=Paper=
{{Paper
|id=Vol-1904/paper6
|storemode=property
|title=About the Attractor-Repeller points during the descent of an asymmetric spacecraft in the atmosphere
|pdfUrl=https://ceur-ws.org/Vol-1904/paper6.pdf
|volume=Vol-1904
|authors=Vladislav V. Lyubimov,Vyacheslav S. Lashin
}}
==About the Attractor-Repeller points during the descent of an asymmetric spacecraft in the atmosphere ==
https://ceur-ws.org/Vol-1904/paper6.pdf
About the attractor-repeller points during the descent of an asymmetric
spacecraft in the atmosphere
V.V. Lyubimov1, V.S. Lashin1
1
Samara National Research University, 34 Moskovskoe Shosse, 443086, Samara, Russia
Abstract
The aim of this study is to analyze the resonant attractor-repeller points during the atmospheric descent of a spacecraft with small asymmetry.
The mathematical simulation of spacecraft rotational motion uses an approximate non-linear system of equations obtained by the method of
integral manifolds. Application of the averaging method and the Lyapunov method makes it possible to obtain realization conditions of
attractor-repeller points on non-resonance parts of the motion. By analyzing of the said conditions, we have identified specific cases when the
principal resonance is either an attractor point or a repeller point.
Keywords: resonance; attractor; repeller; averaging; spacecraft; atmosphere; asymmetry
1. Introduction
Various resonance phenomena in the problem of uncontrolled descent of a spacecraft with a small asymmetry in the
atmosphere are explored in [1-2, etc.]. In particular, the disturbing moments of mass-aerodynamic asymmetry can leads to the
evolution of angular velocity of the spacecraft to the resonance values [3-5]. In this case, the non-resonance evolution of the
angular velocity of the asymmetric spacecraft is the secondary resonance effect [6]. The external stability of resonance is
considered in the problems of perturbed rotational motion of the asymmetric spacecraft in the atmosphere or satellite in orbit
[5,7]. The realization of the main resonance leads to significant increase of the angle of attack. In practice, this can lead to
emergency situation during the deployment of the parachute system of a spacecraft. It is known that the external stability of the
resonance contributes to the evolution of the variables of the dynamic system to resonant values (resonant attractor). Therefore,
the study of the phenomenon of external stability of resonance is an important practical task. This phenomenon arises when the
condition of the external stability of the resonance is satisfied. It should be noted that the investigation of the resonant attractor
and repeller is supposed to be performed in a more general form in comparison with the results presented in [5].
2. Problem statement
Let us assume that spacecraft is a solid body in the form of a cone combined with a spherical surface. Let the axis OX is the
symmetry axis of the cone. In the process of entry into atmosphere, a spacecraft is directed its conical surface along of the air flow.
During the atmospheric descent, the spacecraft engages in the precessional motion. It is known that a spacecraft receives small
angular momentum when undocked from the base orbital module [8]. In this case, the angular momentum lead to the formation of
the components of the angular velocity of the spacecraft ωx(0), ωy(0), ωz(0). These components of the angular velocity are
recorded of the spacecraft body-fixed coordinate system OXYZ. Suppose that these angular velocities are initial when the
spacecraft enters into the atmosphere. The origin of the coordinate system O is located at the center of mass of the spacecraft. In
[9] it is shown that the resonance values of the angular velocity ωx can be determined on the basis of the method of integral
manifolds [10]. The values of angular velocity ωx corresponding to the principal resonance are defined as follows [5]:
rx / (1 I x )1/ 2 . Here ( m zn qSLctg / I )1/ 2 is the angular velocity, mzn is the restoring moment coefficient for the
angle of attack α, q is the dynamic pressure, S is the area of the maximum cross section of a spacecraft, L is the length of
a spacecraft, I x I x / I , I =(Iy +Iz )/2, Ix, Iy, Iz are the principal moments of inertia of a spacecraft. It is known that the principal
resonance has the greatest influence on the evolution of slow variables on the non-resonant parts of the motion, compared with
resonances of higher orders. The aim is to study the realization conditions of resonant attractor and resonant repeller in case of
atmospheric descent of a spacecraft with small aerodynamic-inertial asymmetry. Let the attack angle take arbitrary values. We
apply the method of averaging and the Lyapunov method to the research of attractors and repellers.
