On the Stability and Stabilization of Stationary Motions of Satellites

Cl; THL STABILITY AND STABILIS ATION OF STATIONARY [;lOTIONS OF SATELLITb S \Jl

\0 f-'

Rumjanchev V.V.

To orient, staoilize anu control satellites, rot ore wi th axes fixeu insiue the spacesilip may be uGed. Tr.is problem attracts attention 01' investigators and much literature is uevoteu to it. Presented herein are results of investi!a tions caIrie<1 out by tile author Md his young colleaGues Gn the study of orientation and stabiliz a tion of the motion of the gyrost'-!t f = - in the centra l Newtonian .» I v t~ fieltI0.e. a solid body, with rotors and cavities fillej wit~ homo t enous li qui<1, controlled by the moments of forces applied to the rotors. 1. Let us consider motion in the central Hewtonian 1'iel<1 of the forces of the s a tellite-gyrostav which is a free solid boJy containinG symmetrical, stati~cally and dynamically balanced solid rotors. Let ~ , gravitation centre, be the ori 6 in of the inertial system of coordinate axes O,J15, and 0 , the satellite centre of masses, - the origin of mobile coorJ.inate system OJ:/~~ whose axes are dir e cteu alclOC the main central axes o~ the ~atellit e inertia. Besides, let us introduce another mobile coordinate system OIJ1, with axis z directed along straight line 0,0, axis x - along the straight line orthogonal to axis z anu located in the plane parallel to plane O/SS' and axis 1 - along the straight line orthogonal to the other two and oriented in such a way that system O~!l, like the other two, is right. Let the gyrostat position in coordinate system 0.r7~ be determined by spherical coordinate x •er of mass centre 0 and some three independent variables, for example, Euler's angles IJ ,11, 'f determining

p,

is ru:;LLr i l . ....;.I.t.r ~ i_;.c~'1t f ur cO lU1 1arl.J.ti vc l y ~ Ifli;ill

tue 3 a t e l1i ~e pos it io ll in coordi nat e sy stem OJ.jl, as well as by ililbl c s 0(, of the rotor turns in r e spect to tile b~r03tat body. It should be noted th a t ~ vector:.; L ,j and /:." of axe s x, y and z are colline a r t o t he vecto~ s 0 1 t Le c o or~inate line s of the s pheric al c o ord inat e s ys tem wit h the ori L in at 01. 'rhe above variables are Lat;ran g,ian coorJ i na t e s of system, with n eq ualtlng b + m, wnere m is t ~e number of

--

rotors, and

q~,.I

~I=f,

ql-=Jf., C/j"'U, ~,:e,

r~='r')

:as (£-=1,

r, -==if)

, "") The e quations of the gyrostat motion takes the form of the Lagrang{.an equatiom:

,i0L, _'4').,=Q ( (=1 " "') /1)1. ')~" " • "

gI;t

where tbe

I:agrang~an

function is

L = If\((f·fll;i)'~(yJ.+f·i
it.J;~: +?'

f ' -;r%J~t;AJ~J-tJ

Here Ill, Ji.

J

Vl \,Q I\)

J; and

central moments of the und of inel'tia and "V e ctors axes, respectiv,e ly; W, of tIle instant an Gular ,

,

d

(1.1)

"'.-

~

(1. 3 )

e q ual t o tile beome trical sum of the mome n tlm v e ctors of the rotor rel utive motions, and, t~tefo re, it is p ossible wit h out di minishing th e commOll f e ature to co n1ider onl y one rotor with a xis l, 1D0 oer. t o f inerci a J and an t; ular s peed it. e quivalt·nt to m rotor ::;ys tem, ...,.providing tr,e momentum v e Cl6ij;r o f it s relativ e mOl;ion K is equal t u (1.3). The e quati on of t .le L'otor mo Lion takes tI,e form;

J1t{.x d; 1) ~Q

( 1 .11)

wh ere Generaliz e d f 01' c e Q is a Ill OL1 ent in resp e c t t o

gyrostat inertia, axial moments of the direction of the rotor is tbe projectio~f vector s p eed of the gy rostat body ro-

w

[,J

....

