This paper deals with the modeling and the prediction of the dynamic behavior of uncertain nonlinear systems. An efficient method is proposed to treat these problems. It is based on the Wiener–Haar chaos concept resulting from the polynomial chaos theory and it generalizes the use of the multiresolution analysis well known in the signal processing theory. The method provides a powerful tool to describe stochastic processes as series of orthonormal piecewise functions whose weighting coefficients are identified using the Mallat pyramidal algorithm. This paper shows that the Wiener–Haar model allows an efficient description and prediction of the dynamic behavior of nonlinear systems with probabilistic uncertainty in parameters. Its contribution, compared to the representation using the generalized polynomial chaos model, is illustrated by evaluating the two models via their application to the problems of the modeling and the prediction of the dynamic behavior of a self-excited uncertain nonlinear system.

References

References
1.
Helton
,
J. C.
, and
Davis
,
F. J.
, 2003, “
Latin Hypercube Sampling and the Propagation of Uncertainty in Analyses of Complex Systems
,”
Reliab. Eng. Syst. Saf.
,
81
(
1
), pp.
23
69
.
2.
Sinou
,
J.-J.
, and
Jezequel
,
L.
, 2004, “
Methods to Reduce Nonlinear Mechanical Systems for Instability Computation
,”
Arch. Comput. Methods Eng.
,
11
(
3
), pp.
257
344
.
3.
Wiener
,
N.
, 1938, “
The Homogeneous Chaos
,”
Am. J. Math.
,
60
, pp.
897
936
.
4.
Ghanem
,
R.
, and
Spanos
,
P. D.
, 1991,
Stochastic Finite Elements: A Spectral Approach
,
Springer-Verlag
,
Berlin
.
5.
Cameron
,
H.
, and
Martin
,
W.
, 1947, “
The Orthogonal Development of Nonlinear Functionals in Series of Fourier-Hermite Functional
,”
Ann. Math.
,
48
, pp.
385
392
.
6.
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
.
7.
Williams
,
M. M. R.
, 2006, “
Polynomial Chaos Functions and Stochastic Differential Equations
,”
Ann. Nucl. Energy
,
33
(
9
), pp.
774
785
.
8.
Smith
,
A. H. C.
,
Monti
,
A.
, and
Ponci
,
F.
, 2007, “
Indirect Measurements via a Polynomial Chaos Observer
,”
IEEE Trans. Instrum. Meas.
,
56
(
3
), pp.
743
752
.
9.
Blanchard
,
E. D.
,
Sandu
,
A.
, and
Sandu
,
C.
, 2010, “
Polynomial Chaos-Based Parameter Estimation Methods Applied to a Vehicle System
,”
Proc. Inst. Mech. Eng. Part K: J. Multi-Body Dyn.
,
224
(
1
), pp.
59
81
.
10.
Blanchard
,
E.
,
Sandu
,
A.
, and
Sandu
,
C.
, 2010, “
A Polynomial Chaos-Based Kalman Filter Approach for Parameter Estimation of Mechanical Systems
,”
ASME J. Dyn. Syst., Meas., Control
,
132
(
6
), p.
061404
.
11.
Fischer
,
J.
, and
Bhattacharya
,
R.
, 2008, “
Stability Analysis of Stochastic Systems Using Polynomial Chaos
,” American Control Conference, pp.
4250
4255
.
12.
Crestaux
,
T.
,
Le Maître
,
O.
, and
Martinez
,
J. M.
, 2009, “
Polynomial Chaos Expansion for Sensitivity Analysis
,”
Reliab. Eng. Syst. Saf.
,
94
(
7
), pp.
1161
1172
.
13.
Sudret
,
B.
, 2007, “
Global Sensitivity Analysis Using Polynomial Chaos Expansions
,”
Reliab. Eng. Syst. Saf.
,
93
(
7
), pp.
964
979
.
14.
Hover
,
F. S.
, and
Triantafyllou
,
S.
, 2006, “
Application of Polynomial Chaos in Stability and Control
,”
Automatica
,
42
(
5
), pp.
789
795
.
15.
Nagy
,
Z. K.
, and
Braatz
,
R. D.
, 2006, “
Distributional Uncertainty Analysis Using Power Series and Polynomial Chaos Expansions
,”
J. Process Control
,
17
(
3
), pp.
229
240
.
16.
Fischer
,
J.
, and
Bhattacharya
,
R.
, 2008, “
On Stochastic LQR Design and Polynomial Chaos
,” American Control Conference, pp.
95
100
.
17.
Millman
,
D. R.
,
King
,
P. I.
, and
Beran
,
P. S.
, 2005, “
Airfoil Pitch-and-Plunge Bifurcation Behaviour With Fourier Chaos Expansions
,”
J. Aircr.
,
42
(
3
), pp.
376
384
.
18.
Beran
,
P. S.
,
Pettit
,
C. L.
, and
Millman
,
D. L.
, 2006, “
Uncertainty Quantification of Limit-Cycle Oscillations
,”
J. Comput. Phys.
,
217
(
1
), pp.
217
247
.
19.
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
.
20.
Witteveen
,
J. A. S.
,
Loeven
,
A.
,
Sarkar
,
S.
, and
Bijl
,
H.
, 2007, “
Probabilistic Collocation for Periodic-1 Limit Cycle Oscillations
,”
J. Sound Vib.
,
311
(
1–2
), pp.
421
439
.
21.
Le Maître
,
O. P.
,
Knio
,
M.
