The problem of stabilizing a spin satellite by means of passive dampers is considered. The application of the method of integral manifolds allows us to find conditions for the loss of stability in the analytical form. A lot of work has been devoted to the study of dynamic models of stabilization of satellites with the help of gyroscopic forces. As the main apparatus, the Lyapunov function method and the stability criteria applied to first approximation systems are used. In addition to gyroscopic forces for stabilization, damping devices are used in a number of models to ensure the asymptotic stability of the required modes of satellite motion. In a number of works, passive dampers are considered as such devices. For the case of two co-axial bodies, on each of which one damper is installed, the stabilization problem example, in [1-3]. In this paper, we confine ourselves to the study of a model of a satellite consisting of two bodies, on one of which a damper with a relatively small coefficient of viscous friction is installed. The damper is modeled by a particle of relatively small mass placed in a tube filled with a viscous liquid and attached by a spring. To analyze the system of differential equations, the method of integral manifolds [3, 4] is applied, which allows to significantly reduce the dimensionality of the model and simplify the analysis.
1
small parameter. Some details can be found in [
The system of differential equations under consideration has a manifold of steady states:
Following [
1 = 0}. 1, 2, 1, 1, 1, 2, 1, 1, If for any
= and any neighborhood of zero W in the space of variables 1, 2, 1, 1 we can find a neighborhood of zero 0 of this space such that for any point of this neighborhood the corresponding solution belongs to for ≥ 0.
We will say that
is asymptotically stable with respect to variables . if it is stable with respect to these variables and, in addition, the variables 1, 2, 1, 1 tend to zero with unlimited increase of
Mathematical Modeling / E. Shchepakina, V. Sobolev
We will say that is stabilizable if it is asymptotically stable with respect to variables 1, 2, 1, 1 under → ∞ the solution tends to some point of the manifold .
It follows from the results of [
The differential system under consideration is singularly perturbed one and has a three-dimensional manifold of slow motions:
1 = ( , 1, 2), 1 = ( , 1, 2), the motion along which is described by a system of three scalar differential equations of the form:
̇ = [2 1 − ( + ) 2 ] , 1̇ = − 2 + [ 2( 2 − 2
( + )) + + 2 1 ], 2̇ = 1 + [(− 1 − 1( 1 − 2
( + ) 2) + 12 + 22 − (1 + 1) 1 + 2) + (− 1 + 2 2) + ( − ) 1 − ( 12 + 22)] + 2{[ 2 − (1 + 1) 1] − (1 + 1) 1 + (1 + 1) 1 − (1 + 1)( 12 + 22)} + 3(1 + 1)2( − ) 1.
2
The functions f, g are computed in the usual way [
2( 12 + 22)] , 1̇ = − 2 +
[( − ) 12 2 − 2 ( − )(2 + ) 2 1 + 2 2( + ) 2( 12 + 22) − 1 2 1 ( 12 + 22)], − 2 ( + ) 1 ( 12 + 22) + ( 12 − 2)( 12 + 22)]. 1 1 2̇ = 1 + 2[− ( + )( − )2(1 −
ε ) 1 − 12 ( ( + )( − )2 2 1) + 1 1 2 ( − )(( + ) 12 − 22)
After linearizing the equations on an integral manifold for variables 1, 2 we obtain the linear with respect to 1, 2 subsystem
1̇ = − 2, 2̇ = 1 + 2[− ( + )( − )2(1 − ε 2 1
ε ) 1 − 12 ( ( + )( − )2 2 1) ].
The condition of asymptotic stability with respect to variables 1, 2 is
− ( + )( − )2 < 0.
For the integral manifold of slow motions, the following principle is valid: the variety of stationary states of the initial system is stable (unstable, asymptotically stable with respect to some of the variables, is stabilizable) if and only if it is stable (unstable, asymptotically stable with respect to a part of the variables, stabilizable) the variety of stationary states of a system describing the motion on an integral manifold. It is clear that a violation of the resulting inequality entails a loss o f stability. This is confirmed by the results of numerical experiments. In the figures below, one can see oscillations with increasing amplitude for the variables 1, 2 and ω.
2 1 1 1 2 1 ε 2 1 Mathematical Modeling / E. Shchepakina, V. Sobolev
Fig 2. Solution graph for variable ω.
In the present work, the mathematical model of a satellite stabilized by rotation has been studied by the methods of the geometric theory of singular perturbations. A reduction of the system was carried out, as a result of which, instead of the original system of five differential equations, its projection onto a three-dimensional slow integral manifold was investigated. It should be noted that, due to the validity of the reduction principle for a slow integral manifold, the reduction is carried out correctly, and the reduced system of three differential equations preserves the basic qualitative properties of the original model. An inequality is obtained, in violation of which the satellite loses the required orientation in space.
The study was carried out with the financial support of the RFBR and the Government of the Samara Region within the framework of the scientific project No. 16-41-630524 and the Ministry of Education and Science of the Russian Federation as part of the Samara University's competitiveness increase program (2013-2020).