; "'l, J: d. t.

+W ·l.;; ds f.s\1. 2)

1 denote .nt: W<1SS, main

d'

S Ci -

t" l l it c "" :.Wc.J i L i 1; often i Gn ored i l, 1.1.(: f ir s t ap pr ox ima ti on '11 110n 3 li -'l ited pro blem is C0 l1 3i '.le red, The ;:-utOl' l~ ociolJ influences tu: motiOll of Lw t: j l 'O~tut o od,)' only LJU' J U !;, 11 t"e t;yr03tatic moment

• (

axis

l

of t ue f orc es ap pl i e d to t he rotor.

Ne sna I l di s cuss case s when moment Q i s s uc h tbat: 1) during t ue motion t i,e r e lative an t;u lar s peed of rOLor is c onstant

ta t lon on axlS x.' e p en lng on ~~,q. ,q/<. '--'. ~=",K'=2. . fi. /; are cosine s of the angles form e d bJ axis z and axts x~, Jependinb on q.;~ is the gravitation constant of tile attracting centre, and Q< are generalized n on- g:ravit a tion for c es; let Q[ = O(~:( . .. ,'J The p otential energy of the graVitation forces in (1.2) is put down, as usual, with pr e Cision with up to Gbe memb e rs of tbe order(f60)~included, where t is a cllarac~eristic size of the satellite.

J + ZS.{) -=.N -= c
As can be seen from (1.2), tHe motion of tbe centre of the g yrostat crasses is connected witb its motion around the mass centre. Nevertheless, the influence of tbe latter on the motion of t be mass centre

corresponding to c y clic coordinate e(. If t he central el11:~SOid of t be !,:yrostat inertia is an ellipsoid of rota lon aro uu d axix x , with AI= Al and ~ urostatlc

L

et. '" 6i..= CAM'! . 2) 'I.

(1. 5)

0, with (1.4) havin c t lie first inbe g:ral

(61

ro~ = f!-=~ ocr

moment

k

(1.'1)

is collinear to tilis axis, equati on (1.1)

also admits tbe first inte gral

coL

0rp = Al t.V + K =C :.,;o~ . J

J

( 1 • '3)

expressin~

the construlcy of proj ect ion~ on axis xJof the kin e tic mo tJe nt of the gyrostat t1J. 2. Let us consider first t he li mited

°

proble~,

assuming the mass centre of the gyrostat moves along tbe c ircu 1ar KepJUerian orbit of radius Po with its centre in attracting centre 01 with constant angular sP e ed IN, , with'

W; =l'if.3

\..n

\0 W

f

L, f L.

wher e ~ deno te howogenous functi ons of the power of s from varia bl e s ~i ( l~f,~'J. Since L lloes not depend upon time , the equa ti on ~ of relative gyro st at mo ti on admit t Le Ijeneru !ized enert:;y inte Gral L~ -L. ==:C4mt (2.2) The p o tential ener gy of gravit at in n and centrifu g ial

to rce s,.- t>aA he-

~

1'm>1II

~

b!rrJ( L'

W( ""1, ~ f/=-L ==Zw.(lw A~.-Jwk&-KA)(2.3) " , 2 ' 2. ' "r'" "r' 'I,..

']there A an 1<. , uenotei' project ons on ax£s x of vecI' ~ ltor j o! axis direction !I and t he vec :or of e;yros tatic moment k . Acc ording to t~e principle of permissible displac e laents, t o obtain a relative equilibl'i uGl of the bjTos tat on the circu lar orbit, it is n e cessary and suffici en t tha t

oW =0

t.

jl(~.) ~ 1~ f/ -1) == D) j~ Vi) -= }( rf~ - f) == ~

Ji(s, ,)',J -==.t(!,f.

