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Stability of the stationary motions of a satellite with a gyroscope in a central gravity field
STABILITY OF THE STATIONARY MOTIONS OF A SATELLITE WITH A GYROSCOPE IN A CENTRAL GRAVITY FIELD * R. S. SULIKASHVILI Moscow (Received
17 October
1968)
THE stability of satellites containing rotating parts has been investigated in a
large numberof papers (for a bibliography, see for example 11, 21). In papers by the author[3, 41 some stationary motions of a gyroscope mountedin the satellite on a Cardan suspension in a central Newtonianfield of force, and sufficient conditions for their stability, have been found. The stationary motions of a system consisting of the housing of a satellite and of a gyroscope mountedin it on a Cardan suspension are investigated below: the centres of mass of the satellite housing, the rotor and the rings of the Cardan suspension, and also of the fixed point of the Cardan suspension are the same. It is assumed that the centre of mass of the system moves in a Keplerisn circular orbit and that the angular velocity of the naturalrotation of the rotor is constant. Sufficient conditions for stability are obtained. Comparisonof the results obtained with the results of [41 shows that allowing for the perturbationof the motion of the satellite housing about the centre of mass of the system does not affect the stability conditions of the gyroscope. 1. We consider rectangular systems of coordinate axes: fixed 06~6 with origin at the centre of attraction, and moving Cx,x,x,, Cylyzys, Cylx,z, with origin at the centre of mass C of the mechanical system and axes directed along the principal central axes of inertia, respectively, of the satellite housing, of the rotor with its case, and of the outer ring: it is here assumedthat the axis of the outer ring is directed along the axis x,. * Zh. vychisl.
Mat. mat. Fiz. 9, 2, 479-482,
311
1969.
R. S. Sulikashuili
312
FIG.
We also introduce the orbital coordinate system Cxyz, the z-axis of which is directed along the radius vector OC, the y-axis along the normalto the plane of the orbit, and the ~-axis along the velocity of motion of the centre of mass of the system. We define the position of the gyroscope relative to the system Cx,xzxI by the Eulerian angles 4, 1,4,8 (see Fig. l), and the position of the housing of the satellite relative to the orbital system Cxyz by the cosines PI, /3,, &, yl, yz, ys of the angles formed by the axes of y and z respectively, with the axes x1, x2, x3, where
z
=
p? = V(l-
fJi?- 839,
piy,
p*3 -
+
(I-
y3
fi33)‘hy2
+
=
I(1 -ylz-Y23),
83(1-
Y12 -
W)“3
=
0.
(1.1)
The kinetic and potential energies of the system are of the form I =
(1.3)
‘is IAl(P, + OLdI)’ + Az(P2 + oojw” + A3(P3 + oo133)21+ ‘h(M(pi B3[(ki
+
+ odd 00th)
sin
co9 Ip +
+
(~3
(~3 +
+ 0433)
oo83! cos
sin
$I2 + BZ(i, + PZ+
$I2 +
C&3 +
(pi +
oOPi)
6)0/32)~ cos
+ +
-
Stability
-
of the stationary
motions
313
of a satellite
00 B21 co9 8 + (pi + o&) sin $ sin 8 + sin442+G[(1C)+P2+ (~3 + wfh) cos 11cos elz + Cd(pi + oOBi) sin 0 cos 0 + (pS + oo~s)cos II, cos 8 CS + PZ + o063)sin ei2 + r[G + (~4 + o~~~)cos II, - (p3 + w0&)sin$12 + J[sZ+ (4 + ~2 + w0B2)cos e + (pi + ooBi) sin 43sin e + (pa + ~o~~~)cos $ sin el2 +
(~3+0083)
