Vestnik St Petersburg University-Mathematics最新文献

Rigid-Body Dynamics from the Euler Equations to the Attitude Control of Spacecraft in the Works of Scientists from St. Petersburg State University. Part 2 从欧拉方程到圣彼得堡国立大学科学家作品中的航天器姿态控制的刚体动力学。第二部分

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Vestnik St Petersburg University-Mathematics Pub Date : 2024-05-20 DOI: 10.1134/s1063454124700043

A. A. Tikhonov

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Gyroscopically Coupled In-Plane and Out-of-Plane Vibrations of Rotating Hollow Circular Plate: Case of In-Plane Axis of Rotation 旋转中空圆板的陀螺耦合平面内和平面外振动：旋转轴在平面内的情况

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Vestnik St Petersburg University-Mathematics Pub Date : 2024-05-20 DOI: 10.1134/s1063454124700092

N. F. Morozov, A. V. Lukin, I. A. Popov

{"title":"Gyroscopically Coupled In-Plane and Out-of-Plane Vibrations of Rotating Hollow Circular Plate: Case of In-Plane Axis of Rotation","authors":"N. F. Morozov, A. V. Lukin, I. A. Popov","doi":"10.1134/s1063454124700092","DOIUrl":"https://doi.org/10.1134/s1063454124700092","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>In this work, we construct and study a model of the coupled plane-transverse vibrations of a circular thin plate with a concentric hole under the action of Coriolis and centrifugal inertial forces caused by rotation of the system around an axis located in the plane of the plate. The partial differential equations of oscillations are obtained using the Hamilton–Ostrogradsky variational principle. Under the assumption that the angular velocity of rotation is small relative to the frequency of the operational skew-symmetric bending mode of plate vibrations, an approximate analytical solution is obtained for the radial, circumferential, and transverse components of the displacement field in the free vibration mode. Using the Galerkin projection, the problem was reduced to a system of two second-order linear differential equations for modal coordinates of mutually orthogonal basic skew-symmetric vibration modes of the plate. It is discovered that the regime of initially excited harmonic oscillations in the presence of rotation is transformed into a regime of amplitude-modulated beats. Analytical expressions are derived both for the frequency of the slow beat envelope and for the relative amplitude-modulation factor. We show that it is fundamentally possible to determine the modulus of the projection of the angular-velocity vector onto the plane of the plate from the measured value of the envelope frequency. We consider the problem of choosing the optimal geometric shape of the resonator for maximizing the sensitivity of the system to changes in the angular velocity of rotation. We also address the question of determining the direction of the projection of the angular velocity vector onto the plane of the plate from the measured depth of amplitude modulation of the beat regime.</p>","PeriodicalId":43418,"journal":{"name":"Vestnik St Petersburg University-Mathematics","volume":"38 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2024-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141152940","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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On the Integrability in Quadratures of the Problem of Rolling a Heavy Homogeneous Ball on a Surface of Revolution of the Second Order 论均质重球在二阶旋转面上滚动问题的四次方可积分性

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Vestnik St Petersburg University-Mathematics Pub Date : 2024-05-20 DOI: 10.1134/s1063454124700080

A. S. Kuleshov, A. A. Shishkov

{"title":"On the Integrability in Quadratures of the Problem of Rolling a Heavy Homogeneous Ball on a Surface of Revolution of the Second Order","authors":"A. S. Kuleshov, A. A. Shishkov","doi":"10.1134/s1063454124700080","DOIUrl":"https://doi.org/10.1134/s1063454124700080","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>The problem of rolling a heavy homogeneous ball on an absolutely rough surface of revolution is considered. Usually, when considering this problem, instead of the surface on which the ball rolls, it is customary to explicitly specify the surface along which the ball’s center moves; these surfaces are equidistant. In this case, the solution of the problem reduces to the integration of some second-order linear differential equation with variable coefficients. The coefficients of this equation depend on the characteristics of the surface along which the ball’s center moves, its principal curvatures, and the Lamé coefficients. If the general solution of the corresponding second-order linear differential equation can be found explicitly, the problem of rolling a heavy homogeneous ball on a surface of revolution can be integrated in quadratures. However, it is impossible to find the general solution to the corresponding equation for an arbitrary surface of revolution in an explicit form. Therefore, of interest is the question for which surfaces of revolution can the general solution of a second-order linear differential equation be found explicitly. In this work, it is assumed that the surface along which the center of the ball moves is a nondegenerate surface of revolution of the second order. In this case, the general solution of the second-order linear differential equation, to the integration of which the problem of rolling a heavy homogeneous ball on a surface of revolution is reduced, can be found in an explicit form. Thus, it is proved that, if the center of a heavy ball rolling on a surface of revolution belongs to a nondegenerate surface of revolution of the second order, then the problem of a rolling ball can be integrated in quadratures.</p>","PeriodicalId":43418,"journal":{"name":"Vestnik St Petersburg University-Mathematics","volume":"61 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2024-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141153689","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}

