| Summary: | 碩士 === 國立臺灣海洋大學 === 河海工程學系 === 92 === The purpose of this thesis is to investigate the Helmholtz eigenvalue problems for the non-homogeneous membrane with a radial density distribution. The radial density function is expressed as a polynomial form. The governing equation is a second-order, variable coefficient, differential equation. It will be solved by means of the Frobenius method. The sector membranes, annular-sector membranes and circular membranes with radial constraints are considered in this thesis. In the first two ones, the method of separation of variables and the Frobenius method are employed to obtain the expressions of potential functions. Then by applying the boundary conditions, the natural frequencies and mode shapes are determined. In the third one, three types of constraints respect to their relative positions are also discussed, including a single edge constraint, two opposite edge constraints and an internal constraint. All of these, problems are then solved by using the field-matching method. At first step, the whole domain is divided into circular and annular sub-regions. Secondly, the expressions of potential functions are acquired again by utilizing separation of variables and the Frobenius method. Thirdly, the point collocation is adopted to construct the coupling equations from the border between these two sub-regions. Finally, the natural frequencies and mode shapes can be determined. For homogeneous membranes, the numerical results obtained by the Frobenius method are found in fairly good agreement with that using the Bessel function.
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