{[(pi + OOBI)sin+ 00s 8 + (~3 + woB3)cos II,~0s 8 n =
O(e--e*)2 2
3 + Zwo2(&~i2
+
(4 + pz + uoB2)sin ei2),
-42~2~+ A3W+
(1.3)
c002Pi (yi cos 11,- y3 sin 9)” + B2y22+ B3(y4 sin $ + y3 cos $)“I +
f
3 ~002[(c1+Z)(yic0~~-y3sing)2+(C2+~)(y~sin~sin~+y2cosB+ ~3 sin 8 00s $)2 + (C3+ I) (yl cos 8 sin 11,- y2 sin 8 + y3 cos 8 cos 1+)2]. Here
A,, A2, As; Bi, Bz, Bs; G, C2, C3;
I, J, I are the principal central moments
of inertia of the satellite housing, of the external ring, and of the case and rotor relative to the axes x1, x2, x3; yl, x2, z1 and yl, ~2, y3 respectively; orbital angular velocity of the satellite:
o. is the
pi (i = 1, 2, 3) are the projections on the
axes x1, x2, x3 of the relative angular velocity (relative to Cxyz) of the satellite: fl = 4 is the angular velocity of natural rotation of the rotor, assumed constant: u is the coefficient of stiffness of the spring fixed on the axis of the case:
8*
corresponds to the unstressed state of the spring. Since the kinetic energy T does not depend explicitly on time, the generalized energy integral H = T2 -
To + II = Tz + W = const,
applies where T, is the quadratic part of the kinetic energy and T, is the part independent of the latter:
T from 4, 6, pl, pz, p3,
W is the potential energy of the
gravitational and inertial forces: rV(BI, 83, y1, y2, $7 0) = rII- To,
(1.4)
~‘2 = ‘/g~o~{Aifli~ + (A2 + B2) (I- 61”- P32)+ A31332 + B3( pi sin Ip+ 83 cos $)” + (BI + CI + I) (Pi COS$ - 83 sin 4))2-t
(c, + I) [pi sin 8 sin 9 + (1 - PI2- 82) ‘hcos 8 + p3sin e cos $I2 + (c, + I) [Bi sin q.2cos 8 -
(1 - pi2 - B32)%sin 8 + 83 cos Q cos ei2) I{l/2Q2 + Sho[fii sin* sine + (I- fh2-~p32)‘h~s 8 + ~3cos~sinel).
2.
The stationary motions of the satellite with gyroscope, corresponding to
positions of relative equilibrium of the satellite in the orbital system of coordinates
314
R. S. Sulikashvili
Cxyz, can be found as stationary points of the function W subject to the condition (1.1).
Therefore, introducing the function W, = W + Xx (A is the Lagrange multi-
plier), the equilibrium equation can be written in the form 8W, = 0.
The system of equations (Ll),
(2.1)
(2.1) admits of a solution for which the
principal central axes of inertia of the satellite are directed along the axes of the orbital system of coordinates Cxyz (the axis of the external ring of the gyroscope is directed along the normal to the orbital plane):
pi =
83 =
yi
=
y2 =
0,
and the values of the angles + and 8 are determined from the relations
I3 = ‘;zlc
“4 = 0,
11= 0, 9 = '12%
li =
; =o, 8 =
0,
0
for
8'
=
cOseO
=
'lzn,
52 =
(2.2) (X3)
0,
h = 0 for 0” = 0, h = 0 for O* = 0,. 151
cake
=
---_.,
oo(Cs---2+J-JJ)
(2.4)
h = 0 for 0’ = 00. (2.5)
In the cases (2.2) and (2.3) the axis y1 of the gyroscope case is directed along the velocity of motion of the centre of mass of the satellite,
and the axis
y2 of the rotor, accordingly, along the radius vector of the centre of mass and along the normal to the orbital plane. In the cases (2.4) and (2.5) the axis y1 of the gyroscope case is directed along the radius vector of the centre of mass, and the axis y2 of the rotor is accordingly perpendicular to the orbital plane, and makes a constant angle 8, with the axis of the external ring of the gyroscope. The equilibrium (2.2) is possible
only when the rotor does not rotate (a = 0).