( 2 .1)

The oroita1 sy stem of cooruinates O~jlUniformlY rotatell around axis Z wi th angular sp eed lA. • In this case it is natural to raise a problem about the states of r e l utive e quilibri um or stationary moti on of ti:J.e gyrostat body in tile orbital system of c oordinates and tbeir stability. 'file equations for the l:5yrostat motion will be eL!uations (1.1) for variables~,{i=~S:6J' with Lagrang ian Lmc tion (1.2) in which the terms with muLtiple M are to be c onsi uered constant and equation (2.1) is to be t a ken into account. L0 t us st a rt with t ue case when ( 1 .5) is obse rved. _ , ~_'C , ~ A 1TA-tw. ' ~"'Y''"~~ ~ D.. oeing a constant parameter on whicil the Lat;rangian functi on, obtained by removinf', the additive constance fr om (1.2), depends. This functi on nas tue followin g structure;

/.. ::. 1..1,

i. e. tinding 01' states of relative e~ud.JWPt(J..s reduced to finding stationary values of potential energy W. Monograph (1) VOints to some solutions of equation (2.4). It is important to select variables io respect to which stationary values of Ware expedient to be found. Variables~, and /. are not independent and are connected by three ,~uat.au '

(2 .4)

- 1= 0

(2.5)

Excluding devendeat"~ari!ibleS f' '/' wi th the help of part or all of the ~'1~2.5) or presenting these variables as three independent Lagrangian coordinates ~. \s 4, 5, 6) determining the orientation of the gyrostat body in the orbital system of coordinates, function ~ may be expressed by a lesser number of variables, the minimum number is three. the respective equations of relative et:tUJJUUM, equivalent to (2.4), are considerably non-linear, wnich complicates tneir study. Hence, it is expedient not to consider coordinates qs but to solve the problem using variables ~" taking into account (2.5), and introducing Latirangiun multiple )" (t =1, 2, 1) [21 . Thus, (2.4) is reduced to the problem of unconditional extremum of function

J'i

w.(~.,I.),) ==h/ tt..)..j,

which, in i~8 turn, is reduced to the system of nine equations

'1W,::::(.>.

0A

W

I

-Aw')~ -c.JK· t -). t=O I ' r' . " ~O' I 'dw, _:=},=O

Jt

(2.6)

~==(3wZA 't~ t), X =0 '1)', ' ' 1%. • .. "0' Jr' I (1~'2~) wM'h the same number of unknowns A , Y ~ ~ r , = , 1 l) f -, ,0" " ,( , " Values Ai' k i (i = 1, 2, 3) and We are parameters. Let us assume that the moments of inertia A, and angular speed w. of tne mass centre are fixed, and consider only the influencE of the gyrosta~moment. It should be noted tnat the first six equations (2.6) are linear and have a simple structure, while the rest coincide with equations \2.5)· These equations may be solved at giveo Ko' But it is more interesting to have certain values of ,i.e. to set in advance the direction of the gyrostat body, which must be turned to the

10

attracting centre, and look for respective values fl. ,K, wnicn will allow to realise the relative balance of the spaceship with a given direction to point ~ . Having arbitrary values of /' satisfying tue second equation (2.5) and ~,:::).,. ,and solving (2.6) in respect tOf,l(.,>'z and ~l , it is not difficult to obtain the values of these in relative balance (2J;

=r", ,

),

=

le

-3,;<.tA

3w lz Ay'" -(1 A)lL)~J "S-, A,~.;. - A) ''4", ,.-:;:,'l·(\,- A,w.')~w t),.lOI<" j

qi='1

~,,;.-

L JIJO (

)..1.

l

AJo ====1'

yZ

'-(JW)

0

;'4~

L!:"j

i.,

1.1

~q

('=U:Jj

\01'0.

