The method of proper orthogonal decomposition for dynamical characterization and order reduction of mechanical systems: An overview

Nonlinear Dynamics

Volumn 41, Issue 1-3, 2005, Pages 147-169

  Kerschen, Gaetan ^a,b,d   Golinval, Jean Claude ^a   Vakakis, Alexander F ^b,c   Bergman, Lawrence A ^c  

^a UNIVERSITY OF LIÈGE   (Belgium)

^b NATIONAL TECHNICAL UNIVERSITY OF ATHENS   (Greece)

^c University of Illinois at Urbana Champaign   (United States)

^d UNIVERSITY OF ILLINOIS AT URBANA CHAMPAIGN   (United States)

Author keywords

Dynamic characterization; Order reduction; Proper orthogonal decomposition

Indexed keywords

EIGENVALUES AND EIGENFUNCTIONS; FEATURE EXTRACTION; LINEAR SYSTEMS; MATRIX ALGEBRA; MODAL ANALYSIS; NONLINEAR SYSTEMS; STATISTICAL METHODS;

COVARIANCE MATRIX; DYNAMIC CHARACTERIZATION; ORDER REDUCTION; PROPER ORTHOGONAL DECOMPOSITION (POD);

STRUCTURAL ANALYSIS;

EID: 19944392685     PISSN: 0924090X     EISSN: None     Source Type: Journal    
DOI: 10.1007/s11071-005-2803-2     Document Type: Review

References (89)

