Russian Journal of Nonlinear Dynamics, 2022, vol. 18, no. 4, pp. 639-649. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd221222

NONLINEAR PHYSICS AND MECHANICS

MSC 2010: 37N05, 70K65, 70F15

Spacecraft with Periodic Mass Redistribution: Regular and Chaotic Behaviour

The motion of a spacecraft containing a moving massive point in the central field of Newtonian attraction is considered. Within the framework of the so-called "satellite approximation", the center of mass of the system is assumed to move in an unperturbed elliptical Keplerian orbit. The spacecraft's dynamics about its center of mass is studied. Conditions under which the spacecraft rotates about a perpendicular to the plane of the orbit uniformly with respect to the true anomaly are found. Such uniform rotations are achieved using a specially selected rule for changing the position of a massive point with respect to the spacecraft. Necessary conditions for these uniform rotations are studied numerically. An analysis of the nonintegrability of a special class of spacecraft's rotation is carried out using the method of separatrix splitting. Poincare sections are constructed for certain parameter values. Several linearly stable periodic motions are pointed out and investigated.

Keywords: spacecraft attitude dynamics, spacecraft in an elliptic orbit, spacecraft with variable mass distribution, spacecraft's chaotic oscillations, spacecraft's periodic motions

1. Introduction

As is well known, moving masses play an important role in the dynamics of spacecraft. Such masses can be represented, for example, by liquids, loose objects, crew members, etc. It is clear that early studies on the dynamics of such spacecraft date back to the very beginning of the space age (see, e.g., [1-18]) and this topic still remains of current interest [19-25].

Received December 05, 2022 Accepted December 26, 2022

Alexander A. Burov jtm@narod.ru Vasily I. Nikonov nikon_v@list.ru

Federal Research Center "Computer Science and Control" of Russian Academy of Sciences, ul. Vavilova 40, Moscow, 119333 Russia

Ivan I. Kosenko kosenkoii@gmail.com

Moscow Aviation Institute (National Research University), Volokolamskoe sh. 4, Moscow, 125993 Russia

A. A. Burov, I.I. Kosenko, V. I. Nikonov

Note that, as a rule, it is assumed that a moving mass is significantly smaller than the mass of a spacecraft, and that the center of mass of the spececraft moves in an elliptic orbit. In the case where the moving mass and the mass of the spacecraft are comparable in magnitude, the equations of motion of the spacecraft carrying a moving point mass are neatly derived in the publication [21]. In this publication, both in the case of circular and elliptical orbits of the center of mass, some laws of changing the position of a moving point relative to the spacecraft body were stated, for which the entire system has certain classes of motions. In particular, in the case of an elliptic orbit, special attention is paid to the plane oscillations and rotations of the system about the normal to the orbital plane. The law of motion of a massive point along one of the axes of inertia of a spacecraft is established for which there exists a relative equilibrium of the spacecraft. For this relative equilibrium this axis of inertia is directed along the local vertical. Necessary conditions for the stability of such relative equilibria are investigated1.

The focus of this study is on the equation describing such plane motions of the spacecraft. Conditions under which the rotation angle of the spacecraft changes proportionally to the change in the true anomaly are found. In this paper, such rotations are called rotations uniform in the true anomaly. Necessary conditions for their stability are studied numerically. A narrow region in the parameter space for which the necessary stability conditions are met is found. For those parameter values, Poincaré sections are constructed. With the help of these sections, regions of the phase space presumably containing stable periodic solutions are singled out. These periodic solutions are determined numerically using the methods developed earlier in [34]. The necessary conditions for stability of the found periodic motions are also numerically verified.

It is clear that the chaotic behavior is typical for the class of systems under consideration. Such chaotic behavior is associated, in particular, with the splitting of separatrices. The Poincaré - Melnikov condition, which makes it possible to establish analytically the splitting of separatrices for small nonzero values of eccentricity, is written out.

2. Equations of motion

Following [21], consider a spacecraft of mass M moving in a central Newtonian gravitational field generated by the attracting center N. Let a material point P of mass m move along some invariable curve fixed in the spacecraft. In the moving reference frame Oxyz determined by the principal central axes of inertia of the spacecraft body, the position of the point P is given by the coordinates of the radius vector = (x, y, z)T. If C is the center of mass of the entire system, then

cP=±oP, mM

»=

M m M + m

The analysis of the motion of the spacecraft is carried out within the framework of the so-called "satellite approximation" (c.f. [26]), when it is assumed that the motion of the system around its center of mass does not affect the orbital motion of the center of mass [26]. It is assumed that the orbital motion of the point C which is the center of mass of the whole system is carried out along the elliptic Keplerian orbit. Note that we do not make special assumptions about the ratio of the mass of the moving point and the mass of the spacecraft. In particular, these values can be comparable.

