About the attractor-repeller points during the descent of an asymmetric spacecraft in the atmosphere

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

2. Problem statement

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]:

3. Methods

3.1. Mathematical model

Fa d     tg d  42a dt 22a dt  mA 2a

cos(  1)  - 1,2tg

42a 10 + Ix  x1,2  2 2  Ix  2x  mcos(2  23 ) - 1,2tg

42a tg 2  4 12,2  mcos(2  23 ), m A  (m1A )2  (m2A )2 , m1A    1,22tg 2 Cyп y   2Ixya x1,2 x  1,2tg2  2x x

2a mzп m  I y2z  I 2 , sin 23  I / m , cos 23  I yz / m , Ixy  Ixy / I , Ixz  Ixz / I , I yz  I yz / I , I  I / I are dimensionless moments of inertia of a SC, 1,2  I x2x  a ; x  1,2 is the resonant ratio of frequencies; Сx , Cyп are the aerodynamic 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]. 1 Ix  x  31,2 2 2a mzп

(mzf  Cx y)tg  2a  , sin 1  m1A / mA , cos 1  m2A / m A , a  I 2x2x / 4  2 ;

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]:

ddtx  3  mAg32 g3  (mA g3  )  mA2g3  (m A g3 g2 )   3(mA4g3 )2 (  g2  g2   2 ) m cos(281  23 ) , d dt  3  mAg33 g1   (mA g3 )   mA g3 22  

mA3g3  (mA g3) g1  g12   (mA g3 )2   mA3g2 g3x   (mA g3 )  2mA g3    (m A)22 g3 g1 g3  (m2A)24g32 x 7g2   4 g2    (mA )2 g 2  2   2  m cos(21  23 )  3g4 .

24 3 22 g21  g1     8 Нere g1  2aF1,a2 sin  (x  12,22sina 2  ) , g2  12,2 Isixn2  , g3  2Faa2 , g4  4mzn SLa2 dq .

IFa dt ( 2 ) ( 5 ) ( 6 ) We introduce the function V (x , )  2 . This Lyapunov function can be written as:

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

Thus, the condition of the external stability of the principal resonance [5] has the following form: dx / dt  0 , d/ dt  0 , x dx dt dx / dt  0 , d/ dt  0 , x dx

dt realized; 5) if dx / dt  0 , d/ dt  0 , x   repeller ( 9 ) is realized; 7) if dx / dt  0 , d/ dt  0 , x , rx (0)  x (0)  0 , condition ( 10 ) is fulfilled and attractor ( 9 ) is realized; , rx (0)  x (0)  0 , condition (11) is fulfilled and repeller ( 9 ) is , x (0)  maxrx  0 , condition ( 10 ) is fulfilled and attractor , rx (0)  x (0)  0 , condition (11) is fulfilled and

, x (0)  maxrx  0 , condition ( 10 ) is fulfilled d dt and repeller ( 9 ) is realized; 8) if dx / dt  0 , d/ dt  0 , 0  x (0)  rx (0) , condition (11) is fulfilled and resonant repeller ( 9 ) is realized; 9) if dx / dt  0 , d/ dt  0 , x , x (0)  maxrx  0 , condition (11) is fulfilled and repeller ( 9 ) is realized; 10) if dx / dt  0 , d/ dt  0 , 0  x (0)  rx (0) , the condition ( 10 ) is fulfilled and the resonant attractor ( 9 ) is realized; 11) if dx / dt  0 , d/ dt  0 ,  x , rx (0)  x (0)  0 , condition (11) is fulfilled and repeller ( 9 ) is realized; 12) if dx / dt  0 , d/ dt  0 , x fulfilled and attractor ( 9 ) is realized. , x (0)  maxrx  0 , condition (11) is Similarly we consider the case {x  0,   0} . In this case, the resonant ratio   (1  I x )x  a at I x  2 is 2   x (0)  minrx  0 ,

In this case, the Lyapunov function is ( 8 ). Similar typical twelve cases are following: 13) if dx / dt  0 , d/ dt  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 , (12) , 0  x (0)  rx (0) , condition ( 10 ) is fulfilled and attractor ( 9 ) is realized); 16) if d/ dt  0 , x dx dt   d dt dx / dt  0 , d/ dt  0 , x dx

dt if dx / dt  0 , d/ dt  0 , x dx dt     d dt d dt realized; 23) if dx / dt  0 , d/ dt  0 ,  x is realized; 24) if dx / dt  0 , d/ dt  0 , x ( 9 ) is realized. 4. Numerical results , 0  x (0)  rx (0) , condition (11) is fulfilled and repeller ( 9 ) is realized; 17)

, x (0)  rx (0)  0 , condition ( 10 ) is fulfilled and attractor ( 9 ) is realized; 18) if dx / dt  0 , d/ dt  0 , 0  x (0)  rx (0) , x dx dt d dt is realized; 19) if dx / dt  0 , d/ dt  0 , x  

, condition (11) is fulfilled and resonant repeller ( 9 ) , x (0)  rx (0)  0 , condition ( 10 ) is fulfilled and repeller ( 9 ) is realized;20) if dx / dt  0 , d/ dt  0 , 0  x (0)  rx (0) , condition (11) is fulfilled and resonant repeller ( 9 ) is realized; 21) if dx / dt  0 , d/ dt  0 , x , x (0)  minrx  0 , condition (11) is fulfilled and repeller ( 9 ) is realized; 22) if dx / dt  0 , d/ dt  0 , 0  x (0)  rx (0) , condition ( 10 ) is fulfilled and resonant attractor ( 9 ) is , rx (0)  x (0)  0 , condition (11) is fulfilled and repeller ( 9 ) , x (0)  minrx  0 , condition (11) is fulfilled and attractor

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 , mA  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 , mA  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 aerodynamicinertial 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

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.

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