Satellite Gravitational Stabilization System with Maximum Damping Rate

Satellite Gravitational Stabillzation SYEtem with Maximum Damping

~ ate.

The dynamics of a satellite with a fravitAtional : a t ion s:rstem is inv e sti gate d . Optimal

C!imum durat:ion using

SATELLITE GRAVITATIONAL STABILIZATION SYSTEM WITH MAXIMUM DAMPING RATE

t L~

of the .iampEl '· of the

c oncept of

11

the

": f

:: ion is paiJ to the analysis

p ar·3 P.l t~ t 2 r5

ar e

s :r st ~ m

i rt.' e -.J 1' :.: t ::..t il i t y ". ,') f

s t ~bil i -

prov 1 :1"1 :-l[

.:....i. -

' i ,,', · , :,!: , ~,, · S pi:, ci r~ J

' : !

'

'.

J '~ '

o scillaticn of tne sy st eu. _•.

the orbital plane.

A typical scheme of the satellite's gravitational stabiV.A. Sarychev (USSR)

J.

lization system was suggested iD pape rs [t. 2

:n

th~s scne"'"

(fig. f ) a stabilizer represents two identical b=::: ri c·ic. l ;'

0\ \J1 \J1

connected'with each other with equal

we~Ghts

at the ends. Tni,

stabilizer joiDt the sat;ellite body by means of spherical gim· oal P. ~ and ~ are the centers of mass of the sate lli te and ·'':!!lbilizer. The axes of the coordina :e s ys: en; 4Xz!l~Zz

4x,y.z,

an "

direct along the :najor axes of inertia of L'1e sa t,:" "

:'ite and stabilizer correspondently. The position of the sta .. bilizer relative to the satellite body is fixed by centerinc springs. Li.nearized equations determining oscillati ons of the sa .. :ellite-stabilizer system in a circ'.llar orbit plan e are in :he :'orm [2]:

(8. +-/"Ia,I)dt -I'fa,aa d,

+

3~~ -Ct) - Man W:

c(f +

3Ma,a,~ da

+

+K,(~-d,)+ j1z(a,-riz )- 0,

-l'1a,az ex, + (~+ Ma}) Clz + 3Ma,Qzc.t ci1 + 3[rAz -C,) .. l1a,'jf.J: ri,-

- R. (dt - d,) - iCl. (ri, -riz ) '" 0

.

(t)

Ht,.,. ,

M,.

Here

f"(I+S:+S: »).·+/(,frf+jI')~ -PS,)")A' "{K, ~14}1')" (s, -PS,,)"].,.

,""."Ma

Br, e,

M" A"

and "",. A~. ~.C.l - are masses and the, main central

!!Ioments of inertia of the system body, Cl, , ~ issa

- is a eimbal ab-

coefficient, ~ - is an elasticity co-

ef ficient'~a- is an angular c'~ nt ~ r

velocity of motion of the system's

of mass l.n a ci!'cular orbit, d, , d a - are axis an gles

4 X.I

~ rbit.

with tangent t o the

pect to time t is

iesign ~ t e d

by

ax,

Differentiation with res-

It , is more convenient to introduce dimensionless paramet ers instead of dimension ones:

s,=a,'Ijf P.

0'. \.n

A,-C.

:z - Ba ' Then. at "",t • 'l"

0'.

..

SzaozVi[.

I

z Ba

= ~.

_.&:.et p,- B,

j(,

if.

III = (J.~

}J

GJ:'s,

.. +3(p,-s, )+3j1Sts"a"+I<,(d,-d,, . . ~

(I+S: )ot, - ps, Sad"

)+/(.1(<<' -0',,) .0,

-1'S,s..a.+ll({+S:)d.l+3ps,~a, +3f/(Pz-S;)rIz -J<,(~-d.l) -Ka (d, -a.l) = 0 . Zero solution of system (3) is k,> O.