The stationary motions (2.2)-(2.5) considered are the same, apart from the notation, as the motions (2.6)-(2.9) respectively, of 121 (it is necessary to put Q = 0 in 92.5) ). 3. Sufficient conditions for the stability of these stationary motions of a satellite with a gyroscope with respect to $, 0, 6, 0, Bl, l%, 83, yt, y2, YS can be obtained as the condition of positive definiteness of the second variation of W, on the linear manifold (1.1):
Stability
of the stationary
motions
of a satellite
315
for the solution (2.2) Do,
0+4002(c3-cz+~-~)
B, +
ci + I>
B3 + c2 + 1,
Az+Bz+C3>-4i+B1+Ci, A, > -4, [4(A,+B,-A3-B3)+(C,--2+Z--)][o+40h2(C3--2+Z--)]~~wo~(C~-C~+Z---)~ b-0;
(3.1)
for the solution (2.3) a+IQ2oo-4io4002(C3--2+Z-J)
>07
Bi+Ci
>BB+C3,
JQ++~(A~+B~+C~+Z)>~O(A~+&+C~+~),
A, >A33 o[Z~~o+4a~2(A2+B2--3-Bs) --02(C3-C2+ (3.2) Z--Z)]+W~~[~(A~+B~--~-BS) +3(C3-C2+ I-q[ZPoo -44002(C3-C2 +Z--J)l > 0;
for the solution (2.4) 0+Z&O‘,--02(C3-C2+Z-J) At>As,
>O,
B3fC3
~~+~~.I~(A~+B~+C~+Z-AAS)
~(,jo(A2+B2-A~--B3)
>Bi
+Ci,
>3~0(Bi+C~+Z),
+la+002(A2+B2-Ai--s)1[JQwO(c3-c2+Z--)]>o;
for the solution (2.5) 0+0,,2(C3-C2+Z-Z)-Z2~2(C3-C2+Z-1)-i
>0,
~a2(B3+C2+I-B~-C~--i)+~2Q2(C3-C2+Z-I)-'>0, Ai>A3,
[~(A~+B~+C~-AA~)-~~(B~+CI)+I][B~+C~+ZB,-Ci -Z++'%~O~-~(C~-C~+Z-Z)-~]-~Z~~~WO-~ (B~+C~-B,-CC~)(C~-C~+Z-I)-~ >O,
a[oo2(~3-~2+z-l)-~25/2(C3-C2+Z-J)-i+~o2(A2+B2-A~-B3)
X (3.4) X
[~+‘t,02(C3-C2+Z-1)--12~2(C3-C2+Z-Z)-i]>0.
In the case u = 0, where there is no spring, the last inequality in each of the conditions (3.1M3.4) is considerable simplified, and accordingly assumes the form h(A2+B2--AS----3)
>3(C3-C2+Z-Z),
&+B~>A,+BJ,
4(A2+B2-AAS-B3)
>3(C2--3+1-Z),
Az+Bz>A,+Bs.
In each of the groups of inequalities (3.G(3.4) the first two are the same kpart from the notation) as the corresponding conditions (3.2) and (3.4) (of C41) for the stability of a gyroscope on the assumption of unperturbed motion of the satellite housing. Therefore, for the stationary motions considered the latter
316
R. S. Sulikashvili
assumption does not affect the stability conditions of a gyroscope fixed in the satellite. It is obvious from this that the spring exerts a stabilizing action on equilibrium of the gyroscope. The remaining inequalities in (3.1H3.4) impose definite constraints on the parameters of the satellite housing. The author thanks V. V. Rumyantsev for discussing the paper and for useful comments. Translated by J. Berry REFERENCES 1.
RUMYANTSEV, V. V. The stability of stationary motions of satellites (Ob ustoichivosti stataionarnykh dvizhenii sputnikov). VTs Akad. Nauk SSSR, Moscow, 1967.
2.
KEIN, T. R. and WONG, S. F. Stability of gyroscope mounted on a satellite moving in an orbit in a gravity field. Raketnaya tekhn. kosmon. 3, 7, 98-103, 1965.
3.
SULIKASHVILI, R. S. Stability of the stationary motions of a gyroscope on a satellite in a central gravitational field. Mekhan. tverdogo tela. 6, 28-34, 1968.
4.
