Exact solutions and adiabatic invariants for equations of satellite attitude motion under Coulomb torque

This paper focuses on the attitude dynamics of a defunct axisymmetric satellite under the action of Coulomb forces to enable active space debris removal. Touchless Coulomb interaction occurs between an active spacecraft and the passive satellite at a fixed separation distance. The recently developed multi-sphere method of Schaub and Stevenson allows providing a simplified electrostatic force and torque model between non-spherical space objects. The existence of torques between charged bodies makes it necessary to study the attitude motion of the passive satellite for ensuring the safety of the space debris removal. The goal is first to deduce the equations of motion in the canonical form which is suitable for analytical analysis and then to construct a phase portrait, and to obtain exact solutions using Jacobi elliptic functions. Finally, for the disturbed motion of the system of two bodies, when the distance between the active spacecraft and the defunct satellite (or) and charge voltage changes slowly over time, adiabatic invariants are found in terms of the complete elliptic integrals. In this case, the adiabatic invariants are approximately first integrals of the disturbed system and they remain approximately constant for long time intervals during which the parameters change considerably. For a plane motion, the adiabatic invariants used to obtain an analytical solution for envelope of a deflection angle of the defunct satellite. This work extends the theory to the three-dimensional tumbling motion of a satellite on an orbit. The obtained results can be applied to study an opportunity of the space debris removal by the Coulomb interaction with the active spacecraft as a pusher.

     

Analytical solutions for dynamics of dual-spin spacecraft and gyrostat-satellites under magnetic attitude control in omega-regimes

The attitude dynamics of a dual-spin spacecraft (a gyrostat with one rotor) with magnetic actuators attitude control is considered in the constant external magnetic field at the presence of the spacecraft’s own magnetic dipole moment, which is created proportionally to the angular velocity components (this motion regime can be called as “the omega-regime” or “the omega-maneuver”). The research of the dual-spin spacecraft angular motion under the action of the magnetic restoring torque is fulfilled in the generalized formulation close to the classical mechanics’ task of the heavy body/gyrostat motion in the Lagrange top. Analytical exact solutions of differential equations of the motion are obtained for all parameters in terms of elliptic integrals and the Jacobi functions. New obtained analytical solutions can be classified as results developing the classical fundamental problem of the rigid body and gyrostat motion around the fixed point. The technical application of the omega-regime to the angular reorientation of the spacecraft longitudinal axis along the angular momentum vector is considered.     

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.     

On the chaotic rotation of a liquid-filled gyrostat via the Melnikov-Holmes-Marsden integral

The non-linear motions of a gyrostat with an axisymmetrical, fluid-filled cavity are investigated. The cavity is considered to be completely filled with an ideal incompressible liquid performing uniform rotational motion. Helmholtz theorem, Euler's angular momentum theorem and Poisson equations are used to develop the disturbed Hamiltonian equations of the motions of the liquid-filled gyrostat subjected to small perturbing moments. The equations are established in terms of a set of canonical variables comprised of Euler angles and the conjugate angular momenta in order to facilitate the application of the Melnikov-Holmes-Marsden (MHM) method to investigate homoclinic/heteroclinic transversal intersections. In such a way, a criterion for the onset of chaotic oscillations is formulated for liquid-filled gyrostats with ellipsoidal and torus-shaped cavities and the results are confirmed via numerical simulations.     

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.     

非轴对称陀螺体自激运动的定性分析

研究非轴对称陀螺体在自激控制力矩作用下的姿态运动．将无力矩状态的两个初积分的积分常数T和G^2作为描述自激运动的状态变量建立其微分方程．利用受扰运动在(T，G^2)空间中的相轨迹判断稳态运动的稳定性，并进行了数值验证．     

