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Figure 11

The phase portrait, beam-tip deflection w1 and controller output voltage, Vs, for μ=0.7, ω=3.5158, Vac=0.7 V, and the initial condition (q1=0.86, q̇1=0, Vs=0.266). The small circle and the cross show the location of the stable focus and saddle, respectively.

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Figure 12

The phase portrait, beam-tip deflection w1, and normalized controller output voltage, Vs, for system parameters of μ=0.7, ω=3.5158, Vac=0.9 V, and initial condition of (q1=0.86, q̇1=0, Vs=0.266)

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Figure 13

Bifurcation diagram constructed from a force sweep Vac at fixed nondimensional natural frequency of lower well ωl=1.98

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Figure 14

The phase portraits and fast Fourier transforms using a one-mode model, the system parameters μ=0.7, ω=1.98. (Top) Periodic Vac=4.1 V. (Middle) Period two Vac=4.4 V. (Bottom) Period four Vac=4.7 V.

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Figure 15

The phase portrait and fast Fourier transform of chaotic oscillation. The system parameters are μ=0.7, ω=1.98, and Vac=4.861 V.

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Figure 16

Phase portrait of period 10 in one well once the amplitude of excitation Vac=4.883 and the nondimensional frequency of excitation ω is 1.98

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Figure 17

Bifurcation diagram sweeping the frequency of excitation at fixed Vac of 4 V

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Figure 18

Phase portraits and FFT for the periodic orbit at the (a) natural frequency of the lower equilibrium Ω=ωl=1.98, (b) superharmonic resonance of order two at Ω=ωl/2=0.99, (c) superharmonic resonance of order three at Ω=ωl/3=0.66, and (d) superharmonic resonance of order six at Ω=ωl/6=0.33

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Figure 19

(a) Phase portrait and (b) Lyapunov exponent of the one-well chaotic attractor when Vac=4 and Ω=1.84

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Figure 20

(a) Phase portrait and (b) Lyapunov exponent of the two-well chaotic attractor when Vac=4 V and Ω=1.081

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