Composite materials are increasingly finding use in structures, such as aircraft components, and thus, an accurate method of predicting response is required. Even laminated structures that are considered thin can be significantly affected by transverse shear effects, and as a result, transverse shear should not be neglected. The free vibration response of generally-laminated, thick, skew, trapezoidal plates is investigated as there appears to be a lack of information in this area. In the method developed, Chebychev polynomials are used as displacement functions in the Rayleigh-Ritz method. Various edge supports are considered, and appropriate linear and rotational springs are introduced to approximately satisfy the essential boundary conditions associated with simply-supported and clamped edges. First-order shear theory is used to account for transverse shear effects, and rotary inertia is also included.in the model. Convergence of the solution resulting from changes in spring values and number of terms in the series is investigated. The accuracy of the method is demonstrated by comparing the present method to available results for plates of various quadrilateral shape, material systems, and boundary conditions. Thick laminated plates of both symmetric and unsymmetric construction and of various planforms and boundary conditions are then presented. Cantilever, thick, skew, and trapezoidal plates are then extensively studied, and variations in natural frequencies due to geometric parameter changes, such as taper ratio, sweep angle, and value of the parameter q, are discussed. The parameter, q, is a root length multiplier which determines the length of the quarter-chord line, thus representing a measure of the span. Mode shapes for a number of plates of various planform and support conditions are included. / Master of Science