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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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Front Page Browse CCP CSETS CTR IJRT Other Authors Search Purchase Guide FAQ Contact us	Civil-Comp Proceedings ISSN 1759-3433 CCP: 93 PROCEEDINGS OF THE TENTH INTERNATIONAL CONFERENCE ON COMPUTATIONAL STRUCTURES TECHNOLOGY Edited by: 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 doi:10.4203/ccp.93.259 purchase the full-text of this paper Full Bibliographic Reference for this paper A. Kaveh, L. Shahryari, "Canonical Forms and Graph Theory for Calculating the Eigenfrequencies of Symmetric Space Frames", in , (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 purchase the full-text of this paper (price £20) go to the previous paper go to the next paper return to the table of contents return to the book description purchase this book (price £145 +P&P)
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