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Research Papers

Wiener–Askey and Wiener–Haar Expansions for the Analysis and Prediction of Limit Cycle Oscillations in Uncertain Nonlinear Dynamic Friction Systems

[+] Author and Article Information
Lyes Nechak

e-mail: lyes.nechak@uha.fr

Sébastien Berger

e-mail: sebastien.berger@uha.fr

Evelyne Aubry

e-mail: evelyne.aubry@uha.fr
MIPS Laboratory,
12, Rue des Frères Lumière,
Mulhouse 68093, France

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received April 20, 2012; final manuscript received May 8, 2013; published online September 25, 2013. Assoc. Editor: D. Dane Quinn.

J. Comput. Nonlinear Dynam 9(2), 021007 (Sep 25, 2013) (12 pages) Paper No: CND-12-1059; doi: 10.1115/1.4024851 History: Received April 20, 2012; Revised May 08, 2013

This paper is devoted to the robust modeling and prediction of limit cycle oscillations in nonlinear dynamic friction systems with a random friction coefficient. In recent studies, the Wiener–Askey and Wiener–Haar expansions have been proposed to deal with these problems with great efficiency. In these studies, the random dispersion of the friction coefficient is always considered within intervals near the Hopf bifurcation point. However, it is well known that friction induced vibrations—with respect to the distance of the friction dispersion interval to the Hopf bifurcation point—have different properties in terms of tansient, frequency and amplitudes. So, the main objective of this study is to analyze the capabilities of the Wiener–Askey (general polynomial chaos, multielement generalized polynomial chaos) and Wiener–Haar expansions to be efficient in the modeling and prediction of limit cycle oscillations independently of the location of the instability zone with respect to the Hopf bifurcation point.

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References

Figures

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Fig. 2

Evolution of real and imaginary parts of system eigenvalues

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Fig. 3

Realization of the displacement x1 (t, ξ) for particular samples of the friction coefficient μ

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Fig. 4

Density functions of amplitudes corresponding to the friction dispersion near the Hopf bifurcation point

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Fig. 6

Density functions of the LCO amplitude corresponding to the friction dispersion far from the Hopf bifurcation point estimated with the GPC based model

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Fig. 7

The evolution of the displacement X1 with respect to the friction coefficient in the zone near the Hopf bifurcation point, [0.3, 0.33]

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Fig. 8

The evolution of the displacement X1 with respect to the friction coefficient in the zone near the Hopf bifurcation point, [0.5, 0.55]

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Fig. 9

Limit cycle oscillation corresponding to μ = 0.5 predicted with the ME-GPC model

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Fig. 10

Density functions of the LCO amplitudes corresponding to the friction dispersion far from the Hopf bifurcation point estimated with the ME-GPC based model

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Fig. 11

The evolution of the displacement X1 with respect to the friction coefficient in the zone near the Hopf bifurcation point, [0.5, 0.55] estimated with the ME-LePC

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Fig. 5

Limit cycle oscillation corresponding to μ = 0.5

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Fig. 13

Limit cycle oscillation corresponding to μ = 0.5 predicted with the Wiener–Haar expansion Density functions of the LCO amplitude corresponding to the friction dispersion far from the Hopf bifurcation point estimated with the Wiener–Haar based model

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Fig. 12

Limit cycle oscillation corresponding to μ = 0.5 predicted with the Wiener–Haar expansion

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Fig. 14

The evolution of the displacement X1 with respect to the friction coefficient in the zone near the Hopf bifurcation point, [0.5, 0.55] estimated with the Wiener–Haar expansion (J = 9)

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