This paper presents a study of the interaction between high-speed airflow and the surface of a solid low-melting material in a flowing channel of a model body. Both numerical and experimental approaches are used to solve the problem, which allows one to perform a comprehensive analysis of the processes under study. Numerical simulation conditions correspond to aerodynamic tests in the experimental facility. The unsteady Reynolds-averaged Navier-Stokes (URANS) equations are used to describe a gas flow. When solving the problem, the coupled heat-transfer and turbulence are taken into account. The low-temperature gas-dynamic processes are considered, while the chemical reactions and phase transition are neglected. As a result of numerical simulations, the flow structure and regime in a flowing channel of the model are determined, as well as the pressure and temperature distributions in the near-wall region of a solid combustible material. The gas flow regime corresponds to an underexpanded jet flow with the separation of the boundary layer and the formation of the intense heat-transfer regions at the initial section of the flowing channel. According to the numerical simulation results, in aerodynamic tests with a Mach number of 6, the melting point is attained in the near-wall region of the solid combustible material (polyethylene, polyoxymethylene, and wax). Aerodynamic tests are carried out to validate the obtained results. Experimental results show that the variation in the flowing channel diameter in the thick-wall cylinder made of polyethylene and polyoxymethylene is induced by thermal expansion. In aerodynamic tests with a wax cylinder, the mass reduction and the fusion of the solid-gas interface are revealed.
{"title":"Effect of a gas flow structure in an axisymmetric channel on the inhomogeneous temperature field formation in a low-melting cylinder","authors":"N. P. Skibina, V. V. Faraponov","doi":"10.17223/19988621/81/13","DOIUrl":"https://doi.org/10.17223/19988621/81/13","url":null,"abstract":"This paper presents a study of the interaction between high-speed airflow and the surface of a solid low-melting material in a flowing channel of a model body. Both numerical and experimental approaches are used to solve the problem, which allows one to perform a comprehensive analysis of the processes under study. Numerical simulation conditions correspond to aerodynamic tests in the experimental facility. The unsteady Reynolds-averaged Navier-Stokes (URANS) equations are used to describe a gas flow. When solving the problem, the coupled heat-transfer and turbulence are taken into account. The low-temperature gas-dynamic processes are considered, while the chemical reactions and phase transition are neglected. As a result of numerical simulations, the flow structure and regime in a flowing channel of the model are determined, as well as the pressure and temperature distributions in the near-wall region of a solid combustible material. The gas flow regime corresponds to an underexpanded jet flow with the separation of the boundary layer and the formation of the intense heat-transfer regions at the initial section of the flowing channel. According to the numerical simulation results, in aerodynamic tests with a Mach number of 6, the melting point is attained in the near-wall region of the solid combustible material (polyethylene, polyoxymethylene, and wax). Aerodynamic tests are carried out to validate the obtained results. Experimental results show that the variation in the flowing channel diameter in the thick-wall cylinder made of polyethylene and polyoxymethylene is induced by thermal expansion. In aerodynamic tests with a wax cylinder, the mass reduction and the fusion of the solid-gas interface are revealed.","PeriodicalId":43729,"journal":{"name":"Vestnik Tomskogo Gosudarstvennogo Universiteta-Matematika i Mekhanika-Tomsk State University Journal of Mathematics and Mechanics","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78410883","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A. Ishchenko, V. Burkin, A. S. D’yachkovskiy, Andrey V. Chupashev
This paper considers the investigation of the high-speed motion of specifically-shaped inert bodies (projectiles) in water. The main purpose is to determine the peculiar properties of underwater launching of supercavitating projectiles out of the bore of a test ballistic setup. Preliminary calculations of the inner ballistic parameters of a shot are per-formed for the given launcher and then compared with the corresponding experimental data. Examples of experimentally fired underwater shots are presented. The high-speed high-energy processes associated with the projectile released out of the bore and entering water are described. It is shown that during the underwater launching, the complex gas-dynamic conditions of multi-phase processes following the projectile release out of the bore can significantly affect the trajectory of its further motion in water. Using the experi-mental and calculation methods with the classical loading scheme and an acceleration bore 0.5 mm long or less under given conditions, it has been determined that the pressure at the accelerator exit increases stepwise and can significantly change the flow conditions of the projectile at the start and its further trajectory. Taking into consideration the revealed peculiarities of underwater shots and a set of technical specifications and mass-size parameters, a perspective launching ballistic setup has been designed and used to estimate the motion range of the released supercavitating projectile.