SULIKASHVILI, R. S. Stability of the stationary motions of a gyroscope mounted on a satellite in a central Newtonian field. Mekhan. tuerdogo tela. 2, 1969.
17 October
1968)
THE stability of satellites containing rotating parts has been investigated in a
large numberof papers (for a bibliography, see for example 11, 21). In papers by the author[3, 41 some stationary motions of a gyroscope mountedin the satellite on a Cardan suspension in a central Newtonianfield of force, and sufficient conditions for their stability, have been found. The stationary motions of a system consisting of the housing of a satellite and of a gyroscope mountedin it on a Cardan suspension are investigated below: the centres of mass of the satellite housing, the rotor and the rings of the Cardan suspension, and also of the fixed point of the Cardan suspension are the same. It is assumed that the centre of mass of the system moves in a Keplerisn circular orbit and that the angular velocity of the naturalrotation of the rotor is constant. Sufficient conditions for stability are obtained. Comparisonof the results obtained with the results of [41 shows that allowing for the perturbationof the motion of the satellite housing about the centre of mass of the system does not affect the stability conditions of the gyroscope. 1. We consider rectangular systems of coordinate axes: fixed 06~6 with origin at the centre of attraction, and moving Cx,x,x,, Cylyzys, Cylx,z, with origin at the centre of mass C of the mechanical system and axes directed along the principal central axes of inertia, respectively, of the satellite housing, of the rotor with its case, and of the outer ring: it is here assumedthat the axis of the outer ring is directed along the axis x,. * Zh. vychisl.
Mat. mat. Fiz. 9, 2, 479-482,
311
1969.
R. S. Sulikashuili
312
FIG.
We also introduce the orbital coordinate system Cxyz, the z-axis of which is directed along the radius vector OC, the y-axis along the normalto the plane of the orbit, and the ~-axis along the velocity of motion of the centre of mass of the system. We define the position of the gyroscope relative to the system Cx,xzxI by the Eulerian angles 4, 1,4,8 (see Fig. l), and the position of the housing of the satellite relative to the orbital system Cxyz by the cosines PI, /3,, &, yl, yz, ys of the angles formed by the axes of y and z respectively, with the axes x1, x2, x3, where
z
=
p? = V(l-
fJi?- 839,
piy,
p*3 -
+
(I-
y3
fi33)‘hy2
+
=
I(1 -ylz-Y23),
83(1-
Y12 -
W)“3
=
0.
(1.1)
The kinetic and potential energies of the system are of the form I =
(1.3)
‘is IAl(P, + OLdI)’ + Az(P2 + oojw” + A3(P3 + oo133)21+ ‘h(M(pi B3[(ki
+
+ odd 00th)
sin
co9 Ip +
+
(~3
(~3 +
+ 0433)
oo83! cos
sin
$I2 + BZ(i, + PZ+
$I2 +
C&3 +
(pi +
oOPi)
6)0/32)~ cos
+ +
-
Stability
-
of the stationary
motions
313
of a satellite
00 B21 co9 8 + (pi + o&) sin $ sin 8 + sin442+G[(1C)+P2+ (~3 + wfh) cos 11cos elz + Cd(pi + oOBi) sin 0 cos 0 + (pS + oo~s)cos II, cos 8 CS + PZ + o063)sin ei2 + r[G + (~4 + o~~~)cos II, - (p3 + w0&)sin$12 + J[sZ+ (4 + ~2 + w0B2)cos e + (pi + ooBi) sin 43sin e + (pa + ~o~~~)cos $ sin el2 +
(~3+0083)
{[(pi + OOBI)sin+ 00s 8 + (~3 + woB3)cos II,~0s 8 n =
O(e--e*)2 2
3 + Zwo2(&~i2
+
