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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
IF 0.2 Q4 Mathematics Pub Date : 2024-05-20 DOI: 10.1134/s1063454124700043
A. A. Tikhonov

Abstract

The work attempts to analyze the scientific achievements of the St. Petersburg School of Mechanics in the field of rigid-body dynamics as a continuation of the review dedicated to the 300th anniversary of St. Petersburg State University. This second part of the review describes a fifty-year period ending in 2023. It focuses on general theoretical research carried out by scientists from St. Petersburg State University on both uncontrolled and controlled motions of a rigid body.

摘 要 本著作试图分析圣彼得堡机械学院在刚体动力学领域的科学成就,作为圣彼得堡国立大学 300 周年纪念回顾的延续。回顾的第二部分描述了截至 2023 年的 50 年间的情况。其重点是圣彼得堡国立大学的科学家们就刚体的非受控和受控运动所开展的一般理论研究。
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引用次数: 0
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
IF 0.2 Q4 Mathematics Pub Date : 2024-05-20 DOI: 10.1134/s1063454124700092
N. F. Morozov, A. V. Lukin, I. A. Popov

Abstract

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.

摘要 在这项工作中,我们构建并研究了一个带有同心孔的圆形薄板在科里奥利力和离心惯性力作用下的平面-横向耦合振动模型。振荡偏微分方程是利用 Hamilton-Ostrogradsky 变分原理求得的。假设旋转角速度相对于板振动的偏斜对称弯曲工作模式的频率较小,则可得到自由振动模式下位移场的径向、圆周和横向分量的近似解析解。利用伽勒金投影法,将问题简化为两个二阶线性微分方程系,用于计算板的相互正交基本偏斜对称振动模式的模态坐标。研究发现,在存在旋转的情况下,最初激发的谐波振荡机制转变为振幅调制节拍机制。我们得出了慢节拍包络频率和相对振幅调制因子的分析表达式。我们证明,从根本上可以根据测量到的包络频率值确定角速度矢量在平板平面上的投影模量。我们考虑的问题是选择谐振器的最佳几何形状,使系统对旋转角速度变化的灵敏度最大化。我们还讨论了根据测量到的节拍振幅调制深度确定角速度矢量在板平面上的投影方向的问题。
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引用次数: 0
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
IF 0.2 Q4 Mathematics Pub Date : 2024-05-20 DOI: 10.1134/s1063454124700080
A. S. Kuleshov, A. A. Shishkov

Abstract

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.

摘要 本文考虑了在绝对粗糙的旋转表面上滚动一个重均质球的问题。通常,在考虑这个问题时,不考虑球滚动的表面,而是明确指定球的中心沿其移动的表面;这些表面是等距的。在这种情况下,问题的求解简化为对某个系数可变的二阶线性微分方程的积分。该方程的系数取决于球心沿其移动的表面的特征、主曲率和拉梅系数。如果可以明确地找到相应二阶线性微分方程的一般解,那么在旋转表面上滚动重均质球的问题就可以用四元积分法来解决。然而,对于任意旋转曲面,不可能以显式形式找到相应方程的通解。因此,人们感兴趣的问题是,对于哪些旋转曲面,可以明确地找到二阶线性微分方程的一般解。在本研究中,假定球心沿其移动的曲面是二阶非退化旋转曲面。在这种情况下,二阶线性微分方程的一般解可以用显式求得,而在旋转曲面上滚动重均质球的问题就简化为对该方程的积分。因此,可以证明,如果在旋转曲面上滚动的重球的中心属于二阶非退化旋转曲面,那么滚动球的问题可以用四元数积分。
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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
IF 0.2 Q4 Mathematics Pub Date : 2024-05-20 DOI: 10.1134/s1063454124700079
A. A. Isakov, V. I. Gorikhovskii, M. Yu. Melnik

Abstract

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.

摘要 本文提出了一种解决非平衡气体动力学问题的有效算法,其中考虑到了详细的状态到状态振动动力学。传统方法的问题之一是计算复杂度高,需要大量时间和内存。这项工作探索了利用弛豫速率预测来代替直接计算提高非平衡氧流数值模拟性能的可能性。为此,采用了一种基于非线性回归分析的方法,从而获得了强制谐波振荡器模型中能量交换率系数的高效计算近似公式,同时考虑了自由旋转(FHO-FR),在保持精度的同时显著提高了计算速度,并构建了一个优化模型 FHO-FR-reg。利用获得的回归公式,进行了数值建模,从而验证了入射和反射冲击波后氧气流动问题的模型。由于强迫谐波振荡器(FHO)模型和 FHO-FR 模型的计算复杂度较高,因此无法进行比较。随着通用近似模型的出现,对这些模型的模拟结果进行比较成为可能。数值计算表明,FHO-FR-Reg 模型给出的气体动力参数值与 FHO 模型接近。与其他精度类似的模型相比,所开发的回归模型可以将氧气弛豫模型问题的解决速度提高数倍。
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引用次数: 0
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
IF 0.2 Q4 Mathematics Pub Date : 2024-05-20 DOI: 10.1134/s1063454124700031
M. S. Prokopjeva, H. A. Krayani, V. B. Il’in

Abstract

The development of astronomical research at St. Petersburg University since its foundation is considered. The subjects of works are briefly described and the main achievements of university astronomers in the 18th–19th centuries are highlighted. The most important research carried out at the university in various fields of astronomy in the 20th century are listed. Certain emphasis is made on the mathematical side of the works.

