TECHNICAL PAPERS
Nov 1, 1983

Practical Reduction of Structural Eigenproblems

Publication: Journal of Structural Engineering
Volume 109, Issue 11

Abstract

An evaluation of publications on reduction of structural eigenvalue problems has revealed the following: (1) Most of the published techniques are essentially equivalent to Guyan reduction; (2) published techniques to decrease the error in Guyan reduction require matrix inversions or an iterative process, or both. This paper presents a modified Guyan reduction method which decreases errors in the calculation of the eigenvectors of the system and which does not require matrix inversions whatsoever. It also shows that a simple iterative process could be used to improve further the eigenvalue solution of the system.

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References

1.
Anderson, R. G., Irons, B. M., and Zienkiewicz, O. C., “Vibration and Stability of Plates Using Finite Elements,” International Journal of Solids and Structures, Vol. 4, May, 1968, pp. 1031–1055.
2.
Bathe, Klaus‐Jurgen, and Wilson, E. L., “Solution Method for Eigenvalues Problems in Structural Dynamics,” International Journal for Numerical Methods in Engineering, Vol. 6, 1973, pp. 213–226.
3.
Bathe, K. J., and Wilson, E. L., Numerical Analysis in Finite Element Analysis, First Edition, Prentice Hall, 1976, pp. 259–263, 338–395.
4.
Cedolin, L., and Gallagher, R., “A Frontal‐based Solver for Frequency Analysis,” International Journal for Numerical Methods in Engineering, Vol. 12, 1978, pp. 1659–1666.
5.
Cook, R. D., “A Cost Comparison of Two Static Condensation‐Stress Recovery Algorithms,” International Journal for Numerical Methods in Engineering, Vol. 12, 1978, pp. 581–588.
6.
Cook, R. D., Concepts and Applications of Finite Element Analysis, Wiley, 2nd Ed., 1981, pp. 186–188, 311–315.
7.
Flax, A. H., Comment on “Reduction of Structural Frequency Equations,” American Institute of Aeronautics and Astronautics Journal, Vol. 13, No. 5, May, 1975, pp. 701–702.
8.
Gallagher, R. H., Finite Element Analysis, First Edition, Prentice Hall, 1975, pp. 44–45, 354, 387.
9.
Geradin, M., “Error Bounds for Eigenvalue Analysis by Elimination of Variables,” Journal of Sound and Vibration, Vol. 19, 1971, pp. 111–132.
10.
Gurujee, C. S., and Deshpande, V. L., “An Improved Method of Substructure Analysis,” International Journal of Computers and Structures, Vol. 8, 1978, pp. 147–152.
11.
Guyan, R. J., “Reduction of Stiffness and Mass Matrices,” American Institute of Aeronautics and Astronautics Journal, Vol. 3, No. 2, Feb., 1965, p. 380.
12.
Hensell, R. D., and Ong, J. H., “Automatic Masters for Eigenvalue Economization,” Earthquake Engineering and Structural Dynamics, Vol. 3, Apr., 1975, pp. 375–383.
13.
Huebner, K. H., The Finite Element Method for Engineers, Wiley, 1975, pp. 155–157.
14.
Hughes, T. J. R., “Reduction Scheme for Some Structural Eigenvalue Problems by a Variational Theorem,” International Journal for Numerical Methods in Engineering, Vol. 10, 1976, pp. 845–852.
15.
Irons, B. M., “Structural Eigenvalue Problems: Elimination of Unwanted Variables,” American Institute of Aeronautics and Astronautics Journal, Vol. 3, No. 5, May, 1965, pp. 961–962.
16.
Jennings, Alan, “Mass Condensation and Simultaneous Iteration for Vibration Problems,” International Journal for Numerical Methods in Engineering, Vol. 6, 1973, pp. 543–552.
17.
Johnson, C. P., Craig, R. R., Jr., Yargicoglu, A., and Rajatabhothi, R., “Quadratic Reduction for the Eigenproblem,” International Journal for Numerical Methods in Engineering, Vol. 15, 1980, pp. 911–923.
18.
Kidder, R. L., “Reduction of Structural Frequency Equations,” American Institute of Aeronautics and Astronuatics Journal, Vol. 11, No. 6, June, 1973, p. 892.
19.
Kidder, R. L., “Reply by Author to A. H. Flax,” American Institute of Aeronautics and Astronautics Journal, Vol. 13, No. 5, May, 1975, pp. 702–703.
20.
Leung, Andrew Jee‐Tak, “An Accurate Method of Dynamic Condensation in Structural Analysis,” International Journal for Numerical Method in Engineering, Vol. 12, 1978, pp. 1705–1715.
21.
MacNeal, R. H., and McCormick, C. W., “The Nastran Computer Program for Structural Analysis,” International Journal of Computers and Structures, Vol. 1, 1971, pp. 389–412.
22.
Miller, C. A., “Dynamic Reduction of Structural Models,” Journal of the Structural Division, ASCE, Oct., 1980, pp. 2097–2108.
23.
Miller, C. A., “Errors Resulting from Dynamic Reduction,” Proceedings of the First National Conference on Computers in Civil Engineering, May, 1981, pp. 58–71.
24.
Paz, Mario, Structural Dynamics: Theory and Computation, First Edition, Van Nostrand Reinhold, 1980, pp. 216–225.
25.
Petterson, H., and Popov, E. P., “Substructuring and Equation System Solution in Finite Element Analysis,” International Journal of Computer and Structures, Vol. 7, 1977, pp. 197–206.
26.
Przemieniecki, J. S., Theory of Matrix Structural Analysis, First Edition, McGraw‐Hill, 1968, pp. 147–148, 291–292.
27.
Ramsden, J. N., and Stoker, J. R., “Mass Condensation: A Semi‐automatic Method for Reducing the Size of Vibration Problems,” International Journal for Numerical Methods in Engineering, Vol. 1, 1969, pp. 333–349.
28.
Sen, S. K., and Gould, P. L., “Free Vibration of Shells of Revolution Using FEM,” Journal of the Engineering Mechanics Division, ASCE, Apr., 1974, pp. 283–303.
29.
Sowers, J. D., “Condensation of Free Body Mass Matrices Using Flexibility Coefficients,” American Institute of Aeronautics and Astronautics Journal, Vol. 16, 1978, pp. 272–273.
30.
Vysloukh, V. A., Kandidov, V. P., and Chesnokov, S. S., “Reduction of the Degrees of Freedom in Solving Dynamic Problems by the Finite Element Method,” International Journal for Numerical Methods in Engineering, Vol. 7, 1973, pp. 185–194.
31.
Wilson, E. L., “The Static Condensation Algorithm,” International Journal for Numerical Methods in Engineering, Vol. 8, 1974, pp. 198–203.
32.
Wittrick, W. H., and Williams, F. W., “A General Algorithm for Computing Natural Frequencies of Elastic Structures,” Journal of Mechanics and Applied Mathematics, Vol. 14, Pt. 3, 1970, pp. 263–284.

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Go to Journal of Structural Engineering
Journal of Structural Engineering
Volume 109Issue 11November 1983
Pages: 2591 - 2599

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Published online: Nov 1, 1983
Published in print: Nov 1983

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Mario Paz, M. ASCE
Prof. and Chrm., Civ. Engrg. Dept., Univ. of Louisville, Louisville, Ky.

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