But if

(2.7) I

.Al. -

L

f1l

UJ

J_

which one of the main central axes of gyros tat inertia is collinear to axis may be any values that agree with the first and z, ~'O third equations of (2.5), and the rest unknowns are determined by equations (2.7). As can be seen from (2.7), tue values of (1«: depsnd on-

tu;

lyon (Le ' and Kt. - on (J;g _ 1 ~f'''':'''' Value .A,., being the product of multiplication of~. by the valus of the kinetic moment of the gyros tat in respect to normal to the orbital plane, influences the selection of K. i , and the character of stability, as will be shown hereinafter. The solution~of equations (2.6) may be divided into three classes lJJ. 1. The main central ~s of the gyros tat inertia are collinear to the axts of the orbital coordinate system, i.e. one of the values of and one of the values (1'0 are equal to 1, while the rest are equal to zero. These solutions hold true if the arbitrary gyrostatic moment is collinear to the normal to the orbital plane. 2. One of the main axes of the gyros tat inertia is collinear to axes x or z of tne orbital system, and the other two are turned, in respegt to the orbital axes, by arbitrary angle 8. , which determines the gyrostatic moment. Thus, for

t..

example, if

l

K,o ==Q

(330'=

=2(/:.I.v.nil~ - !(105~"OJ)

-/u ==Jm iJ__

f~c ~~,

(2.8)

=c..ro p~

2W,(A1 -AJ.0n2Q, =KJ,GffJOo -k~,~tnV.

42.9)

~

\ - i 3woIl ,(~ -1I.s )1. '-{<]'Il=O is possible ~~ly if in ~l tive balance

+=""

! A -A,)lJc Yf112{).

J..=f,o c::: 0,

'/2

J< -

')

occurring i f

\0

or if

0;

j

J

\Jl

"30=

!'" = ~6 ==11 IJ~

(-~~

== 0,.

YJq === f .) A =Sf"p,v ., (~ 20 O·

,-,q

'=

CIr.lIJ

f1

For a spacecraft with axis-synur,etrical ellipsoid of the inertia the solutioruof only these two classes are possible. 3. None of the main inertia axes is collinear to tne axes of the orbital coordinate system, e.i. ~ or Y.. , lO (j "are not equal to 1. EaCh possible set of values d'w agrees witn two dynamically equivalent states of balance, differing from each other by tue turn around axis z by angle JT (21. ~'or a spaceship, whicn is a solid body wi tIlOut rotor, with three-axis inertia ellipsoid, only solutions of the first class are possible. Thus, the rotor allows to broaden the class of solutions of equations for balance and orient the spaceShip by any point of its surface to the attracting centre.

At toe same time it snould be uotad that not any orientation of a spacecraft in the orbital system is possible due to the rotor. Thus, for instance, among the solutions of equations (2.6) there are no states of relative bal ance under ,.. niCh one of the main inertia axes of the gyrostat is Co;~\~~~ to the normal to the orbital plane,

two

~o

and the other

the orbital axes.

Now let us turn to the case of freely rotating rotor, when there is integral (1.5). Ignoring, by the ~~ method, the cyclic coordinate ,let us build the function of for a limited problem

ex

Rd

K===R1.+R,+R.~J..-61.N===ifAI,J.~ + iJ 1'11--!,J(,;s 1)2 ,<-

-1

t=,

.I

GJ/

L I1Jy/ - J..) ''''

d'

J

A/L

i-!:i.. Z J

(2.10)

Equations in the RiJti/Ja. form admit energy integral ~ where tne potential energy of the given system is

W"(f'l,)=-R.

=VJo;'U~,A'r'~ -lw,A~f:-K:l;)t I

where moment

K:

R. :""m-t.

l.

