### №2-2022

https://elibrary.ru/contents.asp?id=49026362

**Goloveshkin V.A., Myagkov N.N.**

## Model of interaction of a rigid mesh with a deformable target

An analytical model of the high-velocity interaction of a rigid mesh with a semi-infinite deformable target, which is modeled by a rigid-plastic body, is proposed. We consider the “normal” impact of the mesh on the target: we assume that at the initial moment and subsequent moments of time the mesh is parallel to the target surface, and the mesh velocity vector is perpendicular to the target surface. After the mesh meets the target, the flow pattern of the target has a cellular structure that reflects the geometry of the mesh. Due to the periodic structure of the mesh and the symmetry of the mesh cell, we consider the flow that accompanies the penetration of only 1/8 of the mesh cell. The dependence of the solutions obtained on the geometric parameters of the mesh is characterized by one dimensionless parameter γ equal to the ratio of the wire diameter to the period of the mesh, 01γ≤≤. Analytical formulas are obtained for: the depth of penetration of the mesh into the target; the total mass ejected during the penetration; the total momentum of the ejected mass; the total energy of the ejected mass. Quantitative estimates of the values of these quantities are given for the penetration of a steel mesh into an aluminum target, depending on γ. The effect of amplification of the target momentum is also estimated, which turned out to be the largest for a mesh with a small value of the parameter γ, when the wire diameter is much less than the mesh period.

Pages: 151-174

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**Grishanina T.V., Shklyarchuk F.N.**

## Numerical-analytical method for calculating the oscillations of regular structures

Dynamics of elastic large-sized space structures is of grate interest for design of orbital stations, large radio antennas, radio telescopes, satellites with large solar panels. Special place among space structures is occupied by truss systems consisting of many thousands of elements. They can be used for future large antenna reflectors, platforms, strength frameworks. As a rule, such systems for convenience of assembling in space have a regular structure, i.e. they consist of the same type of sections (modules) connected in series with each other. When calculating the dynamic characteristics of such structures, the finite element method or other similar numerical methods can be used. But when they are used for systems with a large number of sections, difficulties arise due to the large dimension of the tasks being solved. Then the calculation can be very time-consuming. Therefore, it is of interest to develop effective models and methods based on the use of regularity properties of such structures. This paper presents a numerical-analytical method for calculating natural oscillations or harmonic forced oscillations of regular systems, the complexity of which does not depend on the number of modules of the same type and is determined by the number of degrees of freedom of one section. To assess the complexity and accuracy of the proposed calculation method, the problem of the flexural vibrations of a freely supported homogeneons beam is solved and the solution obtained on its basis is compared with the results of the exact solution and the direct solution based on the finite element method. From the calculations given in the article, it can be seen that the described method allows you to get results that are quite close to exact. Moreover, convergence improves with an increase in the number of elements of the same type included in the regular system. Thus, this method can be effective in dynamic calculations of regular structures consisting of a large number of sequentially connected modules of the same type.

Pages: 175-186

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**Belkin A.E.**

## On the use of a genetic algorithm for determining the parameters of anisotropic body providing minimum sound reflection

The problem of determining the values of the density and elastic constants of an anisotropic body, which provide the minimum, in a sense, sound reflection from a given body, is considered. Statements of both the inverse and the direct problem of the diffraction of acoustic waves from a body – an anisotropic cylinder with a rigid core – in an unlimited space filled with an ideal fluid are presented. An algorithm for solving the inverse problem is described, which is a variation of the genetic algorithm. When using this method, the possible values of the desired parameters of the body are sorted out. The difference between the genetic algorithm and the ordinary enumeration method lies in the use of special operations – “crossings” and “mutations” of parameter sets. For each considered set, called a configuration, a direct problem is solved, in connection with which it is considered in detail in the work. For given parameters of the body and the incident wave, the search for a scattered sound field is based on the model of the propagation of small perturbations in an ideal fluid and the linear theory of elasticity. The general equations of motion of a continuous medium are reduced first to a system of equations in partial derivatives, then to a system of ordinary differential equations. The equations are supplemented with boundary conditions on the surface of the body and on the boundary of the anisotropic part with the rigid core. This makes it possible to determine the expansion coefficients of the scattered wave. The degree of sound reflection is defined as a functional on the space of parameters of the body, expressed in terms of the integral of the potential of the velocities of the scattered wave. Several variants of the functional are proposed, which can be used in various variations of the inverse problem. A genetic algorithm is used to minimize this functional. The paper describes in detail the special parameters of the algorithm and their optimal values, the form of data representation in the genetic algorithm and all the main steps.

Pages: 187-202

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**Danilin A.N., Feldstein V.A., Kurdyumov N.N., Shavnya R.A.**

## On modeling the sub-vibrations of two conductors connected by spacers

A system of two conductors connected by spacers in the form of rigid rods is considered. The conductors are affected by the wind flow so that one conductor is in the aerodynamic (satellite) wake of the other, which leads to the emergence of a self-oscillating process. The wake interaction between conductors is modeled with a modified Simpson theory using Blevins and Prices empirical data. Differential equations are derived based on the principle of possible displacements in generalized coordinates, taking into account the nonlinearities of elastic and inertial forces, as well as aerodynamic forces in the wake. For discretization by spatial coordinates, the finite element method is used with the choice of linear and trigonometric shape functions as the basis. The tension force and the longitudinal deformation of the conductor are considered constant values within the element. The dependence of deformation on transverse displacements is determined by a quadratic approximation. To obtain final expressions for aerodynamic forces, polynomial approximations of known experimental data are used, as well as linearization of expressions for these forces written in local (elemental) coordinates.

