Chaotic dynamics of an asymmetrical gyrostat

The chaotic motions of an asymmetrical gyrostat, composed of an asymmetrical carrier and three wheels installed along its principal axes and rotating about the mass center of the entire system under the action of both damping torques and periodic disturbance torques, are investigated in detail in this paper. By introducing the Deprit's variables, one can derive the attitude dynamical equations that are well suited for the utilization of the Melnikov's integral developed by Wiggins and Shaw. By using the elliptic function theory, the homoclinic solutions of the attitude motion of a torque-free asymmetrical gyrostat are obtained analytically, based upon the Wangerin's method developed by Wittenburg. Transversal intersections of the stable and unstable manifolds (typically a necessary condition for chaotic motions to exist) are detected by the techniques of Melnikov's functions. The bifurcation curve between the compound parameters is depicted and discussed. By using a fourth-order Runge–Kutta integration algorithm as a tool of the numerical simulation, the long-term dynamical behavior of the system shows that the technique of the Melnikov's function could successfully be employed to predict the compound physical parameters that correspond to the chaotic dynamical motions of an asymmetrical gyrostat.     

Chaotic attitude motion of gyrostat statellites in a central force field

The nonlinear attitude motion of gyrostat satellites in a central force field is investigated, with particular emphasis on their long-time dynamic behavior for a wide range of parameters. The numbers of equilibrium solutions, as well as their stability, vary with the rotor speed, and bifurcation diagrams have been obtained. Various dynamic behaviors of gyrostat satcllites, e.g. periodic, quasiperiodic, and chaotic, are studied via the Poincaré map technique. It is shown that the rotor speed has a significant effect on the dynamic behavior of gyrostat satellites.     

Stationary solutions of a small gyrostat in the Newtonian field of two bodies with equal masses

The paper deals with the dynamics of a small gyrostat satellite acted upon by the Newtonian forces of two big bodies of equal masses which rotate around their center of mass. The gyrostat’s equations of motion are derived and classes of its stationary solutions, as well as their stability are studied.     

Motion of a one-dimensional system of colliding spheres in a gravitational field

Institute of Mechanical Engineering, Leningrad. Translated from Prikladnaya Mekhanika, Vol. 26, No. 9, pp. 90–95, September, 1990.     

Eulerian equilibria of a triaxial gyrostat in the three body problem: rotational Poisson dynamics in Eulerian equilibria

We consider the noncanonical Hamiltonian dynamics of a triaxial gyrostat in the three body problem. By means of geometric-mechanics methods, we will study the approximated dynamics that arises when we develop the potential in series of Legendre and truncate the series to the second harmonics. Working in the reduced problem, we will study the existence of equilibria that we will denominate of Euler in analogy with classic results on the topic. In this way, we generalize the classical results on equilibria of the three-body problem and many of those obtained by other authors using more classic techniques for the case of rigid bodies. The instability of Eulerian equilibria is proven in this approximate dynamics if the gyrostat is close to the sphere. The rotational Poisson dynamics of the gyrostat placed at an Eulerian equilibrium and the study of the nonlinear stability of some equilibria is considered. The analysis is done in vectorial form avoiding the use of canonical variables and the tedious expressions associated with them.     

Shock wave acceleration in a gravitational field. A special case

A one-dimensional problem of shock wave acceleration in a uniform gravitational field is exactly solved. In front of the shock wave, the medium state is initially in equilibrium and its density decreases according to a power law. The shock wave is generated using a piston moving freely in the gravitational field. The adiabatic index is assumed to be equal to 3. The obtained solution is represented in terms of elementary functions.     

Global motion of a dissipative asymmetric gyrostat

Summary The motion of an asymmetrical gyrostat with external or internal energy dissipation is discussed by use of the qualitative method. The attitude motion of the gyrostat is described by two angular coordinates of the total angular momentum H relative to a body-fixed reference frame. The dynamical equation is written for an autonomous system whose singularity corresponds to the permanent rotation of the gyrostat. The global motion of the gyrostat is determined by the number and distribution of singularities. The type of each singularity varies with the variation of mass geometry and rotor speed, and a bifurcation phenomenon can be observed.

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Das globale Verhalten eines dissipativen unsymmetrischen Kreisels

Übersicht Das Verhalten eines asymmetrischen Kreisels mit innerer oder äußerer Reibung wird qualitativ untersucht. Die Stellung des Kreisels wird durch zwei Kardanwinkel zwischen körperfesten und raumfesten Koordinatensystemen beschrieben. Die sich ergebenden Bewegungsgleichungen gelten für autonome Systeme, deren Singularitäten permanente Drehungen des Kreisels darstellen. Das Globalverhalten des Kreisels ist durch die Anzahl dieser Singularitäten und ihrer Verteilung gegeben. Die Art einer jeden Singularität ändert sich mit der Masseverteilung und der Rotorgeschwindigkeit, und es treten Verzweigungen auf.