引用次数: 0

Regression Models for Calculating State-to-State Coefficients of the Rate of Vibrational Energy Exchanges 计算振动能量交换速率的态对态系数的回归模型

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Vestnik St Petersburg University-Mathematics Pub Date : 2024-05-20 DOI: 10.1134/s1063454124700079

A. A. Isakov, V. I. Gorikhovskii, M. Yu. Melnik

{"title":"Regression Models for Calculating State-to-State Coefficients of the Rate of Vibrational Energy Exchanges","authors":"A. A. Isakov, V. I. Gorikhovskii, M. Yu. Melnik","doi":"10.1134/s1063454124700079","DOIUrl":"https://doi.org/10.1134/s1063454124700079","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>The paper proposes an effective algorithm for solving problems of nonequilibrium gas dynamics taking into account detailed state-to-state vibrational kinetics. One of the problems of traditional methods is their high computational complexity, which requires a lot of time and memory. The work explored the possibilities of using relaxation rate prediction to improve the performance of numerical simulations of nonequilibrium oxygen flows instead of direct calculations. For this purpose, an approach based on a nonlinear regression analysis was used, which made it possible to obtain computationally efficient approximation formulas for the energy exchange rate coefficients in the model of a Forced Harmonic Oscillator, taking into account free rotations (FHO-FR), to significantly increase the calculation speed while maintaining accuracy, and to construct an optimized model FHO-FR-reg. Using the obtained regression formulas, numerical modeling was carried out, which made it possible to validate the model for the problem of oxygen flow behind an incident and reflected shock wave. A comparison between the Forced Harmonic Oscillator (FHO) and the FHO-FR models is not possible due to the high computational complexity of the second model. With the advent of a common approximation model, it became possible to compare simulation results for these models. Numerical calculations have shown that the FHO-FR–reg model gives values of gas-dynamic parameters close to the FHO model. The developed regression models make it possible to speed up the solution to the problem of modeling oxygen relaxation several times compared to other models of similar accuracy.</p>","PeriodicalId":43418,"journal":{"name":"Vestnik St Petersburg University-Mathematics","volume":"48 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2024-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141152904","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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Astronomical Research at the Mathematics Faculty of St. Petersburg University, I 圣彼得堡大学数学系天文学研究，I

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Vestnik St Petersburg University-Mathematics Pub Date : 2024-05-20 DOI: 10.1134/s1063454124700031

M. S. Prokopjeva, H. A. Krayani, V. B. Il’in

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Review of Research on the Qualitative Theory of Differential Equations at St. Petersburg University. I. Stable Periodic Points of Diffeomorphisms with Homoclinic Points and Systems with Weakly Hyperbolic Invariant Sets 圣彼得堡大学微分方程定性理论研究综述。I. 具有同轴点的微分方程稳定周期点和具有弱双曲不变集的系统

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Vestnik St Petersburg University-Mathematics Pub Date : 2024-05-20 DOI: 10.1134/s106345412470002x

N. A. Begun, E. V. Vasil’eva, T. E. Zvyagintseva, Yu. A. Iljin

{"title":"Review of Research on the Qualitative Theory of Differential Equations at St. Petersburg University. I. Stable Periodic Points of Diffeomorphisms with Homoclinic Points and Systems with Weakly Hyperbolic Invariant Sets","authors":"N. A. Begun, E. V. Vasil’eva, T. E. Zvyagintseva, Yu. A. Iljin","doi":"10.1134/s106345412470002x","DOIUrl":"https://doi.org/10.1134/s106345412470002x","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstarct</h3><p>This paper is the first in a series of review publications devoted to the results of scientific research that has been carried out in the Department of Differential Equations of St. Petersburg University over the past 30 years. The current scientific interests of the department staff can be divided into the following directions and topics: studying stable periodic points of diffeomorphisms with homoclinic points, the study of systems with weakly hyperbolic invariant sets, local qualitative theory of essentially nonlinear systems, classification of phase portraits of a family of cubic systems, and the stability conditions for systems with hysteretic nonlinearities and systems with nonlinearities under the generalized Routh–Hurwitz conditions (Aizerman problem). This paper presents recent results on the first two topics outlined above. The study of stable periodic points of diffeomorphisms with homoclinic points is carried out under the assumption that the stable and unstable manifolds of hyperbolic points are tangent at a homoclinic (heteroclinic) point, and the homoclinic (heteroclinic) point is not a point with a finite order of tangency. The research of systems with weakly hyperbolic invariant sets is conducted for the case when neutral, stable, and unstable linear spaces do not satisfy the Lipschitz condition.</p>","PeriodicalId":43418,"journal":{"name":"Vestnik St Petersburg University-Mathematics","volume":"156 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2024-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141153775","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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On the Approximation of the Attraction Field of a Rigid Body by the Attraction Field of Four Material Points of the Same Mass 用四个质量相同的物质点的吸引力场来近似刚体的吸引力场

