Abstract

In this two-part paper we present a novel method for tracking a slowly evolving hidden damage process responsible for nonstationarity in a fast dynamical system. The development of the method and its application to an electromechanical experiment is the core of Part 1. In Part 2, a mathematical model of the experimental system is developed and used to validate the experimental results. In addition, an analytical connection is established between the tracking method and the physics of the system based on the idea of averaging and the slow flow equations for the hidden process. The tracking method developed in this study uses a nonlinear, two-time-scale modeling strategy based on the delay reconstruction of a system’s phase space. The method treats damage-induced nonstationarity as evolving in a hierarchical dynamical system containing a fast, directly observable subsystem coupled to a slow, hidden subsystem. The utility of the method is demonstrated by tracking battery discharge in a vibrating beam system with a battery-powered electromagnetic restoring force. Applications to systems with evolving material damage are also discussed.

Issue Section:
Technical Papers
Keywords:
nonlinear dynamical systems, vibrations, high-speed techniques, stress-strain relations, oscillations
Topics:
Batteries, Damage, Dynamic systems, Errors, Modeling, Phase space, Physics, Vibration, Dimensions, Delays
1.
Qu
,
L.
,
Xie
,
A.
, and
Li
,
X.
,
1993
, “
Study and Performance Evaluation of Some Nonlinear Diagnostic Methods for Large Rotating Machinery
,”
Mech. Mach. Theory
,
28
(
5
), pp.
699
–
713
.
2.
Ma, J., and Li, C. J., 1995, “On Gear Localized Defect Detection by Demodulation of Vibrations—A Comparison Study,” E. Kannatey-Asibu, J., ed., Proc. of Symp. on Mechatronics for Manufacturing in ASME International Mechanical Engineering Congress and Exposition, MED-2-1, pp. 565–576.
3.
McFadden
,
P. D.
, and
Smith
,
J. D.
,
1985
, “
A Signal Processing Technique for Detecting Local Defects in a Gear From the Signal Averaging of Vibration
,”
Proc. Inst. Mech. Eng., Part C: Mech. Eng. Sci.
,
199
(
c4
),
ImechE-1985
ImechE-1985
.
4.
Robert, T. S., and Lawrence, J. M., 1986, “Detection, Diagnosis and Prognosis of Rotating Machinery,” Proc. of the 41st Meeting of the Mech. Failures Prevention Group, Naval Air Test Center, Patuxent River, Maryland.
5.
McFadden
,
P. D.
, and
Wang
,
W. J.
,
1993
, “
Early Detection of Gear Failure by Vibration Analysis–I. Calculation of the Time-Frequency Distribution
,”
Mech. Syst. Signal Process.
,
37
(
3
), pp.
193
–
201
.
6.
Liu
,
B.
,
Ling
,
S.
, and
Meng
,
Q.
,
1997
, “
Machinery Diagnosis Based on Wavelet Packets
,”
J. Vib. Control
,
3
, pp.
5
–
17
.
7.
Baydar, N. Ball, A., and Kruger, U., 1999, “Detection of Incipient Tooth Defect in Helical Gears Using Principal Components,” Starr, A. G., Leung, A. Y. T., Wright, J. R., and Sandoz, D. J., eds. Integrating Dynamics, Condition Monitoring and Control for the 21st Century, A. A. Balkema, Rotterdam, Brookfield, pp. 93–100.
8.
Staszewski
,
W. J.
,
Worden
,
K.
, and
Tomlinson
,
G. R.
,
1997
, “
Time-frequency Analysis in Gearbox Fault Detection Using the Wigner-Ville Distribution and Pattern Recognition
,”
Mech. Syst. Signal Process.
,
11
(
5
), pp.
673
–
692
.
9.
Staszewski
,
W. J.
, and
Tomlinson
,
G. R.
,
1994
, “
Application of the Wavelet Transform to Fault Detection in a Spur Gear
,”
Mech. Syst. Signal Process.
,
8
(
3
), pp.
289
–
307
.
10.
Kim
,
T.
, and
Li
,
C. J.
,
1995
, “
Feedforward Neural Networks for Fault Diagnosis and Severity Assessment of a Screw Compressor
,”
Mech. Syst. Signal Process.
,
9
(
5
), pp.
485
–
496
.
11.
Chancellor
,
R. S.
,
Alexander
,
R.
, and
Noah
,
S.
,
1996
, “
Detecting Parameter Changes Using Experimental Nonlinear Dynamics and Chaos
,”
ASME J. Vibr. Acoust.
,
118
, pp.
375
–
383
.
12.
Isermann
,
R.
,
1984
, “
Process Fault Detection Based on Modeling and Estimation Methods–A Survey
,”
Automatica
,
20
(
4
), pp.
387
–
404
.
13.
Patton, R., Frank, P., and Clark, P., 1989, Fault Diagnosis in Dynamical Systems: Theory and Application, Prinston Hall International (UK) Ltd.
14.
Gertler
,
J.
,
1988
, “
Survey of Model-Based Failure Detection and Isolation in Complex Plants
,”
IEEE Control Syst. Mag.
,
8
(
6
), pp.
3
–
11
.
15.
Nijmeijer, H., and Fossen, T. I., 1999, “New Directions in Nonlinear Observer Design,” Lecture Notes in Control and Information Sciences, Springer-Verlag, London, pp. 351–466.
