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.
Issue Section:
Research Papers
Keywords:
uncertain nonlinear systems,
nonlinear dynamic systems,
mode coupling instability,
random uncertainty,
limit cycle oscillation,
friction-induced vibration,
generalized polynomial chaos-multi-element generalized polynomial chaos,
wiener-haar
Topics:
Bifurcation,
Chaos,
Friction,
Limit cycles,
Modeling,
Oscillations,
Polynomials,
Vibration,
Sliding friction,
Displacement,
Density
References
1.
Ouyang
, H.
, andMottershead
, J. E.
, 2001
, “Unstable Travelling Waves in the Friction-Induced Vibration of Discs
,” J. Sound Vib.
, 248
, pp. 768
–779
.10.1006/jsvi.2001.37202.
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
.10.1016/j.ijmecsci.2005.12.0023.
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
.10.1016/j.ijmecsci.2005.12.0034.
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
.10.1007/s11340-006-9027-35.
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
.10.1016/j.jsv.2009.01.0236.
Rudd
, M. J.
, 1976
, “Wheel/Rail Noise—Part II: Wheel Squeal
,” J. Sound Vib.
, 46
, pp. 381
–394
.10.1016/0022-460X(76)90862-27.
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
.10.1016/j.jsv.2009.08.0178.
Oberst
, S.
, and Lai
, J. C. S.
, 2011
, “Statistical Analysis of Brake Squeal Noise
,” J. Sound Vib.
, 330
(12
), pp. 2978
–2994
.10.1016/j.jsv.2010.12.0219.
Oberst
, S.
, and Lai
, J. C. S.
, 2011
, “Chaos in Brake Squeal Noise
,” J. Sound Vib.
, 330
(5
), pp. 955
–975
.10.1016/j.jsv.2010.09.00910.
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
.10.1115/1.311107911.
Ibrahim
, R. A.
, 1994
, “Friction-Induced Vibration, Chatter, Squeal and Chaos—Part II: Dynamics and Modeling
,” ASME Appl. Mech. Rev.
, 47
, pp. 227
–253
.10.1115/1.311108012.
Spurr
, R. T.
, 1961
, “A Theory of Brake Squeal
,” Proc. Inst. Mech. Eng.
, 1
, pp. 33
–40
.13.
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
.14.
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
.15.
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
.10.1016/j.euromechsol.2007.05.00416.
Antoniou
, S. S.
, Cameron
, A.
, and Gentle
, C. R.
, 1976
, “The Friction-Speed Relation From Stick-Slip Data
,” Wear
, 36
, pp. 235
–254
.10.1016/0043-1648(76)90008-917.
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
.10.1016/0045-7825(85)90009-X18.
Van De
Velde
, F.
, and De
Baets
, P.
, 1998
, “A New Approach of Stick-Slip Based on Quasiharmonic Tangential Oscillations
,” Wear
, 216
, pp. 15
–26
.10.1016/S0043-1648(97)00296-219.
Van De
Velde
, F.
, and De
Baets
, P.
, 1998
, “The Relation Between Friction Force and Relative Speed During the Slip-Phase of Stick-Slip Cycle
,” Wear
, 219
, pp. 220
–226
.10.1016/S0043-1648(98)00213-020.
Kinkaid
, N.
, O'Reilly
, O.
, and Papadopoulos
, P.
, 2003
, “Automotive Disc Brake Squeal
,” J. Sound Vib.
, 267
, pp. 105
–166
.10.1016/S0022-460X(02)01573-021.
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
.10.1504/IJVNV.2005.00752422.
Millner
, N.
, 1978
, “An Analysis of Disc Brake Squeal
,” SAE Technical Paper No. 780332.23.
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
.10.1016/0022-460X(90)90787-Z24.
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
.10.1243/095440701152878925.
Hoffmann
, N.
, and Gaul
, L.
, 2003
, “Effects of Damping on Mode-Coupling Instability in Friction Induced Oscillations
,” ZAMM
, 83
, pp. 524
–534
.10.1002/zamm.20031002226.
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
.27.
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
.28.
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. JRC2010-36229.29.
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
.10.1016/j.euromechsol.2011.03.00230.
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
.10.1016/j.jsv.2012.09.04631.
Nechak
, L.
, Berger
, S.
, and Aubry
E.
, 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
.10.1115/1.400641332.
Nechak
, L.
, Berger
, S.
, and Aubry
E.
, 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
.10.1115/1.400637133.
Beran
, P. S.
, Pettit
, C. L.
, and Millman
, D. L.
, 2006
, “Uncertainty Quantification of Limit-Cycle Oscillations
,” J. Comput. Phys.
, 217
, pp. 217
–247
.10.1016/j.jcp.2006.03.03834.
35.
Ghanem
, R.
, Spanos
, P. D.
, 1991
, Stochastic Finite Elements: A Spectral Approach
, Springer-Verlag
, New York
.36.
Cameron
, H.
, and Martin
, W.
, 1947
, “The Orthogonal Development of Nonlinear Functionals in Series of Fourier-Hermite Functional
,” Ann. Math.
, 48
, pp. 385
–392
.10.2307/196917837.
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
.10.1016/S0045-7825(02)00421-838.
Wan
, X.
, and Karniadakis
, G. E.
, 2006
, “Beyond Wiener–Askey Expansions: Handling Arbitrary PDFs
,” J. Sci. Compt.
, 27
(1–3
), pp. 455
–464
.10.1007/s10915-005-9038-839.
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
.10.1137/S003614290241868040.
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
.10.1137/05064514241.
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
.10.1016/j.ress.2008.10.00842.
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
.10.1016/j.jcp.2005.03.02343.
Mallat
, S.
, 1989
, “A Theory for Multi-Resolution Signal Decomposition: The Wavelet Representation
,” IEEE Trans. Pattern Anal. Mach. Intell.
, 11
(8
), pp. 674
–694
.10.1109/34.19246344.
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
.45.
Hultèn
, J.
, 1993
, “Drum Break Squeal-A Self-Exciting Mechanism With Constant Friction
,” Proceedings of the SAE Truck and Bus Meeting
, SAE Paper No. 932965.46.
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
.10.1115/1.2730536Copyright © 2014 by ASME
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