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Vestnik St Petersburg University-Mathematics Pub Date : 2024-05-20 DOI: 10.1134/s1063454124700110

A. A. Burov, E. A. Nikonova, V. I. Nikonov

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Dynamics of a Double Pendulum with Viscous Friction at the Hinges. I. Mathematical Model of Motion and Construction of the Regime Diagram 铰链处有粘性摩擦的双摆动力学。I. 运动的数学模型和时态图的构建

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Vestnik St Petersburg University-Mathematics Pub Date : 2024-05-20 DOI: 10.1134/s1063454124700109

A. S. Smirnov, I. A. Kravchinskiy

{"title":"Dynamics of a Double Pendulum with Viscous Friction at the Hinges. I. Mathematical Model of Motion and Construction of the Regime Diagram","authors":"A. S. Smirnov, I. A. Kravchinskiy","doi":"10.1134/s1063454124700109","DOIUrl":"https://doi.org/10.1134/s1063454124700109","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>The paper discusses the dynamic behavior of a double mathematical pendulum with identical parameters of links and end loads, which is under the influence of viscous friction at both of its hinges with generally different dissipative coefficients. A linear mathematical model of system motion for small deviations is given, and a characteristic equation containing two dimensionless dissipative parameters is derived. For the case of low damping, approximate analytical expressions are derived that make it possible to evaluate and compare with each other the damping factors during motion of the system in each of the vibration modes. A diagram of dissipative motion regimes is constructed, which arises when the plane of dimensionless parameters is divided by discriminant curves into regions with a qualitatively different character of system motion. It is noted that a dissipative internal resonance can occur in the system under consideration; the conditions for its existence are established in an analytical form, and a graphic illustration of these conditions are also displayed. This publication is the first part of the study of the dynamics of a dissipative double pendulum, the continuation of which will be presented as a separate publication “Dynamics of a Double Pendulum with Viscous Friction at the Hinges. II. Dissipative Vibration Modes and Optimization of the Damping Parameters.”</p>","PeriodicalId":43418,"journal":{"name":"Vestnik St Petersburg University-Mathematics","volume":"61 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2024-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141153692","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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Solution of the Local-Boundary-Value Problem of Control for a Nonlinear Stationary System Taking into Account Computer System Verification 考虑到计算机系统验证的非线性静态系统控制局部边界值问题解法

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Vestnik St Petersburg University-Mathematics Pub Date : 2024-05-20 DOI: 10.1134/s1063454124700055

A. N. Kvitko, N. N. Litvinov

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Determination of Discontinuity Points and the Jump Magnitude of the Original Based on Its Laplace Image 根据拉普拉斯图像确定原点的不连续点和跳跃幅度

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Vestnik St Petersburg University-Mathematics Pub Date : 2024-05-20 DOI: 10.1134/s1063454124700067

A. V. Lebedeva, V. M. Ryabov

{"title":"Determination of Discontinuity Points and the Jump Magnitude of the Original Based on Its Laplace Image","authors":"A. V. Lebedeva, V. M. Ryabov","doi":"10.1134/s1063454124700067","DOIUrl":"https://doi.org/10.1134/s1063454124700067","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>The application of the integral Laplace transform to a wide class of problems leads to a simpler equation relative to the image of the desired original. At the next step, the inversion problem (i.e., the problem of finding the original based on its image) arises. As a rule, this step cannot be carried out analytically, and the problem arises of using approximate inversion methods. In this case, the approximate solution is represented in the form of a linear combination between the image and its derivatives at certain points of the complex half-plane, in which the image is regular. Unlike the image, however, the original may have even discontinuity points. Of undoubted interest is the task of developing methods for determining the possible discontinuity points of the original as well as the magnitudes of the original jump at these points. The suggested methods imply using values of high-order image derivatives in order to obtain a satisfactory accuracy of approximate solutions. The methods for accelerating the convergence of the obtained approximations are given. The results of numerical experiments which illustrate the efficiency of the suggested techniques are demonstrated.</p>","PeriodicalId":43418,"journal":{"name":"Vestnik St Petersburg University-Mathematics","volume":"11 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2024-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141152942","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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