3. Methods
3.1. Mathematical model
The approximate non-linear system of equations of motion of a spacecraft with small aerodynamic-inertial asymmetry,
describing the motion of a spacecraft relative to the center of mass has the form [5]:
dx
Ix m 1,2
2
tg 2 cos 2 23 , (1)
dt
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Mathematical Modeling / V.V. Lyubimov, V.S. Lashin
Fa d tg d mA
cos( 1 )
42a dt 22a dt 2a
1,2 tg
- 10 + Ix x 1,2 2 2 Ix 2x m cos(2 23 ) -
42a
1,2tg
- tg 2 4 1,2
2
m cos(2 23 ), (2)
42a
d
x 1,2 , (3)
dt
d dq
. (4)
dt 2q dt
Here ε is the small parameter, θ=φ-π/2, φ is aerodynamic roll angle; m A , m , 1 , 3 are functions that characterize the values
and relative positions of the aerodynamic and inertial asymmetries of the spacecraft,
1 I x x 31,2 2 2 tg 2
m A ( m1A ) 2 ( m2A ) 2 , m1
A
2a mzп
(myf Cx z )tg 1,2
2a mzп
C yп z
I xz
x 1,2 x 1,2 tg 2 2x x 2a , m2A
1 I 3
x x 1,2 2
(mzf Cx y)tg
2a 2a mzп
1,2 2 tg 2 I xy
C y 2 tg 2a , sin 1 m1A / m A , cos 1 m2A / m A , a I x 2x / 4 2 ;
2 2 2
yп x 1,2 x 1,2 x x
2a mzп a
, sin 23 I / m , cos 23 I yz / m , I xy I xy / I , I xz I xz / I , I yz I yz / I , I I / I are dimensionless
2
m I yz2 I
I x x
moments of inertia of a SC, 1,2 a ; x 1,2 is the resonant ratio of frequencies; Сx , C yп are the aerodynamic
2
coefficients; myf , mzf are the coefficients of small moments caused by asymmetric shape of the spacecraft; y y / L ,
z z / L ; Δy, Δz are small displacements of the center of mass of the spacecraft; Fa Fa (x , , ) is the known function of
slow variables [5]. In equations (1)-(4) we consider the principal resonance, corresponding to the following condition:
x 1,2 0 . There are signs “±” and “ ” in the equations (1)-(3). We assume in the said equations that the upper sign is
selected when ωx> 0, and the lower sign is selected when ωx<0. In the numerical simulation of spacecraft motion, the system of
equations (1)-(4) should be considered together with the system of three differential equations for slowly varying of the center of
mass parameters: the local flight-pass inclination angle (t ) , the spacecraft airspeed V(t) and the spacecraft altitude H(t) [1].
3.2. Averaging and analysis of resonant attractor
After using the method of averaging on non-resonant parts of a spacecraft motion we obtain [5]:
d x m A g g A
A
m g3 A g
3 2 3
( m g 3 ) (m g3 2 )
dt
3
2
m cos(21 23 )
A 2
3(m g3 )2 g 2
( g 2 ) , (5)
4 8
A A 2
d 3 m g 3 g1 2 m g3 g1 g1
A A A A
( m g ) m g ( m g ) ( m g 3
)
2
3 3 3
dt 3 2 3
A A A
m g g A A (m ) 2 g 3 g1 g 3 (m )2 g32 g
3 2 3 (m g3 ) 2m g3 7 g2 4 2
x 2
2 4
x
m cos(21 23 ) 3
A
(m )2 g32 2 2 g1
2
2 g1 g4 . (6)
24 2 8
2a 1,2 sin 1,2
2
sin 2 1,2
2
sin 2 2a 2 4mzn SLa2 dq
Нere g1 (x ) , g2 , g3 , g4 .
