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 quasistatic stability of a rotating drillstring under longitudinal force and torque is analyzed. Constitutive equations are derived, and a technique to solve them is proposed. It is shown that the buckling mode of the drillstring is helical within a section subjected to compressive forces

A mathematical model is proposed to describe the critical quasistatic equilibrium of long rotating drillstrings. The prestress of drillstrings by the gravity and torsion forces, the gyroscopic interaction of rotary and linear motions, and the destabilizing effect of the internal flow of the drilling fluid are taken into account. The phenomena accompanying the drilling to different depths are studied numerically Introduction. One of the most important engineering tasks in modern mining practice is to develop the technology of drilling deep oil and gas wells. The predominant method in this technology is rotary drilling. It makes it possible to drill wells more than 6 km deep; the next target is to reach depths of 7 km and more.The geometry of wells is of two types [6]: (i) rectilinear vertical (conventional) and (ii) curvilinear (penetrating oil-or gas-bearing strata along their stratified structure).The efficiency of rotary deep-hole drilling can be enhanced by revealing critical modes of drillstrings and developing measures to reduce their adverse effect on the process. Such modes may be accompanied by bifurcational buckling and intensive vibrations of drillstrings when their natural frequencies equal the angular speed of rotation. It is important to not only establish the critical speeds of rotation of drillstrings, but also identify the buckling modes, which would allow finding regions of contact interaction between the string and the well wall and to calculate the reactions of such interactions.So far, however, no methods for physical and mathematical simulation of such effects have been developed. This state of affairs is because (i) it is still impossible to visualize the mechanical state of a drillstring during speedup, steady run, stopping, lowering, and pulling; (ii) there are no reliable methods to record dynamic processes in objects of such configuration and dimensions under full-scale conditions; (iii) drillstrings are so flexible that it is difficult to adequately study their mechanics using large-scale physical models (for example, a drill pipe 7 km in length and 0.3 m in diameter can be modeled by a hollow string 7 m in length and 0.3 mm in diameter that spins, is prestressed by gravity and torque, and contains a fluid); and (iv) there are many factors (length, flexibility, longitudinal force, torque, rotary inertia, internal fluid flow) that make the comprehensive mathematical simulation of the static and dynamic processes difficult.The critical parameters of the drilling process can be identified by mathematical modeling, though this would involve severe computational difficulties. They are due to the combination of the geometry of the drillstring and the forces acting on it during drilling. The former factor is more important because the drillstring is geometrically equivalent to a string with relatively low bending and torsional stiffness, which, however, should be calculated by beam theory to correctly describe edge and local effects of deformation. Therefore, applying this theory to strings se...