0
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.

FIGURES IN THIS ARTICLE
<>
Copyright © 2014 by ASME
Your Session has timed out. Please sign back in to continue.

References

Ouyang, H., andMottershead, J. E., 2001, “Unstable Travelling Waves in the Friction-Induced Vibration of Discs,” J. Sound Vib., 248, pp. 768–779. [CrossRef]
Sinou, J. J., Dereure, O., Mazet, F., Thouverez, F., and Jezequel, L., 2006, “Friction-Induced Vibration for an Aircraft Brake System—Part 1: Experimental Approach and Stability Analysis,” Int. J. Mech. Sci., 48, pp. 536–554. [CrossRef]
Sinou, J. J., Thouverez, F., Jezequel, L., Dereure, O., Mazet, F., 2006, “Friction-Induced Vibration fFor an Aircraft Brake System—Part 2: Non-Linear Dynamics,” Int. J. Mech. Sci., 48, pp. 555–567. [CrossRef]
Chevennement-Roux, C., Dreher, T., Alliot, P., Aubry, E., Lainé, J. P., and Jézéquel, L., 2007, “Flexible Wiper System Dynamic Instabilities: Modeling and Experimental Validation,” Exp. Mech., 47, pp. 201–210. [CrossRef]
Hervé, B., Sinou, J.-J., Mahé, H., and Jézéquel, L., 2009, “Extension of the dDestabilization Paradox to Limit Cycle Amplitudes for a Nonlinear Self-Excited System Subject to Gyroscopic and Circulatory Actions,” J. Sound Vib., 323, pp. 944–973. [CrossRef]
Rudd, M. J., 1976, “Wheel/Rail Noise—Part II: Wheel Squeal,” J. Sound Vib., 46, pp. 381–394. [CrossRef]
Coudeyras, N., Nacivet, S., and Sinou, J.-J., 2009, “Periodic and Quasi Periodic Solutions for Multi-instabilities Involved in Brake Squeal,” J. Sound Vib., 328, pp. 520–540. [CrossRef]
Oberst, S., and Lai, J. C. S., 2011, “Statistical Analysis of Brake Squeal Noise,” J. Sound Vib., 330(12), pp. 2978–2994. [CrossRef]
Oberst, S., and Lai, J. C. S., 2011, “Chaos in Brake Squeal Noise,” J. Sound Vib., 330(5), pp. 955–975. [CrossRef]
Ibrahim, R. A., 1994, “Friction-Induced Vibration, Chatter, Squeal and Chaos—Part I: Mechanics of Contact and Friction,” ASME Appl. Mech. Rev., 47, pp. 209–226. [CrossRef]
Ibrahim, R. A., 1994, “Friction-Induced Vibration, Chatter, Squeal and Chaos—Part II: Dynamics and Modeling,” ASME Appl. Mech. Rev., 47, pp. 227–253. [CrossRef]
Spurr, R. T., 1961, “A Theory of Brake Squeal,” Proc. Inst. Mech. Eng., 1, pp. 33–40.
Earles, S. W. E., and Lee, C. K., 1976, “Instabilities Arising From the Frictional Interaction of a Pin- Disc System Resulting in Noise Generation,” Trans. ASME, 1, pp. 81–86.
Earles, S., and Chambers, P., 1987, “Disc Brake Squeal Noise Generation: Predicting Its Dependency on System Parameters Including Damping,” Int. J. Veh. Des., 8, pp. 538–552.
Herve, B., Sinou, J.-J., Mahe, H., and Jezequel, L., 2008, “Analysis of Squeal Noise and Mode Coupling Instabilities Including Damping and Gyroscopic Effects,” Eur. J. Mech. A/Solids, 27, pp. 141–160. [CrossRef]
Antoniou, S. S., Cameron, A., and Gentle, C. R., 1976, “The Friction-Speed Relation From Stick-Slip Data,” Wear, 36, pp. 235–254. [CrossRef]
Oden, J. T., and Martins, J. A. C., 1985, “Models and Computational Methods for Dynamic Friction Phenomena,” Comput. Methods Appl. Mech. Eng., 52, pp. 527–634. [CrossRef]
Van DeVelde, F., and DeBaets, P., 1998, “A New Approach of Stick-Slip Based on Quasiharmonic Tangential Oscillations,” Wear, 216, pp. 15–26. [CrossRef]
Van DeVelde, F., and DeBaets, P., 1998, “The Relation Between Friction Force and Relative Speed During the Slip-Phase of Stick-Slip Cycle,” Wear, 219, pp. 220–226. [CrossRef]
Kinkaid, N., O'Reilly, O., and Papadopoulos, P., 2003, “Automotive Disc Brake Squeal,” J. Sound Vib., 267, pp. 105–166. [CrossRef]
Ouyang, H., Nack, W., Yuan, Y., and Chen, F., 2005, “Numerical Analysis of Automotive Disc Brake Squeal: A Review,” Int. J. Vehicle Noise Vib., 1, pp. 207–231. [CrossRef]
Millner, N., 1978, “An Analysis of Disc Brake Squeal,” SAE Technical Paper No. 780332.
D'Souza, A. F., and Dweib, A. H., 1990, “Self-Excited Vibration Induced by Dry Friction—Part II: Stability and Limit-Cycle Analysis,” J. Sound Vib., 137, pp. 177–190. [CrossRef]
Eriksson, M., and Jacobson, S., 2001, “Friction Behaviour and Squeal Generation of Disc Brakes at Low Speeds,” Proc. Inst. Mech. Eng., 215, pp. 1245–1256. [CrossRef]
Hoffmann, N., and Gaul, L., 2003, “Effects of Damping on Mode-Coupling Instability in Friction Induced Oscillations,” ZAMM, 83, pp. 524–534. [CrossRef]
Meziane, A., and Baillet, L., 2010, “Nonlinear Analysis of Vibrations Generated by a Contact With Friction,” Eur. J. Comput. Mech., 19(1–3), pp. 305–316.
Ragot, P., Berger, S., and Aubry., E., 2008, “Interval Approach Applied to Blades of Windscreens Wiper,” Int. J. Pure Appl. Math., 46(5), pp. 643–648.