(2)

,

of t he cnaracteristic equation are real and equal to one anothe r , It become s possible i f tbe ' equatio'ns

t .. &3

p,.,.)l2Pz -(s,-)lsJ {I+jl2)+{S,-jlSZ)2 •

z

5~ _3 ~ Z Pt ((.sD+p,2 ((+5,')-(S,2+ 5:) + V"f

(+5,1+ 5 :

I<.,

z

Kz

-3 (/+jI')+(s,-jls,)'

are fulfilled. ~o

-

Kz[P, +p'pz -(s,-pSJ.)J+3}/(P,~-PAz- pzS,~) > o. fJ.I-(5,-PS,)S"jp, -p{1+s,(s, -ps" gp, -(s,-)lS")fJ'Sf+SZ)~ 0 To estimate damping rate of system (3)i't is convinient to use the concept of the degree of stability [3J which is represented as an absolute value of of the characteristic equation

min~um

real part of the rOots

f+$f.S;

, -6)

)12(1+ 5,'. s:) 4}:", (/+}lZ)", (sr),Szl"

[ 2~ Z

p,{1+S: )+~(f~S:)-(S: +5:)

1+s:+s;

III

(3)

(4)

z

4Pz -P. Sz-Pz S, ,,0

Q

..7

Here

f

7.

J

r.. - is the system s d.egree of stability.

de t ermine maximum value of the Jegree of s t aoility

l e t us allow th e first two equati ons

p,+jlzp~ -(S,-pSz) > O.

.

choice of free parame ters when all t he r oots

jI ,2 (l+s,l +5,1)

stable asymptotically if [2]

-p,S: -pzS,') ., 0

($")

of stability of the system is shown(3] to be

achi eved a t such

V"f

1<,,, the ' system (~) is in the form: ,

r~ degr~

.2

po int.

i) ).'+3K,[P;+)lp" -(S,-)lSzl]A +

+31<, [p,+)/pz -(s,-)l Sa)2]+ 9}1'(P,~

P in the coordinate systems connect ed with the both bo-

!ie s, K, - is a friction

and

~

+3)1'[pll+S:)+Pz((+S,'ij - 3j1'(.s,z.,.s:

f

2

z

2j1

z

Ln

A =3~'" +(1+ T~'" )S,(s,-PSz)t Tf.,

(6 ) relative to

yl"'S,z+Saz·

f 2 f. 1 ~ )( 2 zy . rJ.q", -"+"3,'" $,-)15,)5z + T~", 1+S,'.s: 2

n _

'

~-

satisfy the inequalities f~p, 6{

f~

anapz'

(7)

}I

For the physically real systems the parameters P1 and

-

Pi

t

-I'" P2

.

~{.

P;z must

(8)

Substitutbg 1n-:o (8) the e:q:>!'essio ns 1'or p, and Pz from (7) at

Let s,"'s.. -: 0

upper sign before the ::-cot we 0btain the system ot 1.nequal1t1••

'- ~ 2t!. ( f+jJV{+sr~s; ) 3+ f!. -'-5,(5,-)15,) , 2 2 ~ 3+f.,

~

pYf+s,'+S;' +1 + s,(s, - )lSz)

~ of t h e stabil i z.e r t h e cons idered

~z=~.

(9)

2r~

i

~

and at lower sie;n before the root

z_

2~!.

3+~:'

{+s,(SrjlS,) ,

2 3 ~~

jI + (s,-pS,)

&,

z'

Y1+s:,+Sz +ji-(s,-jlS1.)Sz

~ he

&

~

Bz ,

s,~

analy sis of the

(10)

s,

i. f' ~ ;

13

!. "l

1):'

C'') :':

'):~

., ' ,f:'

J"

" J

4 : 1'_

P. :' f J : . ~ 1.

<'.:

P2~-1

((I)

2-

~ a;-

jI-2 Pz = }I(f+2)1)

A=I,

c o nsi ~e !'

:-e s pe ct t o f'

3-8

S:;~...:t9!~.

s

(9) and (ID) is ~ te res: in g

,C;!

l ead to a maxi ~u.m v.:ilu e of

..J:

v e ~ cOw-

hi h ' C ~s

:i:,' :.. c"

,

SI. -0 ;

$('10,

5z = 0 ;

s, ~O,

5 z <10

Jr.