,
Najm
,
H. N.
, and
Ghanem
,
R. G.
, 2004, “
Uncertainty Propagation Using Wiener-Haar Expansions
,”
J. Comput. Phys.
,
197
(
1
), pp.
28
57
.
22.
Pettit
,
C. L.
, and
Beran
,
P. S.
, 2006, “
Spectral and Multiresolution Wiener Expansions of Oscillatory Stochastic Process
,”
J. Sound Vib.
,
294
(
4–5
), pp.
752
779
.
23.
Mallat
,
S.
, 1989, “
A Theory for Multi-Resolution Signal Decomposition: The Wavelet Representation
,”
IEEE Trans. Pattern Anal. Mach. Intell.
,
11
(
8
), pp.
674
694
.
24.
Sinou
,
J. J.
,
Dereure
,
O.
,
Mazet
,
F.
,
Thouverz
,
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
(
5
), pp.
536
554
.
25.
Sinou
,
J. J.
,
Dereure
,
O.
,
Mazet
,
F.
,
Thouverz
,
F.
, and
Jezequel
,
L.
, 2006, “
Friction-Induced Vibration for an Aircraft Brake System-Part 2: Nonlinear Dynamics
,”
Int. J. Mech. Sci.
,
48
(
5
), pp.
555
567
.
26.
Chevennement-Roux
,
C.
,
Dreher
,
T.
,
Alliot
,
P.
,
Aubry
,
E.
,
Lainé
,
J.-P.
, and
Jezequel
,
L.
, 2007, “
Flexible Wiper System Dynamic Instabilities: Modeling and Experimental Validation
,”
Springer Exp. Mech.
,
47
(
2
), pp.
201
210
.
27.
Hervé
,
B.
,
Sinou
,
J.-J.
,
Mahé
,
L.
, and
Jezequel
,
L.
, 2007, “
Analysis of Squeal Noise and Mode Coupling Instabilities Including Damping and Gyroscopic Effects
,”
Eur. J. Mech. A/Solids
,
27
(
2
), pp.
141
160
.
28.
Gallina
,
P.
, and
Giovagnioni
,
M.
, 2002, “
Design of a Screw Jack Mechanism to Avoid Self-Excited Vibrations
,”
ASME J. Dyn. Syst., Meas., Control
,
124
(
3
), pp.
477
480
.
29.
Dupont
,
P.
, and
Dunlap
,
D.
, 1995, “
Friction Modeling and PD Compensation at Very Low Velocities
,”
ASME J. Dyn. Syst., Meas., Control
,
117
(
1
), pp.
8
14
.
30.
Kanjuro
,
M.
,
Ecker
,
H.
, and
Dohnal
,
F.
, 2005, “
Stability Analysis of Open-Loop Stiffness Control to Suppress Self-Excited Vibrations
,”
J. Vib. Control
,
11
, pp.
643
669
.
31.
El-Badawi
,
A.
, and
Nasr El-Deen
,
T.
, 2007, “
Quadratic Nonlinear Control of a Self-Excited Oscillator
,”
J. Control Vib.
,
13
(
4
), pp.
403
414
.
32.
Ibrahim
,
R. A.
, 1994, “
Friction-Induced Vibration, Chatter, Squeal and Chaos: Part I-Mechanics of Contact and Friction
,”
ASME Appl. Mech. Rev.
,
47
(
7
), pp.
209
226
.
33.
Ibrahim
,
R. A.
, 1994, “
Friction-Induced Vibration, Chatter, Squeal and Chaos: Part II-Dynamics and Modeling
,”
ASME Appl. Mech. Rev.
,
47
(
7
), pp.
227
253
.
34.
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
,”
ASME J. Vib. Acoustics
,
129
(
3
), pp.
297
306
.
35.
Daubechies
,
I.
, 1988, “
Orthonormal Basis of Compactly Supported Wavelets
,”
Commun. Pure Appl. Math.
,
41
(
7
), pp.
909
996
.
36.
Burrus
,
C. S.
,
Gopinath
,
R. A.
, and
Guo
,
H.
, 1998,
Introduction to Wavelets and Wavelet Transforms
,
Prentice-Hall
,
Upper Saddle River, NJ
.
37.
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
.
38.
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
.
39.
D’souza
,
A. F.
, and
Dweib
,
A. H.
, 1990, “
Self-Excited Vibration Induced by Dry Friction. Part 2: Stability and Limit-Cycle Analysis
,”
J. Sound Vib.
,
137
(
2
), pp.
177
190
.
40.
Eriksson
,
M.
, and
Jacobson
,
S.
, 2001, “
Friction Behaviour and Squeal Generation of Disc Brakes at Low Speeds
,”
Proc. Inst. Mech. Eng.
,
215
(
12
), pp.
1245
1256
.
41.
Hoffmann
,
N.
, and
Gaul
,
L.
, 2003, “
Effects of Damping on Mode-Coupling Instability in Friction-Induced Oscillations
,”
Z. Angew. Math. Mech.
,
83
(
8
), pp.
524
534
.
42.
Hultèn
,
J.
, 1993, “
Drum Break Squeal—A Self Exciting Mechanism With Constant Friction
,” SAE Truck and Bus Meeting, Detroit, MI, SAE Paper No. 932965.
43.
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
.
44.
DeVor
,
R. A.
, 2009, “
Nonlinear Approximation and Its Applications
,”
Multiscale, Nonlinear, and Adaptive Approximation
,
Springer
,
New York
, pp.
169
201
.
You do not currently have access to this content.