Shakeri M. Analytical methods for solving problems of the theory of vibrations for elastic plates of non-canonical form

The dissertation work is devoted to development of the approach to construction of analytical solutions of boundary problems of mathematical physics for areas of noncanonical forms. The boundary problems for the Laplace, Helmholtz, and electroelastic equations for piezoceramic plates are specifically considered. The approach to the construction of analytical solutions is based on the unconventional for classical mathematical physics idea of a general solution of the boundary problem for a given region. The detail description of the content of this concept is presented. The boundary problems for the Laplace and Helmholtz equations are used to illustrate the possibilities of the method and to identify possible difficulties in its practical implementation. Potentially, a wide range of mechanical problems can be considered on the basis of the idea of the general solution of the boundary problem method. A specific implementation illustrating its possibilities is given for the areas whose boundaries are formed by straight line segments. The general solution is constructed in the form of a set of infinite series, each term of which satisfies the basic equation. Such series contains enough arbitrariness to satisfy the boundary conditions on one side of the boundary. Different variants of numerical procedures for determining the coefficients of these series are considered. The possibilities of the method are fully used in the analysis of planar oscillations of parallelogram piezoceramic plates and bending oscillations of parallelogram piezoceramic bimorphs. Spectral characteristics of oscillating plates and geometrical features of their own shapes are determined. Experimental studies have been carried out, which allowed to confirm the reliability of analytical methods for determining the dynamic characteristics of both types of plates in a wide enough range of frequencies.