Vibrations of rectangular plates with arbitrary non-uniform elastic edge restraints

Arbitrary non-uniform elastic edge restraints represent the most general class of boundary conditions for plate problems, and are encountered in many real-world applications. The vibrations of plates with this kind of boundary conditions, however, are rarely studied in the literature perhaps because there is a lack of suitable analytical or numerical techniques. In this investigation, a general analytical method is derived for the vibration analysis of rectangular plates with elastic edge restraints of varying stiffness. Both rotational and translational restraints can be arbitrarily applied to an edge, and their stiffness distributions are generally described in terms of a set of invariants, cosine functions. The displacement solution is sought simply as a linear combination of several one- and two-dimensional Fourier cosine series expansions. All the unknown Fourier coefficients are treated equally as a set of independent generalized coordinates and solved directly from the Rayleigh–Ritz formulation. Unlike the existing techniques, the current method does not require any special procedures or schemes to deal with different boundary conditions. A few "classical" problems involving non-uniform rotational restraints are first solved and used to check the current solution against some of the existing techniques. The modal results are also presented for plates with more complicated boundary conditions in which an edge is no longer completely restrained in the translational direction. The accuracy and reliability of the current method are repeatedly demonstrated through all these examples.