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On the stability behaviour of the rotational motion in a circular orbit for an elastic — rigid body
Mechanics Research Communications, Vol. 26, No. 2, pp. 129-137, 1999 Copyright © 1999 Elsevi~ Science Ltd Printed in the USA. All rights reserved 0093-6413/99IS--see front matter
Pergamon
PII S0093-6413(99)00003-8
On Tile Stability behaviour of the Rotational Motion in a Circular Orbit for an Elastic - Rigid Body
S. A. El Hafeez, M. E. Fares Mathematics Dept., Faculty of science, Mansoura University, Mansoura Egypt (Received 29 July 1998; accepted for print 18 December 1998)
Abstract: In this paper the stability of motion of asymmetric satellite-gyrostat, with Visco-elastic square plate attached to it, around the center of mass in ~l circular orbit under a central gravitational field is studied. The gyrostat has a rotor oriented inside it such that the rotor becomes dynamically as well as statically stable with respect to the whole system. A study of the stability of this system has been done by using the method of separation of motion aml averaging. Furthermore, we have solved the problem where there is no rotor.
Introduction; This paper considers the motion of a symmetric satellite-gyrostat arotmd its center of mass in the inertia central ellipsoid's equatorial plane. The gyrostat has a homogeneous Visco-elastic squa~ plate attached to it. Suppose that tile center of mass of the system moves along the circular orbit under a central gravitational field. Rotation of the symmetric rotor inside the rigid body is statically and dynamically balanced. Assume that the angular momentum tl of the rotor relative to the carrying rigid body is constant and parallel to one of the principal certral inertia axes of satellite. Bending deformations of the viscoelastic square plate accompanied by dissipation of energy are the cause of tile evolution of the rotational motion of the system. Approximate equations describing this evolution are obtained by the method of separation of motion and averaging [1-2]. Averaged equations are written in Andoyer variables [3]. Let the coordinate system Cxyz be connected rigidly with a symmetirc rigid gyrostat ( Cz is the axis of symmetry). Tile square Visco-elastic plate (which is attached to a rigid body) lies in the inertia central ellipsoid's equatorial plane Cxy, where the principal axes of inertia Cx, Cy ofthe rigid body are taken through the center of plate and parallel to its sides. Using linear theory of thin plates, we consider the radius-Vector of an arbitrary point of the plate after bending in the coordinate system Cxyz in the from r' -- r + m(x,y,t)e 3,
Ixl---2, lyJ-<~,
r
-- xe 1 + ye2,
(i)
x2+y2>b2
where m (x,y,t) is the deflection of the plate under bending, e i, (i = 1,2,3) are the units vectors of the axes Cx, Cy and Cz respectively. 129
S.A. EL-HAFEEZ and M. E. FARES
130
Let us introduce tile coordinate system C~1~2~3, which has a translational motion, and C~3 is orthogonal to the orbit plane. The radius vector of the attracting center in the coordinate system C~1~2~3 has projections (R cos toot , R sin mot,o ) where mo is the orbital angular velocity. Let t-t be the gravitational constant of the Newtonian field. Therefore to2 =IaR-3. The functionals of potential energy of elastic deformable plate and of dissipative forces are determined according to the formulas [4]
D' f,.O2to
02to) 2 + 2 ( 1 -
"'" 02to .2
~?2to 02to]}d_2 '
f/
(2) 0 DIdo] =xE[6], (.)=N where, D'
Eh 3
is called the flexural rigidity of a plate, E is young's 12(1- u 2 ) modulus, u is poisson's ratio, h is the thickness of the plate and X. is a constant, characterize the energy dissipation in the plate under bending. The second formula in (2) is due to the assumption that the dissipative functional D [6) ] is proportional to the functional E [d~ ]. E [¢h ] can be obtained by replacing (o by 6~ in the first functional of(2). For variables describing the rotational motion of the gyrostat we choose Andoyer canonical variables Ik, dOk(k = 1,2,3) [5]. To obtain the averaged equations, it is convenient to use the Routh functional R.(I,d~,ch,to, It)and the corresponding Routh equations [3].