Fa 2a Ix Fa IFa dt
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Mathematical Modeling / V.V. Lyubimov, V.S. Lashin
Equations (4) and (5) describe the non-resonant evolution of the angular velocity x and the angle of attack caused by the
Ix
effect of the principal resonance 0 . At the positive values of x resonant ratio is equal to (1 ) x a . Let us
2
assume that the spacecraft has the following ratio of the moments of inertia: I x 2 . Here the resonant ratio is equal to
2x 2 . (7)
We introduce the function V (x , ) . This Lyapunov function can be written as:
2
V (x , ) 2x 2 . (8)
Here x , are determined from equations (5) and (4) respectively. Given the expression of (7), we see that the principal
resonance 0 is realized at
x 0,
(9)
0.
Thus, the condition of the external stability of the principal resonance [5] has the following form:
dV d x d
2x 2 0. (10)
dt dt dt
The condition (10) is a condition of asymptotic stability of a trivial solution (9). The fulfillment of the condition (10)
provides for realization of a resonant attractor (9). On the contrary, the condition of external instability of the principal
resonance is the following:
dV d x d
2x 2 0. (11)
dt dt dt
The condition (11) is the condition of instability of the trivial solution (9). The fulfillment of the condition (11) ensures the
realization of the resonant repeller (9). Let us assume that { x 0, 0} . In this case, asymptotic analysis of conditions (10),
(11) makes it possible to distinguish the following twelve typical cases of resonant attractor or resonant repeller realization: 1) if
d x / dt 0 , d/ dt 0 , x (0) max rx 0 , condition (10) is fulfilled and resonant attractor (9) is realized; 2) if
d x / dt 0 , d/ dt 0 , max rx x (0) rx (0) 0 , condition (11) is fulfilled and resonant repeller (9) is realized; 3) if
d x d r
d x / dt 0 , d/ dt 0 , x , x (0) x (0) 0 , condition (10) is fulfilled and attractor (9) is realized;
dt dt
d x d r
4) if d x / dt 0 , d/ dt 0 , x , x (0) x (0) 0 , condition (11) is fulfilled and repeller (9) is
dt dt
d x d r
realized; 5) if d x / dt 0 , d/ dt 0 , x , x (0) max x 0 , condition (10) is fulfilled and attractor
dt dt
d x d r
(9) is realized; 6) if d x / dt 0 , d/ dt 0 , x , x (0) x (0) 0 , condition (11) is fulfilled and
dt dt
d x d r
repeller (9) is realized; 7) if d x / dt 0 , d/ dt 0 , x , x (0) max x 0 , condition (10) is fulfilled
dt dt
r
and repeller (9) is realized; 8) if d x / dt 0 , d/ dt 0 , 0 x (0) x (0) , condition (11) is fulfilled and resonant repeller
d x d r
(9) is realized; 9) if d x / dt 0 , d/ dt 0 , x , x (0) max x 0 , condition (11) is fulfilled and
dt dt
r
repeller (9) is realized; 10) if d x / dt 0 , d/ dt 0 , 0 x (0) x (0) , the condition (10) is fulfilled and the resonant
d x d r
attractor (9) is realized; 11) if d x / dt 0 , d/ dt 0 , x , x (0) x (0) 0 , condition (11) is fulfilled
dt dt
3rd International conference “Information Technology and Nanotechnology 2017” 37
Mathematical Modeling / V.V. Lyubimov, V.S. Lashin
d x d r
and repeller (9) is realized; 12) if d x / dt 0 , d/ dt 0 , x , x (0) max x 0 , condition (11) is
dt dt
fulfilled and attractor (9) is realized.
