Geometrically Non-linear Free Vibrations of Simply Supported Rectangular Plates Connected to Two Distributions of Rotational Springs at Two Opposite Edges

Abstract

Although the dynamic behavior of rectangular plates has been the subject of much research for many decades, it remains of a crucial importance in various engineering fields and some edge conditions have not yet been treated, especially those involving edges connected to distributed rotational springs and non-linear vibrations. Also, in the practice of Modal Testing, theoretical models are needed for quantitatively estimating the flexibility of the real plate supports. A complementary work is presented here corresponding to plates connected to a distribution of rotational springs at two opposite edges vibrating in the geometrically non-linear regime occurring at large vibration amplitudes. To build the plate trial functions, defined as products of beam functions in the x and y directions, the mode shapes of simply supported beams connected to rotational springs are first calculated. Then, after exposing the general formulation of the non-linear problem, based on Hamilton’s principle and spectral analysis, the plate case is examined. Using the single mode approach, the backbone curves are determined, giving the non-linear frequency-amplitude dependence for plates having different combinations of stiffness and aspect ratios. It is noticed, as may be expected, that the obtained hardening non-linearity effect becomes more accentuated with increasing the rotational spring stiffness.