Formation of mathematical models of stationary dynamics of structures which include thin elastic plates with distributed inertial parameters

Abstract

This paper is devoted to the problems of modeling stationary oscillatory processes in elastic systems that contain thin-walled plates. The described method allows us to include the mathematical formalization of thin-walled plates in oscillatory models of elastic structures with irregular boundaries of computational areas and heterogeneous boundary conditions. The formation of dynamic models of this kind is based on the use of discretization methods, in particular, finite element methods that allows approximating the initial models with distributed inertial and rigid parameters — discrete ones with concentration of parameters at certain points — nodes of a dynamic system. Obviously, such approximations are accompanied by errors, the estimation of which is extremely difficult to perform under the conditions of the mentioned heterogeneities and irregularities even in the interval variants. In the present work, the construction of elements of bending stationary vibrations of thin-walled plates under monoharmonic effects is proposed. Possessing all the properties of a finite element, the proposed element contains the values of the distributed masses as parameters. In contrast to the use of discretized parameters obtained by using the classical finite element, the proposed method eliminates discretization procedures and thus, eliminates the process of forming mathematical models from additional errors associated with discretization procedures. This element of mathematical modeling of the dynamics of bending is called a harmonic element (HaE). The proposed method is based on the development of dynamic compliance methods, previously used to simulate oscillations of beam systems with distributed parameters. The developed mathematical models of stationary oscillations of thin elastic plates with distributed inertial parameters make it possible to include them in discrete-continuous models containing infinite-dimensional flexural elements (plates and beams), material points, solids and concentrated elasticities. Thus, dynamic models of structures subjected to harmonic effects are represented by a set of harmonic elements (HaE), which allow harmonic matching of heterogeneous elements under different boundary conditions.