.. .

f

tn e ~ ':- . '':' .:~~ '_. .

two J. "

'.' . ~.

. .. -

-::.r'

~

-

((3)

yS

Bz "8;

~ nls case Pf& f 'PI."- ( i.e. the sate llit e re~resents crav itational s" abl~ Dars ~~ G w

0

bt a ~n ' ed at

= -3-« 2-

-

1 .:1

1-"

stabilizer - gravi t a tional unstable ones . Solution 2 is wri t t en in t~e f or m of

SL .

~z

spa t ial cases:

r

~ "Z+)J'

A'"

)J(f-2p) 2+)J ,

P1.=f

in the interval

s, - 0

.-: . -1 1 ~ l~

fi

)J

52.

on : y the mo st

.~ ::·: >!~ C

t

~ '" mOX'f ., 31-'

~0 ,

~ equa:i t~ e s

(/2)

-2--

~

-

p l.icateci at ".:hc ar'~~ t:-ary value s :)f : he ? a r=-me'te rs )I, S"

rh at is wny we

t

:;he fix e d va ::' ll e jJ i s le~~:"a : ~-.! ::. "

Nnich de te rmines soiu t i :m2. Sola t i Jn 1 t=ns into 2 and vice versa, when

1

.:;(.J~~ j

t ~· t, ! ' u c: i r, ~

0t

}/(f+2p)

:iere ca.ximll...G1 "all.l~ 0:- :: ::;~

) +fm

1

5'. ~ u- ~ " n

3

when

~ 3+~ ;.dl+S:+S: ~ (p+ Y1+s!+ s:

:,~ ~ {:

C(; ....

') ,"L~~ >

'':: 1'

P . ="_p -1

- {+2)J'

2 Z') ~~.1 ( f-p ...;1+Sf+S, 3+~!. 1-5, ( St-pS,),

--:]

'lIi th

B. -

we obtain tbe one

0\

Cf!~ I T. ~; :'

~J'-(S,-jlSz)Sz

Y{+s:+S;'

wc i ch <.ie t er:::i.nes s o luti on

\J1

• ;1(:

o~Jk..:. 3-8

3+t=..

3+F",

ca se

~ 2-)1 i::l oh e lr.~ ,o rv a l

0,

(p-Y1+S:+S!,> 2~~ ~.fi+(St-jJSz)St 1

Th i :: v n r i;-, nt of

to the cOinc idenc8 o f

4: B.. ..:. 3+{5 - Bt - 2

0 'f:.t_~

., - jJ When

("')

p

l'

=-(

•

Bz 3+v'5' B;~2

Pa.

2+,)1

p(2p-f)

(($)

'

~:a:,::.::''':''::'

c':~

v':!.:!:..:.-=

t:::~

0= ...

.i~ i;:'~~

B-

_ - - - . ",

&-00

~- 1 ' - ' ;

"'

....

)
'. '

-{3

. -::1:

