Dynamic finite element formulations for moderately thick plate vibrations based on the modified Mindlin theory

Abstract

In this paper, the modified Mindlin theory is used for the construction of the dynamic stiffness matrix, the flexibility matrix, and the transfer matrix of a thick plate simply supported at two opposite edges. The modified Mindlin theory operates with bending deflection as the basic variable for the determination of the total (bending+shear) deflection and the angles of rotation. It is shown that the appropriate application of the constructed matrices to various boundary conditions leads to a determined formulation of the eigenvalue problem. As a result, the problem can be treated using ordinary algorithms for linear eigenvalue problems. Therefore the application of the relatively complex Wittrick-Williams algorithm, developed for the transcendental eigenvalue problems with the unusual forms of non-zero determinant is avoided. Using this technique, dynamic finite elements can be obtained in a simpler form than that based on the application of the conventional Mindlin theory. All phenomena related to the dynamic finite element application are investigated in the case of an axial bar vibration in a transparent analytical way. Furthermore, the application of the developed thick plate elements is illustrated through several numerical examples. Examples include using single elements as well as an assembly of elements. In the examples of single elements transcendental eigenfunctions are derived. Also a dynamic beam finite element is considered as a special case of the plate finite strip which exhibits no variation in the transverse direction.