Application of Translational Edge Restraint for Vibration Analysis of Free Edge Kirchhoff’s Plate Including Rigid-Body Modes

A comprehensive theoretical study of closed-form rigid-body modes of a free-free and translationally edge-restrained Euler-Bernoulli beam is presented. Accurate vibrational analysis of a free-free-free-free plate is not possible without the inclusion of degenerate rigid-body beamwise admissible functions. The trivial solution(s) of the beam frequency equation produce(s) a non-trivial modeshape, which (i) satisfies the boundary conditions, (ii) has zero curvature, and (iii) is orthogonal to the other modeshapes. These frequency parameters are “trivial”, i.e. they lead to zero natural frequency, since their modeshapes have no curvature. Mathematically generated orthogonal free-free (classical) beam-wise rigid-body modeshapes, and those generated from non-classical edged beams, have been both separately used as admissible functions in the Rayleigh-Ritz method (RRM) to generate the plate natural frequencies of a free-freefree-free rectangular uniform isotropic Kirchhoff’s plate. With respect to the increasing elastic support, the trifurcation and bifurcation of plate frequencies from the trivial to the flexural frequencies, is investigated. The completely free plate modeshapes are also presented. Also, combination of present closed-form rigid-body modes with polynomial functions, trigonometric functions is also demonstrated.