J =-v,l , R,(I,#,6,to,H), + =
v I a,(l,d),d~,to,H),
(3) d v c b R,(I,~b,~b,to,H) - V m R,(1,d~,cb,to,H)=-Q,,,, dt Qm = -VtbD[&]" where V is the gradient of the functional R. with respect to the corresponding variable, Qto is dissipative force and He 3 is the vector of angular momentmn of the rotor. The Routh functional in Andoyer variables has the form R. = ~1 (G - Oto - He 3,J-l[to](G - Gto - He3)) - 1 Iprh2d.Q + E[to] + H[to] (4) f~ Here G is the angular momentum of the system, G m is the angular mome,mml of the visco-elastic plate, J [to] is the inertia tensor of the system relative to the
STABILITY
OF ORBITING
BODY
131
point C in coordinate system Cxyz and FI [to] is tile potential of gravitational forces and inertia forces of translational motion of the system C~,1~2~3. It is easy to see that [5]
G o , = I [ ( r + t0e3)A tbe3]Pdf~= I ( r A 6~e3)Pd~L
O-la°,r+oe3)ZodV,
(5)
v where o-l(t) is the matrix transformatious from the system C~t~2~3 to the system Cxyz and V is the domain occupied by the gyrostat and non-deformed plate. If the integration takes place through the points of the rigid body, then to = 0 and p is the density of the body, while integrating through the points of the domain V occupied by the plate, then to ~ 0, p is the density of the plate. The components of unit vector O-IR°( YI, Y2, Y3)in Andoyer variables are obtained from [5] as follows: Yl = (cosGt cosdo2 + sinctcos~5! sindo2)cos~l + +[(-coset sindo2 + sinot cos~31 cosdo2)cost~ 2_ - sin ~zsin 61 sin ¢52] sin dO1, Y2 = -(cosot cosdO2 + sinctcos51 sindO2)sindOI + (6) +[(- cos~ sin dO2+ sin ot cos~51 cosdO2) cos~32 - s i n ot sin 61 sin52 ]cosdo 1, 73 = (coset sindO2 - sinotcos81 cosdO2)sin8 2 -sinotsin~51cos5 2 , ct=t00t-dO3. The inertia tensor of the system relative to the frame Cxyz takes the form 161 j-l[0~] = j ~ l _ j ~ l jl[c0]j~l,
J0 =diag{A,A,C},
Jl(°))= J}]), J!})=J(l)=J(217 = 0 12' n -
i=1,2,3 -
32-
(7)
pyo)d~. I
Where A, A, C are principal mon, ents of inertia of the system and J!])(tt)) are the linear components in to of tensor inertia J(to) of the system. The Routh function R* (I k, dOk, ~ , to, H) depends on the small parameter o~2 and the large one D'. When D' ~ 0% the deflection to is zero and when (0 2 = 0the motion of the system is a regular precession, which is assumed to be undisturbed one. In d~is
132
S . A . E L - H A F E E Z and M. E. F A R E S
case the Routh function and canonical equations of motion in Andoyer variables has the form
R0
2 2 12 - 11 -
(I1 - H ) 2
2---A . . ~. . 2 1 ~
'
~b1 = VIIR 0 = ( A - C ) A - I C - I I I
(8)
-CH,
~2 = 12 A - I , {i3 = 0.
