Central Newtonian Force Field Research Articles

Deployment of a space tether in a centrifugal force field with alignment to the local vertical

This study is concerned with a small orbital tether of two bodies to be deployed from a spacecraft so that upon completion of the deployment it turns out to be aligned along the local vertical. The bodies of the tether have equal masses, and the thread connecting the bodies is supposed to be massless. The objective of the study is to build a program law of tether length control taking into account the variation of the angular momentum of the tether under the action of the gravitational torque from the central Newtonian field of forces. The deployment mode of the space tether in a centrifugal force field with its alignment at the conclusion of the deployment along the local vertical is studied. To produce centrifugal forces, the tether is pre-spinned about the orbit binormal. The study consists of two steps. The first step involves the construction of a tether length control law that would provide the planned deployment. At this step, use is made of the tether motion equations written in spherical coordinates for the special case of the tether motion in the orbital plane. A numerical simulation of the tether deployment dynamics is carried out at the second step using the constructed program law of tether length control. Hill-Clohessy-Wiltshire’s equations are used as a mathematical model of the tether. They describe the spatial motion of the tether bodies. These equations do not contain the tether length as a variable in explicit form. Therefore, these equations are modified. The tether tension force appearing in these equations is expressed in terms of the program law of tether length change and its two first time derivatives. The novelty of the study consists in the construction of a program control law that allows the tether to be deployed along the local vertical in a single stage. The study used methods of analytical mechanics, numerical methods, and methods developed by the authors. The obtained results make it possible to find the ranges of values of the deployment system parameters allowing a deployment of this type. The errors of the numerical simulation are estimated. The practical significance of the obtained results consists in the possibility of deploying small tethers in orbit with their alignment at the conclusion of the deployment along the local vertical in a single stage with controlling the tether length without the need for further dumping of libratory oscillations.

Research into the Dynamics of a System of Two Connected Bodies Moving in the Plane of a Circular Orbit by Applying Computer Algebra Methods

Computer algebra methods are used to determine the equilibrium orientations of a system of two bodies connected by a spherical hinge that moves in a central Newtonian force field on a circular orbit under the action of gravitational torque. Primary attention is given to the study of equilibrium orientations of the two-body system in the plane of the circular orbit. By applying symbolic differentiation, differential equations of motion are derived in the form of Lagrange equations of the second kind. A method is proposed for transforming the system of trigonometric equations determining the equilibria into a system of algebraic equations, which in turn are reduced by calculating the resultant to a single algebraic equation of degree 12 in one unknown. The roots of the resulting algebraic equation determine the equilibrium orientations of the two-body system in the circular orbit plane. By applying symbolic factorization, the algebraic equation is decomposed into three polynomial factors, each specifying a certain class of equilibrium configurations. The domains with an identical number of equilibrium positions are classified using algebraic methods for constructing a discriminant hypersurface. The equations for the discriminant hypersurface determining the boundaries of domains with an identical number of equilibrium positions in the parameter space of the problem are obtained via symbolic computations of the determinant of the resultant matrix. By numerical analysis of the real roots of the resulting algebraic equations, the number of equilibrium positions of the two-body system is determined depending on the parameters.

Stationary Motions of a Close to Regular Isosceles Tetrahedron with a Fixed Point in the Central Newtonian Force Field

Stationary Motions of a Close to Regular Isosceles Tetrahedron with a Fixed Point in the Central Newtonian Force Field

Evolution of the rotational motion of a viscoelastic planet with a core on an elliptical orbit

The work is devoted to the study of the evolution of the rotational motion of a planet in the central Newtonian field of forces. The planet is modeled by a body consisting of a solid core and a viscoelastic shell rigidly attached to it. A limited formulation of the problem is considered, when the center of mass of the planet moves along a given Keplerian elliptical orbit. The equations of motion are derived in the form of a system of Routh equations using the canonical Andoyer variables, which are “action-angle” variables in the unperturbed problem and have the form of integro-differential equations with partial derivatives. The technique developed by V.G. Vilke is used for mechanical systems with an infinite number of degrees of freedom. A system of ordinary differential equations is obtained by the method of separation of motions. The system describes the rotational motion of the planet taking into account the perturbations caused by elasticity and dissipation. An evolutionary system of equations for the “action” variables and slow angular variables is obtained by the averaging method. A phase portrait is constructed that describes the mutual change in the modulus of the angular momentum vector G of the rotational motion and the cosine of the angle between this vector and the normal to the orbital plane of the planet’s center of mass. A stationary solution of the evolutionary system of equations is found, which is asymptotically stable. It is shown that in stationary motion, the angular momentum vector G is orthogonal to the orbital plane, and the limiting value of the modulus of this vector depends on the eccentricity of the elliptical orbit. The constructed mathematical model can be used to study the tidal evolution of the rotational motion of planets and satellites. The results obtained in this work are consistent with the results of previous studies in this area.