If the motion of the point P occurs, for example, in the plane Oxy fixed in the spacecraft, then there is a class of motions such that this plane coincides with the plane of the orbit. If J =

1 Sufficient stability conditions are investigated in [27] (see also [28]).

= diag( Jx, Jy, Jz) is the principal central tensor of inertia and p is the angle of deviation of the positive semiaxis Oy from the local descending vertical, then the plane motion of the spacecraft is described by the equation [21]

d_

dt

{[Jz + V {x2 + y2)] {p + V) + n{xy - yx)} =

3k

= —g {(Jy — Jx) sin p cos <p + ¡j, [(.t2 — y2) sin p cos <p + xy cos 2tp\ }. (2.1)

Rc

Here and in what follows, as in [21], the constant k is equal to the product of the gravitational constant by the mass of the attracting center and Rc = (nC, nC)1/2 is the distance from the attracting center to the point C. The coordinates x and y are assumed to be prescribed functions of the independent variable.

If e is the eccentricity of the orbit of the point C, n = where T is the period of rotation of the point C about the attracting center, and v is the true anomaly of this point, then (see, e.g., [29])

n . k n . -o

v =-—r-(l + ecosz/j = f(i/), =-oil + ecosv) .

(1 — e2) Rc (1 — e2) ;

Since 0 ^ e < 1, the function f (v) takes only positive values. Therefore, the true anomaly changes strictly monotonically and can be used as a new independent variable instead of time. We have

— - ft \ —

and Eq. (2.1) can be written as d

— {(1 + ecos vf [[Jz + (x2 + y2)] (<ff + 1) + xy1 - yx1)] } =

= 3(1 + ecos v) {(Jy — Jx) sinpcosp + ¡i [(x2 — y2) sinpcosp + xycos2p }.

After transformations, this equation reads

— 2esinv[[Jz + ¡(x2 + y2)} (p + 1) + ¡(xy1 — yx')] +

+ (1 + ecos v) [2i(xx' + yy')(p' + 1) + [Jz +1 (x2 + y2)] p" + ¡(xy'' — yx'')] =

= ^(Jy — Jx) sinp cos p + ¡[[x2 — y2) sin p cos p + xy cos 2p] }. (2.2)

If, following [21], we assume that the point mass moves along the axis Oy, then x = 0, x' = 0, and Eq. (2.2) takes the form

— 2e sin v [Jz + J (v )](p' + 1) + (1 + e cos v )[J'(v )(p' + 1) + [Jz + J (v)] p'' ] =

= 3(Jy — Jx — J (v ))sin p cos p, J (v )= ¡y2(v). (2.3)

We assume that the value of y depends on the position of the spacecraft in an elliptic orbit, i.e., y = y(v). This determines the dependence J = J(v) determined in the latter formula. Equation (2.3) is the main subject of further discussion.

3. Solutions of Eq. (2.3)

If, as was done in [30], we look for motions in which the rotation angle of the system changes uniformly along with the change in the true anomaly, then p = w(v — v0), and Eq. (2.3) after transformations takes the form

JJ' = gJ + h (3.1)

/ = f(v) = (oj + 1)(1 + ecos z/), g = g(v) = — ^ sin2oj(v — v0) — '2e(oj + 1) sin v

(3 Jy Jx

-aesin2w(z/ — v0) + 2e(oj + 1) sin vj, ae = ———-, ae G (—1, 1).

In general, Eq. (3.1) is solved by the method of variation of constants. The resulting expressions are quite complex and are not given here. However, there are special cases where integration is carried out directly.

3.1. The first special case

For z/0 = 0, oj = e = | one obtains g = 0, and Eq. (3.1) after transformations takes a simple form

(2 + cos v)J' = 2Jz (« + 1) sin v.

and is easy to integrate:

I. J(I/) = J(0)- 2^ + 1)^(^+1^). (3.2)

The graph of the function (3.2) for Jx = 3, Jy = 4 and Jz = 5, i. e., ae = as well as for J(0) = 0, is shown in Fig. 1. Here and below, the Roman numerals on the graphs correspond to numbers of the special cases for which integration is carried out.