{"title":"Underwater launching of a supercavitating projectile out of a ballistic test setup","authors":"A. Ishchenko, V. Burkin, A. S. D’yachkovskiy, Andrey V. Chupashev","doi":"10.17223/19988621/82/8","DOIUrl":"https://doi.org/10.17223/19988621/82/8","url":null,"abstract":"This paper considers the investigation of the high-speed motion of specifically-shaped inert bodies (projectiles) in water. The main purpose is to determine the peculiar properties of underwater launching of supercavitating projectiles out of the bore of a test ballistic setup. Preliminary calculations of the inner ballistic parameters of a shot are per-formed for the given launcher and then compared with the corresponding experimental data. Examples of experimentally fired underwater shots are presented. The high-speed high-energy processes associated with the projectile released out of the bore and entering water are described. It is shown that during the underwater launching, the complex gas-dynamic conditions of multi-phase processes following the projectile release out of the bore can significantly affect the trajectory of its further motion in water. Using the experi-mental and calculation methods with the classical loading scheme and an acceleration bore 0.5 mm long or less under given conditions, it has been determined that the pressure at the accelerator exit increases stepwise and can significantly change the flow conditions of the projectile at the start and its further trajectory. Taking into consideration the revealed peculiarities of underwater shots and a set of technical specifications and mass-size parameters, a perspective launching ballistic setup has been designed and used to estimate the motion range of the released supercavitating projectile.","PeriodicalId":43729,"journal":{"name":"Vestnik Tomskogo Gosudarstvennogo Universiteta-Matematika i Mekhanika-Tomsk State University Journal of Mathematics and Mechanics","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84422587","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A. S. Glazunov, Anastasiya A. Sizova, Irina L. Petrova
In this paper, the problem of choosing and optimizing a program for launching the payload (PL) into a specified circular orbit is considered. Both direct insertion into the circular orbit and the insertion using an intermediate elliptical orbit are examined. The mass range of the PL, which can be placed into the circular orbit with specified parameters, is calculated. The PL is launched using a medium-class launch vehicle (LV), which is proposed to be a hypothetical ballistic missile withdrawn from service. The LV movement program is divided into two sections: the escape from the atmosphere and the launch section. Both sections use a control program with the pitch angle as a parameter. The optimal parameters for the program are determined. Direct insertion into the circular orbit is revealed to be possible for a very limited range of PL masses. It is shown that the range can be significantly extended if an intermediate elliptical orbit is used. The study results prove that the use of the selected LV allows placing the PL in the specified circular and elliptical orbits. For the formation of the orbital trajectories of the PL launch, it is possible to use an almost invariable parametric class of LV flight control programs.