f = m;X ~ = Vs -~

:.:--: :!~-: ~..:. S:':'~~ _ ':':2

-

:::2: __ ~~ =

Jr.,

3.-VS

Ba T,

: "?? = ~~"'?::t;S

:~.i:o

c:=:!·.·i·~ :l-=':"':'!12.~

~~~ol~

~=~'li~ ~:~: :: ~l

;:~'1:~ (' :--4~ '::...;.:-:!

::::1

2

It

c ~~s _

.:.:'

.~~.:-_ !

.:...= -:'1 (:<_:'

[4]

.~ . ?

~-:. ~ :::1

b.,::-:o

~: :L~

;:L,j

"..- -

not':'''::ej

B, - 8% S,

CD

=:'I~

:l·::"3.ti.~.re

acccrii.:J. ~

in.-::c:~ali ty

~

kz

:ot~':"~~l

0~ti~~:

'l~~~AS

of

pl1ysico. l

s~ns e .

is

and for sol u t ion

~E:.:s':r ~tc

2-0

t:,c ::es\.!lts

•

"'

.

-

....

-

-

....

~

,

-

J

..

\

....

..I. I • • ' -

..L.

i:c.C:'·cas~s !''' ;;~.E.y

r..-:Cf;S -

3:;abl~ 'e> =

~ i:::>r.::ll

to the

~ d.ec~ea s~ s

O!l~ (~"-l)

~

0::

at

t;:;"

t

~

• Y'1.B

. Thl;n tiiaxiu:;u:;. 90 int

3(1-

Vi

f<3-VS)4 :: ~3'-21fi

>3+2Y2

Kz
' '' '.~ '~, :

.. ,:'

-t

Max i mu m degree

and th tm

slowl~·

tp.nlir::o to zero

at

.-

~.~

_~

2 vf+s,'

- ....

,~--:

.

~ cc'

s.z~t .:. . ~ :: )I +2(jI ~ 5/)

- . ~ : ::: :

...;r;sr •

P2 =-1

- jI

((7)

( jI-2-11+Sf)(1-s,')

Ba _

a; tl::;~s

case

2(~"S/)

'J :. ,., __ "'...; :) .2.. .L ..:.. . . . . .~

[1.. ...;s::7;Sf - 5,' t2J 5

I+S:

(18)'

-

..;s:;;;

2((+5/)

P1

~ = 2V/+st-+}I

(18)

,

=

(3 - 2 S:) -

So l ution 2

3).1

pz =.jI'Yf"s/(..J(.. st + 21')

P, =i .

,

1:3 achievc-d at

z

,; ~.

8z ~ (3-2st)-~ , 2(/+ sf)

)J

• For this

.

t c"~:: _ ;~.

ma:!s

....'

-:.:..

~ (3- 2s~)--vH;f

f = ma)( ~

and optima l so -

~ ~ 3-21/2'

P • _" ::-

~.~

~:.'

' ~._:_

ex, (a) ~ exz(a) = 0

=-la",

5:>

"":1

::1

b:; r

81

~ ~ -.;(+~ (Vj;Sf + 2J1)

5 r a vi t3t i o n ~ 1

increase s s l owly to maximum which is equa l

~.f(3+fi)

= .'.,:re

. . ___ :.,:.:-:.- : -. .:.. . . :,,".l _

.i.: ~,: :,,·: .:!l

z

-

exi st.

sc l~tion th e valu e

!.iI,8

"":! .c

P,=

o±

the analysis carried

OL B, Ba -~1.(3-·"'5) 2 V:J

•

z.rr;sr -}J

in thein. te r-

81

in the form of the

So l u ti o n 2 ( f ie; . 2 ) ex i st s '.'Ihen

to

::1

=

Bz ~ 3- 2-12

2

. In the i:lter-val

s lowly . ,,'he n

l u tion d:::>es no t

t~at

, and s,:;:;.bili~e r in the fo r m of the F;rav i ta -

(P1=1)

u..·u:: tatlle

:~~cllite

::·o~J.

d i fferent values

\': ,."':;

c ::-.~:;

3jJ

2.

-'2.:'"1-2'",': :'3

:'0 .' . !..:: <

!,:,~/:'; -::

ciz (0)

.< ~ :: ~ " ::

::--_~

'.:. :':

Sz =0

•.• r "'"

an.! at; the end. of th" in.t e rval i':

. " ~ :l.!. C VC~ ~3.:-:':"r.r...!!!l ·.'/hich i : '?(~ual to cor~~S?OndE

:: i3

f~lfield

·· ,·.'o~ c"l····;·~ I. ( .. , . ') ..L.i~_ ~ tn' e in+arva l v ·· '.'c 'n:!:.! '"

*0

