scholarly journals INVESTIGATION OF THE EVOLUTION EQUATIONS OF THE AXISYMMETRIC BODY WITH

2020 ◽  
Vol 69 (1) ◽  
pp. 247-252
Author(s):  
M.Zh. Minglibayev ◽  
◽  
S.B. Bizhanova ◽  
Keyword(s):  
Coordinate System ◽  
Rotational Motion ◽  
Evolution Equations ◽  
Second Harmonic ◽  
Variable Mass ◽  
Dynamic Structure ◽  
Approximate Expression ◽  
Axisymmetric Body ◽  
Osculating Elements ◽  
Two Bodies

In this article we consider mutually gravitating non-stationary two bodies: first body is «central», it is a sphere with a spherical density distribution, the second body is «satellite», which has an axisymmetric dynamic structure and form. Newtonian force interaction is characterized by an approximate expression of the force function, which takes into account the second harmonic. The differential equation of translational-rotational motion of the axisymmetric body is derived with variable mass and variable size in a relative coordinate system. The axes of the own coordinate system for nonstationary two bodies coincides with the main axes of inertia and this position remains unchanged during evolution. The mass of bodies are varied isotropically in the different rates. The problem is investigated by methods perturbation theory. The equations of secular perturbations of translational-rotational motion of satellite are deduced in the analogues osculating elements Delaunay-Andoyer. The solutions of the differential equations of the perturbed motion are obtained by the numerical method and the graphs are constructed using the Wolfram Mathematica package.

scholarly journals TRANSLATIONAL-ROTATIONAL MOTION OF A NONSTATIONARY AXISYMMETRIC BODY

2021 ◽  
Vol 2 (336) ◽  
pp. 131-137
Author(s):  
M. Zh. Minglibayev ◽  
S. B. Bizhanova
Keyword(s):  
Rotational Motion ◽  
Evolution Equations ◽  
Second Harmonic ◽  
First Integral ◽  
Approximate Expression ◽  
Axisymmetric Body ◽  
Order System ◽  
Dynamical Structure ◽  
Symmetric Density ◽  
Principal Axes

A nonstationary two-body problem is considered such that one of the bodies has a spherically symmetric density distribution and is central, while the other one is a satellite with axisymmetric dynamical structure, shape, and variable oblateness. Newton’s interaction force is characterized by an approximate expression of the force function up to the second harmonic. The masses of the central body and the satellite vary isotropically at different rates and do not occur reactive forces and additional rotational moments. The nonstationary axisymmetric body have an equatorial plane of symmetry. Thus, it has three mutually perpendicular planes of symmetry. The axes of its intrinsic coordinate system coincide with the principal axes of inertia and they are directed along the intersection lines of these three mutually perpendicular planes. This position remains unchangeable during the evolution. Equations of motion of the satellite in a relative system of coordinates are considered. The translational- rotational motion of the nonstationary axisymmetric body in the gravitational field of the nonstationary ball is studied by perturbation theory methods. The equations of secular perturbations reduces to the fourth order system with one first integral. This first integral is considered and three-dimensional graphs of this first integral are plotted using the Wolfram Mathematica system.

TRANSLATIONAL-ROTATIONAL MOTION OF THE TRIAXIAL BODY WITH VARIABLE COMPRESSIONS IN THE PRESENCE OF REACTIVE FORCES AND MOMENTS

2020 ◽  
Vol 69 (1) ◽  
pp. 241-246
Author(s):  
M.Zh. Minglibayev ◽  
◽  
O.B. Baisbayeva ◽  
Keyword(s):  
Coordinate System ◽  
Rotational Motion ◽  
Variable Mass ◽  
Spherical Body ◽  
Relative Orientation ◽  
The Body ◽  
Variable Size ◽  
Osculating Elements ◽  
Dynamic Shape ◽  
Central Gravitational Field

In this paper we investigated the translational-rotational motion of a triaxial body of constant dynamic shape and variable mass and size in a non-stationary Newtonian central gravitational field. Differential equations of the translationalrotational motion of the triaxial non-stationary body in the relative coordinate system with the origin at the center of a non-stationary spherical body are derived. The axes of the own coordinate system of the non-stationary triaxial body are directed along the principle axes of inertia of the body and we assumed that in the course of evolution their relative orientation remains unchanged. An analytical expression for the force function of the Newtonian interaction of the triaxial body of variable mass and size with a spherical body of variable size and mass is given. In the presence of reactive forces and moments the equations of translational-rotational motion of a triaxial non-stationary body in osculating elements are obtained in the presence of reactive forces and moments.

MATHEMATICAL MODELS OF ANGULAR MOTION OF SPACE VEHICLES AND THEIR USE IN ORIENTATION CONTROL PROBLEMS

2021 ◽  
Vol 5 ◽  
pp. 124-139
Author(s):  
Viktor Volosov ◽  
◽  
Vladimir Shevchenko ◽  
Keyword(s):  
Coordinate System ◽  
Attitude Control ◽  
Evolution Equations ◽  
General Structure ◽  
Control Synthesis ◽  
Space Vehicles ◽  
Direct Lyapunov Method ◽  
Coordinate Systems ◽  
Advantages And Disadvantages ◽  
Modified Rodrigues Parameters

