Free Vibration Of Quadrilateral Double-plate System

As basic structural units with excellent performance, plates have been generally used in various engineering practices. A good understanding of the dynamics behaviors of this kind of structures is crucial to the design and performance evaluation of mechanical systems. Classical thin plate theory is usually used in the analysis of plate vibration problems. However, the calculated results based on this theory will not be in accordance with the actual case when the thickness of the plate becomes large, because the effects of shear deformation and rotational inertia are all neglected. In order to overcome the limitations of the thin plate theory, Mindlin et al proposed a modified plate theory named the first order shear deformation theory. The vibration plate theory provides a more accurate and reasonable solution, so this paper uses this theory.The transverse free vibrations of a system composed of two plates connected by a homogeneous Winkler elastic layer are investigated by using the powerful Pb-2 Ritz method combined with the first order shear deformation plate theory. First, the functional of the double-plate system is derived. Then the product of a set of complete two dimensional polynomials and a set of basic functions which consist of the boundary shape function is selected as the admissible function. Substituting these displacements functions into the energy functional and setting the first order partial derivatives of the energy functional to be zero, gives a typical eigenvalue equation. Finally, the eigenvalues and eigen vectors of this problem can be solved by a standard eigenvalue solver. Gaussian quadrature is used to calculate the stiffness matrix and the mass matrix of the plate system. For convenience of numerical integration, the arbitrary thick quadrilateral plate is mapped onto a basic square plate through the coordinate transformation.Several plates are selected as the testing examples to verify the applicability and the effectiveness of the present method for solving the free vibration problem of moderately double-plate system, by convergence studies and comparing the present results with other existing results. The accuracy and efficiency of the proposed method are demonstrated through several numerical examples by comparison and convergency studies. A lot of numerical results for reasonable natural frequency parameters of double-plate system with different combinations of boundary conditions are presented. The effects of the spring stiffness and shear deformation of the plate on the frequency parameters are carefully studied. These numerical results can be used as a reference to the designers and researchers in this area.