(4 + pz + uoB2)sin ei2),
-42~2~+ A3W+
(1.3)
c002Pi (yi cos 11,- y3 sin 9)” + B2y22+ B3(y4 sin $ + y3 cos $)“I +
f
3 ~002[(c1+Z)(yic0~~-y3sing)2+(C2+~)(y~sin~sin~+y2cosB+ ~3 sin 8 00s $)2 + (C3+ I) (yl cos 8 sin 11,- y2 sin 8 + y3 cos 8 cos 1+)2]. Here
A,, A2, As; Bi, Bz, Bs; G, C2, C3;
I, J, I are the principal central moments
of inertia of the satellite housing, of the external ring, and of the case and rotor relative to the axes x1, x2, x3; yl, x2, z1 and yl, ~2, y3 respectively; orbital angular velocity of the satellite:
o. is the
pi (i = 1, 2, 3) are the projections on the
axes x1, x2, x3 of the relative angular velocity (relative to Cxyz) of the satellite: fl = 4 is the angular velocity of natural rotation of the rotor, assumed constant: u is the coefficient of stiffness of the spring fixed on the axis of the case:
8*
corresponds to the unstressed state of the spring. Since the kinetic energy T does not depend explicitly on time, the generalized energy integral H = T2 -
To + II = Tz + W = const,
applies where T, is the quadratic part of the kinetic energy and T, is the part independent of the latter:
T from 4, 6, pl, pz, p3,
W is the potential energy of the
gravitational and inertial forces: rV(BI, 83, y1, y2, $7 0) = rII- To,
(1.4)
~‘2 = ‘/g~o~{Aifli~ + (A2 + B2) (I- 61”- P32)+ A31332 + B3( pi sin Ip+ 83 cos $)” + (BI + CI + I) (Pi COS$ - 83 sin 4))2-t
(c, + I) [pi sin 8 sin 9 + (1 - PI2- 82) ‘hcos 8 + p3sin e cos $I2 + (c, + I) [Bi sin q.2cos 8 -
(1 - pi2 - B32)%sin 8 + 83 cos Q cos ei2) I{l/2Q2 + Sho[fii sin* sine + (I- fh2-~p32)‘h~s 8 + ~3cos~sinel).
2.
The stationary motions of the satellite with gyroscope, corresponding to
positions of relative equilibrium of the satellite in the orbital system of coordinates
314
R. S. Sulikashvili
Cxyz, can be found as stationary points of the function W subject to the condition (1.1).
Therefore, introducing the function W, = W + Xx (A is the Lagrange multi-
plier), the equilibrium equation can be written in the form 8W, = 0.
The system of equations (Ll),
(2.1)
(2.1) admits of a solution for which the
principal central axes of inertia of the satellite are directed along the axes of the orbital system of coordinates Cxyz (the axis of the external ring of the gyroscope is directed along the normal to the orbital plane):
pi =
83 =
yi
=
y2 =
0,
and the values of the angles + and 8 are determined from the relations
I3 = ‘;zlc
“4 = 0,
11= 0, 9 = '12%
li =
; =o, 8 =
0,
0
for
8'
=
cOseO
=
'lzn,
52 =
(2.2) (X3)
0,
h = 0 for 0” = 0, h = 0 for O* = 0,. 151
cake
=
---_.,
oo(Cs---2+J-JJ)
(2.4)
h = 0 for 0’ = 00. (2.5)
In the cases (2.2) and (2.3) the axis y1 of the gyroscope case is directed along the velocity of motion of the centre of mass of the satellite,
and the axis
y2 of the rotor, accordingly, along the radius vector of the centre of mass and along the normal to the orbital plane. In the cases (2.4) and (2.5) the axis y1 of the gyroscope case is directed along the radius vector of the centre of mass, and the axis y2 of the rotor is accordingly perpendicular to the orbital plane, and makes a constant angle 8, with the axis of the external ring of the gyroscope. The equilibrium (2.2) is possible
only when the rotor does not rotate (a = 0).
