Through an extension of Gorman's superposition techniques, a detailed investigation of the title problem is undertaken. In Part A, a critical assessement of the available literature dealing with this problem is provided. The classical theory of thin plate motion is then briefly discussed in Chapter 1 along with the method of superposition as it relates to dynamic problems of rectangular plates. In Part B, the problem of the non-rectangular quadrilateral plates is then discussed in Chapter 2 where it is shown how any polygonal plate can be divided into two sets of rectangular and right angle triangular regions. Also in Chapter 2, the basic building blocks used throughout this investiga- tion are studied. Consequently, solutions for the triangular and rectangular elements mentioned above are given. It is then shown how these solutions are combined to arrive at a solution to the problem at hand. It is in Part C that this technique is applied, in Chapter 3 to the solution of the simply sup- ported symmetrical trapezoidal plate, and in Chapter 4 to the fully clamped plate, resulting in a highly accurate analytical solution for these important engineering problems, and therefore establishing a systematic approach to the solution of a very large family of non-rectangular quadrilateral plates. Numerical results along with some modal shapes are also provided in Chapter 5, and compared with previously available data where good agreement is shown. Ex- cellent agreement with experimental results, in the case of the fully clampėd plate is reported. Finally, a brief but comprehensive discussion on the validity and accuracy of the technique is provided in Part D. Some discussion on the effect of rotatory inertia and shear on the flexural motion of plates is briefly given along with few words on internal damping effects.