Keywords: multi-symmetry, trusses, decomposition, eigenfrequencies, graph.

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 the efficient solution of such problems, especially when their corresponding 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 are studied in references [1,2], and the applications to vibrating mass-spring systems and frame structures are developed in [3,4], respectively. These forms are also applied to calculating the buckling load of symmetric frames [5].

Consider a structural system with two translational degrees of freedom (DOFs) per node which has two axes of symmetry. Suppose each DOF is parallel to one of the axes and is perpendicular to the other axis. One can find matrices in canonical forms, and using the symmetry relationships twice, one can find four submatrices. The union of the eigenvalues for these four submatrices results in the eigenvalues of the original matrix.

In this paper, the region in which the structural system is situated is divided into upper, lower, left and right subregions. The stiffness matrix of the entire system is formed and then using the existing direct and reverse symmetries, relationships between the entries of the matrix are established.

1

Kaveh A., Sayarinejad M.A., "Eigensolutions for matrices of special patterns", Communs Numer. Methods Eng., 19, 125-136, 2003.

2

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

3

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

4

Kaveh A., Salimbahrami B., "Eigensolutions of symmetric frames using graph factorization", Communs Numer. Methods Eng., 20, 889-910, 2004. doi:10.1002/cnm.711

5

Kaveh A., Rahami H., "New canonical forms for analytical solution of problems in structural mechanics", Communications in Numerical Methods in Engineering, No. 9, 21, 499-513, 2005. doi:10.1002/cnm.763

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