Automated Truncation of Ritz Vector Basis in Modal Transformation
Publication: Journal of Engineering Mechanics
Volume 116, Issue 11
Abstract
The number of load‐dependent Ritz vectors required for an accurate dynamic analysis, using the mode superposition method, depends on both the spatial distribution and the spectral (frequency) content of the loading. In this paper we demonstrate that, by monitoring two modal truncation criteria for spectral content and spatial representation of the loading, in the progress of algorithm we can predetermine the accuracy of subsequent dynamic analysis. In the case of earthquake excitation, the loading spatial representation truncation criterion is a participating mass. The spectral content truncation criterion exploits the analogy to the static correction method and unique properties of the load‐dependent vector algorithm. The numerical effort required for the calculation of modal truncation criteria is slight compared to the numerical effort of generating the Ritz vectors. Therefore, monitoring truncation criteria should become a standard step in the load‐dependent vector algorithm. In addition, it provides a possibility for automated truncation of Ritz vector basis in the modal dynamic analysis of complex structures.
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References
1.
Agnostopoulos, S. A. (1982). “Wave and earthquake response of offshore structure; Evaluation of modal solution.” J. Struct. Div., ASCE, 108(10), 2175–2191.
2.
API recommended practice for planning, designing and constructing fixed offshore platforms. (1980). American Petroleum Institute, Washington, D.C.
3.
Clough, R. W., and Penzien, J. (1975). Dynamics of structures. McGraw‐Hill Inc., New York, N.Y.
4.
Clough, R. W., and Wilson, E. L. (1979). “Dynamic analysis of large structural systems with local nonlinearities.” Comput. Methods Appl. Mech. Engrg., 18, 107–129.
5.
Cornwell, R. E., Craig Jr., R. R., and Johnson, C. P. (1983). “On the application of the mode‐acceleration method to structural engineering problems.” Earthquake Engrg. Struct. Dyn., 11, 679–688.
6.
Hansten, O. E., and Bell, K. (1979). “On the accuracy of mode superposition analysis in structural dynamics.” Earthquake Engrg. Struct. Dyn., 7, 405–411.
7.
Hudson, D. E. (1979). “Reading and interpreting strong motion accelerograms.” Report, Earthquake Engrg. Res. Inst., California Institute of Technology, Pasadena, Calif.
8.
Ibrahimbegovic, A., and Wilson, E. L. (1989). “Simple numerical algorithms for mode superposition analysis of linear structural systems with non‐proportional damping.” Comput. Struct., 33, 523–531.
9.
Ibrahimbegovic, A. et al. (1990). “Ritz method for dynamic analysis of large discrete linear systems with non‐proportional damping.” Earthquake Engrg. Struct. Dyn.
10.
Ibrahimbegovic, A., and Wilson, E. L. (1990). “A methodology for dynamic analysis of large structure‐foundation systems with local nonlinearities.” Earthquake Engrg. Struct. Dyn.
11.
Leger, P., and Wilson, E. L. (1987). “Generation of load dependent ritz transformation vectors in structural dynamics.” Engrg. Comput., 4(4), 309–318.
12.
Lanczos, C. (1950). “An iteration method for the solution of the eigenvalue problem of linear differential and integral operator.” J. Res. Nat. Bur. Stand., 45(4), 255–282.
13.
Lin, Y. K. (1967). Probabilistic theory of structural dynamics. McGraw Hill Inc., New York, N.Y.
14.
Maddox, N. R. (1975). “On the number of modes necessary for accurate response and resulting forces in dynamic analysis.” J. Appl. Mech., 42, 516–517.
15.
Nour‐Omid, B., and Clough, R. W. (1984). “Dynamic analysis of structures using lanczos coordinates.” Earthquake Engrg. Struct. Dyn., 12, 565–577.
16.
Nour‐Omid, B., and Clough, R. W. (1985). “Block lanczos method for dynamic analysis of structures.” Earthquake Engrg. Struct. Dyn., 13, 271–275.
17.
Paige, C. C. (1971). “The computation of eigenvalues and eigenvectors of very large sparse matrices,” thesis presented to the University of London, at London, England, in partial fulfillment of the requirements for the degree of Doctor of Philosophy.
18.
Parlett, B. N. (1980). The symmetric eigenvalue problem. Prentice‐Hall, Englewood Cliffs, N.J.
19.
Parlett, B. N. (1982). “Two monitoring schemes for the Lanczos algorithm.” Computing Methods in Applied Science and Engineering, R. Glowinski and J. L. Lions, eds., Elsevier Science Publisher, New York, N.Y.
20.
Parlett, B. N., and Nour‐Omid, B. (1985). “The use of a refined error bound when updating eigenvalues of tridiagonal.” Linear Alg. Appl., 68, 179–219.
21.
Wilson, E. L. (1977). “Numerical methods for dynamic analysis.” Int. Symp. On Numer. Methods in Offshore Engrg., Swansea, United Kingdom.
22.
Wilson, E. L., Yuan, M. W., and Dickens, J. M. (1982). “Dynamic analysis by direct superposition of Ritz vectors.” Earthquake Engrg. Struct. Dyn., 10, 813–821.
23.
Wilson, E. L., and Itoh, T. (1983). “An eigensolution strategy for large systems.” Comput. Struct., 16, 259–265.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 116 • Issue 11 • November 1990
Pages: 2506 - 2520
Copyright
Copyright © 1990 ASCE.
History
Published online: Nov 1, 1990
Published in print: Nov 1990
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