{"title":"Optimization of a program for launching the payload on a specified trajectory","authors":"A. S. Glazunov, Anastasiya A. Sizova, Irina L. Petrova","doi":"10.17223/19988621/80/6","DOIUrl":"https://doi.org/10.17223/19988621/80/6","url":null,"abstract":"In this paper, the problem of choosing and optimizing a program for launching the payload (PL) into a specified circular orbit is considered. Both direct insertion into the circular orbit and the insertion using an intermediate elliptical orbit are examined. The mass range of the PL, which can be placed into the circular orbit with specified parameters, is calculated. The PL is launched using a medium-class launch vehicle (LV), which is proposed to be a hypothetical ballistic missile withdrawn from service. The LV movement program is divided into two sections: the escape from the atmosphere and the launch section. Both sections use a control program with the pitch angle as a parameter. The optimal parameters for the program are determined. Direct insertion into the circular orbit is revealed to be possible for a very limited range of PL masses. It is shown that the range can be significantly extended if an intermediate elliptical orbit is used. The study results prove that the use of the selected LV allows placing the PL in the specified circular and elliptical orbits. For the formation of the orbital trajectories of the PL launch, it is possible to use an almost invariable parametric class of LV flight control programs.","PeriodicalId":43729,"journal":{"name":"Vestnik Tomskogo Gosudarstvennogo Universiteta-Matematika i Mekhanika-Tomsk State University Journal of Mathematics and Mechanics","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79818655","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper presents the analysis of a mechanical and theoretical model of a mi-cromechanical RL-type gyroscope (MMG). The behavior of the resonator in a free oscil-lation mode is studied by solving the problem in a linear formulation. The main part of the paper is devoted to the mathematical model development for a dual-mass MMG with a disk-shaped resonator, which is fixed on the elastic leg on a movable base. The important condition of the problem implies the arbitrary angular velocity of the base Ω. The derived equations are analyzed in terms of orbital coordinates (r, k, θ, χ). The equa-tions determining the precession angle (θ) of the considered gyroscope are obtained. A brief analysis of the proposed dependences and the corresponding conclusions about the behavior of the system under free oscillations are presented.
{"title":"Dynamics of a micromechanical dual-mass gyroscope of RL-type in a free oscillation mode","authors":"E. Antonov, O. Gribova","doi":"10.17223/19988621/82/5","DOIUrl":"https://doi.org/10.17223/19988621/82/5","url":null,"abstract":"This paper presents the analysis of a mechanical and theoretical model of a mi-cromechanical RL-type gyroscope (MMG). The behavior of the resonator in a free oscil-lation mode is studied by solving the problem in a linear formulation. The main part of the paper is devoted to the mathematical model development for a dual-mass MMG with a disk-shaped resonator, which is fixed on the elastic leg on a movable base. The important condition of the problem implies the arbitrary angular velocity of the base Ω. The derived equations are analyzed in terms of orbital coordinates (r, k, θ, χ). The equa-tions determining the precession angle (θ) of the considered gyroscope are obtained. A brief analysis of the proposed dependences and the corresponding conclusions about the behavior of the system under free oscillations are presented.","PeriodicalId":43729,"journal":{"name":"Vestnik Tomskogo Gosudarstvennogo Universiteta-Matematika i Mekhanika-Tomsk State University Journal of Mathematics and Mechanics","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87770724","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A priori estimates are established for solutions of one-dimensional inhomogeneous hyperbolic equations with an integral load in the main part, which has the form a(s) = sp, for p = 1, 0.5 and -1, with inhomogeneous initial and homogeneous boundary conditions. Here s is the integral over the space variable of the square of the modulus of the derivative of the solution of the equation with respect to x. Examples of linearization of loaded equations by substituting the right-hand sides of the estimates for a(s) are given.
