Date of Award


Degree Type


Degree Name

Doctor of Philosophy


The Ritz approach is used in this thesis to study the free structural vibration problem of a wide class of plates, shallow shells and solids. The boundaries and/or internal supports of the systems considered are described by polynomials in Cartesian coordinates. The geometric boundary conditions at these boundaries or internal supports are enforced by including in a simple polynomial trial series the polynomial equation describing the position of the support raised to the appropriate power. For plates and shallow shells, in some instances, the displacement field is discretized into several domains or elements, as in the standard finite element method, and continuity conditions along the connecting boundaries are enforced by using a penalty method approach. Complicating effects such as point masses, stepped geometry, cut-outs and internal point supports are also considered, the geometric conditions at the point supports being enforced again by using a penalty method approach.;First, the natural frequencies and mode shapes of a number of isotropic rectangular plates with straight or curved internal line supports are obtained. Isotropic and orthotropic plates of more general planform, including circular, elliptical and hypocycloidally shaped plates are then considered. The effect of complications such as internal point and line supports, stepped geometry, internal cut-outs and point masses is also considered in some cases.;The approach is next extended to the study of the free vibration of shallow shells and planform describable by general polynomial expressions. A number of shallow shells, many not previously considered in the literature, are then treated, including shallow shells of circular, elliptical and rectangular planform. A number of curvatures are considered, including spherical, hyperbolic and cylindrical, and complicating effects, such as stepped thickness, discontinuous curvatures and curved internal line supports, are included in the analysis.;Finally, the Ritz approach is applied to the study of the free vibration of a class of solids bounded by the x, y and z planes as well as by a fourth surface describable by a polynomial expression in the cartesian coordinates x, y and z. By exploiting symmetry a number of problems are considered these being a sphere, a circular cylinder, an elliptical cylinder, a cone and a prismatic solid of more general cross section.



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