Pages: 203-222

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**Birykov D.R.**

## On the problem of a spherical sound wave diffraction by an elastic inhomogeneous anisotropic ball with an absolutely solid inclusion

A mathematical formulation of the problem of diffraction of a spherical sound wave by a linearly elastic radially inhomogeneous transversely isotropic ball with an absolutely solid inclusion is presented. The ball is characterized by density, elastic constants – components of the elasticity tensor – and external and internal radii. The sphere described above is placed in a three-dimensional unlimited space filled with an ideal fluid with certain values of density and speed of sound. The statement describes the input data and some of their limitations. An algorithm for solving the posed diffraction problem is presented. The algorithm is partly analytical, partly numerical. An incident spherical wave, a sound wave scattered by a ball, and elastic waves propagating inside an elastic ball are represented as infinite sums. The definition of the wave scattered by the ball is reduced to the determination of the coefficients of the expansion of the scattered wave field into an infinite sum. To determine these coefficients, a boundary value problem is solved. The differential equations in this boundary value problem are ordinary differential equations describing waves in an elastic ball and obtained from the general equations of motion of a continuous medium. These differential equations are supplemented by boundary conditions on the surfaces of an elastic ball. On the outer surface, the boundary conditions are the continuity of the velocity, normal and shear stresses. On the inner surface – continuity of displacements. The solution of the boundary value problem with the given conditions makes it possible to calculate the displacements inside the ball during the propagation of the wave u. through them, the coefficients of the sound wave scattered by the body. To demonstrate the solution of the problem with the help of software implementation, the results of numerical studies for some particular input data are given.

Pages: 223-234

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**Skleznev Andrey A., Vasiliev Valery V.**

## Liner stability of a cylindrical composite pressure vessel

The article considers the problem of stability of the inner sealing shell (liner) of a composite pressure cylinder, considered as an infinitely long isotropic cylindrical shell, located in an absolutely rigid environment, simulating the composite layer of the balloon and compressing the shell so that it can lose stability. Using the equations of the nonlinear theory of cylindrical shells, an exact solution is obtained that determines the critical pressure or the limiting value of the subcritical deformation of the shell. It has been established that the critical pressure and deformation depend on the connection between the inner isotropic and outer composite shells. Two limiting cases are studied – shells rigidly connected to each other, and shells unilaterally connected only in the radial direction in the absence of friction. Some possible intermediate variants are considered – shells connected on a part of the section contour. The result obtained earlier is refined, based on the assumption that the form of buckling of the inner shell is described by nonlinear equations of the theory of shallow shells. The obtained solution is compared with the published experimental results.

Pages: 235-246

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**Dolbin I.V., Karnet Yu.N., Kozlov G.V., Vlasov A.N.**

## Comparative analysis of the efficiency of reinforcing polymer nanocomposites with 2d nanofillers using the example of polyimide/na+-montmorrillonite and polyvinyl alcohol/graphene oxide

To realize the extremely high characteristics of polymer/2D nanofiller nanocomposites, it is necessary to create an optimal structure of the nanofiller in the polymer matrix. The aggregation degree and the aspect ratio of the tactoids of the 2D nanofiller are determined by the ratio of the nominal elasticity moduli of the nanofiller and the matrix polymer. The effective elasticity modulus of the nanofiller in the polymer matrix of the nanocomposite is determined by its rigidity.

Pages: 247-254

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**Dmitriev L.D., Kondratov A.P., Yamilinets S.Yu.**

## Deformation and changes in the strength properties of multilayer fabrics as a result of swelling in mixtures of organic solvents of different lipophylicity

A technique is proposed, and a study of deformation during swelling in a liquid of multilayer organofilament in a free and linearly isomeric state is carried out using direct video recording of changes in the thickness of layers in cross-section and gravimetry of samples in contact with the liquid. It is proposed to use the values of the regression coefficients of the kinetic curve of the swelling deformation of various layers of the composite and the limiting value of the geometric dimensions of the composite material as lyophilicity parameters. A synergistic effect of swelling deformation of multiporous cushioning organofilament in a mixture of multipolar solvents in a free and linearly isomeric state has been established. It is shown that the adhesive layer of rubber and the cellular layer of rubber, which is subjected to increased disproportionate compression up to monolithization in a linearly isomeric state, has the least physical and chemical resistance of organofilament used in printing. Deformation of multilayer organofilament in liquid leads to loss of mechanical and operational properties as a result of plucking of the upper monolithic layer of elastomer during printing.

Pages: 255-273

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**Russkikh S.V.**

## Nonlinear dynamics of a plane elastic rod system in a reduced quasi-static bending formulation

The nonlinear dynamics of a flat elastic rod system is considered, which consists of an arbitrary number of elastic inextensible rods connected at the ends by elastic-viscous hinges allowing large relative angles of rotation. The displacements of each rod are described by its final rotation as a rigid body relative to a straight line connecting two adjacent hinge nodes, and a bend with a small transverse displacement. Active control of the system is carried out with the help of horizontal and vertical forces applied in the hinge nodes. The equations of motion of a composite system with an arbitrary number of core elements in a fixed coordinate system are based on the principle of possible displacements and are presented in the form of finite formulas convenient for numerical integration using standard programs and algorithms implemented in computer algebra languages. The reduction of the initial system of equations is carried out by quasi-static bending by neglecting the inertia of the bending forms of the movement of the rods and excluding generalized coordinates representing these forms, which are the angles between the tangent to the curved axis of the rod and its undeformed axis. Thus, “fast variables” are excluded from the equations of motion of the system. An algorithm for converting the initial equations into reduced system equations for an arbitrary number of core elements of the system is presented. An example of a numerical solution of the problem of the reaction of a rod system to an arbitrary perturbing pulse in full and reduced formulations is considered. Comparisons are made and estimates of the accuracy and complexity of numerical integration are given when considering a complete system of nonlinear differential equations and equations of a reduced system.