"File second equation of (3) describes the bending oscillations of the plate i~ perturbed motion and can be rewritten as 192o c~ d -I p ~ - + O'(l + Z~-~) A Ao~ + ~ [ J (o~)(G - Go, - lie3) A ripe 3 + +2(G-Gt
o - He3,Vojj-l(o~)(G-Go~
-lle3)-
(9)
-31o2 p ( O - I R 0 , r + t o ) ( o - l R ° , e 3 ) = 0 we will find the solution of(9) using the method of motion's separation under condition that Andoyer canonical variables correspond to undisturbed problem(8). We try an expression for this solution in tile form co = Igt.oI +1~2~2+..., ( I O)
where oO
o~ I =
X~' (_X)n /,id n=0
Onto lO( r, c3 t n
t)
In the first approximation co = E;t.t)1 = 8(fo10 -ZcblO), where COl0 is the solution of equation (9) when X = 0. Then COl0 is the solution of the following equation AA col0 = g l ( t ) x + g2(t)y, where ( 1 1) gl(t) = -gsin~bl + 3c002P)' 1)'2, g2(t) = - g c o s ~ l + 3~20 PY2)'3,
o -I2 g=
A2 C
[(2A - C)I 1 - 2AH] = const
When the edges of plate are free and the hole is clamped the solution of equation(l l) can be obtained approximatelly by using the Ritz's Variational method. Assume that co l0 has the form: COl0 = V ( x ) ~ ( y ) [ {Ao(t ) + A l ( t ) ( x s g n x - a)4+...} sgn x + +{Bo(t) + Bl(t ) ( y s g n y - a)4+...} sgny] where from [6] we have
(12)
STABILITY OF ORBITING BODY x5 ~(x)-!20 +(b 5 a2b 3 30 24
a2 x3 a 3 2 a2b 2 +~-~x sgnx 48 + ( 16 a3b 2 t---48~) sgn x'
133 b4 24
a3b, ~)x+
y5 a 2 y3 a 3 y2 a2b 2 b 4 a 3b +(Y)= 120 48 +~,8 s g n y + ( - i ( ~ - 2 4 - ,.-,)Y+ b 5 a2b 3 a3b 2 +(~ 24 + - ~ - - ) sgn y, (13) Here the functions ql (x), ~ (y) represent the deflections under bending of two bars, which along the principal axes Cx, Cy are arranged. The ends of these bars x = _+b, y =+ b are connected rigidlly with a symmetric gyrostat, while the other ends x = +_2' 2 are fiee. The 1st approximation ofequatkm _ a Y = _+a(12) takes the form: o~10 = ~l'(x) d~(Y)[A0 (t) sgn x + B0 (t)sgny ]. The coefficients A0(t ), B0(t) are determined fiom the following conditions of Ritz's variational method:
J
'L(0~ 10)~V(x)d~(Y) sgn x d~2 = 0, (14)
~L(o~ 10)xV(x)d~(y) sgny dr2 = 0. where Then
L(~Ol0) = V2V2t°10 - (xgl(t) + yg2(t)). A0(t), B0(t)are given by
(15)
Ao(t ) = kg2(t), Bo(t) = kgl(t) and k is a positive constant which can be deduced from (14). Then the solution of equation (11) is given by • 0~10 = k~lJ(x)dp(y)[gl (t) sgn y + g2 (t) sgn x]. Consequently we confine ourselves to the first two terms in (10) considering
o~! = COl0- ~6~10 = k~(x)~b(y)[(g 2 - ~g2) sgn x + (gl - ;~g-I)sgn Y] By substtituting o~ = ~ 0 1 in (3) we obtain the following equations of the perturbed motion:
S.A. EL-HAFEEZ and M. E. FARES
134
|l =+
I
,J-l(c0)(G-Go, -He3) +3¢02[(A-C)y3 a+-i + 0
(l(,)
c(~010 - )UblO)~-+l(Y3YlX + Y3Y2y)pdL)],
Jl = 3mg[(A_ C)'r3~$~-+ OY3 f g(to i 0 - Xcb 10) ~qj2~-a(y 3y ix + ~,3y2Y)pdf2] ' D
i3 = -3co2[(A _ C)"~,3~-+ 073 I ~(ml0-)~olO)~§cz(Y3Ylx . 0 ~Y3Y2y)pd~)], et = (00t - ~3 Now we carry out the averaging of the right hand side of equations (16) with respect to fast angular variables ~l,~2,ct,which is similar to that in [3,51. Putting co002= 0 in equations (16), gives the following equations for the variable ! 1 : il =
gkpcS~/i2-1~ [ (II-H) X -- C--- {COSqbl(-Sill+l + )~1COS~l) -
_ sin ~bl(_COS~bl _ XO01sin qbl)} _ cos~bl(+ 1 sin,~l _ )(+2 cos~,l ) _
(I 7)
-sin*l(-+ 1cos+ 1 - X,~b2 sin~bl)], S = IX,l,(X)t~(y)sgnydf2 = I y q , ( x ) , ( y ) s g n x
dr2.