3.2. The second special case

For z/0 = 0, oj = e = —|ae one obtains h = 0, and Eq. (3.1) after transformations takes the form

(1 + e cos v) J' = (2e — 1) sin vJ. As a result of its integration, one obtains

/11 \ 1/e-2

II. JM = m m^) p.3)

It is clear that the solution thus found has a physical meaning only when the inequality

— 1 < a ^ 0

holds true.

For Jx = 5, Jy = 3 and Jz = 4 we have e = The graph of the function (3.3) at these values, as well as at J(0) = 1, is shown in Fig. 1.

27T v 3TT 0 Fig. 1

3.3. The third special case

For z/0 = 7T, uj = e = ^ae one also obtains h = 0. In this case, Eq. (3.1) after transformations takes the form

(1 + e cos v) J' = (2e + 1) sin vJ. As a result of its integration, the obtained expression reads:

III. J (v) = J (n)

1 - e

l/e+2

(3.4)

1 + e cos v t

It is clear that the solutions found have a physical meaning only when the inequality

0 < a < 1

holds true.

For Jx = 3, Jy = 5 and Jz = 4 we have e = The graph of the function (3.4) at these values, as well as at J(n) = 1, is shown in Fig. 1.

Note that for Jx = Jy there is always exactly one of the second and third special solutions.

3.4. The fourth special case

Let now the equalities v0 = 0, uj = —e = |ae be fulfilled. Then we also have h = 0. In this case, Eq. (3.1) after transformations takes the form

(1 + e cos v) J' = (3 + 2e) sin vJ.

As a result of its integration, the expression

( 1 + e \2+3/e

IV. J(y) = J{ 0) (3.5)

v 7 w \1 + e cos v) v 7

is obtained. It is clear that the solutions found have a physical meaning only when the inequality

2

0 ^ ae < -3

is fulfilled. If, for example, Jx = 3, Jy = 5 and Jz = 4, one obtains e = |. The graph of the function (3.4) at these values, as well as at J(n) = 1, is shown in Fig. 1.

3.5. The fifth special case

Let the equalities u0 = 7r, oj = — e = — |ae be fulfilled. One also obtains h = 0. In this case, Eq. (3.1) after transformations takes the form

(1 + e cos v) J' = (3 — 2e) sin vJ.

As a result of its integration, the expression

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( 1- e \2-3/e

V. J{v) = J(vr) ——— (3.6)

v ' K ' V1 + e cos v J y '

is obtained. It is clear that the solutions found have a physical meaning only when the inequality

2

-- < ae ^ 0 3

is fulfilled. Thus, for Jx = 5, Jy = 3 and Jz = 4 we have e = |. The graph of the function (3.6) at these values, as well as at J(n) = 1, is shown in Fig. 1.

Note that for Jx = Jy there is always exactly one of the fourth and fifth special solutions.

4. Necessary conditions of stability

To study the necessary conditions of stability of the found rotations, the linearization of the equations of motion in their vicinities is carried out. We put

p = w(v — v0) + 5p.

Then

p' = w + bp', p'' = bp", and after the transformations, the linearized equation of motion takes the form

(1 + e cos v )(Jz + J (v ))bp'' + ((1 + e cos v )J '(v) — 2e sin v (Jz + J (v )))bp' —

— 3(Jy — Jx — J (v)) cos 2w(v — v0)bp = 0.

Below, the necessary stability conditions are investigated for specially selected partial values of the parameters.

It turned out that only for the third special case there exists an interval of values of the parameter s, for which the absolute value of the trace Tr(A) of the monodromy matrix does not exceed two: for 0.4032324 < s < 0.5683768 one obtains Tr(A) e [—2, 2].

5. Chaotic behavior of the system

It is natural to expect that chaotic behavior will be observed in the case under consideration. Since in the first special case the eccentricity of e = ^ is not close to zero, the method of separatrix splitting (see, e.g., [32]) turns out to be inapplicable. Therefore, for the values Jx = 3, Jy =4 and Jz = 5, a Poincare section is constructed numerically (see Fig. 2).