Rotation of Isosceles Tetrahedron in Central Newtonian Force Field: Staude Cone

Rotation of Isosceles Tetrahedron in Central Newtonian Force Field: Staude Cone

Symbolic-Analytic Methods for Studying Equilibrium Orientations of a Satellite on a Circular Orbit

This paper proposes two simple methods for determining equilibrium orientations of a satellite moving in a central Newtonian force field along a circular orbit under the action of the gravitational torque. The first method uses linear algebra approaches, while the second one employs computer algebra algorithms. The equilibrium orientations of the satellite in the orbital coordinate system for given values of principal central moments of inertia are determined by the roots of a system of nonlinear algebraic equations. To determine the equilibrium solutions, the system of algebraic equations is decomposed using linear algebra methods and algorithms for Grobner basis construction. The equilibria of the satellite are determined by analyzing the real roots of the algebraic equations from the Grobner bases constructed. Using the proposed approach, it is shown that the satellite with unequal principal central moments of inertia has 24 equilibrium orientations on a circular orbit.

Mathematical Simulation of Satellite Motion with an Aerodynamic Attitude Control System Influenced by Active Damping Torques

The dynamics of a satellite moving in a central Newtonian force field in a circular orbit under the influence of aerodynamic and active damping torques depending on projections of the satellite’s angular velocity is studied. A method for determining all equilibrium positions (equilibrium orientations) of the satellite in the orbital coordinate system given the values of aerodynamic torque, damping coefficients, and the principal central moments of inertia is proposed. In the case when the axes of the coordinate system attached to the satellite coincide with the axes of the orbital coordinate system, necessary and sufficient conditions for the asymptotic stability of the corresponding zero equilibrium position are obtained using the Routh–Hurwitz criterion. The domains with satisfied asymptotic stability conditions for the zero equilibrium position are analyzed depending on various dimensionless parameters of the problem. The damping of spatial oscillations of the satellite is numerically studied for various values of aerodynamic torque and damping coefficients.

On stability of planar periodic motions of a satellite in vicinity of resonant rotation

The stability of planar resonant periodic rotations of a rigid body around its centre of mass in a central Newtonian field of forces is considered. The orbit of the centre of mass of the satellite is assumed to be an elliptical one, the body’s inertia ellipsoid is general. Resonant rotations examined mean that the body makes 3 turns around its centre of mass during 2 revolutions of the centre of mass on its orbit (Mercury-type resonance 3:2). For the set of eccentricity value e ε [0,0.25] typical for planets’ ones the regions of stability in the first approximation and regions of unstability are built in three-dimensional space of parameters (eccentricity and two inertial parameters).

Эволюция вращательного движения спутника с гибкими вязкоупругими стержнями на эллиптической орбите

The rotational motion of an artificial satellite in the central Newtonian force field is investigated. The satellite is modeled as a symmetric solid with rigid viscoelastic rods rigidly attached along the symmetry axis and with a spherical inertia tensor. A limited statement of the problem is considered: the center of mass of the satellite moves along a given Keplerian elliptic orbit. As is known, some artificial Earth satellites (AES) have whip radio antennas for a length of hundreds of meters. The problem considered in this paper is a model for studying the rotational motion of this type of satellites. To solve this problem, the method of separation of motions and averaging is used for mechanical systems with an infinite number of degrees of freedom. We obtain an averaged system of differential equations in the Andoyer variables, which describes the evolution of the satellite's rotational motion. Partial solutions of the evolutionary system of the equations are found. For each such class of solutions phase portraits are built. It is shown that in the process of dissipative evolution caused by energy dissipation in rods, the angle between the vector of the angular momentum of the satellite's rotational motion and the normal to the orbital plane decreases to zero. Besides, new classes of stationary motions are found: the axis of symmetry of the satellite makes an arbitrary angle with the normal to the orbit plane, or the module of the angular velocity of the satellite rotation does not depend on the orbital angular velocity of the radius vector of its center of mass.