T( _ 7)2

( 2 • 11 )

+2-(,)• J'J'r:

denotes projections on axes

;. =

~,

of the gyrostatic

Ne

(2.12 )

The states of relative balance of the gyrostat in this case are determined from the condition

~W;=O

\J1

(2.13)

Since for relative balance of the gyros tat body and~~~7 rotation of the rotor, at q,=-D {[=;,.,.5.: () and 0< =01. between the vectors of the gyrostatic moments the ratio

;~ = J. +JIV'V-+l)t

\0 \J1

(2.14 )

nolds true, it is clear tnat (2.11) coincides with (2.4). Therefore, the state of relative balance of the gyrostat in (1.1.) and (1.5), ill case (2.14) is observed, coincide [lJ. Finally, let us consider the limited problem of the gyrostat stationary movement in respect to the orbital coordinate system. Let the central ellipsoid of the gyros tat inertia be the ellipsoid of rotation around axis 'JCJ with a gyrostatic moment collinear so that A,=A2 , Kt =K2 =0 • The orientation of the gyros tat body is determined by the Euler angles. The moment of forces ~ applied to the rotor and determining, in accordance with (1.4), value Q( , is considered to be an arbitrary continuous function which, in particular, may correspond to cases (1.5) or (1.6). The first integral (1.8) is in agreement with cyclic coordinate ~

F,,,,,t.... /?UIC-#. ':i R == /... - r;G = A, ['f'~.h,.·B t OJf2cv, (rj;J.i..BC9JOvny + UJ.h..l"J]+Grm 0where the changed potential energy is - V(~, t;I)

~unction R coincides with that for the case with a solid body, and the problem of the movement of the reduced system for tne dynamically symmetrical gyros tat is equivalent to a similar problem for a dynamically symmetrical solid body. Without dwelling upon the solutions, let us note that under Changing guroscopic moment k, =~ It) toe angular speed of the gyros tat rotation cf ' obtained from integral (1.7), will, in contrast to cases with one solid body or gyrostat with KJ =t4,.yt, depend on the time by the function tj (t) . Thus, in this case too, the rotor allows to broaden tne class of solutions that may be used for controlling a syaceship. 3. Now let us consider a problem taking into account the mutual influence of the centre mass motion around the mass centre. Ignoring cyclic coordinate 61 , which is in agreement with the first integral (1.7), by constructing the ~ function we shall find the potential energy for

(1.5):

wC}" X'f' ,f.) =2~{k-t k.~Ar -;'hfJj +1!.1A.(Sy2'J' ", (/, U (3.1) ;J

r (] fJ, S-h-. 0 vo 't'

(2.15)

/

wnere the inertia moment of tne gyrostat in reSi-ect to axis ~ is ~ == f ~ and I' a , denote the cosines of the angles formed by axes ,;r, , wi th axis '7 . 'J ariables ~, aad are connected by three ratio&:

{

A,l.·

$ MflVQLJ(

f.

~(fd =1 (i;,t/ -fj=Q i(f.r=lr:fr.~- fj ~~ 5

.£ ((J, 'f.'~-=f!'.f.

-.f1,.)l

Constructing the tunction

i

V( B,'/f) == -j ~/[3 (AJ -A,)(XTJLg fll,sj"zP~Ly]+

/-,

w'If',f ~

X) tC ( ;

r'~

"J

.) .) t.

=0

(3.2)

1

=117 +L >.J. i-I

the equations of sta ionary motions of the gyros tat will be:

~

1:.' Mflf =: -

C4JLX

9 +;.11-z:f~ Adf.'- -1) =0

cow. = (fr1Sl.ZtO.J. j?.k O~ J

J

-,))

~

~;c.= 0

./

'=J. t)..3/. == 0 ~ '0'\,

<-

~ ==(J., -A)2JI'

~~ 'u[.- = (.3 Aw.L .

t

0

-.QK,

.A2 )ry t), Cl'

I

Ll =

.1(-.

0

(l-=I2,J) J

0, (3.3)

'

Q=J(K-.t~'fJ

whe:e W. is determined by equation (2.1) and denote the constant angular speed of the mass centre of the spacecraft in tne stationary motion at constant values:

f ==f· >)( ==XJ/f,- -=f·" '" r -=f..

I .\,

-:::.\ LV

(e'-: 1.1. .?)

