Stability of the cylindrical precession of a satellite in an elliptic orbit

We study the linear problem on the stability of rotation of a dynamically symmetric satellite about the normal to the plane of the orbit of its center of mass. The orbit is assumed to be elliptic, and the orbit eccentricity is arbitrary. We assume that the Hamiltonian contains a small parameter characterizing the deviation of the satellite central ellipsoid of inertia from the sphere. This is a resonance problem, since if the small parameter is zero, then one of the frequencies of small oscillations of the symmetry axis in a neighborhood of the unperturbed rotation of the satellite about the center of mass is exactly equal to the frequency of the satellite revolution in the orbit. We indicate a countable set of values of the angular velocity of the unperturbed rotation for which the resonance is even double. The stability and instability domains are obtained in the first approximation with respect to the small parameter.     

To the problem of plane periodic rotations of a satellite in an elliptic orbit

We study the motion of a satellite (a rigid body) with respect to its center of mass in an elliptic orbit of small eccentricity. We analyze the nonlinear problem of the existence and stability of periodic (in the orbital coordinate system) rotations of the satellite with a period multiple of the period of revolution of its center of mass in the orbit. We study the direct and reverse rotations. In particular, we find and investigate the set of bifurcation values of the satellite dimensionless inertial parameter near which the branching of the periodic reverse rotations occurs. We consider three specific examples of application of the obtained general theoretical conclusions. In one of these examples, we prove the stability of the direct resonance rotations of Mercurial type. In the other two examples, we consider the branching problem for reverse rotations with a period whose ratio to the period of motion of the center of mass in the orbit is equal to 1 or 2.     

Stability and control of tethered satellite with chemical propulsion in orbital plane

The tethered satellite with chemical propulsion has broad application prospects in the disposal of abandoned satellites, the orbital rescue of spacecrafts, and the transportation of space supplies, which is completely different from the traditional applications of tethered satellites. Therefore, new research on its dynamics, stability, and control becomes useful and interesting. In this article, based on a dumbbell model of tethered satellite, the dynamics equations of tethered system in orbital maneuvering are established. Furthermore, according to the definitions of transversal and radial propulsive coefficients, analytical solutions of the equilibrium position for librational angle are derived during maneuvering in orbital plane; meanwhile, the effects of propulsive coefficients on librational stability are analyzed, which provides a basis for a selection of expected attitude trajectory. Then, a method of hierarchical sliding-mode tension control is presented to track the expected in-plane angle. This method can address the underactuated problem of tethered systems without either complex coordinate transformation for the system state model or constraint equation restrictions. During orbital flight, in-plane and out-of-plane angles are decoupled, so the tether tension control cannot be conducted to inhibit the out-of-plane angle. To solve this problem, the binormal component of thrust acceleration normal to the orbital plane is adopted as a control variable, and a feedback linearization-based thrust controller is designed to damp out the out-of-plane angle. Afterwards, orbital transfer cases between two circular orbits are studied to demonstrate the effectiveness of the tethered satellite with chemical propulsion. Numerical simulation results indicate that the stability of librational angles has a close relation to propulsive coefficients, and distributions of stable centers and unstable saddle points are totally different on both sides of bifurcation point. In addition, tracking control requirements for tethered satellite are guaranteed by designed controllers, which ensure flight safety in orbital maneuvering.     

Control of chaotic oscillations of a satellite

Analytical conditions and practical methods of their realization are proposed to solve a problem of a command signal tracking for a nonlinear disturbed system.Non- linear disturbed plants consisting of linear dynamic block and nonlinear block in feedback are considered.Nonlinear part of the plant and disturbance are unknown and bounded. The paper illustrates a possibility of applications of proposed algorithms to control libra- tion angle of satellite.     

Nonsmall liquid oscillations in a right circular cylinder

In this study we consider certain nonlinear effects which occur during oscillations of a liquid partially filling a right circular cylinder. The problem of nonlinear oscillations of a liquid in a circular cylinder has been considered in [1, 2]. The same problem has been solved in [3, 4] for arbitrary cavities by a somewhat different method.In the present paper we investigate the stability of forced oscillations of a liquid in a cylinder when the latter performs small harmonic oscillations in a plane passing through its axis.     

