![]() ![]() The use of the well-developed matrix theory for certain structured matrices, such as Jacobi or band matrices, as applied to inverse problems has also been investigated (de Boor and Golub 1978 Erra and Philippe 1997 Biegler-König 1981b Boley and Golub 1987). Dias de Silva, de Oliveira, and others have studied how multiplying the discrete system by another matrix can affect its structure (Downing and Householder 1956 Dias da Silva 1986 de Oliveira 1972). Modifying the matrix through the addition of another matrix has also been considered in (Morel 1976 Bohte 1968 Pereyra et al. Structuring matrices by prescribing specific entries has been studied in (Chu 1992 Friedland et al. According to Chu, these methods can be distinguished by the types of procedures used in imposing structure to the matrix (Chu 1998). Various methods can be used to impose such structure. Since PΛP − 1 is the trivial solution, pre-conditioning of the P matrix is required so that any given structural requirements of the system can be satisfied. Additionally, any arbitrary invertible matrix P can be used to obtain another solution (matrix) with the same spectrum, namely PΛP − 1. In most cases, the solution to the inverse problem begins by placing the given desired eigenvalues along the main diagonal entries of a diagonal matrix Λ. Although interesting, this approach cannot be used for novel engineering design to construct a system having a specific spectrum without another system on which to base the design. This clearly indicates that a system usually exists and that it can be tested to obtain data required for the inverse problem of mathematically reconstructing the system. One of the most common techniques in inverse eigenvalue problems is to use/measure the system’s spectrum and then constrain the system in some fashion in order to obtain a second spectrum (Hochstadt 1967 Hald 1976 Boley and Golub 1978 de Boor and Saff 1986 Gladwell 1984). Particularly, it appears that most of the literature focuses on system identification. Gladwell takes a more direct route in which he considers specific inverse problems and matrix structures related to mechanical vibrations (Gladwell 2004). This has been made clear by a thorough review of the topic by Chu and Golub (Chu 1998 Chu and Golub 2005). Much focus has been applied to the study of discrete inverse eigenvalue problems. This approach presents greater value since many numerical and analytical tools already exist for the solution of discrete problems and many engineering systems are often modelled as discrete systems. For the application of inverse eigenvalue theory to the field of vibrations, this would involve the use of discrete matrix representations of real systems. ![]() While continuous inverse theories have been studied, such as the classical Sturm-Liouville problem (Chadan 1997 Gel’fand and Levitan 1951 Gantmakher and Kreĭn 2002), a more interesting approach for the purpose of design is to use discrete theory. A rather broad field covering many subjects, such as control systems, structural analysis, particle physics and vibrations, inverse eigenvalue problems have an interesting and large field of application. Although not currently used for such purpose, the theory could potentially be applied to such design problems. One area that seeks to solve these difficulties and which potentially holds great promise for addressing the problem of design for frequency spectrum is that of inverse eigenvalue problems. This is because a single set of natural frequencies can be produced by multiple systems and thus multiple solutions are possible. A better approach would be to design the system directly from the natural frequencies.įrom a mathematical point of view, this problem is ill-posed. This approach is both time consuming and indirect. If specific natural frequencies are sought, empirical or iterative methods are used to modify the system’s physical parameters until the desired eigenvalues are obtained. Generally, the problem begins by defining the system’s physical parameters and then calculating the natural frequencies using eigenvalue theory. These design problems can be considered as eigenvalue problems, since the eigenvalues are used to determine the natural frequencies (frequency spectrum) of the system. In mechanical and structural system design, engineers are often faced with the task of designing systems which either have natural frequencies which must fall outside a specific range or operate at exactly certain frequencies. ![]()
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