A finite generalized integral transform was applied to three general classes of problems in the vibrations of continuous media. For its kernel the eigenfunction of an associated eigenvalue problem was used and the result was denoted the eigentransform. The eigentransform was applied to (1) continuous media with both non-uniform stiffness and mass distributions; (2) continuous media with uniform stiffness but non-uniform mass distribution; and (3) to problems with time-dependent boundary conditions.

A general method was presented for treating vibrations of continuous media with non-uniform stiffness and mass distributions. The eigentransform was applied to the governing partial differential equation and subsequently the transformed displacement was found to satisfy an infinite set of coupled ordinary differential equations similar to those encountered in the vibrations of discrete masses. These equations led to a matrix eigenvalue problem from which approximate eigenvalues and eigenvectors were obtained. The differential equations were uncoupled using a transformation matrix of the eigenvectors and then were solved for the generalized time function. Finally, the inversion series for the transform was used to obtain the solution for the dynamic response. To illustrate the method, the first four frequencies and mode shapes were determined for the longitudinal vibration of a tapered rod.

The eigentransform was used to develop a general procedure for treating continuous media with uniform stiffness but non-uniform mass distribution. These results, similar to those for the general non-uniform problem, reduced this problem to a matrix eigenvalue problem. The mode shapes were determined by summation using the eigenvectors and mode shapes for the uniform continuous media. Several problems for beams and plates with concentrated masses were solved as examples. This approach demonstrated definite computational advantages for plates over past treatments where frequency equations were determined as infinite series.

Vibrations of continuous media with time-dependent boundary conditions were then treated using the eigentransform. One dimensional media were considered first and next isotropic and orthotropic plates. One-dimensional continuous media were treated in a general way by specifying a differential operator of even order in the spatial derivatives. Applications to rods and beams were presented. The vibration of isotropic plates for an arbitrary shape was treated by expressing the equations in normal and tangential coordinates. The eigentransform of the governing equation was performed using an identity and theorem of vector analysis. The time-dependent boundary conditions were allowed to have an arbitrary variation along the boundary. Detailed applications were then made to rectangular and circular isotropic plates. The eigentransform of the orthotropic plate equation was performed by integration by parts. Again the boundary conditions were permitted to vary arbitrarily around the boundary. The response of a simply supported plate with an arbitrary edge displacement was determined as an illustration.

This investigation demonstrated that the eigentransform is a logical generalization of other finite integral transforms. The concept of a generalized finite integral transform extends the advantages of integral transforms to a much broader class of boundary value problems.