Ix
Similarly we consider the case { x 0, 0} . In this case, the resonant ratio (1 ) x a at I x 2 is
2
2x 2 . (12)
In this case, the Lyapunov function is (8). Similar typical twelve cases are following: 13) if d x / dt 0 , d/ dt 0 ,
x (0) min rx 0 , condition (10) is fulfilled and resonant attractor (9) is realized; 14) if d x / dt 0 ,
d/ dt 0 , min rx x (0) rx (0) 0 , condition (10) is fulfilled and repeller (9) is realized; 15) if d x / dt 0 ,
d x d r
d/ dt 0 , x , 0 x (0) x (0) , condition (10) is fulfilled and attractor (9) is realized); 16) if
dt dt
d x d r
d x / dt 0 , d/ dt 0 , x , 0 x (0) x (0) , condition (11) is fulfilled and repeller (9) is realized; 17)
dt dt
d x d r
if d x / dt 0 , d/ dt 0 , x , x (0) x (0) 0 , condition (10) is fulfilled and attractor (9) is realized;
dt dt
r d x d
18) if d x / dt 0 , d/ dt 0 , 0 x (0) x (0) , x , condition (11) is fulfilled and resonant repeller (9)
dt dt
d x d r
is realized; 19) if d x / dt 0 , d/ dt 0 , x , x (0) x (0) 0 , condition (10) is fulfilled and repeller
dt dt
r
(9) is realized;20) if d x / dt 0 , d/ dt 0 , 0 x (0) x (0) , condition (11) is fulfilled and resonant repeller (9) is
d x d r
realized; 21) if d x / dt 0 , d/ dt 0 , x , x (0) min x 0 , condition (11) is fulfilled and repeller
dt dt
r
(9) is realized; 22) if d x / dt 0 , d/ dt 0 , 0 x (0) x (0) , condition (10) is fulfilled and resonant attractor (9) is
d x d r
realized; 23) if d x / dt 0 , d/ dt 0 , x , x (0) x (0) 0 , condition (11) is fulfilled and repeller (9)
dt dt
d x d r
is realized; 24) if d x / dt 0 , d/ dt 0 , x , x (0) min x 0 , condition (11) is fulfilled and attractor
dt dt
(9) is realized.
4. Numerical results
Numerical results obtained from solve of the equations (1)-(4) confirmed fulfillment of the twenty-four cases discussed
above. In particular, Fig. 1 shows the dependence of the Lyapunov function on slow variables x and ω when realization of
resonant attractor. This numerical result corresponds to a typical case 5). Fig. 2 shows the dependence of the Lyapunov function
on slow variables x and ω when realization of resonant repeller. The numerical result shown in Fig. 2 corresponds to case 2).
The following parameters of the spacecraft and initial conditions of motion were used in the construction of Figs. 1-2: m =70 kg;
S= 0.1 m2, L = 0.54 m, m 0.02 , m A 0.05 , 1 3 , I = 1 kgm2, Ix =0.3 kgm2, V(0) is the initial value of the spacecraft
velocity, V(0) = 3400 m/s, (0) is the initial value of the local flight-pass inclination angle, (0) = -0.087 rad, H(0) is the initial
value of spacecraft altitude, H(0) = 100 km, (0) =0, (0) 0.05 rad, x =10 s-1 (Fig.1); m 0.005 , m A 0.05 , 1 3 0 ,
x =15 s-1 (Fig. 2). Direction of non-resonant evolution of the corresponding variables is indicated in Figs. 1-2 by arrows.
5. Conclusion and results
Thus, the use of the method of averaging and Lyapunov's second method made it possible to carry out an asymptotic analysis
of the non-resonant evolution of slow variables during the atmospheric descent of the spacecraft with small aerodynamic-
inertial asymmetry. By doing so, we obtained conditions for realization of the resonant attractor and resonant repeller at arbitrary
angles of attack. In addition, we identified ten typical cases of resonant attractor realization and fourteen typical cases of
resonant repeller realization. The approximate analytical results of the study correspond to the results of the numerical
simulation. The conditions presented in this study indicate that the resonant attractor can become the resonant repeller. It is also
possible for a reverse transition. These transitions can occur due to the change of sign of the angular velocity x . By analyzing
of the stability conditions, we assumed that the asymmetry parameters take constant values. It should be noted that the descent of
a spacecraft with variable asymmetry into the atmosphere presents of a practical interest. For example, the variable asymmetry in
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Mathematical Modeling / V.V. Lyubimov, V.S. Lashin
the considered dynamical system can lead to a transition from the resonant attractor to the resonant repeller. Research of such
transient modes falls beyond the scope of this study and may be detailed in the following papers.
Fig.1. Lyapunov’s function and angular velocities when resonant attractor. Fig.2. Lyapunov’s function and angular velocities when resonant repeller.
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