对于具有非齐次初始边界条件和齐次边界条件的主体部分具有积分载荷的一维非齐次双曲方程，在p = 1、0.5和-1条件下，方程的形式为A (s) = sp，建立了先验估计。这里s是对空间变量对方程解的导数的模的平方的积分。给出了通过代入a(s)的估计的右边来线性化加载方程的例子。
{"title":"On linearization of hyperbolic equations with integral load in the main part using an a priori estimate of their solutions","authors":"O. Boziev","doi":"10.17223/19988621/80/2","DOIUrl":"https://doi.org/10.17223/19988621/80/2","url":null,"abstract":"A priori estimates are established for solutions of one-dimensional inhomogeneous hyperbolic equations with an integral load in the main part, which has the form a(s) = sp, for p = 1, 0.5 and -1, with inhomogeneous initial and homogeneous boundary conditions. Here s is the integral over the space variable of the square of the modulus of the derivative of the solution of the equation with respect to x. Examples of linearization of loaded equations by substituting the right-hand sides of the estimates for a(s) are given.","PeriodicalId":43729,"journal":{"name":"Vestnik Tomskogo Gosudarstvennogo Universiteta-Matematika i Mekhanika-Tomsk State University Journal of Mathematics and Mechanics","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77928667","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Investigation of an approximate solution of the integral equation of the exterior Dirichlet boundary value problem for the Helmholtz equation in the two-dimensional space","authors":"E. H. Khalilov","doi":"10.17223/19988621/82/4","DOIUrl":"https://doi.org/10.17223/19988621/82/4","url":null,"abstract":"","PeriodicalId":43729,"journal":{"name":"Vestnik Tomskogo Gosudarstvennogo Universiteta-Matematika i Mekhanika-Tomsk State University Journal of Mathematics and Mechanics","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74937824","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, a representation of a linear functional in the weighted Sobolev space is obtained. The space is normed without use of pseudodifferential operators. The norm contains partial derivatives of all intermediate orders of the test function. The Sobolev space is considered to be of non-Hilbert type. First, we deduce the representation of linear functional via a boundary element of the test space. The boundary element corresponds to the given functional. This way, referring to Clarkson’s inequalities, we prove the uniqueness of the boundary element. Then, to obtain a condition for the boundary element, we differentiate the function built based on the norm. The result leads to a representation of an arbitrary linear functional via the boundary element. When considering the boundary element as unknown, the representation performs as a nonlinear differential equation. Second, we consider a finite linear functional. The extreme function of such a functional was built in our earlier papers. The extreme function is expressed via convolution of the fundamental solution of a linear partial differential equation with a given functional. The functional performs as a distribution in the convolution. Convolution exists if the linear functional is finite. Using this fact, we prove that the representation of a finite linear functional via the boundary element is identical to the representation via the extreme function.
{"title":"Linear finite functional in the weighted Sobolev space","authors":"Igor K. Korytov","doi":"10.17223/19988621/81/2","DOIUrl":"https://doi.org/10.17223/19988621/81/2","url":null,"abstract":"In this paper, a representation of a linear functional in the weighted Sobolev space is obtained. The space is normed without use of pseudodifferential operators. The norm contains partial derivatives of all intermediate orders of the test function. The Sobolev space is considered to be of non-Hilbert type. First, we deduce the representation of linear functional via a boundary element of the test space. The boundary element corresponds to the given functional. This way, referring to Clarkson’s inequalities, we prove the uniqueness of the boundary element. Then, to obtain a condition for the boundary element, we differentiate the function built based on the norm. The result leads to a representation of an arbitrary linear functional via the boundary element. When considering the boundary element as unknown, the representation performs as a nonlinear differential equation. Second, we consider a finite linear functional. The extreme function of such a functional was built in our earlier papers. The extreme function is expressed via convolution of the fundamental solution of a linear partial differential equation with a given functional. The functional performs as a distribution in the convolution. Convolution exists if the linear functional is finite. Using this fact, we prove that the representation of a finite linear functional via the boundary element is identical to the representation via the extreme function.","PeriodicalId":43729,"journal":{"name":"Vestnik Tomskogo Gosudarstvennogo Universiteta-Matematika i Mekhanika-Tomsk State University Journal of Mathematics and Mechanics","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76313043","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
M. Bubenchikov, D. Mamontov, S. Azheev, A. A. Azheev
In the problem of the inertial rotation of molecular objects, only kinematic relations for the nodes of the molecular structure are evolutionary. These relations determine the position of the atoms of a supermolecule depending on the instantaneous angular velocity of the object in inertial motion. All other relations are algebraic, since they are integrals of the equations of rotational motion. The latter relations include both the projections of the angular velocities of the molecule and the instantaneous coordinates of the atoms. Within the framework of the fourth-order Runge-Kutta scheme, each time step is divided into four positions. Initially, in each of these positions, new values of coordinates are determined or the initial coordinates of atoms at the first position of the first time step are used. After the coordinates are found, in the same position, the projections of angular velocities of the supermolecule are obtained from conservation relations for the projections of the angular momentum. Based on the values of the coordinates in four positions, the coordinates on a new time layer are recalculated. After that, solving the system of three linear algebraic equations according to Cramer's rule, the projections of angular velocities at a new time step are determined. Then, the cycle is repeated. During the inertial rotation, the kinetic energy of an object is conserved. Verification of the calculated kinetic energy shows that the result is obtained with machine accuracy. Further, the constructed calculation scheme is used to study the Louis Poinsot instability. The full range of the considered instability for a fullerene C100 (C1 symmetry) is presented.