(3.4)

determined by equations (3.3). At given values ~,. and/; " (.11 satisfying the second equation (3.2), constant!" and are related by the ratio _ 2 52 2 (.Q) x. --

2.

vJo

_.1 /11 4-A (~L 2 -r;-: -==- ' w

1'1' ,-,

_1) J

tne Lagrangian multiples, angle of tne gyrostatic moment are ~here

J 3 [213 0 == ~ ~ AJ<~ -(LA fi._ ~)2J Il • w l

Vl

t

\0

0\

L::::--,

(. :"1

J

\ ::::- - 3 woL l -;;- A 2 "Yw

..1\20

0

r..=I

)('0

-

1

$'

.A.~O Si>.)(

J

Sin dfJ

:::::

).JO M.Q

:to

l

1?:K

' value 01 the order of e~,. The possibility o of sniftin tne or~ ital plane from tne ~tracting centre was stated by Roberson, R. E. {5j and Stepanov;' S. Ya. ~6J; the latter found all possible solutions for tne type~ I n case 01' a freely rotating rotor, wllen int et;ral (1.6) ho lds true, equations of t~e stationary mowements of the gyros tat coinCide wi tn equa ti ons (j. 3) tor tne case (1.5), if constant K of the area integrals (1.1) is in both cases .... the same , and tne E;yro static cr,OU."[Jt:..; k. "'ld /(' a r e conne cted by tne ratio (2 .14) {7J. 4. Now l et us eX"im ine toe st .' 1bility of relative oal"nC8S of the gyrostat in a limited problem for (l .~ ). On the basis

)

Q)

~~ [\c{c. r (J.,o -A)2 l)fi ,1

with

J'r:

~
and prOjections

'1

j

C1n X o

)

(3.5)

K.

balances in tne Hmi tad problem. In case ;'J,' =1) , the difis the value of the angular speed of tne mass centre, but tile orientation of the gyros tat body in tue orbital system ie tne same as in respective solutions of cla13ses 1 and 2 of the limited problem, i.e. the main axes of inertia of tne spaceship are either collinear to the axes of the orbital system or one of the main axes of the spacecraft is col11near t o axis 2;, while the two axes are turned to the orbital ones. It Sllou lCl be not ed tr.at tne statiot,ary movements during wnich tne spaceship is turned to the tangent of the orbit, the attracting centre is in L~e plane of the orbit, are possible only for an aWymetriclll spacecraft [11; for instance, the movement for which f~cence

(3.6)

}.30/ 0

~

;

fio and under (3.2).

~J

~

+)10)(,. constants A are ,-co

(3f.{,'-,{ Jo

= 0 ,

(3.7)

found from equations

of tne Lagrangian t heorem, suf:!.'iciCfJ i

Proceeding from (3.5), tne rati o of the square of the true angular speed of tile mass centre to the square of the Kepler's angular speed differs from the unit by a very small

(i'if.r .

value of the order If .\)0 =1=0 , angle .)(.0 ' formed by radius-vector of the m:lSS centre with the illane Ci{S' is d~ft'erent from 2; ero but is very small of the order ~

,.I
The values )~cJ ' ).30 ' lco ' and K" differ from respective ones (2.7) 1n the limited problem by the same value. Hence, stati onary motion (3.4) of the gyrostat is very little different from its respective 11

).~O

le(f,)

= 0

;

{.; (;!.o.i t i() .~1 0

~cr

QalCi n ct?

are th" conditions of t he '"111ilLU!L of functio" 'il l Wilici. are obtained ~s conditio!~s of Ccrta iIJ p o sitlvc:J~8~ 0f t!, ~ se cohf wariatlon of function W. These c u nu j t lG ~~ taM E the folluwing form Ll}: t ! - 6 - )' for solutioJls of the first (;la89, w~en t(~" -I .3: -0 ~J ="', .U - I

y-a

A

('U

--,y - t .,' -OJC -

A,

+

J'-'-

v"

>A, >A,

A

2.

