Oscillations of a rectangular plate on a single-sided elastic base under pulse loading

Abstract

Oscillations of a rectangular plate with hinged support of the edges, which is supported by a unilateral elastic base and subjected to a forceful pulse loading, are considered. Rectangular pulse shape in time is selected. When exposed to an external load, the linear-elastic base resists compression, but does not perceive decompression, with the result that the unloaded plate detaches from the base. The study showed that the amplitude of the plate deflection after separation from the base may be greater than the amplitude of movement in the direction of the pulse. This ratio also applies to bending stresses in the plate, which is called the dynamic effect of the asymmetry of the elastic characteristic of the system. Conditions are established when the specified effect occurs. The conditions are related to the elastic and mass characteristics of the oscillatory system and the duration of the pulse, but they do not include the magnitude of the applied pressure to the plate, which is a consequence of the piecewise linear force characteristics of the system. The movement of the plate is divided into two stages. On the first of these, the plate is in contact with the base, and at the second stage this contact is absent and the unsupported plate carries out a free oscillatory motion. Analytical solutions of the equation of dynamic plate deformation at both stages are constructed. This was made possible by choosing a special pressure distribution over the plate surface. The distribution is chosen so that at the first stage of the movement the deformed middle surface of the plate is like the form of free oscillations at the first frequency. Analytical solutions were fitted, as a result of which compact formulas were obtained for calculating changes in deflections and bending stresses over time. The formula for the ratio of the amplitudes of the plate deflections thru both sides of the static equilibrium position is also derived. The conditions has had clarified when this ratio is greater than unity, which corresponds to the manifestation of the dynamic effect of an asymmetric elastic characteristic of the system. Examples of calculations using analytical solutions and numerical integration of the differential equation of motion on a computer are given. The complete correspondence of the numerical results to which the indicated methods lead is shown.