Stability of a flow in a circular tube

Many studies, both theoretical and experimental, have been dedicated to the stability of flow in a circular tube (see, for example, review [1]). In every case mathematical investigation has not succeeded in obtaining an expression for hydrodynamic instability of such a flow for disturbances of sufficiently low amplitude. (An exception is [2].) Experiment also indicates the stability of such a flow [3], with a laminar mode being extended to Reynolds numbers of the order of tens of thousands. These facts are the basis for the assumption that the flow of a viscous incompressible liquid in a circular tube is stable for small perturbations. However, there is no analytical or even numerical proof of this hypothesis. Moreover, some studies, for example [2], indicate the instability of such a flow in relation to three-dimensional nonaxiosymmetric perturbations. The analysis of hydrodynamic stability with respect to three-dimensional disturbances of flow within a circular tube conducted in this study showed the stability of the flow over a wide range of wave numbers and Reynolds numbers.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 20–24, January–February, 1973.     

Motion and oscillations of a circular perturbed vortex ring

The long bending distortions of the central line of a perturbed circular slender vortex ring with an axial velocity component are studied with the equation of motion of Callegari and Ting [1] rather than with the equation due to a cut-off method as in Widnall and Sullivan [2]. The link between the evolution of the perturbation and the inner structure of the vortex is then shown and the study of the perturbations of small amplitudes gives an analytical expression of the period of oscillation for a circular perturbed inviscid vortex. A numerical code that solves the full nonlinear filament evolution equation has been developed and validated by the results of the linear analysis. In the small amplitude limit, the numerical simulations and the analytical approach are in excellent agreement both for an inviscid and a viscous vortex. The evolution of a perturbation of finite amplitude is also given.     

Optimal control of a rotational motion of a gyrostat on circular orbit

A control scheme is proposed to guarantee an optimal stabilization of a given rotational motion of a symmetric gyrostat on circular orbit. The gyrostat controlled by the control action generated by rotating internal rotors. In such study the asymptotic stability of this motion is proved using Barbachen and Krasovskii theorem's and the optimal control law is deduced from the conditions that ensure the optimal asymptotic stability of the desired motion. As a particular case, the equilibrium position of the gyrostat, which occurs when the principal axes of inertia coincide with the orbital axes, is proved to be asymptotically stable. The present method is shown to more general than previous ones.     

基于北斗导航卫星信号的低轨卫星自主定轨（英文）

利用北斗卫星导航信号对低轨卫星自主定轨进行了研究。首先对卫星轨道动力学方程进行了推导,建立一个考虑地球J2摄动下的卫星动力学模型,并利用4阶龙格-库塔数值积分法进行轨道传播。选用北斗卫星导航信号的伪距和伪距率作为定轨的测量方程。然后,对北斗卫星可见星的个数进行了分析。最后,采用SRUKF算法对低轨CHAMP卫星轨道状态进行估计,并通过STK8.1生成仿真数据对SRUKF、UKF和EKF算法的跟踪性能进行对比分析。仿真结果表明,在跟踪过程中最少能接收到18颗北斗卫星信号,能够满足定轨要求。在相同条件下,SRUKF算法的跟踪精度优于UKF、EKF算法,SRUKF算法能减少89%的位置误差和90%的速度误差。     

Relative equilibrium stability of a mechanical system with deformable elements in a circular orbit

The relative equilibrium stability is analyzed for a mechanical system in an orbit. The system consists of two rigid bodies connected by a thin inextensible elastic rod. The problem of stability of steady motions is reduced to the minimization problem for the system’s potential energy consisting of the potential energy of elastic, gravitational, and centrifugal forces.     

Diffraction of a plane wave by a reinforced circular cavity in a massif

State University, Donetsk. Translated from Prikladnaya Mekhanika, Vol. 28, No. 3, pp. 19–22, March, 1992.     

Hamiltonian structure for rigid body with flexible attachments in a circular orbit

Hamiltonian structure of a rigid body in a circular orbit is established in this paper. With the reduction technique, the Hamiltonian structure of a rigid body in a circular orbit is derived from Lie-Poisson structure of semidirect product, and Hamiltonian is derived from Jacobi's integral. The above method can be generalized to establish the Hamiltonian structure of a rigid body with a flexible attachment in a circular orbit. At last, an example of stability analysis is given. The project supported by National Natural Science Foundation of China and Aeronautic Science Foundation.