1. Berkooz, G., Holmes, P., and Lumley, J. L., 'The proper orthogonal decomposition in the analysis of turbulent flows', Annual Review of Fluid Mechanics 25, 1993, 539-575.
2. Karhunen, K., 'Über Lineare Methoden in der Wahrscheinlichkeitsrechnung', Annals of Academic Science Fennicae, Series Al Mathematics and Physics 37, 1946, 3-79.
3. Kosambi, D., 'Statistics in function space', Journal of Indian Mathematical Society 7, 1943, 76-88.
4. Loeve, M., 'Fonctions Aléatoires du Second Ordre', in Processus stochastiques et mouvement Brownien, P. Levy (ed.), Gauthier-Villars, Paris, 1948.
5. Obukhov, M. A., 'Statistical description of continuous fields', Transactions of the Geophysical International Academy Nauk USSR 24, 1954, 3-42.
6. Pougachev, V. S., 'General theory of the correlations of random functions', Izvestiya Akademii Nauk USSR 17, 1953, 1401-1402.
7. Berkooz, G., 'Observations on the proper orthogonal decomposition', in Studies in Turbulence, Springer, New York, 1992, pp. 229-247.
8. Jolliffe, I. T., Principal Component Analysis, Springer, New York, 1986.
9. Pearson, K., 'On lines and planes of closest fit to systems of points in space', Philosophical Magazine 2, 1901, 559-572.
10. Hotelling, H., 'Analysis of a complex of statistical variables into principal components', Journal of Educational Psychology 24, 1933, 417-441, 498-520.
11. Watanabe, S., 'Karhunen-Loeve expansion and factor analysis theoretical remarks and applications', in Proceedings of the 4th Conference on Information Theory, Prague, Czech Republic, 1965.
12. Mees, A. I., Rapp, P. E., and Jennings, L. S., 'Singular value decomposition and embedding dimension', Physical Review A 36, 1987, 340-346.
13. Ravindra, B., 'Comments on "On the physical interpretation of proper orthogonal modes in vibrations'", Journal of Sound and Vibration 219, 1999, 189-192.
14. Liang, Y. C., Lee, H. P., Lim, S. P., Lin, W. Z., Lee, K. H., and Wu, C. G., 'Proper orthogonal decomposition and its applications, Parti: Theory', Journal of Sound and Vibration 252, 2002, 527-544.
15. Wu, C. G., Liang, Y. C., Lin, W. Z., Lee, H. P., and Lim, S. P., 'A note on equivalence of proper orthogonal decomposition methods', Journal of Sound and Vibration 265, 2003, 1103-1110.
16. Holmes, P., Lumley, J. L., and Berkooz, G., Turbulence, Coherent Structures, Dynamical Systems and Symmetry, Cambridge, New York, 1996.
17. Wax, M. and Kailath, T., 'Detection of signals by information theoretic criteria', IEEE Transactions on Acoustics, Speech and Signal Processing 33, 1985, 387-392.
18. Graham, M. D. and Kevrekedis, I. G., 'Alternative approaches to the Karhunen-Loeve decomposition for model reduction and data analysis', Computers and Chemical Engineering 20, 1996, 495-506.
19. Bayly, P. V., Johnson, E. E., Wolf, P. D., Smith, W. M., and Ideker, R. E., 'Predicting patterns of epicardial potentials during ventricular fibrillation', IEEE Transactions on Biomedical Engineering 42, 1995, 898-907.
20. Epureanu, B. I., Hall, K. C., and Dowell, E. H., 'Reduced-order models of unsteady viscous flows in turbomachinery using viscous-inviscid coupling', Journal of Fluids and Structures 15, 2001, 255-273.
21. Barnston, A. G. and Ropelewski, C. F., 'Prediction of ENSO episodes using canonical correlation analysis', Journal of Climate 5, 1992, 1316-1345.
22. Leen, T. K., Rudnick, M., and Hammerstrom, R., 'Hebbian feature discovery improves classifier efficiency', in Proceedings of the IJCNN, IEEE, Piscataway, NJ, 1990, pp. 51-56.
23. Fitzsimons, P. M. and Rui, C., 'Determining low dimensional models of distributed systems', in Advances in Robust and Nonlinear Control Systems, ASME DSC 53, 1993.
24. Cusumano, J. P. and Bai, B. Y., 'Period-infinity periodic motions, chaos and spatial coherence in a 10-degree-of-freedom impact oscillator', Chaos, Solitons and Fractals 3, 1993, 515-535.
25. Cusumano, J. P., Sharkady, M. T., and Kimble, B. W., 'Experimental measurements of dimensionality and spatial coherence in the dynamics of a flexible-beam impact oscillator', Philosophical Transactions of the Royal Society of London 347, 1994, 421-438.
26. Kreuzer, E. and Kust, O., 'Analysis of long torsional strings by proper orthogonal decomposition', Archive of Applied Mechanics 67, 1996, 68-80.
27. Azeez, M. F. A. and Vakakis, A. F., 'Proper orthogonal decomposition of a class of vibroimpact oscillations', Journal of Sound and Vibration 240, 2001, 859-889.
28. Al-Dmour, A. S. and Mohammad, K. S., 'Active control of flexible structures using principal component analysis in the time domain', Journal of Sound and Vibration 253, 2002, 545-569.
29. Benguedouar, A., 'Proper Orthogonal Decomposition in Dynamical Modeling: A Qualitative Dynamic Approach', PhD thesis, Boston University, Boston, MA, 1995.
30. Epureanu, B. I., Tang, L. S., and Paidoussis, M. P., 'Coherent structures and their influence on the dynamics of aeroelastic panels', International Journal of Non-Linear Mechanics 39, 2004, 977-991.