16.
Ashton, S. A., and Shields, D. N., 1999, “Fault Detection Observer for a Class of Nonlinear Systems,” Nijmeijer, H., and Fossen, T. I., eds., New Directions in Nonlinear Observer Design, Lecture Notes in Control and Information Sciences, Springer-Verlag, London, pp. 353–373.
17.
Jeon
,
Y. C.
, and
Li
,
C. J.
,
1995
, “
Non-linear ARX Model Based Kullback Index for Fault Detection of a Screw Compressor
,”
Mech. Syst. Signal Process.
,
9
(
4
), pp.
341
–
358
.
18.
McGhee
,
J.
,
Henderson
,
I. A.
, and
Baird
,
A.
,
1997
, “
Neural Networks Applied for the Identification and Fault Diagnosis of Process Valves and Actuators
,
J. of Int. Measr. Confederation
,
20
(
4
), pp.
267
–
275
.
19.
Allessandri, A., and Parisini, T., 1997, “Model-Based Fault Diagnosis Using Nonlinear Estimators: A Neural Approach,” Proceedings of the American Control Conference, 2, pp. 903–907.
20.
Li, C. J., and Fan, Y., 1997, “Recurrent Neural Networks for Fault Diagnosis and Severity Assessment of a Screw Compressor,” DSC ASME Dynamic Systems & Control Division, 61, pp. 257–264.
21.
Cusumano
,
J. P.
, and
Kimble
,
B.
,
1995
, “
A Stochastic Interrogation Method for Experimental Measurements of Global Dynamics and Basic Evolution: Application to a Two-Well Oscillator
,”
Nonlinear Dyn.
,
8
, pp.
213
–
235
.
22.
Kimble
,
B. W.
, and
Cusumano
,
J. P.
,
1996
, “
Theoretical and Numerical Validation of the Stochastic Interrogation Experimental Method
,”
J. Vib. Control
,
2
, pp.
323
–
348
.
23.
Takens, F., 1981, “
Detecting Strange Attractor in Turbulence,” Rand, D. A., and Young, L. S., eds. Dynamical Systems and Turbulence, Warwick, Springer Lecture Notes in Mathematics, Springer-Verlag, Berlin, pp. 336–381.
24.
Sauer
,
T.
,
Yorke
,
J. A.
, and
Casdagli
,
M.
,
1991
, “
Embedology
,”
J. Stat. Phys.
,
65
(
3-4
), pp.
579
–
616
.
25.
Fraser
,
A. M.
, and
Swinney
,
H. L.
,
1986
, “
Independent Coordinates for Strange Attractors from Mutual Information
,”
Phys. Rev. A
,
33
(
2
), pp.
1134
–
1140
.
26.
Kennel
,
M. B.
,
Brown
,
R.
, and
Abarbanel
,
H. D. I.
,
1992
, “
Determining Embedding Dimension for Phase-Space Reconstruction Using a Geometric Construction
,”
Phys. Rev. A
,
45
(
6
), pp.
3403
–
3411
.
27.
Ott, E., 1993, Chaos in Dynamical Systems, Cambridge, New York.
28.
Moon
,
F. C.
, and
Holms
,
P.
,
1979
, “
A Magnetoelastic Strange Attractors
,
J. Sound Vib.
,
65
(
2
), pp.
275
–
296
.
29.
Feeny, B. F., Yuan, C. M., and Cusumano, J. P., 2000, “Parametric Identification of an Experimental Magneto-Elastic Oscillator,” J. Sound Vib. (accepted).
30.
Friedman
,
J. H.
,
Bentley
,
J. L.
, and
Finkel
,
R.
,
1977
, “
An Algorithm for Finding Best Matches in Logarithmic Expected Time
,”
ACM Trans. Math. Softw.
,
3
(
2
), pp.
209
–
226
.
31.
Golnaraghi, M., Lin, D., and Fromme, P., 1995, “Gear Damage Detection using Chaotic Dynamics Techniques: A Preliminary Study,” Proc. of the 1995 ASME Des. Tech. Conf., Symposium on Time-varying Systems and Structures, Boston, MA, Sep 17–20, pp. 121–127.
32.
Craig
,
C.
,
Neilson
,
D.
, and
Penman
,
J.
,
2000
, “
The Use of Correlation Dimension in Condition Monitoring of Systems with Clearance
,”
J. Sound Vib.
,
231
(
1
), pp.
1
–
17
.
33.
Jiang
,
J. D.
,
Chen
,
J.
, and
Qu
,
L. S.
,
1999
, “
The Application of Correlation Dimension in Gearbox Condition Monitoring
,”
J. Sound Vib.
,
223
(
4
), pp.
529
–
541
.
34.
Logan
,
D. B.
, and
Mathew
,
J.
,
2000
, “
Using the Correlation Dimension for Vibration Fault Diagnosis of Rolling Element Bearing–I. Basic Concepts
,”
Mech. Syst. Signal Process.
,
10
(
3
), pp.
241
–
250
.
35.
Rosenstein
,
M. T.
,
Collins
,
J. J.
,and
DeLuca
,
C. J.
,
1993
, “
A Practical Method for Calculating Largest Lyapunov Exponents From Small Data Sets
,”
Physica D
,
65
(
1-2
), pp
117
–
134
.
Copyright © 2002
 by ASME
You do not currently have access to this content.