Block circulant matrices and applications in free vibration analysis of cyclically repetitive structures
- First Online:
In this paper, block circulant matrices and their properties are investigated. Basic concepts and the necessary theorems are presented and then their applications are discussed. It is shown that a circulant matrix can be considered as the sum of Kronecker products in which the first components have the commutativity property with respect to multiplication. The important fact is that the method for block diagonalization of these matrices is much simpler than the previously developed methods, and one does not need to find an additional matrix for orthogonalization. As it will be shown not only the matrices corresponding to domes in the form of Cartesian product, strong Cartesian product and direct product are circulant, but for other structures such as diamatic domes, pyramid domes, flat double layer grids, and some family of transmission towers these matrices are also block circulant.
Unable to display preview. Download preview PDF.
- 4.Hasan, M.A., Hasan, J.A.K.: Block eigenvalue decomposition using nth roots of identity matrix. In: 41st IEEE Conference on Decision and Control, vol. 2, pp. 2119-2124 (2002)Google Scholar
- 5.Aghayere, A.O.: Structural systems with polar symmetry: solution by quasi-circulant matrices. M.Sc. thesis, Massachusetts Institute of Technology (1983)Google Scholar
- 6.Kaveh, A., Nemati, F.: Eigensolution of rotationally repetitive space structures using a canonical form. Commun. Numer. Methods Eng. doi:10.1002/cam.1265, 20 May 2010
- 7.Zloković G.M.: Group Theory and G-vector Spaces in Engineering Structures, Vibration, Stability and Statics. Ellis Horwood Limited, Chichester (1989)Google Scholar
- 9.Zingoni, A.: Group-theoretical applications in solid and structural mechanics: a review. In: Topping, B.H.V., Bittnar, Z. (eds.) Chapter 12 in Computational Structures Technology. Saxe-Coburg, UK (2002)Google Scholar