Summary
The eigen characteristics of a dynamical system offer a vector sub-space suitable for performing canonical transformations on the system of equations of the structural dynamics problems being considered. While the concept is mathematically fully developed for both damped and undamped systems, practitioners — at all levels — tend to indiscriminately use the system’s normal modes as a basis for applications containing nonproportional damping. Such a practice in most cases is a reasonable approximation and results in small, if not infinitesimal, errors. However, with the increase of sophistication and accuracy requirements in certain applications of modal analysis, these approximations must be fully analyzed and understood.
In this paper, emphasis is directed to some specific applications for which it is generally a common practice to use normal, or undamped, modes as a vector subspace for use with nonproportionally damped system. These are:
-
a)
Expansion of measured mode shapes for use in model updating procedures.
-
b)
Computation of normal modes from identified complex mode shapes.
-
c)
Response calculations and modal filtering.
Detailed background on complex modes is presented. Commonly used transformations are examined and error models are derived and quantified.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Caughey, T. K., O’Kelly, M. M. J. (1965), Classical normal modes in damped linear dynamic systems, Trans. ASME, J. Applied Mech. 32, 583–588.
Mitchell, L. (1990), Complex modes: a review, Proc. 8th IMAC, Kissimmee, FL.
Lang, G. F. (1989), Demystifying complex modes, Sound and Vibration, 23, 36–40.
Liang, Z., Tong, M., Lee, G. C. (1992), Complex modes in damped linear dynamic systems, Int. J. of Analytical and Experimental Modal Analysis, 7, 1–20.
Deblauwe, F., Allemang, R. J. (1986), A possible origin of complex modal vectors, Proceedings of the 11th International Seminar on Modal Analysis, paper A2-3, Katholic University of Leuven.
Leuridan, J., Brown, D. and Allemang, R. J. (1982), Direct system parameter identification of mechanical structures with application to modal analysis, AIAA paper 82-0767, 548–556.
Ibrahim, S. R. (1982), Dynamic modeling of structures from measured complex modes, AIAA Journal, 21, 898–901.
Ibrahim, S. R. (1983), Computation of normal mode from identified complex modes, AIAA Journal, 21, 446–451.
Niedbal, N. (1984), Analytical determination of real normal modes from measured complex responses, SAE Paper No. 84-0095, Palm Springs, CA.
Zhang, Q, Lallement (1985), New method of determining the eigensolutions of the associated conservative structure from the identified eigensolutions, Proceedings of the 3 rd International Modal Analysis Conference, 322–328.
Natke, H. G. and Robert, D. (1985), Determination of normal modes from identified complex modes, Z. Flugwiss. Weltraumforsch, g., 82–88.
Zhang, Q., Lallement, G. (1987), Comparison of normal eigenmodes calculation methods based on identified complex eigenmodes, J. of Spacecraft, 69–73.
Wei, M. L., Allemang, R. J., Brown, D. L. (1987), Real-normalization of measured complex modes, Proceedings of the 5th International Modal Analysis Conference, London, 708–712.
Wei, M. L., Zhang, Q., Allemang, J. and Brown, D. L. (1987), A time domain subspace iteration method for the normalization of measured complex modes, Proc. 12th International Seminar on Modal Analysis, paper No. S2-3, Katholic University of Leuven, Belgium.
Ibrahim, S. R., Fullekrug, U. (1990), Investigation into exact normalization of incomplete complex modes by decomposition transformation, Proc. 8th IMAC, Las Vegas.
Placidi, F., Poggi, F., Sestieri, A. (1991), Real modes computation from identified modal parameters with estimation of generalized damping, Proc. 9th IMAC, Florence, 1991.
Imregun, M. and Ewins, D. J. (1993), Realization of Complex Modes, Proc. 11th IMAC, 1303–1309.
Sestieri, A. and Ibrahim, S. R. (1994), Analysis of errors and approximations in the use of modal coordinates, Journal of Sound and Vibrations, 177(2), 145–157.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1999 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Ibrahim, S.R. (1999). Existence and Normalization of Complex Modes for Post Experimental Use in Modal Analysis. In: Silva, J.M.M., Maia, N.M.M. (eds) Modal Analysis and Testing. NATO Science Series, vol 363. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-4503-9_21
Download citation
DOI: https://doi.org/10.1007/978-94-011-4503-9_21
Publisher Name: Springer, Dordrecht
Print ISBN: 978-0-7923-5894-7
Online ISBN: 978-94-011-4503-9
eBook Packages: Springer Book Archive