Canonical Forms Applied to Structural Mechanics
The main objective of this chapter is to illustrate different applications of the canonical forms in structural mechanics with particular emphasis on calculating the buckling load and eigenfrequencies of the symmetric structures.
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Canonical Forms Applied to Structural Mechanics
7.1
Introduction
The main objective of this chapter is to illustrate different applications of the canonical forms in structural mechanics with particular emphasis on calculating the buckling load and eigenfrequencies of the symmetric structures. In the first part, the problem of finding eigenvalues and eigenvectors of symmetric mass–spring vibrating systems is transferred into calculating those of their modified subsystems. This decreases the size of the eigenvalue problems and correspondingly increases the accuracy of their solutions and reduces the computational time [1]. In the second part, a methodology is presented for efficient calculation of buckling loads for symmetric frame structures. This is achieved by decomposing a symmetric model into two submodels followed by their healing to obtain the factors of the model. The buckling load of the entire structure is then obtained by calculating the buckling loads of its factors [2]. In the third part, the graph models of planar frame structures with different symmetries are decomposed, and appropriate processes are designed for their healing in order to form the corresponding factors. The eigenvalues and eigenvectors of the entire structure are then obtained by evaluating those of its factors. The methods developed in this part simplify the calculation of the natural frequencies and natural modes of the planar frames with different types of symmetry [3]. In the fourth part, methods are presented for calculating the eigenfrequencies of structures. The first approach is graph theoretical and uses graph symmetry. The graph models are decomposed into submodels, and healing processes are employed such that the union of the eigenvalues of the healed submodels contains the eigenvalues of the entire model. The second method has an algebraic nature and uses special canonical forms [4]. In the fifth part, general forms are introduced for efficient eigensolution of special tri-diagonal and five-diagonal matrices. Applications of these forms are illustrated using problems from mechanics of structures [5]. A. Kaveh, Optimal Analysis of Structures by Concepts of Symmetry and Regularity, DOI 10.1007/978-3-7091-1565-7_7, © Springer-Verlag Wien 2013
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7 Canonical Forms Applied to Structural Mechanics
In the sixth part, the decomposability conditions of matrices are studied. Matrices that can be written as the sum of three Kronecker products are studied; examples are included to show the efficiency of this decomposition approach [6]. In the seventh part, canonical forms are used to decompose the symmetric line elements (truss and beam elements) into sub-elements of less the number of degrees of freedom (DOFs). Then the matrices associated with each sub-element are formed, and finally the matrices associated with each subsystem are combined to form the matrices of the prime element [7]. In the final part, an efficient eigensolution is presented for calculating the buckling load and free vibration of rotationally cyclic s
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