The stationary motions (2.2)-(2.5) considered are the same, apart from the notation, as the motions (2.6)-(2.9) respectively, of 121 (it is necessary to put Q = 0 in 92.5) ). 3. Sufficient conditions for the stability of these stationary motions of a satellite with a gyroscope with respect to $, 0, 6, 0, Bl, l%, 83, yt, y2, YS can be obtained as the condition of positive definiteness of the second variation of W, on the linear manifold (1.1):
Stability
of the stationary
motions
of a satellite
315
for the solution (2.2) Do,
0+4002(c3-cz+~-~)
B, +
ci + I>
B3 + c2 + 1,
Az+Bz+C3>-4i+B1+Ci, A, > -4, [4(A,+B,-A3-B3)+(C,--2+Z--)][o+40h2(C3--2+Z--)]~~wo~(C~-C~+Z---)~ b-0;
(3.1)
for the solution (2.3) a+IQ2oo-4io4002(C3--2+Z-J)
>07
Bi+Ci
>BB+C3,
JQ++~(A~+B~+C~+Z)>~O(A~+&+C~+~),
A, >A33 o[Z~~o+4a~2(A2+B2--3-Bs) --02(C3-C2+ (3.2) Z--Z)]+W~~[~(A~+B~--~-BS) +3(C3-C2+ I-q[ZPoo -44002(C3-C2 +Z--J)l > 0;
for the solution (2.4) 0+Z&O‘,--02(C3-C2+Z-J) At>As,
>O,
B3fC3
~~+~~.I~(A~+B~+C~+Z-AAS)
~(,jo(A2+B2-A~--B3)
>Bi
+Ci,
>3~0(Bi+C~+Z),
+la+002(A2+B2-Ai--s)1[JQwO(c3-c2+Z--)]>o;
for the solution (2.5) 0+0,,2(C3-C2+Z-Z)-Z2~2(C3-C2+Z-1)-i
>0,
~a2(B3+C2+I-B~-C~--i)+~2Q2(C3-C2+Z-I)-'>0, Ai>A3,
[~(A~+B~+C~-AA~)-~~(B~+CI)+I][B~+C~+ZB,-Ci -Z++'%~O~-~(C~-C~+Z-Z)-~]-~Z~~~WO-~ (B~+C~-B,-CC~)(C~-C~+Z-I)-~ >O,
a[oo2(~3-~2+z-l)-~25/2(C3-C2+Z-J)-i+~o2(A2+B2-A~-B3)
X (3.4) X
[~+‘t,02(C3-C2+Z-1)--12~2(C3-C2+Z-Z)-i]>0.
In the case u = 0, where there is no spring, the last inequality in each of the conditions (3.1M3.4) is considerable simplified, and accordingly assumes the form h(A2+B2--AS----3)
>3(C3-C2+Z-Z),
&+B~>A,+BJ,
4(A2+B2-AAS-B3)
>3(C2--3+1-Z),
Az+Bz>A,+Bs.
In each of the groups of inequalities (3.G(3.4) the first two are the same kpart from the notation) as the corresponding conditions (3.2) and (3.4) (of C41) for the stability of a gyroscope on the assumption of unperturbed motion of the satellite housing. Therefore, for the stationary motions considered the latter
316
R. S. Sulikashvili
assumption does not affect the stability conditions of a gyroscope fixed in the satellite. It is obvious from this that the spring exerts a stabilizing action on equilibrium of the gyroscope. The remaining inequalities in (3.1H3.4) impose definite constraints on the parameters of the satellite housing. The author thanks V. V. Rumyantsev for discussing the paper and for useful comments. Translated by J. Berry REFERENCES 1.
RUMYANTSEV, V. V. The stability of stationary motions of satellites (Ob ustoichivosti stataionarnykh dvizhenii sputnikov). VTs Akad. Nauk SSSR, Moscow, 1967.
2.
KEIN, T. R. and WONG, S. F. Stability of gyroscope mounted on a satellite moving in an orbit in a gravity field. Raketnaya tekhn. kosmon. 3, 7, 98-103, 1965.
3.
SULIKASHVILI, R. S. Stability of the stationary motions of a gyroscope on a satellite in a central gravitational field. Mekhan. tverdogo tela. 6, 28-34, 1968.
4.
SULIKASHVILI, R. S. Stability of the stationary motions of a gyroscope mounted on a satellite in a central Newtonian field. Mekhan. tuerdogo tela. 2, 1969.