By averaging the right hand side of equation (17)with respect to the angles qbl,~ and by using equation (8) we obtain:
=-nl( 12 - 12)[(A-C)I I-AIII[(2A-C)I I -2AII]? where 111 -
kp2cxS A5 C3 > 0
(I 8)
It follows from equation (18), that the stationary values are All 11=12 , I I - A _ C
(19)
The stationary solution I 1 = 12 stable if ( A - C) I 1 < AH. (20) In the case 11 = I2 the variable I1 has an extrumum, therefore I1 has not any evolution. Now we find the evolution of variables 12 and 13, when condition(20) is satisfied. The sationary solution I 1 = I2 means that the vector G of angular momentum conincides with the symmetry axis Cz, therefore Cosc~ 2 = 1, sin6 2 = 0,
where
Cos8 2 = lll~ 1,
STABILITY OF ORBITING BODY
135
and by substituting in (6), we obtain: ~'1 = cosetcosd~ + sinot cos51 sin ~b, ~'2 = - c o s o t s i n O + s i n ~ c o s d~cos61, 3'3 = -sin~tsin61,
where
12 - H
= ¢1 +
+-----C-'
cos l = 131 1.
Averaging of the system (16), gives#,a = - n2112(I 2 - I2)[(I2 - 11)(! + 31~ 122) - 4(~0131~1C], (21) + , a = - 4 n 2 1 2 2 ( I 2 -12)[(13 - 1 t 131] I) - C o 9 o ( l + 12 1~2)1, where 91o40 p2~ ~kS n2 8C >0. Putting the right hand side of (21) equal to zero, we find stationary points, which are the solutions corresponding to variables 12 and 13, we have I2--- 13 = I. In this case the variational equations for system (21) are given by:
= -8n2r
- q ) [ ( t - l-l)
oCl,
il = - 8 n 2 1 - i (~, - q) [(I - 1t) - 2 030C], Then - fi = - 8 n 2 I-1(~ _ q)c00C. Consequently, the motion under condition (20) is stable. The stationary solution I 1= 12 = 13 is corresponding to the rotational motion of the system around axis of symmetry, which coincides with normal to the orbit. Anofl~er case of stationary solutions will be considered in the next paper. Special case:
Now we will consider the case in which there is 11o rotor i. e. H = 0. Then from equation (18), we obtain = - n
• 1113 I 3(12 1 - i2),. ,It
l't'l-
~
~
(22) ,It
kp2E 7~ S ( 2 A - C)2(A - C) *5 ,3 ' AC
Where A*, A ' a n d C ' a r e the principal moments of inertia of the systetn (without rotor) relative to Cx, Cy, Cz, axes respectively. Since 2 A - C > 0, then the sign of n i coincides with sign o f ( A * - C*). If A * > C * , then I 1 tends to zero and this means that G lies in the equatorial's plane of the body. But if A * < C * , then I 1 tends to 12 which means that G coincides with the symmetry
136
S.A. EL-HAFEEZ and M. E. FARES axis of the system. Now we will consider tile case A > C , I I = 0 and find the evolution of variables 12 and 13. In this case without loss of generality we lake qbI = 0. The averaging equations for 12 and 13 take the form .-1
<12 >~b2,~ = 9{°8 P2~,16g k s A
[8Am013121 _ 412{1 + 12122)],
(23) ,-1 <]3 >+2,or =
9m8 p27~gkSA 16
[ A m 0 { 5 + 6 1 2. 122 - 31431 2-4 )-8131.
From (23) the stationary solution is 12 = 13 =A0} 0 . rl'he stability of the motion of the system can be deduced fi'om the following variational equations
=-8n~ ~,
fi =-8n~ '1, ,-1
9~o4 p 2 z g k S A n~z = 16 This stationary solution, means that the vector G coincides with C~3 and tile satellite rotates with a regular angular velocity {t)0 around the axis C~3. rlo determine the position of this system with respect to tile fiame C~1~2~3, wc introduce angle [3 between the unit vector 0 -1 p,0 and tile axis of symmetry CZ, i.e.