摘要 阐述了圣彼得堡大学自建校以来天文学研究的发展情况。文章简要介绍了圣彼得堡大学的研究课题,并重点介绍了该大学天文学家在 18-19 世纪取得的主要成就。列出了 20 世纪该大学在天文学各个领域开展的最重要研究。在一定程度上强调了作品的数学方面。
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引用次数: 0
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
IF 0.2 Q4 Mathematics Pub Date : 2024-05-20 DOI: 10.1134/s106345412470002x
N. A. Begun, E. V. Vasil’eva, T. E. Zvyagintseva, Yu. A. Iljin

Abstarct

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.

Abstarct 本文是专门介绍圣彼得堡大学微分方程系过去 30 年科研成果的系列评论出版物中的第一篇。该系教职员工目前的科研兴趣可分为以下几个方向和主题:研究具有同轴点的差分变形的稳定周期点、研究具有弱双曲不变集的系统、本质上非线性系统的局部定性理论、立方系统家族的相位肖像分类、具有滞后非线性的系统和具有广义 Routh-Hurwitz 条件下非线性的系统(Aizerman 问题)的稳定条件。本文介绍了上述前两个课题的最新成果。在双曲点的稳定流形和不稳定流形相切于同曲(异曲)点,且同曲(异曲)点不是有限阶相切点的假设下,对具有同曲点的衍射稳定周期点进行了研究。针对中性、稳定和不稳定线性空间不满足 Lipschitz 条件的情况,对具有弱双曲不变集的系统进行了研究。
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引用次数: 0
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
IF 0.2 Q4 Mathematics Pub Date : 2024-05-20 DOI: 10.1134/s1063454124700110
A. A. Burov, E. A. Nikonova, V. I. Nikonov

Abstract

In this paper, we study the problem of approximating the gravitational potential of a rigid body by the potential of a system of four identical point masses. Considering the potential as an expansion in terms of a parameter characterizing the ratio of the average size of a body to the distance to a test point in space, we propose an approach to constructing an approximate expression up to terms of the third order of smallness. This approach is used to build a model of the gravitational field of the nucleus of comet 67P/Churyumov–Gerasimenko.

摘要 在本文中,我们研究了用四个相同点质量系统的势来近似刚体引力势的问题。我们将引力势视为一个参数的扩展,该参数表征了一个物体的平均尺寸与到空间测试点的距离之比,我们提出了一种方法来构建一个小到三阶的近似表达式。这种方法被用于建立 67P/Churyumov-Gerasimenko 彗星核的引力场模型。
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引用次数: 0
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
IF 0.2 Q4 Mathematics Pub Date : 2024-05-20 DOI: 10.1134/s1063454124700109
A. S. Smirnov, I. A. Kravchinskiy

Abstract

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.”

摘要 本文讨论了具有相同链节参数和末端载荷的双数学摆的动态行为,该摆在其两个铰链处受到粘性摩擦的影响,其耗散系数通常不同。给出了小偏差时系统运动的线性数学模型,并推导出包含两个无量纲耗散参数的特征方程。在低阻尼情况下,推导出近似分析表达式,从而可以评估和比较系统在每种振动模式下运动时的阻尼系数。当无量纲参数平面被判别曲线划分为系统运动性质不同的区域时,就会产生耗散运动状态图。我们注意到,在所考虑的系统中可能会出现耗散内部共振;我们以分析的形式确定了其存在的条件,并展示了这些条件的图解。本出版物是耗散双摆动力学研究的第一部分,其续篇将作为单独出版物 "铰链处有粘性摩擦的双摆动力学 "介绍。II.耗散振动模式和阻尼参数优化"。
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引用次数: 0
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
IF 0.2 Q4 Mathematics Pub Date : 2024-05-20 DOI: 10.1134/s1063454124700067
A. V. Lebedeva, V. M. Ryabov

Abstract

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.

摘要 在许多问题中应用积分拉普拉斯变换,可以得到一个相对于所需原点图像的较简单方程。下一步就会出现反演问题(即根据原图的图像找到原图的问题)。通常情况下,这一步无法通过分析来完成,于是就出现了使用近似反演方法的问题。在这种情况下,近似解以图像及其在复半平面上某些点的导数的线性组合形式表示,图像在复半平面上是规则的。然而,与图像不同的是,原图甚至可能存在不连续点。毫无疑问,开发确定原图可能的不连续点以及原图在这些点上的跳变幅度的方法是一项令人感兴趣的任务。所建议的方法意味着使用高阶图像导数值,以获得令人满意的近似解精度。还给出了加速所获近似值收敛的方法。数值实验结果表明了所建议技术的效率。
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引用次数: 0
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
IF 0.2 Q4 Mathematics Pub Date : 2024-05-20 DOI: 10.1134/s1063454124700055
A. N. Kvitko, N. N. Litvinov

Abstract

In this work, an algorithm is suggested to construct the control function that provides the transition of a broad class of stationary nonlinear controllable systems of ordinary differential equations from the initial state to the origin of coordinates taking into account the possibility of verification of the operability of computer systems. Constructive sufficient conditions ensuring the existence of a solution to the set problem are found. The efficiency of the algorithm is shown by the numerical modeling of a specific practical problem.

摘要 在这项工作中,考虑到验证计算机系统可操作性的可能性,提出了一种构建控制函数的算法,该函数提供了一大类静态非线性可控常微分方程系统从初始状态到坐标原点的过渡。找到了确保集合问题解存在的建设性充分条件。通过对一个具体实际问题的数值建模,展示了该算法的效率。
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引用次数: 0
期刊
Vestnik St Petersburg University-Mathematics
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