Fig. 2

In the figure, one can observe what is called "developed chaos". However, in this "sea of chaos" there are "islands of regularity"2. In the publication [34], devoted to the Meissner equation, a numerically analytical iterative method for finding periodic solutions "surrounded" by islands of regularity was proposed. The calculations performed now have shown the effectiveness of the same method in relation to Eq. (2.3). The application of the above-mentioned method to the considered parameter values allowed us to identify three linearly stable motions. These motions, denoted as (a), (b) and (c), are depicted in Figs. 3-5.

6. Analytical treatment of nonintegrability

In the publication [21], special attention was paid to the question of the existence of relative equilibria of the system at any values of the eccentricity of the orbit. As was shown, if the point P moves along the axis Oy in such a way that (see [21])

J (v ) = -Jz + (JZ + J (0))

1 + £

1 + £ cos v

then the differential equation (2.3) of the plane motions of a spacecraft in the elliptic orbit takes the form

d2(f i (s - 3)(1 + ecosz/)

I 7Z ! To

dv 2

3

(1 + e)2

1 + e cos v

Jx-J +J( 0) an „ore »> = 0, 8 = 3 Jz + J(0) . (6.1)

'This terminology is borrowed from V. V.Beletsky (see, e.g., [31]).

2

div

2

1

8TT —6n -47r -2TT 0 V/-2-3

Fig. 5. Solution (c). Initial condition is (—0.00005, —1.061975). Period is 6n _RUSSIAN JOURNAL OF NONLINEAR DYNAMICS, 2022, 18(4), 639-649

In the case of a circular orbit, when e = 0, the relative equilibria p = 0 and p = n are stable if s = 0. Under the same condition, the relative equilibria p = ±-| are unstable. The hyperbolic singular points corresponding to the latter equilibria are interconnected in the phase space by unsplit separatrices. Splitting separatrices, at least for small nonzero values of eccentricity, allows us to establish analytically the origin of chaotic behavior.

Before studying the splitting of separatrices, let us introduce ^ = 2p as a new variable. This allows us to rewrite the equation of motion (6.1) up to the linear terms of the parameter e as

d2 ^ .2

dv 2

+ w2(1 + ef (v)) sin é = 0,

6 /6

w2 = S, f{v) = —T-2-(—T-l) cos V w2 \w2 '

or in the form of Hamilton's equations

^ = -^2(l + e/M)sin0, (6.2)

with the Hamiltonian function

H = ^pl - w2( 1 + ef(v)) cos 0

(cf. [33]). We assume that the system is close to the integrable one (0 < e ^ 1). When e = 0, the system (6.2) is integrable: the Hamiltonian function

H0 = ^pI - u)2 cos 0 (6.3)

is the first integral of the equations of motion. The general solution of Eq. (6.2) in this case is written out in elliptic functions; the double asymptotic solutions connecting unstable equilibrium positions with themselves are described in hyperbolic functions (see, e.g., [33]):

(6.4)

These solutions filling in the extended phase space Rl(p^) x S 1(x Sl(v) nonsplit sepa-ratrix surfaces are used in the Poincaré - Melnikov integral, the calculation of which allows us to conclude that the equations of motion are not integrable.

To determine the integrand in the Poincaré - Melnikov integral, let us differentiate the function (6.3) by virtue of the system (6.2). We have

{Ho, H}

dp 2 (lé 1 ' • sin

(6.2)

2

prip—+UJ sinf

dv dv

The Poincaré - Melnikov function takes the form

-w2ef (v)p^ sin

<x

J (a) = J f (v + a)p^ sin é(v ) dv.

Here the integral is taken along two asymptotic solutions (6.4) of an unperturbed system. After

lengthy, but standard calculations, we find

= sinh_1 sina"

At w2 = 6 the function J (a) has isolated zeros, so the separatrices not only split, but also intersect each other. In this case, the equations of motion (6.2) turn out to be nonintegrable.

The values of oj = ±\/6 are exceptional: at these values, consideration of perturbations of the first order of smallness is insufficient to establish the splitting of separatrices. In this case, it is necessary to use the Poincare - Melnikov method of a higher order.

7. Concluding remarks

Certain advances in solving problems of the dynamics of orbital systems with variable mass distribution allow us to hope for the detection of more complex motions, as well as solutions to problems of control theory, in particular, the swing and overturn problems.

Conflict of interest

The authors declare that they have no conflict of interest.

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