Gravity-oriented satellite dynamics subject to gravitational and active damping torques

The dynamics of the rotational motion of a satellite moving in the central Newtonian field of force over a circular orbit under the effect of gravitational and active damping torques, which depend on the satellite angular velocity projections, has been investigated. The paper proposes a method of determining all equilibrium positions (equilibrium orientations) of a satellite in the orbital coordinate system for specified values of damping coefficients and principal central moments of inertia. The conditions of their existence have been obtained. For a zero equilibrium position where the axes of the satellite-centered coordinate system coincide with the axes of the orbital coordinate system, the necessary and sufficient conditions for asymptotic stability are obtained using the Routh–Hurwitz criterion. A detailed analysis of the regions where the conditions of the asymptotic stability of a zero equilibrium position are fulfilled have been obtained depending on three dimensionless parameters of the problem, and the numerical study of the process of attenuation of satellite’s spatial oscillations for various damping coefficients has been carried out. It has been shown that there is a wide range of damping parameters from which, by choosing the necessary values, one can provide the asymptotic stability of satellite’s zero equilibrium position in the orbital coordinate system.

On the Stability of Periodic Mercury-type Rotations

We consider the stability of planar periodic Mercury-type rotations of a rigid body around its center of mass in an elliptical orbit in a central Newtonian field of forces. Mercurytype rotations mean that the body makes 3 turns around its center of mass during 2 revolutions of the center of mass in its orbit (resonance 3:2). These rotations can be 1) symmetrical 2π- periodic, 2) symmetrical 4π-periodic and 3) asymmetrical 4π-periodic. The stability of rotations of type 1) was investigated by A.P.Markeev. In our paper we present a nonlinear stability analysis for some rotations of types 2) and 3) in 3rd- and 4th-order resonant cases, in the nonresonant case and at the boundaries of regions of linear stability.

Single-axis equilibrium orientations to the attracting center of a symmetrical prolate orbital gyrostat with an elastic rod

Under study in the restricted formulation is the motion of a symmetrical prolate stationary gyrostat along a Keplerian circular orbit in a central Newtonian field of forces. An elastic homogeneous rod, rectilinear in the undeformed state, is rigidly clamped by one end in the body of gyrostat along its axis of symmetry. There is a point mass at the free end of the rod. The inextensible elastic rod, for simplicity of constant circular cross-section, performs infinitesimal space oscillations in the process of system motion. In this case, we neglect the terms in the system’s tensor of inertia which are nonlinear with respect to displacements of the points of the rod.We consider the following (so-called semi-inverse) problem: Under what kinetic momentumof the flywheel, among the relative equilibriums of the system (the states of rest relative to the orbital coordinate system) does there exist an equilibrium such that the axis, arbitrarily chosen in the coordinate system associated with the gyrostat, is collinear with the local vertical? In the discretization of the problem, we present the values of the Lagrange coordinates that define the deformation of the rod for these equilibria and the value of gyrostatic moment providing the presence of the equilibrium in question.

Satellite dynamics due to gravity and constant torques

We consider the dynamics of the rotational motion of a satellite moving in the central Newtonian force field in a circular orbit due to gravity and constant torques. We propose a method for determining all positions of equilibrium (equilibrium orientations) of a satellite in the orbital coordinate system given the values of the vector of constant torque and the major central moments of inertia; we obtained the conditions for the existence of these positions. We classify the domains with an equal number of equilibrium positions using algebraic methods for the construction of discriminant hypersurfaces. The results obtained in this study can be used to construct gravitational control systems for the orientation of artificial Earth satellites.

Satellite dynamics under the influence of gravitational and aerodynamic torques. A study of stability of equilibrium positions

The dynamics of the rotational motion of a satellite, moving in the central Newtonian force field under the influence of gravitational and aerodynamic torques, is investigated. The paper proposes a method for determining all equilibrium positions (equilibrium orientations) of a satellite in the orbital coordinate system for specified values of aerodynamic torque and the major central moments of inertia; the sufficient conditions for their existence are obtained. For each equilibrium orientation the sufficient stability conditions are obtained using the generalized energy integral as the Lyapunov function. The detailed numerical analysis of the regions where the stability conditions of the equilibrium positions are satisfied is carried out depending on four dimensionless parameters of the problem. It is shown that, in the general case, the number of satellite’s equilibrium positions, for which the sufficient stability conditions are satisfied, varies from 4 to 2 with an increase in the value of the aerodynamic torque magnitude.

