Geometric characteristics of the deformation state of the shells with orthogonal coordinate system of the middle surfaces

The aim of this work is to receive the geometrical equations of strains of shells at the common orthogonal not conjugated coordinate system. At the most articles, textbooks and monographs on the theory and analysis of the thin shell there are considered the shells the coordinate system of which is given at the lines of main curvatures. Derivation of the geometric equations of the deformed state of the thin shells in the lines of main curvatures is given, specifically, at monographs of the theory of the thin shells of V.V. Novozhilov, K.F. Chernih, A.P. Filin and other Russian and foreign scientists. The standard methods of mathematic analyses, vector analysis and differential geometry are used to receive them. The method of tensor analysis is used for receiving the common equations of deformation of non orthogonal coordinate system of the middle shell surface of thin shell. The equations of deformation of the shells in common orthogonal coordinate system (not in the lines of main curvatures) are received on the base of this equation. Derivation of the geometric equations of deformations of thin shells in orthogonal not conjugated coordinate system on the base of differential geometry and vector analysis (without using of tensor analysis) is given at the article. This access may be used at textbooks as far as at most technical institutes the base of tensor analysis is not given.

POSTULATION AND BUILDING OF A NUMERICAL ALGORITHM FOR SOLVING THE PROBLEMS OF THE DYNAMICS OF THE THEORY OF CONICAL SHELLS IN NONORTHOGONAL COORDINATE SYSTEM

The paper presents the formulation and numerical algorithm for solving problems of the dynamics of the theory of conical shells in a non-orthogonal coordinate system. The object of the study are conical shells, the equations of which are represented in non-orthogonal coordinate system. Purpose of the work is to formulate and construct a numerical algorithm for solving the problems of the dynamics of conical shells in a non-orthogonal coordinate system. The methods of research include the basic principles of the theory of shells to Tymoshenko's type and numerical methods. The formulation of problems and a numerical algorithm for studying the dynamic behavior of conical shells in a non-orthogonal coordinate system are considered. The results obtained in the work can be used in the design of elements of shell structures in the rocket, aircraft and shipbuilding industries. KEYWORDS: CONIC SHELL, DYNAMIC PROCESSES, NON-ORTHOGONAL COORDINATE SYSTEM, NUMERICAL METHODS

Determination of Fiber-Structure for the Effective Reinforcement of Planar Composite-Material Constructions

To describe the behaviour of a composite material reinforced by fibers along curvilinear trajectories, a resolving system of differential equations has been obtained. I developed a structural model within the framework of the two-dimensional non-uniform linear elasticity problem. The reinforcement is performed on the basis of three approaches: according to the grid lines of the orthogonal coordinate system determined by the given conformal representation; according to trajectories isogonal to the given curves; according to the helical trajectories in the axisymmetric formulation of the problem. Particular solutions of the corresponding boundary problems have been obtained. Recommendations for the effective reinforcement of planar constructions have been presented.

On the breakup of spiralling liquid jets

The generation of drops from a jet spiralling out of a spinning device, under the action of centrifugal force, is considered for the case of small perturbations introduced at the inlet. Close to the inlet, where the disturbances can be regarded as small, their propagation is found to be qualitatively similar to that of a wave propagating down a straight jet stretched by an external body force (e.g. gravity). The dispersion equation has the same parametric dependence on the base flow, but the base flow is, of course, different. Further down the jet, where the amplitude of the disturbances becomes finite and eventually resulting in drop formation, the flow appears to be quite complex. As shown, for the regular/periodic process of drop generation, the wavelength corresponding to the frequency at the inlet, increasing as the wave propagates down the stretching jet, determines, in general, not the volume of the resulting drop but the sum of volumes of the main drop and the satellite droplet that follows the main one. The proportion of the total volume forming the main drop depends on how far down the jet the drops are produced, i.e. on the magnitude of the inlet disturbance. The volume of the main drop is found to be a linear function of the radius of the unperturbed jet evaluated at the point where the drop breaks away from the jet. This radius, and the corresponding velocity of the base flow, have to be found simultaneously with the jet’s trajectory by using a jet-specific non-orthogonal coordinate system described in detail in Shikhmurzaev & Sisoev (J. Fluid Mech., vol. 819, 2017, pp. 352–400). Some characteristic features of the nonlinear dynamics of the drop formation are discussed.

EXTENDED CLARKE TRANSFORMATION FOR n-PHASE SYSTEMS

The article describes how to convert space vectors written in a stationary multiphase system, consisting of a number of phases where n > 3, to the stationary alfa beta orthogonal coordinate system. The transformation of vectors from a stationary n-phase system to the stationary alfa beta orthogonal coordinate system is defined The inverse transformation of a vector written in the orthogonal coordinate system to a stationary n-phase system is also defined. The application of the extended Clarke transformation allows control calculations to be performed in both stationary alfa beta or rotating dq orthogonal coordinate systems. This gives the possibility of performing different control strategies. It has a practical application for drive systems with five-phase, six-phase or dual three-phase motors.

Ferromagnetic Flow of Viscous Fluid in a Slot between Fixed Surfaces of Revolution

In this paper the steady laminar flow of viscous incompressible ferromagnetic fluid is considered in a slot between fixed surfaces of revolution having a common axis of symmetry. The boundary layer ferromagnetic equations for axial symmetry are expressed in terms of the intrinsic curvilinear orthogonal coordinate system x, θ ,y.The method of perturbation is used to solve the boundary layer equations. As a result, the formulae defining such parameters of the flow as the velocity components vx, vy, and the pressure , were obtained.

Numerical modelling of sea currents and tidalwaves

We consider a mathematical model of sea currents and tidal waves based on the marine dynamics primitive equations. The equations are written in the orthogonal coordinate system on sphere with arbitrary position of the poles. It makes it possible to increase horizontal resolution due to placement of a pole into vicinity of the considered sub-area. Two problems are solved: (1) joint computation of wind-generated, baroclinic and tidal currents in the Black and Azov Seas; (2) simulating mesoscale variability of coastal currents in the Black Sea. The second problem is solved with increased horizontal resolution in the coastal zone of Gelendzhik.