万有引力场中陀螺体的混沌运动

研究万有引力场中沿圆轨道运行的非对称陀螺体的姿态运动,引入Ｄｅｐｒｉｔ正则变量建立系统的Ｈａｍｉｌｔｏｎ结构,利用Ｍｅｌｎｉｋｏｖ方法证明在万有引力短作用的昆体产生混沌运动的可能性。对Ｐｏｉｎｃａｒｅ截面的数值计算表明提高陀螺体的转子转速可对混沌起抑制作用。     

A new type of non-Noether exact invariants and adiabatic invariants of generalized Hamiltonian systems

For a generalized Hamiltonian system with the action of small forces of perturbation, the Lie symmetries, symmetrical perturbation, and adiabatic invariants is presented. Based on the invariance of equations of motion for the system under general infinitesimal transformation of Lie groups, the Lie symmetrical determining equations, and exact invariants of the system are given. Then the determining equations of Lie symmetrical perturbation and adiabatic invariants of the disturbed systems are obtained. Furthermore, in the special infinitesimal transformations, two deductions are given. At the end of the paper, one example is given to illustrate the application of the method and result.     

Topological Invariants and Solutions with a High Complexity for Scalar Semilinear PDE

In this paper topological invariants of dynamics of scalar parabolic equations are defined. Conservation of these invariants causes persistence of complex spatially chaotic patterns in the dynamics. It implies, in particular, the existence of extremely many stable essentially different spatially complex solutions of scalar semilinear parabolic, elliptic and hyperbolic equations in the situation when coefficients of equations depend on the spatial variable strongly enough and the domain is large or unbounded, the number of different solutions exponentially depends on the volume of domain. Another implication of the conserved quantities is persistence of complexity of solutions in the dynamics.     

Effect of elasticity on the dynamics of a superconducting rotor rotating in a magnetic field

A spherical shape of the outer surface of rotors of some types of noncontact gyroscopes gives rise to conditions, where the force field ensures the stability of the center of mass relative to the base and has an insignificant effect on the angular motion of the rotor. However, there are some effects (for instance, the Barnett—London effect), which lead to emergence of moments of mechanical forces even for spherical bodies. The effect of rotor elasticity on the motion of a superconducting deformable spherical solid body in a magnetic field is studies. It is shown that the moment of mechanical forces acting on the body in the magnetic field is proportional in the first approximation to the angular velocity squared. The effect of this moment on the dynamics of angular motion of the rotor is studied.     

Lie symmetrical perturbation and a new type of non-Noether adiabatic invariants for disturbed generalized Birkhoffian systems

For a generalized Birkhoffian system with the action of small disturbance, the Lie symmetrical perturbation and a new type of non-Noether adiabatic invariants are presented. On the basis of the invariance of disturbed generalized Birkhoffian system under general infinitesimal transformation of group, the determining equation of Lie symmetrical perturbation of the system is constructed. Based on the definition of higher-order adiabatic invariants of a mechanical system, a new type of non-Noether adiabatic invariants of a disturbed generalized Birkhoffian system is obtained by investigating the Lie symmetrical perturbation. Then, a new type of exact invariants of non-Noether type is given, furthermore adiabatic invariants and exact invariants of non-Noether type are obtained under the special infinitesimal transformation of group. Finally, an example is given to illustrate the application of the method and results.     

Lie symmetries,symmetrical perturbation and a new adiabatic invariant for disturbed nonholonomic systems

For a nonlinear nonholonomic constrained mechanical system with the action of small forces of perturbation, Lie symmetries, symmetrical perturbation, and a new type of non-Noether adiabatic invariants are presented in general infinitesimal transformation of Lie groups. Based on the invariance of the equations of motion for the system under general infinitesimal transformation of Lie groups, the Lie symmetrical determining equations, constraints restriction equations, additional restriction equations, and exact invariants of the system are given. Then, under the action of small forces of perturbation, the determining equations, constraints restriction equations, and additional restriction equations of the Lie symmetrical perturbation are obtained, and adiabatic invariants of the Lie symmetrical perturbation, the weakly Lie symmetrical perturbation, and the strongly Lie symmetrical perturbation for the disturbed nonholonomic system are obtained, respectively. Furthermore, a set of non-Noether exact invariants and adiabatic invariants are given in the special infinitesimal transformations. Finally, one example is given to illustrate the application of the method and results.     