~~e

Kz ·N;o.ic;o. C3.:1 :'.ot

c:::e~·fic: ~ :,,:

fo r solution f

0

." "': o!. ~: ~ 3.-2-./2 :.,~ :'~':s

i~s

to

=20°,

...i~::..~l3

-

0\ Vl

.... _" ....... .- . . - 3.::.:

c:i. (O)

b~: ::':0 .:~_· :' ·, ::- o ,",:

_ _ ~ ::-

, pz = 1

_ 2 . .'::- -:

.,',=:.

:''':''':': :':':'::'""":

S::.:.17 only t::'e s::':;:n of ':he

~o

s~~~:! ~~

, :'-: ~ . ; - '~ J

-( ' Pz = f

s a::'.s.:'y :::= :.::.::i':"-;i'::''::'s ;:;.:' s:1.bi:i::, (4). It is

=2...:":'= ~ -e =s

:: 3.:::--,/

:';"'-::-J.b~ ·~

':":-.-:02 an.! ,,: ':.c;: '''~ ~2a , 'N ~-?:--~

s,=SIl.=O ~s ,:: :-.s':'~ ' ::- " l ':':0 ':'':''':~~~:l:_, .. _

case p.~

J:; - _ ___ ~ _

-- ... f

k • . P" P2

p.e_(

=:' :·.1':-: :'::

.~ :

= 1 ~~ S

pz. ~o~e

=-1. • When

Sf

to

-

..f3

J"-W

com?lioatcd. fo::m .

)/(f+2s,') - 2 (j.J2_ s,Z)w;:s:' P, = _r;-:-;:r • Pz = -( 2 ,1+5,2 + j1

(Ig)

L:1 the fOll o '::i!1!_:

.) 0 ~ s/.L

1:. ' ~$

2

z ) 2?!~5,"'1 .2

3) s,

>

~ B~

0

in solution

:1:);-," :

1'('

3+Y5 (/+st)

.<.

Bt -

2

& ~ (3S"-2)-~

0'<'

Z{f+sn

- B. -

2

S,

t ~c

€

t " ",;x~=W : " _ ·" ,1

A c-1

~2 I

a

3~ '

a;-

0"\ V1

'-0

~

--2- (I +

~

c

v,:

3(I-s:>

sn

•

~ :,

,

v a~T .

j .~ ~~ ~ (: ~:

: ". e

h -

ac._~eve,i when

[.i.. --.fSS;-i:s,. )

.-i)~-

.51---:..

at

+f

s/ '" f

f ~Jr , ~'ff

]1-1

(23)

~ • In oh~s case

and maxl::lUID value

p,=Pz·',

i =if

is

,

The c o.::di t ion of

positivenes~

ent kz lead to the inequal1t7 04 ~(3+2Y2)(f+

t-

of the

s:)

s;) !: ~ l-

1'. ass 01 of t he

c a rry out : ~e

ela~t ic ity

coeffi c~-

(24)

~.2_

':-.' ' :" ;' L 'l :\r l ·'l!"'~t

,... : ::..:: ~ - - ~ .

'J.:-:

_~

.s,..,.O , s~ l' 0

'1::1. , :-1

~!:: a.t

S ;J.j f, ~ l' .1tl O n

of

s: .

. o r mala e (17-25)·

_0

1S "NUy

All.:'!

.!....::l

~ : :~

-e r'.-! ~ 31

30-

-:-!

(9) '''': : (10) ~.s

5 ~~ t~~5

i~

:~ s ~

~;.'

~ " .' :, .::. .

~ t. -:

,:' 1. ~-?

:.- n ~ c:' :- ~ l

:;'4;

- ~ .~

';

:-. ~. ::C.: .

!-j

~ .! t

L!1 '2~v.a il:Y

';" :;'" .1. !. -

:10~;

:' 3

Vi'

}","'W

~ul!~ll ~ d. ':" _; a:".. 5

'1I~, ~ :-1

a,

.a~

t;he

-: e::'

::' =1..1..

(12)

(3s:--2)+iSSt~-/,'

*' -8-,- ~ -'--'-2='(i-.(:-~

SI. (3s,"-2)+-vs;r:;; If; = 2(f+S:)

:..;: :asy cO show tt-at !c.:::'eved

f+

:"l....'1 '],

-

.:.: :::':n= value of t he ..le ~l" ee of !:tabili !":.

)I

~ a:. e d

is a!s o

Pz= ;.df+St(Yt+Sf-2j/) Bz

'h 'c:

In C0 :1.c:" '...l s i c ::. .:.. -· t

(2f)

(2~ +}I)(f-S:)

2 (f +?, )

_.'

.il.;::lcult t :, s r.ow -:!'~s. t

01..

c ~·

c o rres;:>onc.en::1 Y '" it;h soJ. ut1. on 2 and f

lution of t he 1.:-1e quali r y

