Frequency Analysis of a Linear Elastic Structure Carrying a Chain of Oscillators
Publication: Journal of Engineering Mechanics
Volume 125, Issue 5
Abstract
In this technical note we analyze the free vibration of M undamped oscillators attached to an arbitrarily supported, linear elastic structure. Using the assumed-modes method with N component modes, the frequency equation governing the free vibration for this combined system is typically obtained as the characteristic determinant of a generalized eigenvalue problem of size (N + M) × (N + M). In this note we will show that by algebraically manipulating the generalized eigenvalue problem associated with free vibration, we can reduce it to a simple secular equation consisting of the sum of N terms, the roots or natural frequencies of which can be obtained either numerically or graphically. In addition, the resultant secular equation lends itself to the solution of an inverse problem that cannot be easily solved by analyzing the original generalized eigenvalue problem.
Get full access to this article
View all available purchase options and get full access to this article.
References
1.
Bunch, J. R., Nielsen, C. P., and Sorensen, D. C. ( 1987). “Rank-one modification of the symmetric eigenproblem.” Numer. Math., 31, 31–48.
2.
Cha, P. D., and Pierre, C. ( 1999). “Imposing nodes to the normal modes of a linear elastic structure.” J. Sound and Vibration, 219(4), 669–687.
3.
Dongarra, J. J., and Sorensen, D. C. ( 1987). “A fully parallel algorithm for the symmetric eigenvalue problem.” SIAM J. Sci. Stat. Comput., 8, 139–154.
4.
Dowell, E. H. ( 1979). “On some general properties of combined dynamical systems.” J. Appl. Mech., 46(1), 206–209.
5.
Ercoli, L., and Laura, P. A. A. ( 1987). “Analytical and experimental investigation on continuous beams carrying elastically mounted masses.” J. Sound and Vibration, 114(3), 519–533.
6.
Gürgöze, M. ( 1996). “On the eigenfrequencies of a cantilever beam with attached tip mass and a spring-mass system.” J. Sound and Vibration, 190(2), 149–162.
7.
Harmon, T. L., Dabney, J., and Richert, J. ( 1997). Advanced engineering mathematics using MATLAB V.4. PWS Publishing, Boston.
8.
Kukla, S., and PosiadaŁa, B. ( 1994). “Free vibrations of beams with elastically mounted masses.” J. Sound and Vibration, 175(4), 557–564.
9.
Kukla, S. ( 1997). “Application of Green functions in frequency analysis of Timoshenko beams with oscillators.” J. Sound and Vibration, 205(3), 355–363.
10.
Lueschen, G. G. G., Bergman, L. A., and McFarland, D. M. ( 1996). “Green's functions for uniform Timoshenko beams.” J. Sound and Vibration, 194(1), 93–102.
11.
Meirovitch, L. ( 1967). Analytical methods in vibrations. Macmillan, New York.
12.
Nicholson, J. W., and Bergman, L. A. (1986). “Free vibration of combined dynamical system.”J. Engrg. Mech., 112(1), 1–13.
13.
Rossi, R. E., Laura, P. A. A., Avalos, D. R., and Larrondo, H. ( 1993). “Free vibrations of Timoshenko beams carrying elastically mounted, concentrated masses.” J. Sound and Vibration, 165(2), 209–223.
Information & Authors
Information
Published In
History
Published online: May 1, 1999
Published in print: May 1999
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.