Proper orthogonal decomposition (POD) is studied in an effort to increase its applicability as a modal analysis tool. A modification is proposed to make better use of spatial resolution and to accommodate arbitrary spacing in the discretization. The theory for this modification is rooted in the discrete approximation of the integral orthogonality condition for continuous normal modes. The modified POD is applied to a finite element beam and an experimental beam sensed with accelerometers, and the resulting proper orthogonal modes (POMs) are compared to the theoretical modes of the beam. The POMs are used as a basis for decomposing the signal ensemble into proper modal coordinates. The proper modal coordinates are used to evaluate the POMs and to match modes with modal frequencies and damping.
Berkooz, G. , Holmes, P., and Lumley, J. L., 1993, “The proper orthogonal decomposition in the analysis of turbulent flows,”Annual Review of Fluid Mechanics25, 539-575.
2.
Cusumano, J. P. and Bai, B. Y., 1993, “Period-infinity periodic motions, chaos and spatial coherence in a 10 degree of freedom impact oscillator,”Chaos, Solitons and Fractals3(5), 515-535.
3.
Cusumano, J. P. , Sharkady, M. T., and Kimble, B. W., 1993, “Spatial coherence measurements of a chaotic flexible-beam impact oscillator”, in Aerospace Structures: Nonlinear Dynamics and System Response ASME AD-Vol. 33, pp. 13-22.
4.
Davies, M. A. and Moon, F. C., 1997, “Solitons, chaos, and modal interactions in periodic structures”, in Nonlinear Dynamics: The Richard Rand 50th Anniversary Volume, A. Guran, ed., pp. 119-143, World Scientific, Singapore.
5.
Ewins, D. J. , 1984, Modal Testing: Theory and Practice, Research Studies Press, Letchworth, Hertfordshire, England.
6.
Feeny, B. F. , 1997, “Interpreting proper orthogonal modes in vibrations,” in Proceedings of the ASME DETC, Sacramento, CA, CD-ROM.
7.
Feeny, B. F. and Kappagantu, R., 1998, “On the physical interpretation of proper orthogonal modes in vibrations,”Journal of Sound and Vibration211(4), 607-616.
8.
Fitzsimons, P. and Rui, C., 1993, “Determining low-dimensional models of distributed systems”, in Advances in Robust Nonlinear Control Systems ASME DSC Vol. 53, pp. 9-15.
9.
Forsythe, G. E. , Malcolm, M. A., and Moler, C. B., 1977, Computer Methods for Mathematical Computations, Prentice Hall, Englewood Cliffs, NJ.
10.
Georgiou, I. T. and Schwartz, I. B., 1996, “A proper orthogonal decomposition approach to coupled mechanical systems”, in Nonlinear Dynamics and Controls ASME DE-Vol. 91, pp. 7-12.
11.
Georgiou, I. T. , Schwartz, I. B., Emaci, E., and Vakakis, A. F., 1999, “Interaction between slow and fast oscillations in an infinite degree-of-freedom linear system coupled to a nonlinear system: Theory and experiment,”ASME Journal of Applied Mechanics66(2), 448-459.
12.
Harris, C. M. and Crede, C. E., 1996, Shock and Vibration Handbook, McGraw-Hill, New York.
13.
Kappagantu, R. V. and Feeny, B. F., 2000, “Part 2: Proper orthogonal modal modeling of a frictionally excited beam,”Nonlinear Dynamics23(1), 1-11.
14.
Kreuzer, E. and Kust, O., 1996, “Proper orthogonal decomposition—An efficient means of controlling self-excited vibrations of long torsional strings”, in Nonlinear Dynamics and Controls ASME DE-Vol. 91, pp. 105-110.
15.
Kust, O. , 1997, “Modal analysis of long torsional strings through proper orthogonal decomposition,”ZAMM77(S1), S183-S184.
16.
Lenaerts, V. , Kerschen, G., and Golinval, J. C., 2000, “Parameter identification of nonlinear mechanical systems using proper orthogonal decomposition,” in Proceedings of the IMAC XVIII, San Antonio, TX.
17.
Lumley, J. L. , 1967, “The structure of inhomogeneous turbulent flow”, in Atmospheric Turbulence and Radio Wave Propagation, A. M Yaglom and V. I. Tatarski, eds., pp. 166-178, Nauka, Moscow.
18.
Lumley, J. L. , 1970, Stochastic Tools in Turbulence, Academic Press, New York.
19.
Ma, X. , Azeez, M.A.F., and Vakakis, A. F., 2000, “Nonlinear normal modes and nonparametric system identification of nonlinear oscillators,”Mechanical Systems and Signal Processing14(1), 37-48.
20.
Ma, X. and Vakakis, A. F., 1999, “Nonlinear transient localization and low dimensional models of a flexible system with a clearance,” in Proceedings of the ASME Design Engineering Technical Conferences, Las Vegas, NV, CD-ROM.
21.
Ravindra, B. , 1999, “Comments on 'On the physical interpretation of proper orthogonal modes in vibrations”', Journal of Sound and Vibration219(1), 189-192.
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Riaz, M. S. and Feeny, B. F., 1999, “Proper orthogonal decomposition of an experimental cantilever beam,” in Proceedings of the ASME DETC, Las Vegas, NV, CD-ROM.
23.
Yasuda, K. and Kamiya, K., 1997, “Experimental identification technique of nonlinear beams in time domain,” in Proceedings of the ASME Design Engineering Technical Conferences, Sacramento, CA, CD-ROM.