31. De Boe, P. and Golinval, J. C., 'Principal component analysis of a piezo-sensor array for damage localization', Structural Health Monitoring 2, 2003, 137-152.
32. Feldmann, U., Kreuzer, E., and Pinto, F., 'Dynamic diagnosis of railway tracks by means of the Karhunen-Loève transformation', Nonlinear Dynamics 22, 2000, 183-193.
33. Turner, I. Y., Wood, K. L., and Busch-Vishniac, I. J., 'Monitoring of signals from manufacturing processes using K-L transform', Mechanical Systems and Signal Processing 14, 2000, 1011-1026.
34. Georgiou, I. T. and Schwartz, I. B., 'Dynamics of large scale coupled structural-mechanical systems: A singular perturbation proper orthogonal decomposition approach', SIAM Journal of Applied Mathematics 59, 1999, 1178-1207.
35. Georgiou, I. T., 'Invariant manifolds, nonclassical normal modes, and proper orthogonal modes in the dynamics of the flexible spherical pendulum', Nonlinear Dynamics 25, 2001, 3-31.
36. Kappagantu, R. and Feeny, B. F., 'Part 1: Dynamical characterization of a frictionally excited beam', Nonlinear Dynamics 22, 2000, 317-333.
37. Kappagantu, R. and Feeny, B. F., 'Part 2: Proper orthogonal modal modeling of a frictionally excited beam', Nonlinear Dynamics 23, 2000, 1-11.
38. Ma, X. and Vakakis, A. F., 'Nonlinear transient localization and beat phenomena due to backlash in a coupled flexible system', Journal of Vibration and Acoustics 123, 2001, 36-44.
39. Alaggio, R. and Rega, G., 'Characterizing bifurcations and classes of motion in the transition to chaos through 3D-Tori of a continuous experimental system in solid mechanics', Physica D 137, 2000, 70-93.
40. Rega, G. and Alaggio, R., 'Spatio-temporal dimensionality in the overall complex dynamics of an experimental cable/mass system', International Journal of Solids and Structures 38, 2001, 2049-2068.
41. Alaggio, R. and Rega, G., 'Exploiting results of experimental nonlinear dynamics for reduced-order modeling of a suspended cable', in Proceedings of the 18th Biennal Conference on Mechanical Vibration and Noise - ASME DETC, Pittsburgh, PA, 2001.
42. Hemez, F. M. and Doebling, S. W., 'Review and assessment of model updating for non-linear, transient dynamics', Mechanical Systems and Signal Processing 15, 2001, 45-73.
43. Lenaerts, V., Kerschen, G., and Golinval, J. C., 'Identification of a continuous structure with a geometrical non-linearity, Part II: Proper orthogonal decomposition', Journal of Sound and Vibration 262, 2003, 907-919.
44. Feeny, B. F., 'On proper orthogonal co-ordinates as indicators of modal activity', Journal of Sound and Vibration 255, 2002, 805-817.
45. Han, S. and Feeny, B. F., 'Application of proper orthogonal decomposition to structural vibration analysis', Mechanical Systems and Signal Processing 17, 2003, 989-1001.
46. Quaranta, G., Mantegazza, P., and Masarati, P., 'Assessing the local stability of periodic motions for large multibody non-linear systems using proper orthogonal decomposition', Journal of Sound and Vibration 271, 2004, 1015-1038.
47. Azeez, M. F. A. and Vakakis, A. F., 'Numerical and experimental analysis of a continuous overhang rotor undergoing vibro-impacts', International Journal of Non-Linear Mechanics 34, 1999, 415-435.
48. Kappagantu, R. and Feeny, B. F., 'An optimal modal reduction of a system with factional excitation', Journal of Sound and Vibration 224, 1999, 863-877.
49. Liang, Y. C., Lin, W. Z., Lee, H. P., Lim, S. P., Lee, K. H., and Sun, H., 'Proper orthogonal decomposition and its applications, Part II: Model reduction for MEMS dynamical analysis', Journal of Sound and Vibration 256, 2002, 515-532.
50. Ma, X., Vakakis, A. F., and Bergman, L. A., 'Karhunen-Loève modes of a truss: Transient response reconstruction and experimental verification', AIAA Journal 39, 2001, 687-696.
51. Ma, X. and Vakakis, A. F., 'Karhunen-Loève decomposition of the transient dynamics of a multibay truss', AIAA Journal 37, 1999, 939-946.
52. Steindl, A. and Troger, H., 'Methods for dimension reduction and their application in nonlinear dynamics', International Journal of Solids and Structures 38, 2001, 2131-2147.
53. Friswell, M. and Inman, D. J., 'Sensor validation for smart structures', Journal of Intelligent Material Systems and Structures 10, 1999, 973-982.
54. Kerschen, G., De Boe, P., Golinval, J. C., and Worden, K., 'Sensor validation using principal component analysis', Smart Materials and Structures 14, 2005, 36-42.
55. Ghanem, R. and Spanos, P., Stochastic Finite Elements: A Spectral Approach, Springer, Heidelberg, Germany, 1991.
56. Li, R. and Ghanem, R., 'Adaptive polynomial chaos expansions applied to statistics of extremes in nonlinear random vibration', Probabilistic Engineering Mechanics 13, 1998, 125-136.
57. Sarkar, A. and Ghanem, R., 'Mid-frequency structural dynamics with parameter uncertainty', Computer Methods in Applied Mechanics and Engineering 191, 2002, 5499-5513.
58. Schenk, C. A. and Schueller, G. I., 'Buckling analysis of cylindrical shells with random geometric imperfections', International Journal of Non-Linear Mechanics 38, 2003, 1119-1132.