D--mot + 2 -+2 -+3" By using (3), (4) and when tl = 0, we obtain ' " Z (sin213~j] B = - A , - I 1 2 + O 0 + ~3A ,-1 m o2 p 2 gkS[(sln2[3)-
(24)
Since Y1 = sin[3, ]'2 = 0, ")'3 = cos[3, then we have
3
2
4*
"
[2 = 2 m v ( A - C ) s i n 2 ~ 3 - 9 m v 9 2
ekS(
1
sin 213- Z D cos213) cos2l)
and from (24) we obtain ,-1
~ + 9 0 } 8 9 2 z~;kSA
,
,
~cos2 213.t_~m0(A3 2 -C)A
,-I
sin2[3 = 0 ( 2 5 )
From (25) it is obvious when the motion is stationary the angle I3 takes the values (rtk')k: ----oo =+m O! {2 + /tk'tk=+°° ~k' = - m 'and k' is an integer. It is cleat that the first set of points, the motion is stable (i. e. axis ofsymmett3, of the satellite coincides with the unit vector 0-1R0), but at the second set of points, tile motion is unstable.
STABILITY OF ORBITING BODY
However in the second case 11 = 12,A*
4" ~Q) N/ A FIGURE OF THE SYSTEM WI'IH "IHE AX ;I.LS NOTE: (I]TIIE RJ(~ID BOOY (1) tHE ROTOK (~) I~lE VISCOELA~'I IC PLA I"E (4) 111EClRCIJLAR ORBI'r ($) ~ ' E R
OF GRAvrrY
(61CJ~.NTEIt OF MASS 4?)~I~OJ~'~I(~N OF G ONCXY PLANE (I) CN-NODIELLINE
137
Pergamon
PII S0093-6413(99)00003-8
On Tile Stability behaviour of the Rotational Motion in a Circular Orbit for an Elastic - Rigid Body
S. A. El Hafeez, M. E. Fares Mathematics Dept., Faculty of science, Mansoura University, Mansoura Egypt (Received 29 July 1998; accepted for print 18 December 1998)
Abstract: In this paper the stability of motion of asymmetric satellite-gyrostat, with Visco-elastic square plate attached to it, around the center of mass in ~l circular orbit under a central gravitational field is studied. The gyrostat has a rotor oriented inside it such that the rotor becomes dynamically as well as statically stable with respect to the whole system. A study of the stability of this system has been done by using the method of separation of motion aml averaging. Furthermore, we have solved the problem where there is no rotor.
Introduction; This paper considers the motion of a symmetric satellite-gyrostat arotmd its center of mass in the inertia central ellipsoid's equatorial plane. The gyrostat has a homogeneous Visco-elastic squa~ plate attached to it. Suppose that tile center of mass of the system moves along the circular orbit under a central gravitational field. Rotation of the symmetric rotor inside the rigid body is statically and dynamically balanced. Assume that the angular momentum tl of the rotor relative to the carrying rigid body is constant and parallel to one of the principal certral inertia axes of satellite. Bending deformations of the viscoelastic square plate accompanied by dissipation of energy are the cause of tile evolution of the rotational motion of the system. Approximate equations describing this evolution are obtained by the method of separation of motion and averaging [1-2]. Averaged equations are written in Andoyer variables [3]. Let the coordinate system Cxyz be connected rigidly with a symmetirc rigid gyrostat ( Cz is the axis of symmetry). Tile square Visco-elastic plate (which is attached to a rigid body) lies in the inertia central ellipsoid's equatorial plane Cxy, where the principal axes of inertia Cx, Cy ofthe rigid body are taken through the center of plate and parallel to its sides. Using linear theory of thin plates, we consider the radius-Vector of an arbitrary point of the plate after bending in the coordinate system Cxyz in the from r' -- r + m(x,y,t)e 3,
Ixl---2, lyJ-<~,
r
-- xe 1 + ye2,
(i)
x2+y2>b2
where m (x,y,t) is the deflection of the plate under bending, e i, (i = 1,2,3) are the units vectors of the axes Cx, Cy and Cz respectively. 129
S.A. EL-HAFEEZ and M. E. FARES
130
Let us introduce tile coordinate system C~1~2~3, which has a translational motion, and C~3 is orthogonal to the orbit plane. The radius vector of the attracting center in the coordinate system C~1~2~3 has projections (R cos toot , R sin mot,o ) where mo is the orbital angular velocity. Let t-t be the gravitational constant of the Newtonian field. Therefore to2 =IaR-3. The functionals of potential energy of elastic deformable plate and of dissipative forces are determined according to the formulas [4]
D' f,.O2to
02to) 2 + 2 ( 1 -
"'" 02to .2
~?2to 02to]}d_2 '
f/
(2) 0 DIdo] =xE[6], (.)=N where, D'
Eh 3
is called the flexural rigidity of a plate, E is young's 12(1- u 2 ) modulus, u is poisson's ratio, h is the thickness of the plate and X. is a constant, characterize the energy dissipation in the plate under bending. The second formula in (2) is due to the assumption that the dissipative functional D [6) ] is proportional to the functional E [d~ ]. E [¢h ] can be obtained by replacing (o by 6~ in the first functional of(2). For variables describing the rotational motion of the gyrostat we choose Andoyer canonical variables Ik, dOk(k = 1,2,3) [5]. To obtain the averaged equations, it is convenient to use the Routh functional R.(I,d~,ch,to, It)and the corresponding Routh equations [3].
