ON DEGENARATE CASE OF THREE-DIMENTIONAL PROBLEM OF THICK ELASTIC PLATESʼ DYNAMICS

The three-dimensional problem of elastic membrane dynamics of arbitrary boundary outline with a given superficially distributed external force effect on it, is solved. The three-dimensional field of elastic dynamic displacements of membrane points is described by a differential model, which is a degenerate case of the previously constructed equations of the spatial theory of elasticity concerning thick elastic plates. It is assumed that the latter are supplemented with the information on the initial-boundary condition of the membrane, which in the general case is not of a force nature, and its amount is not sufficient for a mathematically correct formulation of the initial-boundary value problem. The solution of these incompletely defined initial-boundary problems of membrane dynamics became possible after root-mean square mathematical modeling of existing spatial-dynamic observations for the initial condition and current dynamics of its outline by a system of discretely and continuously defined modeling functions. The field of elastic-dynamic displacements of membraneʼ points constructed in the work, being an accurate solution of its differential mathematical model, with the available information on the initial-boundary condition of the latter, is consistent with the root-mean square criterion. Estimations of the accuracy of such modeling are made and conditions for its uniqueness are formulated in the paper. Cases of purely transverse and purely planar oscillations of the membrane are considered. The numerical realization of the obtained considered problemsʼ solution is conducted even with a minimum number of observations for the membraneʼs condition confirms the simplicity and correctness of the approach to these mathematically complex problemsʼ research of elastic-dynamics of thick elastic plates.