Stability of the steady rotations of a satellite with internal damping in a central gravitational field

The dynamics of two bodies connected by a hinge joint, and moving in a plane under the action of a central gravitational force field is analyzed. Each body is modeled as a rigid massless link with a point mass at one end; their other ends are connected together by a hinge joint. The equations of motion of the connected bodies include the equations for the orbital motion of the bodies, the orientation (attitude) of the assembly, and the relative orientation (shape) of the bodies with respect to each other. Dynamic coupling between these degrees of freedom give rise to a complex dynamical system. Relative equilibria, corresponding to circular orbits of fixed radius, are obtained from these equations of motion. The free dynamics has a symmetry due to the cyclic coordinate representing the true anomaly. Routh reduction is carried out to eliminate this coordinate and obtain the reduced dynamics. We carry out stability analysis for the relative equilibria. Numerical simulations using a symplectic integrator are carried out for perturbations from these relative equilibria, to confirm their stability properties. These numerical simulations also suggest the use of shape change to alter the overall orientation and orbit of the assembly.

Multibody systems in planar motion are modelled as two or more rigid components that are connected and can move relative to each other. The dynamics of such multibody systems in planar motion in a central gravitational force field is analysed. The equations of motion of the system include the equations for the orbital motion of the bodies, the orientation (attitude) of the assembly, and the relative orientation (shape) of the bodies with respect to each other. Dynamic coupling between these degrees of freedom gives rise to complex dynamical systems that are usually not integrable. Relative equilibria, corresponding to circular orbits of the multibody system, are obtained. The free dynamics has a symmetry due to a cyclic coordinate. Routh reduction is carried out to eliminate this coordinate and obtain the reduced dynamics. The stability of the relative equilibria is analysed using the Routh stability criterion when it is applicable; an expansion of the Hamiltonian in normal form is used otherwise. We apply the general results to a multibody system consisting of two hinged planar bodies, each modelled as a rigid massless link with a point mass at one end with their other ends connected by a hinge joint. We obtain the relative equilibria of this model, and carry out a stability analysis for the relative equilibria. Numerical simulations using a symplectic integrator are carried out for perturbations to these relative equilibria, to confirm their stability properties.

In the framework of the model of M.A. Lavrentiev, the effect of internal elastic and dissipative forces on the rotational motion of the planet in a central gravitational field in a circular orbit is studied. The averaged equations of the rotational motion of the planet are derived. The stability of plane rotations is investigated. The analysis of the evolution of rotational motion depending on the values of the parameters and the initial conditions is carried out.

In this paper we study the problem of the motion of a two-gyrostat chain about a fixed point in a central gravitational field. We assume that the mass distribution of each gyrostat is analogous to the one of a Lagrange top, the gyrostatic moment of each gyrostat is constant relative to its carrier, and the center of a spherical joint connecting the gyrostats belongs to their dynamic symmetry axes. We establish and analyze sufficient conditions for stability of the chain’s permanent rotations about a vertical axis. Our findings extend corresponding results in the dynamics of a single gyrostat to a case of the two-gyrostat chain as well as generalize some of the known properties of permanent rotations in the many-body dynamics.

Exact solutions of an f(R) -theory (of gravity) in a static central (gravitational) field have been studied in the literature quite well, but, to find and study exact solutions in the case of a non-static central field are not easy at all. There are, however, approximation methods of finding a solution in a central field which is not necessarily static. It is shown in this article that an approximate solution of an f(R)-theory in a general central field, which is not necessary to be static, can be found perturbatively around a solution of the Einstein equation in the general theory of relativity. In particular, vacuum solutions are found for f(R) of general and some special forms. Further, applications to the investigation of a planetary motion and light’s propagation in a central field are presented. An effect of an f(R)-gravity is also estimated for the SgrA*–S2 system. The latter gravitational system is much stronger than the Sun–Mercury system, thus the effect could be much stronger and, thus, much more measurable.