J =-v,l , R,(I,#,6,to,H), + =
v I a,(l,d),d~,to,H),
(3) d v c b R,(I,~b,~b,to,H) - V m R,(1,d~,cb,to,H)=-Q,,,, dt Qm = -VtbD[&]" where V is the gradient of the functional R. with respect to the corresponding variable, Qto is dissipative force and He 3 is the vector of angular momentmn of the rotor. The Routh functional in Andoyer variables has the form R. = ~1 (G - Oto - He 3,J-l[to](G - Gto - He3)) - 1 Iprh2d.Q + E[to] + H[to] (4) f~ Here G is the angular momentum of the system, G m is the angular mome,mml of the visco-elastic plate, J [to] is the inertia tensor of the system relative to the
STABILITY
OF ORBITING
BODY
131
point C in coordinate system Cxyz and FI [to] is tile potential of gravitational forces and inertia forces of translational motion of the system C~,1~2~3. It is easy to see that [5]
G o , = I [ ( r + t0e3)A tbe3]Pdf~= I ( r A 6~e3)Pd~L
O-la°,r+oe3)ZodV,
(5)
v where o-l(t) is the matrix transformatious from the system C~t~2~3 to the system Cxyz and V is the domain occupied by the gyrostat and non-deformed plate. If the integration takes place through the points of the rigid body, then to = 0 and p is the density of the body, while integrating through the points of the domain V occupied by the plate, then to ~ 0, p is the density of the plate. The components of unit vector O-IR°( YI, Y2, Y3)in Andoyer variables are obtained from [5] as follows: Yl = (cosGt cosdo2 + sinctcos~5! sindo2)cos~l + +[(-coset sindo2 + sinot cos~31 cosdo2)cost~ 2_ - sin ~zsin 61 sin ¢52] sin dO1, Y2 = -(cosot cosdO2 + sinctcos51 sindO2)sindOI + (6) +[(- cos~ sin dO2+ sin ot cos~51 cosdO2) cos~32 - s i n ot sin 61 sin52 ]cosdo 1, 73 = (coset sindO2 - sinotcos81 cosdO2)sin8 2 -sinotsin~51cos5 2 , ct=t00t-dO3. The inertia tensor of the system relative to the frame Cxyz takes the form 161 j-l[0~] = j ~ l _ j ~ l jl[c0]j~l,
J0 =diag{A,A,C},
Jl(°))= J}]), J!})=J(l)=J(217 = 0 12' n -
i=1,2,3 -
32-
(7)
pyo)d~. I
Where A, A, C are principal mon, ents of inertia of the system and J!])(tt)) are the linear components in to of tensor inertia J(to) of the system. The Routh function R* (I k, dOk, ~ , to, H) depends on the small parameter o~2 and the large one D'. When D' ~ 0% the deflection to is zero and when (0 2 = 0the motion of the system is a regular precession, which is assumed to be undisturbed one. In d~is
132
S . A . E L - H A F E E Z and M. E. F A R E S
case the Routh function and canonical equations of motion in Andoyer variables has the form
R0
2 2 12 - 11 -
(I1 - H ) 2
2---A . . ~. . 2 1 ~
'
~b1 = VIIR 0 = ( A - C ) A - I C - I I I
(8)
-CH,
~2 = 12 A - I , {i3 = 0.
