Keywords: symmetry, frames, graphs, factors, decomposition, eigenvalues, eigenvectors, natural frequencies, natural modes.

Symmetry has been widely studied in science and engineering. Large eigenvalue problems arise in many scientific and engineering problems. While the basic mathematical ideas are independent of the size of the matrices, the numerical determination of eigenvalues and eigenvectors becomes more complicated as the dimensions and the sparsity of the matrices increases. Special methods are required for efficient solution of such problems.

Methods are developed for decomposing the graph models of structures in order to calculate the eigenvalues of matrices with special patterns, references [1,2,3,4]. The application of these methods is extended to the vibration of mass-spring systems [5] and frame structures [6]. Symmetry is also employed in the stability analysis of frames [7,8].

In this paper, graph models are associated with frame structures. Decomposition approaches are employed to form matrices of special patterns for such systems. The problem of finding eigenvalues and eigenvectors of symmetric frames is transferred into calculating those of their factors. The factors of a symmetric model are obtained by a decomposition followed by a new healing process. This results in efficient methods for evaluating the natural frequencies and natural modes of symmetric frames. Different forms of the symmetry are accompanied by simple illustrative examples.

Decomposition and healing processes presented in this article reduce the dimensions of the matrices for dynamic analysis of the symmetric frames. Therefore, for large-scale problems the accuracy of calculation increases and the cost of computation decreases. The present method can also be applied to similar eigensolution problems such as stability analysis of symmetric frames for calculating the critical loads of the frames.

1

Kaveh, A. and Sayarinejad, M.A., "Eigensolutions for matrices of special patterns", Communications in Numerical Methods in Engineering, 19:125-136, 2003.

2

Kaveh, A. and Sayarinejad, M.A., "Eigensolution of specially structured matrices with hyper-symmetry", International Journal for Numerical Methods in Engineering, 67(7):1012-1043, 2006. doi:10.1002/nme.1660

3

Kaveh, A., Structural Mechanics: Graph and Matrix Methods, Research Studies Press (John Wiley), Exeter, U.K., 3rd edition, 2004.

4

Kaveh, A. and Sayarinejad, M.A., "Eigensolutions for factorable matrices of special patterns", Communications in Numerical Methods in Engineering, 20:133-146, 2004. doi:10.1002/cnm.656

5

Kaveh, A. and Sayarinejad, M.A., "Graph symmetry in dynamic systems", Computers and Structures, 82(23-26):2229-2240, 2004. doi:10.1016/j.compstruc.2004.03.066

6

Kaveh, A. and Salimbahrami, B., "Eigensolutions of symmetric frames using graph factorization", Communications in Numerical Methods in Engineering, 20:889-910, 2004. doi:10.1002/cnm.711

7

Kaveh, A. and Dadfar, B., "Eigensolution for stability analysis of planar frames, by Graph Symmetry", Computer-Aided Civil and Infrastructure Engineering, 22:367-275, 2007. doi:10.1111/j.1467-8667.2007.00493.x

8

Kaveh, A. and Salimbahrami, B., "Buckling load of symmetric frames using canonical forms", Computers and Structures, 85(11): in press, 2007. doi:10.1016/j.compstruc.2006.08.082

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