p{2p- I+S:)

(3s,"-2)-Y$st-,,'

f = ma X ~ •

~Bz

1 ! ,,. -. ,,1'; Ba B, = -2 (3+,g)(1+Sn

(2..r;;;j+~)-{1;Sf

r·. ~ 3. ~:i.;~ · ... ~.i

,_ ~l c i.de

?~ '3" v,:, ous

:'or::,:.:l a (20) .

A= (.

~ tfS/~f

8,

,,1.1'-'- 2 c

c ')!'_struc tl Oll

= ::' ~ ':'::l .: d

~ '" ~w;:;r-2}J) N::"n

,- ,,"

p,c-f, Plo"

3+YS'

_ ~--:"::r :"5 ,;.e -:: .2

(20J

Pz =1

2}1-V(+s; 81.

S,

.) !1

,S,f."O. This variant of const:-uct i c n corre s -

?,) nd s :0 :::e c 01.ncide nce 01 :h e c e r.ter

;; ~ u:: : 1

)1' ::--.e

, ;

(25)

~~ ~(3-2..rz)(I+Sn

l ~ te h ~ "~l ~ ~ e <: ~nbal ~os 1 ti o n P . ~o r ~ h e c u ~si~~r c d ca s e so l u-

3+Y5 z ~ 8, ~-2- (1+5,)

2(1+5,1.) -:~ ,

- B, -

z)

a.

(35,"-2)+~

1,

2 (f+st)

and to the inequality

in solution 2. Let Sf: 0

(3S~-2P~" ~<'. 3+i5'(t+

'

t

-: !1o.~

'" 0,

-.::.

.: : !"'.:.': ;-::-

.., ..-

::.? :... . :. ": ___"7'. S '"

:; :

~ ~':!. r. .:

.,

.. .

-: : "~

~

S - ....) l • .J...:. l .....;.·

,\ , .

:....: h l ~ ecr_: ·_" r-.: ..i ~ !..il.:Ja l

l ::"' s.'ia': l L 2 3. -:--.: - ::.:.:~

-~ : ", ~

:: . "3..~ :" ...

_.:. z ".: r.

. ~ 0r.!

LIT ERA T U R E 1-

r. Okhots1rnsky, D.E., Sarychev, V.A.,"!. Gravitatlonal Stab i-

sateetite

2- sta8dize-z

lization System of the Artificial Satellite$, Col. Arti-

2

ficial Earth Satellite(U.S.S.R. Academy of Sciences,

cente'Zit1fj s/YZin[JS

p- sphe'Z icae [Jim8at:

/

Moscow,I963)v.I6,p.5-9.

3-

/ /

2. Sarychev, V.A., Investigation of the Dynamics of a Gra/

vitational Stabilization System, Col. Artificial

~arth

/

Satellite(U.S.S.R. Academy of Science6,Moscow,I963)v.I6,

/j!!2 !i /',

p.IO-3'. 0\ 0\

o

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Linear System,NISO,N9,I946.

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DISCUSSION

Q.

How is the degree of stability calculated with the characteristic equation?

A. This can be done by various means, for example by using the random search method, but in this paper a new method was thought out for working out the degree of stability which is not connected with random search and which makes it possible to obtain this degree of stabilization very accurately with the existence of analytic correlations. The idea of this method is the following. We assume certain hypothesis on the character of the distribution of roots on the plane.

Q. You mentioned the equilibrium attitude. equilibrium positions as well?

Doesn't there exist other possible

A. Others are possible but I have studied the small variations of the satellite around a determined equilibrium attitude.

663