59. Sirovich, L., 'Turbulence and the dynamics of coherent structures, Part I: Coherent structures', Quarterly of Applied Mathematics 45, 1987, 561-571.
60. Meirovitch, L., Computational Methods in Structural Dynamics, Sijthoff and Noordhoff, Alphen a/d Rijn, The Netherlands, 1980.
61. Morrison, D. F., Multivariate Statistical Methods, McGraw-Hill, New York, 1967.
62. Golub, G. H. and Van Loan, C. F., Matrix Computations, The Johns Hopkins University Press, London, 1996.
63. Otte, D., 'Development and Evaluation of Singular Value Analysis Methodologies for Studying Multivariate Noise and Vibration Problems', PhD thesis, Katholieke Universiteit Leuven, Belgium, 1994.
64. Staar, J., 'Concepts for Reliable Modeling of Linear Systems With Application to On-Line Identification of Multivariate State Space Descriptions', PhD thesis, Katholieke Universiteit Leuven, Belgium, 1982.
65. Oja, E., 'A simplified neuron model as a principal component analyzer', Journal of Mathematical Biology 15, 1982, 267-273.
66. Oja, E., 'Neural networks, principal components and subspaces', International Journal of Neural Systems 1, 1989, 61-68.
67. Sanger, T. D., 'Optimal unsupervised learning in a single linear feedforward neural network', Neural Networks 2, 1989, 459-473.
68. Baldi, P. and Hornik, K., 'Neural networks and principal component analysis: Learning from examples without local minima', Neural Networks 2, 1989, 53-58.
69. Kramer, M. A., 'Nonlinear principal component analysis using autoassociative neural networks', AIChE Journal 37, 1991, 233-243.
70. North, G. R., 'Empirical orthogonal functions and normal modes', Journal of the Atmospheric Sciences 41, 1984, 879-887.
71. Davies, M. A. and Moon, F. C., 'Solitons, chaos and modal interactions in periodic structures', in Nonlinear Dynamics: The Richard Rand 50th Anniversary Volume, World Scientific, Singapore, 1997.
72. Feeny, B. F. and Kappagantu, R., 'On the physical interpretation of proper orthogonal modes in vibrations', Journal of Sound and Vibration 211, 1998, 607-616.
73. Feeny, B. F., 'On the proper orthogonal modes and normal modes of continuous vibration systems', Journal of Vibration and Acoustics 124, 2002, 157-160.
74. Feeny, B. E and Liang, Y., 'Interpreting proper orthogonal modes of randomly excited vibration systems', Journal of Sound and Vibration 265, 2003, 953-966.
75. Kerschen, G. and Golinval, J. C., 'Physical interpretation of the proper orthogonal modes using the singular value decomposition', Journal of Soundand Vibration 249, 2002, 849-865.
76. Lin, W. Z., Lee, K. H., Lu, P., Lim, S. P., and Liang, Y. C., 'The relationship between eigenfunctions of Karhunen-Loève decomposition and the modes of distributed parameter vibration system', Journal of Soundand Vibration 252, 2002, 527-544.
77. Vakakis, A. F., Manevitch, L. I., Mikhlin, Y. V., Pilipchuk, V. N., and Zevin, A. A., Normal Modes and Localization in Nonlinear Systems, Wiley, New York, 1996.
78. Emaci, E., Azeez M. A. F., and Vakakis, A. F., 'Dynamics of trusses: Numerical and experimental results', Journal of Sound and Vibration 214, 1998, 953-964.
79. Kerschen, G., Feeny, B. F., and Golinval, J. C., 'On the exploitation of chaos to build reduced-order models', Computer Methods in Applied Mechanics and Engineering 192, 2003, 1785-1795.
80. Ma, X., 'Order Reduction, Identification and Localization Studies of Dynamical Systems', PhD thesis, University of Illinois at Urbana-Champaign, Urbana, IL, 2000.
81. Jordan, C., 'Essai sur la géométrie à n dimensions', Bulletin de la Société mathématique 3, 1875, 103-174.
82. De Cock, K., Principal Angles in System Theory, Information Theory and Signal Processing, PhD thesis, Katholieke Universiteit Leuven, Belgium, 2002.
83. Kambhatla, N., 'Local Models and Gaussian Mixture Models for Statistical Data Processing', PhD thesis, Oregon Graduate Institute of Science & Technology, OR, 1996.
84. Sohn, H., Worden, K., and Farrar, C. R., 'Statistical damage classification under changing environmental and operational conditions', Journal of Intelligent Material Systems and Structures 13, 2002, 561-574.
85. Kerschen, G. and Golinval, J. C., 'Non-linear generalisation of principal component analysis: From a global to a local approach', Journal of Sound and Vibration 254, 2002, 867-876.
86. Kerschen, G. and Golinval, J. C., 'A model updating strategy of non-linear vibrating structures', International Journal for Numerical Methods in Engineering 60, 2004.
87. Kerschen, G. and Golinval, J. C., 'Feature extraction using auto-associative neural networks', Smart Materials and Structures 13, 2004, 211-219.
88. Hyvarinen, A., Karhunen, J., and Oja, E., Independent Component Analysis, Wiley, New York, 2001.
89. Roan, M. J., Erling, J. G., and Sibul, L. H., 'A new, non-linear, adaptive, blind source separation approach to gear tooth failure detection and analysis', Mechanical Systems and Signal Processing 16, 2002, 719-740.