{"title":"Rotation of supermolecules around an intermediate axis of inertia","authors":"M. Bubenchikov, D. Mamontov, S. Azheev, A. A. Azheev","doi":"10.17223/19988621/80/5","DOIUrl":"https://doi.org/10.17223/19988621/80/5","url":null,"abstract":"In the problem of the inertial rotation of molecular objects, only kinematic relations for the nodes of the molecular structure are evolutionary. These relations determine the position of the atoms of a supermolecule depending on the instantaneous angular velocity of the object in inertial motion. All other relations are algebraic, since they are integrals of the equations of rotational motion. The latter relations include both the projections of the angular velocities of the molecule and the instantaneous coordinates of the atoms. Within the framework of the fourth-order Runge-Kutta scheme, each time step is divided into four positions. Initially, in each of these positions, new values of coordinates are determined or the initial coordinates of atoms at the first position of the first time step are used. After the coordinates are found, in the same position, the projections of angular velocities of the supermolecule are obtained from conservation relations for the projections of the angular momentum. Based on the values of the coordinates in four positions, the coordinates on a new time layer are recalculated. After that, solving the system of three linear algebraic equations according to Cramer's rule, the projections of angular velocities at a new time step are determined. Then, the cycle is repeated. During the inertial rotation, the kinetic energy of an object is conserved. Verification of the calculated kinetic energy shows that the result is obtained with machine accuracy. Further, the constructed calculation scheme is used to study the Louis Poinsot instability. The full range of the considered instability for a fullerene C100 (C1 symmetry) is presented.","PeriodicalId":43729,"journal":{"name":"Vestnik Tomskogo Gosudarstvennogo Universiteta-Matematika i Mekhanika-Tomsk State University Journal of Mathematics and Mechanics","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74070861","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Complex experimental and theoretical studies of shock-wave characteristics and penetration parameters are conducted to develop physical and mathematical models of deformation and fracture of ceramic and composite materials under high-velocity loading. The purpose of this work is to investigate the fracture of ceramic and composite targets and their efficiency in ceramic-containing structures. The experiments are performed using a ballistic test stand (Research Institute of Applied Mathematics and Mechanics, National Research Tomsk State University). The deformation and fracture of ceramic (aluminum oxide Al2O3, aluminum oxide (KVP-98, corundum), zirconium dioxide ZrO2, silicon carbide SiC) and composite (TiC-NiCr and TiB2-B4C) materials are studied experimentally in the impact velocity range of 0.5 - 6.5 km/s. The dynamics of the fracture of ceramic plates (silicon carbide and aluminum oxide) is analyzed. The fragments of the plates and their size distribution are given. The study of the impact resistance of ceramic and composite samples reveals two groups of materials with essentially different levels of efficiency. For both groups, the dependence of the efficiency parameter on the impact velocity demonstrates a decrease in the velocity range from 0.5 to 3.0 - 4.5 km/s and a subsequent increase when the impact velocity tends to 6.5 km/s. The obtained experimental results may be used during the validation and verification of numerical and analytical models, approaches, and software packages.