     

Stress state of an elastic ball in a spherically symmetric gravitational field

We consider a spherically symmetric static problem of general relativity whose solution was obtained in 1916 by Schwarzschild for a metric form of a special type. This solution determines the metric coefficients of the exterior and interior Riemannian spaces generated by a gravitating solid ball of constant density and includes the so-called gravitational radius r _g. For a ball of outer radius R=r _g, the metric coefficients are singular, and hence the radius r _g is traditionally assumed to be the radius of the event horizon of an object called a black hole. The solution of the interior problem obtained for an incompressible ideal fluid shows that the pressure at the ball center increases without bound for R=9/8r _g, which is traditionally used for the physical justification of the existence of black holes. The discussion of Schwarzschild’s traditional solution carried out in this paper shows that it should be generalized with respect to both the geometry of the Riemannian space and the elastic medium model. In this connection, we consider the general metric form of a spherically symmetric Riemannian space and prove that the solution of the corresponding static problem exists for a broad class of metric forms. A special metric form based on the assumption that the gravitation generating the Riemannian space inside a fluid ball or an elastic ball does not change the ball mass is singled out from this class. The solution obtained for the special metric form is singular with respect to neither the metric coefficients nor the pressure in the fluid ball and the stresses in the elastic ball. The obtained solution is compared with Schwarzschild’s traditional solution.     

Abnormal dynamics in cascading model with gravitational effect

Nonlinear Dynamics - In actual networks, the distance between nodes has an effect on load flow. Nodes closer to a node are more attractive, as if there is some kind of gravity between the nodes....     

Forecasting Hamiltonian dynamics without canonical coordinates

Conventional neural networks are universal function approximators, but they may need impractically many training data to approximate nonlinear dynamics. Recently introduced Hamiltonian neural networks can efficiently learn and forecast dynamical systems that conserve energy, but they require special inputs called canonical coordinates, which may be hard to infer from data. Here, we prepend a conventional neural network to a Hamiltonian neural network and show that the combination accurately forecasts Hamiltonian dynamics from generalised noncanonical coordinates. Examples include a predator–prey competition model where the canonical coordinates are nonlinear functions of the predator and prey populations, an elastic pendulum characterised by nontrivial coupling of radial and angular motion, a double pendulum each of whose canonical momenta are intricate nonlinear combinations of angular positions and velocities, and real-world video of a compound pendulum clock.

     

不规则小行星引力场内的飞行动力学

小行星探测是当前深空探测的主要方向之一,具有重要的科学意义.绝大多数小行星引力场极不规则,探测器在小行星附近运动形态复杂多样.由于同时受到中心引力、快速自旋的不规则形状摄动力、以及光压摄动等作用,探测器容易与小行星发生碰撞或逃逸.概述小行星研究现状和不规则引力场建模方法.重点介绍不规则引力场内动力学特性,包括引力平衡点、局部流形、自然周期轨道和悬停探测轨道等,尝试提出新的研究方向.     

<非线性随机动力学与控制——Hamilton理论体系框架>评介

自然、工程、与社会中广泛存在非线性随机动力学现象.随机运动是大量样本运动的集合,从各个样本运动来看,变化似无规律可言;但从总体来看,却有一定的统计规律性.非线性可使系统的运动规律变得十分复杂、丰富多彩.关于非线性随机动力学的研究始于上世纪中叶,经历了几十年的发展,已经有了长足进步,但离认识的‘自由王国’尚路途遥远.     

Effect of a gravitational field on the stratification of suspensions in vertical flows

Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, No. 1, pp. 29–33, January–February, 1991.     

Cavitation streamline flow of lamina in transverse gravitational field

The problem of cavitation streamline flow located on the linear base of a lamina in a gravity solution current is solved by the systems of Ryabushinskii and Zhukovskii-Roshko. The method of fragment-continuum approximation of the boundary condition at the free boundary was used, in which this condition is exactly satisfied at a finite number of points. In this way the original problem comes down to a solution of a system of nonlinear equations whose solvability can be shown by the method of V. N. Monakhov [1]. The main consideration in the present work was given to a numerical solution of this system of equations on a computer. The problem is similar to the type for large Froude numbers, when the effect of weight on the flow is small, studied in [2-5]. In [6, 7] the flow problems were solved by the method of finite differences. The approximations of the boundary condition at the free boundary used earlier are based on the use of the smallness of these or other characteristics of flow. Thus, for example, the linearization of Levi-Chivit [8] is rightly used in the assumption of smallness of the change in the modulus and angle of inclination of the velocity at the free flow line; a stronger linearization is based on the requirement of smallness of additional velocities caused by an obstacle in comparison with the velocity of the undisturbed current [9]. In the given work the problems studied lead to a range of cavitation and Froude numbers when the gravitational force exerts a considerable effect on the main characteristics of the flow. As an example of one of the possible applications of the calculation, the solution of the problem of choice of the form of a body of zero buoyancy with a zone of constant pressure is given.Translated from Zhurnal Prikladnoi Mekhanik i Tekhnicheskoi Fiziki, No. 5, pp. 132–136, September–October, 1971.