+..s.;..'> A ..i; fv\~,

J

(4 . 1.)

)

for solution of the second class, vhen

~ le == C¥.l D., ~l' ==/0. =-1'0 -= 0, J'l':= t

A~

A

A

k.

+~. W).! u. > 1

J<~

t

:z

j

W? /,

(3 t(NllQ,)

(,)

~

I

=~Oq

tq

)

the balance of one solid body

>A

~ 7Al

JJ

Q '

.3(AI-A$){A~-A.I)+{Al-AJ + hlo('~(/J(A1WltU. + + A Sio- do.-.A and in case I. _y ~ s) ::::> 0 " _

(4 . 2)

are obtained from (4.1) at Kz. =0 . Comparing conditions (4.6) with those (4.1) and (4.5) ve come to a conclusion that rotating rotor produce both stabilizing and destabilizing effectslIt may be shown {2J, i f A,.?A.??Aj ,that large circles

1

1<.,

r"-o'~ -OJ

Al t~W,(,q)l~, >A.1 , J

fJQ=ji.==V'Y.JIfJ"

A,-1J?"

l

t.'-AJ

[,tn'I}.>

f,.--!'. 0

=!if>,~

.

(4.')

(A, - AlJ{A. . - A.. )¥i..lO. +(~ - A, t r/:'I,) (A, - A,lfh.ll. - A.J WILd,,»O for soluti ons of the tbird class

v\:>

(4.6 )

7'A.l

YA

I.

f2J

-Al13 ± (A,-Al }'1 =0

.j; f.;

divide sphere -= 1 into four regions. In the regions vi th points JJ i 1, Q,> 0, while in the regions wUh points if. == t-/ 4 ·.::.0, with the exception of points y. and !' - ) OJ (, ::::! f ,at which ~ 0. At A, = AI. , these circles coincide with circle )j 0 and throughout the sphere, w1 th the exception of thf~ circle and points l:s ==.± 1, A> o. At A.1 -= Al the circles coincide with circle 0, throughout the sphere, with the exception of this circle and points f , A,'" o. In the regions wbars a> 0, the balances are stable at Atv">). and unstable sercularly at Alo -<). , as vell as in the regions vhere A-<. 0, at any ).'0 . The balance is unstable at ~,o?). and at in the regions where Q.":: 0 and at ). >)" > V (I _ ~n the regions ....here Cl> O. Let us consider the case when integral (1.6) exists. It was shown in (7) that for a certain position of the gyrostat balance function W(~ ,Y,;.) has t~e minimum, and for the same position of balance furl'ction W .ji) also has the minimum . Hence, according to the Lagrangian theor~ the

=

0, .A >.:A (2,,). == / +(fL..../;pc)

(4.4 )

'0

=+"

2

Vl

\0 --l

waere a, band c are determined by the equation (2.5) of the work [2J . Inequalities (4.1) - ., (4.4) are also necessary and sufficient conditions for secular stability; if these conditions are not observed (a sign of any of the inequalities is changed), relative balance vill be secularly unstable. Energy dissipation, needed for stabilization of the positions of relative balance till the asymptotic stability, may be introduced into the system in many ways. One of the simplest .... ays is application ot viscous liquid filling a cavity in the spaceship's body to capacity. 1J The above reeul ts hold true if liquid is presenl iB all .ae ,laRes I l l . Changing the signs in some of the inequalities (4.1) - (4.4), if i t leads to the uneve.n degree of stability. makes respective relative balances of tne gyrostat unstable. Thus. for instance, the solution of class 1 will be unstable when solving one of the four groups of inequalities:

A.l +4~ >A.l:?/1, I A~ + {,; .>A,; A.l >A, >A.l r ~

,

.J?L

W,

J

.4.3::::>AJ. +/L

K A,?AJ. t :: I A,?A.l , A.l+ .t.>A A~f .KL>A;>A>A+-K.(4.5) o

4 rJ.:l J

H.

f

:.l

.

