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Research Papers

Analysis of the van der Pol System With Coulomb Friction Using the Method of Multiple Scales

[+] Author and Article Information
Hiroshi Yabuno

Department of Mechanical Engineering, Faculty of Science and Technology, Keio University, Yokohama 223-8522, Japanyabuno@mech.keio.ac.jp

Yota Kunitho, Takuma Kashimura

Graduate School of Systems and Information Engineering, University of Tsukuba, Tsukuba 305-8573, Japan

J. Vib. Acoust 130(4), 041008 (Jul 11, 2008) (7 pages) doi:10.1115/1.2890401 History: Received June 15, 2007; Revised October 24, 2007; Published July 11, 2008

The effect of Coulomb friction on the nonlinear dynamics of a van der Pol oscillator is presented. A map from the magnitude of a peak to that of the succeeding valley in the time history is analytically described by considering both the exponential growth due to negative viscous damping and the switching condition due to Coulomb friction, which is a function of the sign of the velocity of the system. The steady states and their stability are clarified and the difference from those in the case without Coulomb friction is revealed. The addition of Coulomb friction makes the trivial equilibrium, which is an unstable focus in the system without friction, into a locally asymptotically stable equilibrium set. The branch of stable nontrivial steady states is not bifurcated from the trivial steady state by the effect of Coulomb friction and is different from the branch in the case without Coulomb friction, which is bifurcated from the trivial steady state through Hopf bifurcation. Furthermore, experiments are conducted and the theoretically predicted dynamics due to Coulomb friction is confirmed.

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Copyright © 2008 by American Society of Mechanical Engineers
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Figures

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Figure 1

Relationship between response amplitude Xn* and solution of amplitude equation an(t*)

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Figure 2

Unstable fixed points without nonlinear effect. (Dashed line denotes unstable fixed points of Plin(Xn*). In the region right of the dashed and dotted lines, the oscillation decays owing to the positive viscous damping. In the left region, the system has negative viscous damping. Even with negative damping, when the initial displacement is in the hatched region, the self-excited oscillation cannot be produced by the effect of Coulomb friction.)

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Figure 3

Stable and unstable fixed points (solid line: stable fixed points; dashed line: unstable fixed points; dotted line: nontrivial steady state amplitude in the case without Coulomb friction; hatched region: when the initial displacement is in this region, oscillation decays even when the system has negative viscous damping) (μn=1.25, T*=4.62×10−2, fk=2.60×10−3)

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Figure 4

Mechanical van der Pol oscillator

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Figure 5

Experimental setup

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Figure 6

Experimental unstable fixed points without nonlinear effect and dependence of occurrence of self-excited oscillation on initial displacement. (Dashed line is for unstable fixed points. When initial displacement is in the white region and in the hatched region, self-excited oscillation can and cannot be produced, respectively.)

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Figure 7

Time history of free oscillation (μn=0, x(0)=4mm)

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Figure 8

Time history of linear system with negative viscous damping under relatively large initial displacement (μl=−0.02, x(0)=3.4mm)

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Figure 9

Time history of linear system with negative viscous damping under relatively small initial displacement (μl=−0.02, x(0)=2mm)

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Figure 10

Experimental stable and unstable fixed points with nonlinear effect. (Dashed and solid lines indicate unstable and stable fixed points, respectively. When the initial displacement is in the white region, self-excited oscillation is produced and the amplitude converges to a stable fixed point. When the initial displacement is in the hatched region, self-excited oscillation cannot be produced even when the system has a negative viscous damping effect.)

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Figure 11

Time history of nonlinear system with negative viscous damping under relatively small initial displacement (μl=−0.08, x(0)=0.66mm)

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Figure 12

Time history of nonlinear system with negative viscous damping under relatively large initial displacement (μl=−0.08, x(0)=0.5mm)

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