A general structure of the kinematic equations for attitude evolution of a spacecraft (SC) (coordinate system associated with a spacecraft (SCS)) relative to the reference coordinate system (RCS) is proposed. It is assumed that the origins of the coordinate systems coincide and are located at an arbitrary point of the spacecraft. Each of the coordinate systems rotates at an arbitrary absolute angular velocity (relative to the inertial space) specified by the projections on their axes. Attitude parameters can be the Euler–Krylov angles, Rodrigues–Hamilton parameters, and modified Rodrigues parameters. It is shown that the well-known representations of the attitude evolution equations of the SCS relative to the RCS using the Rodrigues-Hamilton parameters (components of normalized quaternions) can be simply obtained from the solution of the Erugin problem of finding the entire set of differential equations with a given integral of motion. The advantages and disadvantages of use for each of the specified attitude parameters are considered. A method of attitude control synthesis is proposed which is common for all these equations and based on the decomposition of the original problem into kinematic and dynamic ones and the use of well-known generalizations of the direct Lyapunov method for their solution. The property of structural roughness according to Andronov–Pontryagin [27–29] of the obtained algorithm is illustrated with the help of computer simulation. Particularly, a specific example illustrates the possibility for even a structurally simplified algorithm of stabilizing a specified constant spacecraft attitude to track the program of its change with sufficient accuracy. The tracking task is typical for the control of spacecraft docking, spacecraft de-orbiting, and performing route surveys of the Earth's surface.

The Bargmann Constraint and Integrable System for a Second-Order Spectral Problem

2015 ◽  
Vol 70 (11) ◽  
pp. 913-917
Author(s):  
Wei Liu ◽  
Yafeng Liu ◽  
Shujuan Yuan
Keyword(s):  
Coordinate System ◽  
Spectral Problem ◽  
Integrable System ◽  
Evolution Equations ◽  
Lagrange Equations ◽  
Lax Pairs ◽  
Infinite Dimensional ◽  
Hamilton System ◽  
Finite Dimensional ◽  
Dimensional Soliton

AbstractIn this article, the Bargmann system related to the spectral problem (∂2+q∂+∂q+r)φ=λφ+λφx is discussed. By the Euler–Lagrange equations and the Legendre transformations, a suitable Jacobi–Ostrogradsky coordinate system is obtained. So the Lax pairs of the aforementioned spectral problem are nonlinearised. A new kind of finite-dimensional Hamilton system is generated. Moreover, the involutive solutions of the evolution equations for the infinite-dimensional soliton system are derived.

scholarly journals NEW FORMS OF THE PERTURBED MOTION EQUATION

REPORTS ◽  
2020 ◽  
pp. 5-13
Author(s):  
M.Zh. Minglibayev ◽  
Ch.T. Omarov ◽  
A.T. Ibraimova
Keyword(s):  
Model Problem ◽  
Variable Mass ◽  
Initial Approximation ◽  
Motion Equation ◽  
Mutual Distance ◽  
Celestial Bodies ◽  
Gravitating Systems ◽  
Osculating Elements ◽  
Geometric Elements ◽  
Dynamic Element

Real celestial bodies are neither spherical nor solid. Celestial bodies are unsteady, in the process of evolution their masses, sizes, shapes and structures are changes. The paper considers a model problem proposed as an initial approximation for the problems of celestial mechanics of bodies with variable mass. Based on this model problem, perturbation theory methods are developed and new forms of the perturbed motion equation are obtained. The model problem as the problem of two bodies with variable mass in the presence of additional forces proportional to speed and mutual distance is a class of intermediate motions. This class of intermediate motions describes an aperiodic motion along a quasiconical section. In this paper, on the basis of this class of aperiodic motion over a quasiconical section, various new forms of the perturbed motion equation in the form of Newton's equations are obtained. Based on the known equations of perturbed motion for the osculating geometric elements p, e, , i, , in the form of the Newton equation, we obtained the equations of perturbed motion for the following system of osculating elements p, e, i, , , and a, e, i, , , . Oscillating variables involving a dynamic element are suitable in the general case. A system of variables, where instead of the dynamic element is introduced - the average longitude in orbit is used in the quasielliptic case . The obtained new forms of the equation of perturbed motion, in the form of Newton's equations, in various systems of osculating variables can be effectively used in the study of the dynamics of non-stationary gravitating systems.

On Constructing the Inertial System of High Accuracy by VLBI Methods

1975 ◽  
Vol 26 ◽  
pp. 381-393
Author(s):  
C. A. Krasinsky
Keyword(s):  
Coordinate System ◽  
Rotational Motion ◽  
High Accuracy ◽  
Inertial System ◽  
Inertial Coordinate System ◽  
Angular Measurements ◽  
Celestial Sphere

AbstractImprovement of the accuracy of angular measurements to values of, which theoretically may be achieved by VLBI methods, demands an inertial coordinate system of the corresponding quality. Such a system cannot depend on parameters of the Earth’s orbital and rotational motion and must be based on other principles than the adopted system. An approach which seems most natural was developed by E.P. Fedorov who proposed construction of the inertial system by determining the mutual angular distances between objects on the celestial sphere. In this paper, a method of synchronous and quasi-synchronous observations is developed which may realize this idea.

scholarly journals Exact Periodic Seismic Waves in the Nikolaevskii Model

2014 ◽  
Vol 44 (3) ◽  
pp. 91-106
Author(s):  
Ognyan Y. Kamenov
Keyword(s):  
Periodic Solutions ◽  
Nonlinear Evolution Equation ◽  
Seismic Waves ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Dynamic Structure ◽  
Integrable Evolution ◽  
The Common ◽  
Spatial Displacements ◽  
First Time

Abstract In the present paper three different in their structure families, of exact periodic solutions of the nonlinear evolution equation of Nikolaevskii, have been obtained. The common dynamic structure of these families of periodic solutions has been shown as well as the spatial displacements, typical of the non-integrable evolution equations, for each separate harmonics. These exact solutions are published for the first time

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