In this paper the extraction of left eigenvectors from measured frequency response functions is presented and discussed. It is shown that contrary to the extraction of right eigenvectors from measured response data, the extraction of left eigenvectors is inherently ill-posed when the same measured information is being used. The ill-posedness can result in a spatially oscillating additive component (short wavelength) appearing in the identified left eigenvectors, a property which is not described in existing literature on structural dynamics. Some well-known regularization methods are hence used to reduce the inaccuracies appearing in the extracted left eigenvectors. It is also shown that a further improvement in the accuracy of the solution can be achieved by making use of an approximation of the mass distribution combined with regularization. Throughout this paper, an attempt is made to provide a physical insight to the various aspects related to the left eigenvectors and their extraction. This includes (a) the interpretation of left eigenvectors as internal force distributions; (b) the inherent reason for the ill-conditioning, and (c) the spatial filtering of short wavelength achieved by regularization, yielding an improved estimate of the left eigenvectors.

Issue Section:
Technical Papers
Topics:
Eigenvalues, Wavelength, Approximation, Filtration, Frequency response, Structural dynamics
1.
Ben-Israel, A., and Greville, T. N. E., 1974, Generalized Inverses: Theory and Applications, John Wiley and Sons, New York.
2.
Bucher
I.
, and
Braun
S. G.
,
1993
, “
The Structural Modification Inverse Problem: an Exact Solution
,”
Mechanical Systems and Signal Processing
, Vol.
7
, pp.
217
–
238
.
3.
Bucher
I.
, and
Braun
S. G.
,
1994
, “
Structural Response Optimization for Minimizing Vibratory Response via Redesign or Modification, Part I: Theory
,”
Journal of Sound and Vibration
, Vol.
175
, pp.
433
–
454
.
4.
Ewins, D. J., 1984, Modal Testing, Reseach Studies Press.
5.
Gladwell, G. M. L., 1986, Inverse Problems in Vibration, Martinus Nijhoff.
6.
Glasko, V. B., 1988, Inverse Problems of Mathematical Physics, American Institute of Physics, New York.
7.
Hansen
P. C.
,
1987
, “
The Truncated SVD as a Method for Regularization
,”
B.I.T.
, Vol.
28
, pp.
543
–
553
.
8.
Inman, D. J., 1989, Vibration: With Control, Measurement, and Stability, Prentice-Hall, Englewood Cliffis, NJ.
9.
Lallement, G., Link, M., and Cesar, B., 1991, “Procedures for Updating Dynamic Mathematical Models,” European Space Agency Contract Report, No. BE5463.
10.
Lancaster, P., 1969, Theory of Matrices, Academic Press, New York.
11.
Lawson, C. L., and Hanson, D. J., 1974, Solving Least Squares Problems, Prentice-Hall, Englewood Cliffs, NJ.
12.
Meirovitch, L., 1980, Computational Methods in Structural Dynamics, Sijhoff and Noordhoff.
13.
Meirovitch
L.
,
1985
, “
The Implementation of Modal Filters for Control of Structures
,”
Journal of Guidance Control and Dynamics
, Vol.
8
, No.
6
, pp.
707
–
716
.
14.
Ram, 1989, “Modal Analysis in Truncates Systems,” Ph.D. Thesis, Technion Israel Institure of Technology Haifa, Israel.
15.
Ram
Y. M.
, and
Braun
S. G.
,
1990
, “
Structural Dynamic Modifications Using Truncated Data: Bounds for the Eigenvalues
,”
Mechanical Systems and Signal Processing
, Vol.
4
, No.
1
, pp.
39
–
52
.
16.
Shelley, S. J., and Allemange, R. J., 1992, “Calculation of Discrete Modal Filters Using the Modified Reciprocal Vector Method,” Proceedings of the International Modal Analysis Conference, San Diego, CA.
17.
Zhang, Q., Allemange, R. J., and Brown, D. L., 1992, “Calculation of Discrete Modal filters Using the Modified Reciprocal Modal Vector Method,” Proceedings of the International Modal Analysis Conference, San Diego, CA.
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