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An inverse eigenvalue problem is one where a set or subset of (generalized) eigenvalues is specified and the matrices that generate it are sought. Many methods for solving inverse eigenvalue problems are only applicable to matrices of a specific type. In this chapter, two recently proposed methods for structured (direct) solutions of inverse eigenvalue problems are presented. The presented methods are not restricted to matrices of a specific type and are thus applicable to matrices of all types. For the first method, the Cayley–Hamilton theorem is developed for the generalized eigenvalue vibration problem. For a given (desired) frequency spectrum, many solutions are possible. Hence, a discussion of the required information and suggestions for including structural constraints are given. An algorithm for solving the inverse eigenvalue problem using the generalized Cayley–Hamilton theorem is then demonstrated. An algorithm for solving partially described systems is also specified. The Cayley–Hamilton theorem algorithm is shown to be a good tool for solving inverse generalized eigenvalue problems. Examples of application of the method are given. A second method, referred to as the inverse eigenvalue determinant method, is also introduced. This method provides another direct approach to the reconstruction of the matrices of the generalized eigenvalue problem, given knowledge of its eigenvalues and various physical parameters. As for the first method, there are no restrictions on the type of matrices allowed for the inverse problem. Examples of application of the method are also given, including application-oriented examples.

As a musical instrument construction material, wood is both musically and aesthetically pleasing. Easy to work and abundant, it has traditionally been the material of choice. Unfortunately, wood is also a very inconsistent material. Due to great environmental and climatic variations, wooden specimens present large variations in their mechanical properties, even within species of a similar region. Surprisingly, an industry based entirely on acoustics has done very little to account for these variations. For this reason, manufactured wooden guitars are acoustically inconsistent. Previous work has shown that varying the dimensions of a guitar soundboard brace is a good method for taking into account variations in the mechanical properties of the wooden soundboard plate. In this thesis, the effects of a scalloped-shaped brace on the natural frequencies of a brace-plate system have been studied and tools have been developed in order to calculate the dimensions of the brace required to account for variations in the mechanical properties of the plate. It has been shown that scalloped braces can be used to modify two natural frequencies of a brace-plate system simultaneously. Furthermore, the most important criteria in modifying any given frequency of a brace-plate system is the mass and stiffness properties of the brace at the antinode of the given frequency’s associated modeshape. Subsequently, designing a brace for desired system natural frequencies, by taking into account the mechanical properties of the wooden plate, is an inverse eigenvalue problem. Since few methods exist for solving the inverse eigenvalue problem of general matrices, a new method based on the generalized Cayley-Hamilton theorem was proposed in the thesis. A further method, based on the determinant of the generalized eigenvalue problem was also presented. Both methods work well, although the determinant method is shown to be more efficient for partially described systems. Finally, experimental results were obtained for the natural frequencies of simply supported wooden plates, with and without a brace, as well as the inverse eigenvalue determinant method. Good correlation was found between theoretical and experimental results.

Although many improvements in the manufacturing of musical instruments have been made recently, one aspect that has often been overlooked is that of the acoustic consistency of the final manufactured product. The aim of this thesis is to create a method in which a soundboard can be frequency matched to a brace in order to meet a set standard after assembly. A simple analytical model is created in order to study the effect of the plate's stiffness and brace thickness on the combined system. The assumed shape method is used in the analysis. Results show that by adjusting the thickness of the brace in order to compensate for the stiffness of the plate, one of the natural frequencies can be adjusted to meet a certain value. However, matching multiple natural frequencies cannot be done with a rectangular brace. Therefore modifications to the shape of the brace are suggested.