7H~)

:L~",.

The sufficient conditions of stability in respect to

=

!r::-

f,:::t

=ii

.

\."::')1 ,

(f•

c:;fAdc;:.0ns .1 6!::'bili ty of po si tions ~

Fesi.gi~alagQe of the gyros tat in case of (1.6) will be broader than the conditions of stability in case of (1.5). This is interesting for applications, moreover since conditions (1.6) are more simple to be realised than conditi ons

(1.5) . Without touching upon the condi tions of stability for

all possible pOSitions of balance [7, 2], let us consider onl¥ the conditions of stability for the solutions of the first class, when the ruain inertia axes are collinear to the axes of the orbital system, and in the position of balance the !"otor does not rotate, Le. 01.=0 , and its axis is directed arbitrarily in the gyrostat body; the constant of integral (1. 6) is IV' =Jw,tl • r h e sufficient condit ions of stablli t¥ are ine~ualities

A~ f}(':>At>AJ ) 4(Al--A:)(A;. tJt, l_~)fJf1-;4,)~~O(4.7) It is evident, that co nditions (4.7) of stability of relative balance of the body with the rotor, that does not rotate 1n tile position of balance in r e spect to tbe body, are somewhat broader than cond itions (4. 6 ) of stability for one solid body with t i.e same mass distribut i on. ~'or instance, if the rotor axis is colli near to axis .:J:., 1, l>- == -= 0) , conditions (4.7 ) are reduc ed t o ine~ualities

(e, =

Vl

\0 CD

G

A.l -::>A t - ]... A, > Al ~ A./.>AJ

(4.8)

whi ch may be fulfilled i n cas e A>- <: 11" when in the positi on of balance t h e central axis of tbe inertia ellipsoid of the gyrostat is directed along the normal to the orbital pl a ne. If the rotor axis is collinear to axis .1:, Ol X2. condi tio ns (4.7) are reduced to ille~uali ties (4.6). If t h e cond itions o f stability of stationary motions are conSId ered for the full problem, i t may be said that (1, 2J they differ from respective c onditions of stability for li-

problem for real sattellites - gyrostats, whose characteristic size is mu~h less than the distancs to the attracting centre. The availability of rotors broadens the ~ class of motions and allows to control, orient and stabiliz e the spacecraft.

REl.<'ERENCES

1. Rumyantsev, V.V., 1967. Ob ustoychivosti sta tsiona rnych dvizneniy sputnikov, VTs AN S8SR, Moscow. 2. Stepanov, S.Ya., 1969 . 0 mnOZhestve statSi on arnich dvizh eniy sputnika-girostata v tsentralnonI nyu t onovsk om pole sil i ich ustoych1vost1, P~, t.33, vyp.4, 7 37-744. 3. Morozov, V.M.,1969. Ob ustoychivosti otnoSit e lnogo ravno vesi¥a sputnika pri deystvii gravitatSio n nogo, magnithOgO i aerodinamicheskogo nIoment ov. Kosmicheskiye issledovaniya, t. VH, Vyp . 3, 395-401.

4. Kolesnikov, N.N., 19 66. Ob uHtoycnivosti 8vob odn ogo gir o stata, Vestnik~ MGU, No. 3. 5. Roberson, R.E., 1968. Circular orbits of non-infi n itessimal material bodies of inverse s~uare fields . T. of Astronautical 3ci . , v.15, No. 2. 6 . Stepanov, S.Ya., 1969. 0 statsionarnych dvizheniy a ch sputnika-giros t ata. PMM, t.33, vyp.l, 127-131. 7. Rumyantsev, V.V., 1968 . Ob ustoychivosti otno s itelnych ravnovesiy 1 stat s ionarnych dvizheniy sputnika-girostata. Inzhenerniy 7.hurna1. ~l ecba n ika tverdogo tela, No.4, 15-21.