view Abstract Citations (14) References (11) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Time-dependent fluid flow in a central gravitational field. Cheng, A. F. Abstract The time evolution of spherically symmetric self-similar polytropic flows in a central gravitational field is analyzed. The required similarity transformation is performed, boundary conditions are discussed, and some examples of solutions to the equations of motion are given. These describe the evolution of an accretion flow for a polytropic index of 3/2, accretion flows with a polytropic index of 3, and the decay of an explosive flaring process into a steady stellar-wind outflow for an index of 3/2. Phase plane analysis is used to examine the qualitative behavior of the solutions at large values of dimensionless hydrodynamic time, and several general theorems governing this behavior are obtained. Publication: The Astrophysical Journal Pub Date: April 1977 DOI: 10.1086/155185 Bibcode: 1977ApJ...213..537C Keywords: Fluid Flow; Gravitational Fields; Polytropic Processes; Shock Wave Propagation; Stellar Winds; Time Dependence; Hydrodynamics; Magnetohydrodynamic Flow; Steady Flow; Stellar Mass Accretion; Stellar Models; Unsteady Flow; Astrophysics full text sources ADS |

The equilibrium positions of a satellite carrying a three-degree-of-freedom powered gyroscope in a central gravitational field

The impact of internal dissipation on the rotational motion of a satellite in a central gravitational field is investigated. A satellite is modeled as a system of two solid bodies: shell and spherical damper. For a dynamically symmetric satellite in a circular orbit the stability of stationary rotations in dependence on values of damping factor and angular velocity of a satellite is analyzed.

Bifurcation and stability of the steady motions and relative equilibria of a rigid body in a central gravitational field

The dynamics of a dumbbell shaped spacecraft are modeled as two identical mass particles connected by a linear elastic spring. The equations of motion of the spacecraft in a planar orbit in a central gravitational field are presented. The equations of motion characterize orbit, attitude, and shape (or elastic deformation) degrees of freedom and the coupling between them. Relative equilibria, corresponding to circular planar orbits, are obtained from these equations of motion. Linear equations of motion that describe perturbations from these relative equilibria are presented. New dynamics and control problems are introduced for these linear equations. Controllability results are presented for various actuation assumptions, based on the linear equations.

The asymptotic properties of the motions of satellites in a central field due to internal dissipation

In the present work we consider asymmetric gyrostat which has a homogeneous viscoelastic disc and two bars attached to it. Furthermore, the gyrostat has a rotor oriented inside it such that the rotor is statically and dynamically balanced. This sytem has a rotational motion around its center of mass in a circular orbit under a central gravitational field. Bending vibrations of the bars and the disc are accompanied by dissipation of energy, which is the cause of the evolution of the system's rotational motion. Using the method of separation of motion and averaging, the approximate equations describing the evolution of rotational motion in terms of Andoyer canonical variables are obtained. The stationary motions for the system are deduced, together with the conditions of its stability.

For a satellite with a ball damper, the effect of internal dissipation on rotational motion in the central gravitational field is studied. The equations of rotational motion of a dynamically symmetric satellite with a spherical damper in an elliptical orbit are obtained. For the case of a circular orbit, the spatial resonance rotations of a dynamically symmetric satellite with a ball damper were investigated using the averaging method.

C of optimal trajectories requires the solution of the so-called adjoint equations. These are a system of ordinary linear differential equations derived from the original system. The components of the solution vector of this system are often called Lagrangian multipliers. In a recent note Eckenwiler has demonstrated that the adjoint equations allow a closed-form solution during coasting periods of an optimal trajectory. The multipliers were given in terms of the state variables of the original system, which are determined by the Kepler laws. Four different cases had to be considered in accordance with the kind of orbit of the mass-point. The importance of such a closed-form representation also was pointed out in this paper. The purpose of this note is 1) to disclose the general background for the existence of closed-form Lagrangian multipliers, 2) to show that the formulas given by Eckenwiler may be simplified by the introduction of the Kepler time equation, and 3) to extend the results to the case of an inversesquare central gravitational field which is perturbed by an additional term, inverse-proportional to the third power of the center distance. In a general, not necessarily central gravitational field, only one integral of the adjoint equations during coasting periods can be set up immediately. This integral is given by the energy theorem. In a central gravitational field, however, two solution vectors exist in a closed form. They can be expressed in terms of the original state variables. The existence of a second closed-form solution is due to the angular momentum theorem. A fundamental matrix of the adjoint

In this paper, discusses the motion in a central gravitational field under considering space-time expansion, and realize the unification of structure of big scope space-time and physical phenomena of small scope, and set up a new mechanism of the formation and evolution of galaxies and celestial bodies, which is different from the mechanism given by big bang and is based on the continual generation of matter inside celestial bodies or galaxies. Point out that galaxies and celestial bodies are growing up in the course of spacetime’s expansion.