On the rotational motion of a gyrostat about a fixed point with mass distribution

The perturbed rotational motion of a gyrostat about a fixed point with mass distribution near to Lagrange’s case is investigated. The gyrostat is subjected under the influence of a variable restoring moment vector, a perturbing moment vector, and a third component of a gyrostatic moment vector. It is assumed that the angular velocity of the gyrostat is sufficiently large, its direction is close to the axis of dynamic symmetry, and the perturbing moments are small as compared to the restoring ones. These assumptions permit us to introduce a small parameter. Averaged systems of the equations of motion in the first and second approximations are obtained. Also, the evolution of the precession angle up to the second approximation is determined. The graphical representations of the nutation and precession angles are presented to describe the motion at any time.     

Adiabatic invariants for dynamical systems with one degree of freedom

Adiabatic invariants for dynamical systems with one degree of freedom, whose equation of motion is (1), and where the existence of the corresponding Hamilton action integral is not imposed, are established. The adiabatic invariants may vary according to their structure. Using the theory a few particular problems, including non-autonomous Duffing and Van der Pol oscillators, are analysed. Finally, it is indicated how the adiabatic invariants can be used for finding approximate solutions, stability analyses and for plotting phase curves.     

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.

     

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.     

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.     

带偏心转子陀螺体的混沌运动

本文讨论了无力矩条件下带有质量偏心轴对称转子的非对称陀螺体的运动。利用动量变量列写动力学方程,并将系统化作受周期微扰作用下的Ｅｕｌｅｒ－Ｐｏｉｎｓｏｔ运动。应用Ｍｅｌｎｉｋｏｖ方法预测系统存在Ｓｍａｌｄ马蹄意义下的混沌运动,此结论与Ｐｏｉｎｃａｒｅ截面的数值计算相符。从Ｐｏｉｎｃａｒｅ截面的相图也可明显看出转子对于双自旋卫星的姿态稳定作用。     

On the stability behaviour of the rotational motion in a circular orbit for an elastic — rigid body

In this paper the stability of motion of a symmetric satellite-gyrostat, with Visco-elastic square plate attached to it, around the center of mass in a circular orbit under a central gravitational field is studied. The gyrostat has a rotor oriented inside it such that the rotor becomes dynamically as well as statically stable with respect to the whole system. A study of the stability of this system has been done by using the method of separation of motion and averaging. Furthermore, we have solved the problem where there is no rotor.     

Chaotic motion in a class of asymmetrical Kelvin type gyrostat satellite

This work is an investigation on the roots of chaotic attitudinal motion in a class of asymmetrical gyrostat satellites. The result shows that for a class of Kelvin type gyrostat satellite, there is an equivalent rigid spinning satellite with the same attitude dynamics. Finding some constants of motion and eliminating the cyclic coordinates, the rotational kinetic energy is changed to a quadratic form and using Jordan canonical form of the associated inertia tensor and transforming the coordinate system, the Hamiltonian has been changed to those of a rigid satellite. The Hamiltonian has been split into integrable and non-integrable parts. Using Deprit canonical transformation and Andoyer variables the integrable part has been reduced to a one-dimensional form. The reduced Hamiltonian shows that the regular dynamics of the satellite can be chaotic, under the influence of gravitational effects. To demonstrate various attitudinal dynamics of the satellite, a second-order Poincaré map is employed. This research shows firstly, that the attitudinal dynamics of Kelvin type gyrostat satellites and rigid satellites follow the same dynamical patterns, secondly, for non-linear analysis of dynamics of gyrostat satellite based on the perturbation methods, there is a preferable form for Hamiltonian of the system in the near-integrable fashion and thirdly the chaotic motion is originated from the gravitational field effects that can be suppressed by increasing the attitudinal energy of the satellite in comparison with the translational energy.