"File second equation of (3) describes the bending oscillations of the plate i~ perturbed motion and can be rewritten as 192o c~ d -I p ~ - + O'(l + Z~-~) A Ao~ + ~ [ J (o~)(G - Go, - lie3) A ripe 3 + +2(G-Gt
o - He3,Vojj-l(o~)(G-Go~
-lle3)-
(9)
-31o2 p ( O - I R 0 , r + t o ) ( o - l R ° , e 3 ) = 0 we will find the solution of(9) using the method of motion's separation under condition that Andoyer canonical variables correspond to undisturbed problem(8). We try an expression for this solution in tile form co = Igt.oI +1~2~2+..., ( I O)
where oO
o~ I =
X~' (_X)n /,id n=0
Onto lO( r, c3 t n
t)
In the first approximation co = E;t.t)1 = 8(fo10 -ZcblO), where COl0 is the solution of equation (9) when X = 0. Then COl0 is the solution of the following equation AA col0 = g l ( t ) x + g2(t)y, where ( 1 1) gl(t) = -gsin~bl + 3c002P)' 1)'2, g2(t) = - g c o s ~ l + 3~20 PY2)'3,
o -I2 g=
A2 C
[(2A - C)I 1 - 2AH] = const
When the edges of plate are free and the hole is clamped the solution of equation(l l) can be obtained approximatelly by using the Ritz's Variational method. Assume that co l0 has the form: COl0 = V ( x ) ~ ( y ) [ {Ao(t ) + A l ( t ) ( x s g n x - a)4+...} sgn x + +{Bo(t) + Bl(t ) ( y s g n y - a)4+...} sgny] where from [6] we have
(12)
STABILITY OF ORBITING BODY x5 ~(x)-!20 +(b 5 a2b 3 30 24
a2 x3 a 3 2 a2b 2 +~-~x sgnx 48 + ( 16 a3b 2 t---48~) sgn x'
133 b4 24
a3b, ~)x+
y5 a 2 y3 a 3 y2 a2b 2 b 4 a 3b +(Y)= 120 48 +~,8 s g n y + ( - i ( ~ - 2 4 - ,.-,)Y+ b 5 a2b 3 a3b 2 +(~ 24 + - ~ - - ) sgn y, (13) Here the functions ql (x), ~ (y) represent the deflections under bending of two bars, which along the principal axes Cx, Cy are arranged. The ends of these bars x = _+b, y =+ b are connected rigidlly with a symmetric gyrostat, while the other ends x = +_2' 2 are fiee. The 1st approximation ofequatkm _ a Y = _+a(12) takes the form: o~10 = ~l'(x) d~(Y)[A0 (t) sgn x + B0 (t)sgny ]. The coefficients A0(t ), B0(t) are determined fiom the following conditions of Ritz's variational method:
J
'L(0~ 10)~V(x)d~(Y) sgn x d~2 = 0, (14)
~L(o~ 10)xV(x)d~(y) sgny dr2 = 0. where Then
L(~Ol0) = V2V2t°10 - (xgl(t) + yg2(t)). A0(t), B0(t)are given by
(15)
Ao(t ) = kg2(t), Bo(t) = kgl(t) and k is a positive constant which can be deduced from (14). Then the solution of equation (11) is given by • 0~10 = k~lJ(x)dp(y)[gl (t) sgn y + g2 (t) sgn x]. Consequently we confine ourselves to the first two terms in (10) considering
o~! = COl0- ~6~10 = k~(x)~b(y)[(g 2 - ~g2) sgn x + (gl - ;~g-I)sgn Y] By substtituting o~ = ~ 0 1 in (3) we obtain the following equations of the perturbed motion:
S.A. EL-HAFEEZ and M. E. FARES
134
|l =+
I
,J-l(c0)(G-Go, -He3) +3¢02[(A-C)y3 a+-i + 0
(l(,)
c(~010 - )UblO)~-+l(Y3YlX + Y3Y2y)pdL)],
Jl = 3mg[(A_ C)'r3~$~-+ OY3 f g(to i 0 - Xcb 10) ~qj2~-a(y 3y ix + ~,3y2Y)pdf2] ' D
i3 = -3co2[(A _ C)"~,3~-+ 073 I ~(ml0-)~olO)~§cz(Y3Ylx . 0 ~Y3Y2y)pd~)], et = (00t - ~3 Now we carry out the averaging of the right hand side of equations (16) with respect to fast angular variables ~l,~2,ct,which is similar to that in [3,51. Putting co002= 0 in equations (16), gives the following equations for the variable ! 1 : il =
gkpcS~/i2-1~ [ (II-H) X -- C--- {COSqbl(-Sill+l + )~1COS~l) -
_ sin ~bl(_COS~bl _ XO01sin qbl)} _ cos~bl(+ 1 sin,~l _ )(+2 cos~,l ) _
(I 7)
-sin*l(-+ 1cos+ 1 - X,~b2 sin~bl)], S = IX,l,(X)t~(y)sgnydf2 = I y q , ( x ) , ( y ) s g n x
dr2.
