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Civil-Comp Proceedings
ISSN 1759-3433
CCP: 93
PROCEEDINGS OF THE TENTH INTERNATIONAL CONFERENCE ON COMPUTATIONAL STRUCTURES TECHNOLOGY
Edited by: B.H.V. Topping, J.M. Adam, F.J. Pallarés, R. Bru and M.L. Romero
Paper 259

Canonical Forms and Graph Theory for Calculating the Eigenfrequencies of Symmetric Space Frames

A. Kaveh and L. Shahryari

Centre of Excellence for Fundamental Studies in Structural Engineering, Iran University of Science and Technology, Tehran, Iran

Full Bibliographic Reference for this paper
A. Kaveh, L. Shahryari, "Canonical Forms and Graph Theory for Calculating the Eigenfrequencies of Symmetric Space Frames", in B.H.V. Topping, J.M. Adam, F.J. Pallarés, R. Bru, M.L. Romero, (Editors), "Proceedings of the Tenth International Conference on Computational Structures Technology", Civil-Comp Press, Stirlingshire, UK, Paper 259, 2010. doi:10.4203/ccp.93.259
Keywords: three-dimensional frames, symmetry, graphs, decomposition and healing, canonical forms, eigenfrequencies.

Summary
Symmetry has been widely used in science and engineering. Many eigenvalue problems arise in many scientific and engineering problems. While the basic mathematical ideas are independent of the size of matrices, the numerical determination of eigenvalues and eigenvectors becomes more complicated as the dimensions of matrices increase. Special methods are beneficial for efficient solution of such problems, especially when their matrices are highly sparse.

Methods are developed for decomposing and healing the graph models of structures, in order to calculate the eigenvalues of matrices and graph matrices with special patterns. The eigenvectors corresponding to such patterns for the symmetry of Form I, Form II and Form III were studied in References [1], and the applications to free vibration and stability of frame structures were developed in [2,3].

For dynamic analysis of structure, the calculation of eigenvalues and eigenvectors is required. When the structural models are symmetric, such calculation can be simplified using some concepts of graph theory. In this paper, two methods are presented for eigensolution of space frames. The first uses a graph model and employs a decomposition and healing process for the formation of the factors of the graph model and calculating the natural frequencies and natural modes. The second uses the canonical forms for the construction of submatrices, from which the eigenvalues can be obtained. Both methods lead to identical results.

References
1
A. Kaveh, M.A. Sayarinejad, "Graph symmetry in dynamic systems", Computers and Structures, 82(23-26), 2229-2240, 2004. doi:10.1016/j.compstruc.2004.03.066
2
A. Kaveh, B. Salimbahrami, "Buckling load of symmetric frames using canonical forms", Computers and Structures, 85(11), 1420-1430, 2007. doi:10.1016/j.compstruc.2006.08.082
3
A. Kaveh, L. Shahryari, "Buckling load of planar frames with semi-rigid joints using weighted symmetric graphs", Computers and Structures, 85, 1704-1728, 2007. doi:10.1016/j.compstruc.2007.02.011

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