The equations of nonaxisymmetric vibrations of sandwich cylindrical shells with discrete core under nonstationary loading are presented. The components of the elastic structure are analyzed using a refined Timoshenko theory of shells and rods. The numerical method used to solve the dynamic equations is based on the integro-interpolation method of constructing finite-difference schemes for equations with discontinuous coefficients. The dynamic problem for a sandwich cylindrical shell under distributed nonstationary loading is solved with regard for the discreteness of the core Keywords: three-layer cylindrical shell, ribbed core, nonstationary loading, Timoshenko theory, numerical algorithm, forced vibrationsThe stress-strain state and strength of multilayer shells under dynamic loads depend on a number of geometrical and physicomechanical parameters. Various assumptions are usually made considering specific relations between these parameters and the purpose and required accuracy of the analysis. Since the mechanical models of multilayer shells are very complex, and essentially different kinematic and static hypotheses are used, there are a great variety of design models and equations [1]. Two types of mathematical models of multilayer shells are distinguished: one is based on unified hypotheses applied to the whole stack of layers [3] and the other is based on hypotheses that account for the kinematic and static features of each layer [2]. Such approaches also apply to sandwich shells with ribbed core: an orthotropic model and a model accounting for the discreteness of the core. The latter model was used in [5] to study the axisymmetric vibrations of sandwich shells with ribbed core under nonstationary loads. The forced axisymmetric vibrations of reinforced conical shells were investigated in [7]. The nonaxisymmetric vibrations of multilayer cylindrical shells with discrete ribs under distributed nonstationary loads were examined in [6].In this paper, we will address sandwich shells with cross-ribbed core. It is obvious that second-type models of multilayer shells apply here. We will use the vibration equations from the quadratically nonlinear Timoshenko theory of shells and rods. The equations will be integrated using an explicit finite-difference scheme. We will analyze, as a numerical example, the forced nonaxisymmetric vibrations of an inhomogeneous shell structure under distributed nonstationary loading.

The nonstationary behavior of reinforced compound structures is analyzed within the framework of the geometrically nonlinear Timoshenko theory of shells and rods. The numerical method used to solve the problem is based on the finite-difference approximation of the original differential equations. The dynamic behavior of a reinforced compound structure under nonstationary loading is demonstrated by way of a numerical example. The numerical results are compared with experimental data Keywords: reinforced discrete shell, Timoshenko shell theory, nonstationary loading, numerical method Introduction. Reinforced compound shells are complex, spatially inhomogeneous, elastic structures that include discrete inclusions and areas where the geometrical and material parameters change. These factors complicate problem formulations. In particular, the partial differential equations describing the stress-strain state of the original structures include discontinuities of the first kind of the strain and stress components. This is why special numerical algorithms have been developed to adequately simulate wave processes in shells with singularities.The handbook [4] presents algorithms and programs for design of compound shells under stationary and dynamic loads. The dynamic behavior of compound shells under nonstationary loads is analyzed in the monograph [5]. The nonstationary behavior of reinforced shells of revolution with discrete ribs is studied in [2]. The axisymmetric nonlinear vibrations of discretely reinforced conical shells are examined in [9]. Dynamic problems for reinforced ellipsoidal shells are solved in [13]. Initial deflections are accounted for in [11] in solving dynamic problems for discretely reinforced cylindrical shells under nonstationary loads. The nonaxisymmetric forced vibrations of sandwich cylindrical shells with ribbed core are studied in [12].Here we formulate dynamic problems for compound shells and outline a numerical algorithm for solving them. The algorithm is further used to solve model problems.1. Problem Formulation. Governing Equations. Consider a compound inhomogeneous structure consisting of a compound shell and ring ribs rigidly fixed to it along contact lines (the ribs are supposed to be placed along the coordinate line α 2 ) [1]. Forced vibrations of the structure are modeled by a hyperbolic system of nonlinear differential equations of the Timoshenko theory of shells and curvilinear rods [2]. The variation of the displacements of the shell components across the thickness is described in the coordinate frame ( , ) s z by the formulas

A dynamic problem for a cylindrical shell on an elastic foundation is formulated. A numerical algorithm for solving this problem is outlined. The results obtained are analyzed Keywords: nonlinear theory of shells, dynamic problems, elastic foundation, numerical methods Introduction. The dynamic interaction of cylindrical shells with elastic media is a problem of current importance because cylindrical shells are widely used in modern engineering structures such as tunnels, tanks, pipelines, and drill and casing pipes. Mathematically, such problems are quite complicated, in both formulation and solution (use of the theory of elasticity and the theory of shells, formulation of medium-shell interface conditions, development of a numerical algorithm, etc.) [1,5,6]. There are more simple approaches to analyze the interaction between a structure and an ambient medium [2,4,9]. An example is the Winkler model [3,9]. The dynamic interaction of spherical and cylindrical shells with bilateral and unilateral Winkler foundations was analyzed in [4] in the axisymmetric case. Forced vibrations of an infinite cylindrical shell in an elastic medium under nonaxisymmetric loading were studied in [3] considering Winkler and Pasternak foundations. A number of dynamic problems for one-and three-layer beams and rods on an elastic foundation were addressed in [13,14]. Static stress-strain problems for plates and shells on an elastic foundation are solved in [15,16].The present paper addresses nonstationary problems of dynamic deformation of a cylindrical shell in an elastic medium. We will examine the cases where the medium is described by the equations of continuum and is modeled by a Winkler foundation. The results obtained will be compared.1. Cylindrical Shell on Elastic Foundation. The dynamic behavior of a cylindrical shell is described by the vibration equations of the Timoshenko theory of shells [6] including terms modeling the elastic foundation. The vibration equations are derived using a geometrically nonlinear version of the theory (Timoshenko theory of shells of the second order). These equations have the form