Mathematical Modeling of Three-Dimensional Fields of Transverse Dynamic Displacements of Thick Elastic Plates
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MATHEMATICAL MODELING OF THREE-DIMENSIONAL FIELDS OF TRANSVERSE DYNAMIC DISPLACEMENTS OF THICK ELASTIC PLATES V. A. Stoyana† and K. V. Dvirnychuka‡
UDC 517.95:519.86:539.3
Abstract. We investigate the dynamics of an elastic plate of finite dimensions. The three-dimensional field of dynamic transverse displacements of the plate is constructed as a solution of two-dimensional differential equations parametrically dependent on the transverse coordinate. We consider the cases of discrete and continuous sets of the initial and boundary conditions that are satisfied by the mean square criterion. We describe the features of the solution of the problems in unbounded space–time domains. Keywords: thick elastic plates, dynamics of distributed parameter systems, mathematical simulation. INTRODUCTION The theoretical fundamentals of the analysis of the dynamics of elastic structures such as plates and shells, which were developed in the second half of the last century, were constructed [1–3] on the assumption that the thickness of the plates and shells is small as compared with their main geometrical dimensions. The resolving differential equations of such structures, as a rule, two-dimensional, are considered only for special [4–6] classically defined initial-boundary conditions. The analysis of the dynamics of plates and shells of finite thickness always involves [7–9] certain numerical and analytical problems, which arise both in creating the mathematical model of the object and forming the initial-boundary observations, and in solving the problem related to both mathematical and computing complexity. There are many [9, 10] approaches to setting up specified dynamic equations of plates of finite thickness; however, they had certain mechanical models. The hypothesis-free problem solution proposed by Lur’e [11] was limited only to the static case. A generalization of the results obtained in [11] has allowed developing [12] a semi-three-dimensional mathematical model of the dynamic processes that take place in the axisymmetrically loaded elastic layer. The two-dimensional differential equations, which parametrically depend on the degenerate coordinate of the layer, completely describe the three-dimensional field of its elastic dynamic strains. Under some constraints, the results of [12] were propagated [13, 14] to the dynamics of the elastic layer referred to the Cartesian coordinate system. Applying the models to the analysis of the dynamics of thick elastic plates of finite dimensions remained unresolved problems. The present paper solves the initial–boundary-value problems of the dynamics of elastic plates restricted in the plan. Applying the technique [15, 16] of the mathematical modeling of the initial-boundary external-dynamic impacts of the spatially distributed dynamic system on the mechanical objects described by models [13, 14] has allowed solving these problems without constraints on the form of the object and the amount and quality of the information about its initial-boundary state. In the paper, we will construc
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