Description of Plates with the Matrix Klein–Gordon Equation

The matrix Klein–Gordon equation (MKGE) describes waveguides with a discrete structure of their cross section. It may also be considered as a continuous waveguide model constructed by applying the waveguide finite element method. In this paper, the possibility of describing the bending vibrations of a thin plate with MKGE is studied. The problem is not trivial, as MKGE contains second-order time and coordinate derivatives, and bending vibrations are described by an equation with a fourth-order coordinate derivative. In this paper, matrix Klein–Gordon equations of different dimensions are derived for a plate. It is shown that a linear approximation of plate deformation over a cross section leads to overestimated plate stiffness values, but more complicated models give correct values.