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TECHNICAL PAPERS

Dynamics of a Quasiperiodically Forced Rayleigh Oscillator

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

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

L. K. Kana

Doctoral School of Electronics and Information Technology, UDÉTI, Faculty of Science, University of Dschang, P.O. Box 67 Dschang, Cameroon

I. Moussa

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

K. Kyamakya

Institut f. Informatik-Systeme, University of Klagenfurt, Universitaetsstr. 65, A-9020 Klagenfurt, Austriakyamakya@isys.uni-klu.ac.at

A. Laurent

 Université Blaise Pascal, Clermont Ferrand II, IUT-LEM, GEII, Avenue Aristide BRIAND BP 2235, 03100 Montluçon, Francelaurent@moniut.univ-bpclermont.fr

1

Corresponding author.

J. Dyn. Sys., Meas., Control 128(3), 600-607 (Nov 21, 2005) (8 pages) doi:10.1115/1.2232684 History: Received November 21, 2003; Revised November 21, 2005

This paper studies the dynamics of a self-excited oscillator with two external periodic forces. Both the nonresonant and resonant states of the oscillator are considered. The hysteresis boundaries are derived in terms of the system’s parameters. The stability conditions of periodic oscillations are derived. Routes to chaos are investigated both from direct numerical simulation and from analog simulation of the model describing the forced oscillator. One of the most important contributions of this work is to provide a set of reliable analytical expressions (formulas) describing the system’s behavior. These are of great importance to design engineers. The reliability of the analytical formulas is demonstrated by a very good agreement with the results obtained by both the numeric and experimental analyses.

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Copyright © 2006 by American Society of Mechanical Engineers
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Figures

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Figure 1

Graphical representation of the hysteresis domains (black regions) and nonhysteresis domains (white regions) in the σk1 plane (Σ1, Σ1′, and Σ2) and k1k2 plane (Σ3): Σ1: (k2=C1ste and l1=(8∕9)ε12η3); Σ1′: (k2=C2ste and l1′=(8∕9)ε12η3(2−1)); Σ2: [k2=C3ste,l5=(8∕9)η3ε12+9σ2η−(η2ε14∕3−3σ2ε1−(2∕3))3∕2 and l2=(8∕9)η3ε12+9σ2η+(η2ε14∕3−3σ2ε1−(2∕3))3∕2−(8∕9)η3ε12+9σ2η−(η2ε14∕3−3σ2ε1−(2∕3))3∕2]; (Σ3): (σ=0,l3=(4∕9)ε1 and l4=(∣ω2−ω22∣∕ω2)(2∕3)(1−[9k1∕4ε1](2∕3)))

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Figure 2

Effects of σ and k1 on the resonant frequency—response curves (ω=1, ω2=2.3, ε1=0.13112, and k2=1). Analytical results (solid lines) and numerical results (stars) for k1=0.015; k1=0.025; k1=0.030; and k1=0.050. Stability boundary (squares) and unstable region (shaded).

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Figure 5

Bifurcation diagram showing the coordinate x of the attractor in the Poincaré cross section versus k1 (ω=1.000, k2=0.000, ε1=2.300, and F1=0.040)

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Figure 6

Numerical phase portraits of the oscillator with the parameters of Fig. 5 for k1=1.850 (torus) and k1=2.398 (chaos)

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Figure 7

Bifurcation diagram showing the coordinate x of the attractor in the Poincaré cross section versus F1 (ω=1.000, k2=0.000, ε1=1.390, and k1=3.328)

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Figure 11

Experimental phase portraits of the oscillator with the parameters of Fig. 8: P3:F1=0.0410Hz (period-1), P4:F1=0.0385Hz (period-2), P5:F1=0.0360Hz (period-3), and P6:F1=0.0275Hz (chaos). Voltage scales: X-input: 1V/div. and Y-input: 0.5V/div.

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Figure 10

Experimental phase portraits of the oscillator with the parameters of Fig. 6: P1:R7=5400Ω(k1=1.850) (torus) and P2:R7=4166Ω(k1=2.398) (chaos). Voltage scales: X-input: 1V/div. and Y-input: 1V/div.

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Figure 9

Schematic of the electronic simulator

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Figure 8

Numerical phase portraits of the oscillator with the parameters of figure 7 for (a) F1=0.0410 (period-1), (b) F1=0.0385 (period-2), (c) F1=0.0360 (period-3), and (d) F1=0.0275 (chaos)

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Figure 4

Effects of k1, k2 and σ on the resonant amplitude-response curves (ω=1.000, ε1=0.015, and ω2=2.300): Analytical results (solid lines) and numerical results (stars) for k1=0.0032 (curves C1, C1*, and C1**), k1=0.0050 (curves C2, C2*, and C2**), k1=0.0065 (curves C3, C3*, and C3**), and k1=0.0080 (curves C4, C4*, and C4**). Stability boundary (squares) and unstable region (shaded).

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Figure 3

Effects of k1 and σ on the resonant amplitude—response curves (ω=1, ω2=2.3, ε1=0.015, and k2=0.0465): Analytical results (solid lines) and numerical results (stars) for σ=0.000; σ=0.002; σ=0.003; and σ=0.005 Stability boundary (squares) and unstable region (shaded).

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