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TECHNICAL PAPERS

Behavior of a Self-Sustained Electromechanical Transducer and Routes to Chaos

[+] Author and Article Information
J. C. Chedjou1

 International Centre for Theoretical Physics (ICTP), Strada Costiera 11, 34014 Trieste, Italy, IUT-LEM, 03100 Montluçon Cedex, France, and Department of Physics, Faculty of Science, University of Dschang, BP 67, Dschang, Cameroonchedjou@ictp.it; chedjou@ant.uni-hannover.de; chedjou@moniut.univ-bpclermont.fr

K. Kyamakya

Chair of Computer Science in Transportation, Institut fü. Informatik-systeme, University of Klagenfurt, Universitaetsstr. 65, A-9020 Klagenfurt, Austriakya@ant.uni-hannover.de

I. Moussa

Department of Physics, Faculty of Science, University of Yaoundé-I, BP 812, Yaoundé, Cameroonmoussaildoko@yahoo.fr

H.-P. Kuchenbecker

Institut für Allgemeine Nachrichtentechnik, Univeristät Hannover, Appelstr. 9A, 30167, Hannover, Germanyku@ant.uni-hannover.de

W. Mathis

Institut für Theoretische Elektrotechnik und Hochfrequenztechnik, University of Hannover, Appelstr. 9A, 30167, Hannover, Germanymathis@tet.uni-hannover.de

1

Corresponding author.

J. Vib. Acoust 128(3), 282-293 (Nov 16, 2005) (12 pages) doi:10.1115/1.2172255 History: Received July 11, 2003; Revised November 16, 2005

This paper studies the dynamics of a self-sustained electromechanical transducer. The stability of fixed points in the linear response is examined. Their local bifurcations are investigated and different types of bifurcation likely to occur are found. Conditions for the occurrence of Hopf bifurcations are derived. Harmonic oscillatory solutions are obtained in both nonresonant and resonant cases. Their stability is analyzed in the resonant case. Various bifurcation diagrams associated to the largest one-dimensional (1-D) numerical Lyapunov exponent are obtained, and it is found that chaos can appear suddenly, through period doubling, period adding, or torus breakdown. The extreme sensitivity of the electromechanical system to both initial conditions and tiny variations of the coupling coefficients is also outlined. The experimental study of̱the electromechanical system is carried out. An appropriate electronic circuit (analog simulator) is proposed for the investigation of the dynamical behavior of the electromechanical system. Correspondences are established between the coefficients of the electromechanical system model and the components of the electronic circuit. Harmonic oscillatory solutions and phase portraits are obtained experimentally. One of the most important contributions of this work is to provide a set of reliable analytical expressions (formulas) describing the electromechanical system behavior. These formulas are of great importance for design engineers as they can be used to predict the states of the electromechanical systems and respectively to avoid their destruction. The reliability of the analytical formulas is demonstrated by the very good agreement with the results obtained by both the numeric and the experimental analysis.

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Copyright © 2006 by American Society of Mechanical Engineers
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Figures

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Figure 1

Representation of the eigenvalues in the complex plane (Im(η),Re(η)) (the parameters used are defined in the text)

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Figure 2

The steady states A0 and B0 versus the detuning parameter σ: analytical results (solid lines); numerical results (stars); and experimental results (dots). (The corresponding parameters are defined in the text.)

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Figure 3

The steady states A0 and B0 versus the damping coefficient ϵ2 for three values of σ: analytical results (solid lines (stable stationary state) and broken lines (unstable stationary state)); numerical results (stars); and experimental results (dots in (a) and (b)). (The corresponding parameters are defined in the text.)

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Figure 4

Numerical phase portraits for the oscillators x(Pi1) and y(Pi2): P11 and P12: c2=0.090, P21 and P22: c2=0.450, P31 and P32: c2=0.700, P41 and P42: c2=0.818. (The corresponding parameters are defined in the text.)

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Figure 5

Bifurcation diagrams showing the coordinates x and y of the attractors in the Poincaré cross section versus c2. (The corresponding parameters are defined in Fig. 4.)

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Figure 6

Basins of attraction for chaotic solutions (black regions) and regular solutions (white regions). x0=0 and ẋ0=0 (the corresponding parameters are defined in the text).

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Figure 7

Schematic of the complete electronic simulator

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Figure 8

Experimental phase portraits for the oscillators x(Pi1) and y(Pi2): P11 and P12: R11=426,800Ω, P21 and P22: R11=85,360Ω, P31 and P32: R11=54,870Ω, P41 and P42: R11=46,960Ω.

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