Dynamics of a gyrostat satellite subjected to the action of gravity moment. Equilibrium attitudes and their stability

We study the dynamics of the rotational motion of a gyrostat satellite moving in the central Newtonian force field along a circular orbit. We propose a method for determining the equilibrium attitudes (equilibrium orientations) of a gyrostat satellite in the orbital coordinate system for given values of the gyrostatic moment vector and principal central moments of inertia, and obtain their existence conditions. For each equilibrium orientation, sufficient conditions for stability are obtained using a generalized energy integral such as a Lyapunov function. We conduct a detailed numerical analysis of domains where the stability conditions for equilibrium attitudes are satisfied depending on four dimensionless parameters of the problem. It is shown that the number of equilibrium attitudes of a gyrostat satellite for which the sufficient conditions of stability are satisfied in the general case varies from four to two with an increase in the magnitude of a gyrostatic moment. The results obtained in this paper can be used for constructing gravitational systems of control over the orientation of the Earth's artificial satellites.

Dynamics of an axisymmetric gyrostat satellite. Equilibrium positions and their stability

The dynamics of an axisymmetric gyrostat satellite in a circular orbit in the central Newtonian force field is investigated. All the equilibrium positions of the gyrostat satellite in the orbital system of coordinates are determined, and the conditions for their existence are analysed. All the bifurcation values of the system parameters at which the number of equilibrium positions changes are found. It is shown that, depending on the values of the parameters of the problem, the number of equilibrium positions of a gyrostat satellite can be 8, 12 or 16. The evolution of regions where the sufficient conditions for stability of the equilibrium positions hold is investigated.

Bifurcations of the relative equilibria of a gyrostat satellite for a special case of the alignment of its gyrostatic moment

The bifurcations of the equilibria of a gyrostat satellite with a centre of mass moving uniformly in a circular Kepler orbit around an attracting centre are investigated. It is assumed that the axis of rotation of a statically and dynamically balanced flywheel rotating at a constant relative angular velocity is fixed in the principal central plane of inertia of the gyrostat containing the axis of its mean moment of inertia and that it is not collinear with any principal central axis of inertia of the system. The problem is solved in a direct formulation, that is, the whole set of equilibria with respect to the orbital system of coordinates of the gyrostat satellite is determined using the given moments of inertia, the value of the gyroscopic moment and the direction cosines of the axis of rotation of the flywheel and the changes in this set are investigated as a function of the bifurcation parameter, that is, the magnitude of the gyrostatic moment of the system. A parametric analysis of the relative equilibria of the three possible classes of equilibria for a system in a circular orbit in a central Newtonian force field is carried out using computer algebra facilities.

Evolution of motion of two viscoelastic planets in the field of forces of their mutual attraction

Translational-rotational motion of two viscoelastic planets in a gravitational force field is studied. The planets are modeled by homogeneous isotropic viscoelastic bodies. In their natural undeformed state each of the planets represents a sphere. We investigate a specific case when the planet’s centers of mass move in a fixed plane, the axis of rotation for each planet being directed along the normal to this plane. An equation describing the evolution of a slow angular variable (perihelion longitude) is derived. The observed displacement of the perihelion of Mercury is compared with the results obtained in the considered model problem about motion of two viscoelastic planets. Quite important is the fact that the planet of smaller mass (Mercury) moves not in a central Newtonian field of forces, but rather in the gravitational field of a rotating viscoelastic planet (Sun).

Motion of a Rigid Body in a Newtonian Field of Force Exerted by Three Attracting Centers

In this paper, a nonsymmetric rigid body rotating around a fixed point under the action of a central Newtonian field of force exerted by three centers of attraction is considered. The angular momentum principle is applied to deduce the equations of motion of this body. These equations represent a simple autonomous system of twelve nonlinear ordinary differential equations, which describe the motion of the body. The first integrals for such a system are obtained. Euler, Lagrange, and the kinetic symmetry cases are obtained as special cases from this problem. The numerical solution for this system is obtained by using the fourth-order Runge-Kutta method. The aim is to find the influence of the characteristic parameters of the body on the motion. Two cases of study are given. The first occurs when the three attracting centers lie on the fixed axes OX, OY, and OZ, and the second occurs when these centers are reduced to one attracting center that lies on the vertical OZ-axis. A comparison between the solutions...

Dynamics of an axisymmetric gyrostat satellite under the action of gravitational moment

Dynamics of an axisymmetric gyrostat satellite moving in the central Newtonian force field along a circular orbit is investigated. All equilibrium positions of the gyrostat satellite in the orbital coordinate system are determined, and the sufficient stability conditions of equilibrium positions are derived.