Abstract
The dynamic stiffness matrix of a rectangular plate for the most general case is developed by solving the bi-harmonic equation and finally casting the solution in terms of the force–displacement relationship of the freely vibrating plate. Essentially the frequency dependent dynamic stiffness matrix of the plate when all its sides are free is derived, making it possible to achieve exact solution for free vibration of plates or plate assemblies with any boundary conditions. Previous research on the dynamic stiffness formulation of a plate was restricted to the special case when the two opposite sides of the plate are simply supported. This restriction is quite severe and made the general purpose application of the dynamic stiffness method impossible. The theory developed in this paper overcomes this long-lasting restriction. The research carried out here is basically fundamental in that the bi-harmonic equation which governs the free vibratory motion of a plate in harmonic oscillation is solved in an exact sense, leading to the development of the dynamic stiffness method. It is significant that the ingeniously sought solution presented in this paper is completely general, covering all possible cases of elastic deformations of the plate. The Wittrick–Williams algorithm is applied to the ensuing dynamic stiffness matrix to provide solutions for some representative problems. A carefully selected sample of mode shapes is also presented.
Highlights
The free vibration analysis of plates and plate assemblies is a topic which has continually inspired researchers for well over two centuries
An exact dynamic stiffness (DS) matrix has been developed for a rectangular plate for the most general case
This has been achieved by obtaining an infinite series general solution which satisfies the governing differential equations exactly
Summary
The free vibration analysis of plates and plate assemblies is a topic which has continually inspired researchers for well over two centuries. A general procedure to develop the dynamic stiffness matrix of a structural element can be briefly summarized as follows (i) Seek a closed form general solution of the governing differential equations describing the free vibratory motion of the structural element in an exact sense in terms of the unknown coefficients appearing in the general solution. (iii) Eliminate the unknown coefficients by relating the amplitudes of the harmonically varying forces to those of the corresponding displacements and thereby generating the frequency dependent dynamic stiffness matrix As it is well known, the closed form solution for free vibration analysis of a rectangular thin plate has been widely reported only for the special case when the opposite sides of the plate are supported. Once the dependency between sub-vectors d~kj and ~f kj is determined, which defines the dynamic stiffness matrix K kj for each case arising from the symmetry, it becomes possible with the help of Eq (18) to derive the overall dynamic stiffness matrix K for the complete plate, which relates the vectors d~ and ~f
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