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TECHNICAL BRIEFS

On the Proper Orthogonal Modes and Normal Modes of Continuous Vibration Systems

[+] Author and Article Information
B. F. Feeny

Department of Mechanical Engineering, Michigan State University, East Lansing, MI 48824

J. Vib. Acoust 124(1), 157-160 (Aug 01, 2001) (4 pages) doi:10.1115/1.1421352 History: Received February 01, 1999; Revised August 01, 2001
Copyright © 2002 by ASME
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References

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 Fractals, 3, No. 5, pp. 515–535.
Davies, M. A., and Moon, F. C., 1997, “Solitons, Chaos, and Modal Interactions in Periodic Structures,” Nonlinear Dynamics: The Richard Rand 50th Anniversary Volume, A. Guran, ed., World Scientific, Singapore, pp. 119–143.
Kust,  O., 1997, “Modal Analysis of Long Torsional Strings Through Proper Orthogonal Decomposition,” Zeitschrift für angewandte Mathematik und Mechanik,77, S1, pp. S183–S184.
Feeny,  B. F., and Kappagantu,  R., 1998, “On the Physical Interpretation of Proper Orthogonal Modes in Vibrations,” J. Sound Vib., 211(), pp. 607–616.
Forsythe, G. E., Malcolm, M. A., and Moler, C. B., 1977, Computer Methods for Mathematical Computations, Prentice Hall, Englewood Cliffs.
Feeny, B. F., and Liang, Y., 2001, “Interpreting Proper Orthogonal Modes of Randomly Excited Linear Vibration Systems,” submitted.
Ma, X., Azeez, M. A. F., and Vakakis, A. F., 1998, “Nonparametric Nonlinear System Identification of a Nonlinear Flexible System Using Proper Orthogonal Mode Decomposition,” Seventh Conference on Nonlinear Vibrations, Stability, and Dynamics of Structures, Blacksburg, VA.
Boivin,  N., Pierre,  C., and Shaw,  S. W., 1995, “Non-Linear Normal Modes, Invariance, and Modal Dynamics Approximations of Non-Linear Systems,” Nonlinear Dyn., 8, pp. 315–346.
Meirovitch, L., 1997, Principles and Techniques of Vibrations, Prentice Hall, Upper Saddle River.

Figures

Grahic Jump Location
The first two discretized linear normal modes of a cantilevered beam are plotted with a solid line. The corresponding POMs are plotted with circles.
Grahic Jump Location
The first two discretized linear normal modes of a hinged-hinged beam are plotted with a solid line. The corresponding POMs are plotted with circles.
Grahic Jump Location
An animation of vibration in the first nonlinear normal mode of a nonlinear beam
Grahic Jump Location
The two most dominant POMs for a nonlinear beam vibrating in the first nonlinear normal mode. The dominant POM corresponds to 99.8 percent of the signal power.
Grahic Jump Location
The deflection u1(t) at x=0.0825 versus the deflection u6(t) at x=0.5 during vibration in the first nonlinear normal mode. The straight line is the projection of the dominant POM.
Grahic Jump Location
The deflections of linear modal coordinate q3, versus the first linear modal coordinate, q1, of a nonlinear beam during vibration in the first nonlinear normal mode. The straight line is the projection of the dominant POM.

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