{"title":"Analysis of impact resistance for two groups of ceramic and composite materials","authors":"S. Zelepugin, V. Tolkachev, Il’ya M. Tyryshkin","doi":"10.17223/19988621/80/8","DOIUrl":"https://doi.org/10.17223/19988621/80/8","url":null,"abstract":"Complex experimental and theoretical studies of shock-wave characteristics and penetration parameters are conducted to develop physical and mathematical models of deformation and fracture of ceramic and composite materials under high-velocity loading. The purpose of this work is to investigate the fracture of ceramic and composite targets and their efficiency in ceramic-containing structures. The experiments are performed using a ballistic test stand (Research Institute of Applied Mathematics and Mechanics, National Research Tomsk State University). The deformation and fracture of ceramic (aluminum oxide Al2O3, aluminum oxide (KVP-98, corundum), zirconium dioxide ZrO2, silicon carbide SiC) and composite (TiC-NiCr and TiB2-B4C) materials are studied experimentally in the impact velocity range of 0.5 - 6.5 km/s. The dynamics of the fracture of ceramic plates (silicon carbide and aluminum oxide) is analyzed. The fragments of the plates and their size distribution are given. The study of the impact resistance of ceramic and composite samples reveals two groups of materials with essentially different levels of efficiency. For both groups, the dependence of the efficiency parameter on the impact velocity demonstrates a decrease in the velocity range from 0.5 to 3.0 - 4.5 km/s and a subsequent increase when the impact velocity tends to 6.5 km/s. The obtained experimental results may be used during the validation and verification of numerical and analytical models, approaches, and software packages.","PeriodicalId":43729,"journal":{"name":"Vestnik Tomskogo Gosudarstvennogo Universiteta-Matematika i Mekhanika-Tomsk State University Journal of Mathematics and Mechanics","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80261726","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, a group model for a real extension of the de Sitter plane is pro-posed. This group contains a group of special matrices, which is a subgroup of the general linear group. It is established that there exists a left-invariant contact metric structure on this group, which is normal and, therefore, para-Sasakian. The basis vector fields of the Lie algebra of infinitesimal automorphisms are found. The Lie group of automorphisms has the maximum dimension and, in addition to the Levi-Civita connection, it also retains a contact metric connection with skew-symmetric torsion. In this connection, all structural tensors of the para-Sasakian structure, as well as the torsion and curvature tensors, are covariantly constant. Using a nonholonomic field of orthonormal frames adapted to the contact distribution, an orthogonal projection of the Levi-Civita connection onto the contact distribution is found, which is a truncated connection. Passing to natural coordinates, differential equations of geodesics of the truncated connection and Levi-Civita connection are found. Thus, the Levi-Civita contact geodesic connections coincide with the truncated connection geodesics. This means that through each point in each contact direction there is a unique Levi-Civita geodesic connection tangent to the contact distribution. The Levi-Civita connection, like the contact metric connection, is consistent with the contact distribution.
{"title":"Left-invariant para-Sasakian structure on the group model of the real extension of the de Sitter plane","authors":"V. I. Pan’zhenskii, Yulia V. Dyranova","doi":"10.17223/19988621/82/2","DOIUrl":"https://doi.org/10.17223/19988621/82/2","url":null,"abstract":"In this paper, a group model for a real extension of the de Sitter plane is pro-posed. This group contains a group of special matrices, which is a subgroup of the general linear group. It is established that there exists a left-invariant contact metric structure on this group, which is normal and, therefore, para-Sasakian. The basis vector fields of the Lie algebra of infinitesimal automorphisms are found. The Lie group of automorphisms has the maximum dimension and, in addition to the Levi-Civita connection, it also retains a contact metric connection with skew-symmetric torsion. In this connection, all structural tensors of the para-Sasakian structure, as well as the torsion and curvature tensors, are covariantly constant. Using a nonholonomic field of orthonormal frames adapted to the contact distribution, an orthogonal projection of the Levi-Civita connection onto the contact distribution is found, which is a truncated connection. Passing to natural coordinates, differential equations of geodesics of the truncated connection and Levi-Civita connection are found. Thus, the Levi-Civita contact geodesic connections coincide with the truncated connection geodesics. This means that through each point in each contact direction there is a unique Levi-Civita geodesic connection tangent to the contact distribution. The Levi-Civita connection, like the contact metric connection, is consistent with the contact distribution.","PeriodicalId":43729,"journal":{"name":"Vestnik Tomskogo Gosudarstvennogo Universiteta-Matematika i Mekhanika-Tomsk State University Journal of Mathematics and Mechanics","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80863896","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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