Lee, H. W., Sandu, C., and Holton, C., 2010, “Wheel-Rail Dynamic Model and Stochastic Analysis of the Friction in the Contact Path,” ASME Conf. Proc./2010 Joint Rail Conference, Paper No. JRC2010-36229.
Nechak, L., Berger, S., Aubry, E., 2011, “A Polynomial Chaos Approach to the Robust Analysis of the Dynamic Behavior of Friction Systems,” Eur. J. Mech. A/Solids, 30(4), pp. 594–607. [CrossRef]
Nechak, L., Berger, S., and Aubry, E., 2013, “Non-Intrusive Generalized Polynomial Chaos for the Stability Analysis of Uncertain Dynamic Friction Systems,” J. Sound Vib., 332(5), pp. 1204–1205. [CrossRef]
Nechak, L., Berger, S., and AubryE., 2012, “Prediction of Random Self Friction Induced Vibrations in Uncertain Dry Friction Systems Using a Multi-Element Generalized Polynomial Chaos Approach,” J. Vibr. Acoust., 134(4), pp. 041015–041029. [CrossRef]
Nechak, L., Berger, S., and AubryE., 2012, “Wiener-Haar Expansion for the Modelling and Prediction of the Dynamic Behaviour of Nonlinear Uncertain Systems,” J. Dyn. Syst. Meas. Control, 134(5), pp. 051011–051022. [CrossRef]
Beran, P. S., Pettit, C. L., and Millman, D. L., 2006, “Uncertainty Quantification of Limit-Cycle Oscillations,” J. Comput. Phys., 217, pp. 217–247. [CrossRef]
Wiener, N., 1938, “The Homogeneous Chaos,” Am. J. Math., 60, pp. 897–936. [CrossRef]
Ghanem, R., Spanos, P. D., 1991, Stochastic Finite Elements: A Spectral Approach, Springer-Verlag, New York.
Cameron, H., and Martin, W., 1947, “The Orthogonal Development of Nonlinear Functionals in Series of Fourier-Hermite Functional,” Ann. Math., 48, pp. 385–392. [CrossRef]
Xiu, D., and Karniadakis, G. E., 2002, “Modeling Uncertainty in Steady State Diffusion Problems via Generalized Polynomial Chaos,” Comput. Methods. Appl. Mech. Eng., 191(43), pp. 4927–4948. [CrossRef]
Wan, X., and Karniadakis, G. E., 2006, “Beyond Wiener–Askey Expansions: Handling Arbitrary PDFs,” J. Sci. Compt., 27(1–3), pp. 455–464. [CrossRef]
Babuska, I., Tempone, R., and Zouraris, G. E., 2004, “Galerkin Finite Element Approximation of Stochastic Elliptic Partial Differential Equations,” SIAM J. Numer. Anal., 42(2), pp. 800–825. [CrossRef]
Babuska, I., Nobile, F., and Tempone, R., 2007, “A Stochastic Collocation Method for Elliptic Partial Differential Equations With Random Input Data,” SIAM J. Numer. Anal., 45(3), pp. 1005–1034. [CrossRef]
Crestaux, T., Le Maitre, O., and Martinez, J. M., 2009, “Polynomial Chaos Expansion for Sensitivity Analysis,” Reliab. Eng. Syst. Saf., 94(7), pp. 1161–1172. [CrossRef]
Wan, X., and Karniadakis, G., 2005, “An Adaptive Multi-Element Generalized Polynomial Chaos Method for Stochastic Differential Equations,” J. Comput. Phys., 209(2), pp. 617–642. [CrossRef]
Mallat, S., 1989, “A Theory for Multi-Resolution Signal Decomposition: The Wavelet Representation,” IEEE Trans. Pattern Anal. Mach. Intell., 11(8), pp. 674–694. [CrossRef]
Sinou, J.-J., and Jezequel, L., 2007, “Mode Coupling Instability in Friction Induced Vibrations and its Dependency on System Parameters Including Damping,” Eur. J. Mech. A/Solids., 26(1), pp. 107–122.
Hultèn, J., 1993, “Drum Break Squeal-A Self-Exciting Mechanism With Constant Friction,” Proceedings of the SAE Truck and Bus Meeting, SAE Paper No. 932965.
Sinou, J. J., Fritz, G., and Jezequel, L., 2007, “The Role of Damping and Definition of the Robust Damping Factor for a Self-Exciting Mechanism With Constant Friction,” J. Vib. Acoust., 129(3), pp. 297–307. [CrossRef]

Figures

Grahic Jump Location
Fig. 2

Evolution of real and imaginary parts of system eigenvalues

Grahic Jump Location
Fig. 3

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

Grahic Jump Location
Fig. 4

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

Grahic Jump Location
Fig. 5

Limit cycle oscillation corresponding to μ = 0.5

Grahic Jump Location
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

Grahic Jump Location
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]

Grahic Jump Location
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]

Grahic Jump Location
Fig. 9

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

Grahic Jump Location
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

Grahic Jump Location
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

Grahic Jump Location
Fig. 12

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

Grahic Jump Location
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

Grahic Jump Location
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)

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In