By averaging the right hand side of equation (17)with respect to the angles qbl,~ and by using equation (8) we obtain:
kp2cxS A5 C3 > 0
(I 8)
It follows from equation (18), that the stationary values are All 11=12 , I I - A _ C
(19)
The stationary solution I 1 = 12 stable if ( A - C) I 1 < AH. (20) In the case 11 = I2 the variable I1 has an extrumum, therefore I1 has not any evolution. Now we find the evolution of variables 12 and 13, when condition(20) is satisfied. The sationary solution I 1 = I2 means that the vector G of angular momentum conincides with the symmetry axis Cz, therefore Cosc~ 2 = 1, sin6 2 = 0,
where
Cos8 2 = lll~ 1,
STABILITY OF ORBITING BODY
135
and by substituting in (6), we obtain: ~'1 = cosetcosd~ + sinot cos51 sin ~b, ~'2 = - c o s o t s i n O + s i n ~ c o s d~cos61, 3'3 = -sin~tsin61,
where
12 - H
= ¢1 +
+-----C-'
cos l = 131 1.
Averaging of the system (16), gives
= -8n2r
- q ) [ ( t - l-l)
oCl,
il = - 8 n 2 1 - i (~, - q) [(I - 1t) - 2 030C], Then - fi = - 8 n 2 I-1(~ _ q)c00C. Consequently, the motion under condition (20) is stable. The stationary solution I 1= 12 = 13 is corresponding to the rotational motion of the system around axis of symmetry, which coincides with normal to the orbit. Anofl~er case of stationary solutions will be considered in the next paper. Special case:
Now we will consider the case in which there is 11o rotor i. e. H = 0. Then from equation (18), we obtain = - n
• 1113 I 3(12 1 - i2),. ,It
l't'l-
~
~
(22) ,It
kp2E 7~ S ( 2 A - C)2(A - C) *5 ,3 ' AC
Where A*, A ' a n d C ' a r e the principal moments of inertia of the systetn (without rotor) relative to Cx, Cy, Cz, axes respectively. Since 2 A - C > 0, then the sign of n i coincides with sign o f ( A * - C*). If A * > C * , then I 1 tends to zero and this means that G lies in the equatorial's plane of the body. But if A * < C * , then I 1 tends to 12 which means that G coincides with the symmetry
136
S.A. EL-HAFEEZ and M. E. FARES axis of the system. Now we will consider tile case A > C , I I = 0 and find the evolution of variables 12 and 13. In this case without loss of generality we lake qbI = 0. The averaging equations for 12 and 13 take the form .-1
<12 >~b2,~ = 9{°8 P2~,16g k s A
[8Am013121 _ 412{1 + 12122)],
(23) ,-1 <]3 >+2,or =
9m8 p27~gkSA 16
[ A m 0 { 5 + 6 1 2. 122 - 31431 2-4 )-8131.
From (23) the stationary solution is 12 = 13 =A0} 0 . rl'he stability of the motion of the system can be deduced fi'om the following variational equations
=-8n~ ~,
fi =-8n~ '1, ,-1
9~o4 p 2 z g k S A n~z = 16 This stationary solution, means that the vector G coincides with C~3 and tile satellite rotates with a regular angular velocity {t)0 around the axis C~3. rlo determine the position of this system with respect to tile fiame C~1~2~3, wc introduce angle [3 between the unit vector 0 -1 p,0 and tile axis of symmetry CZ, i.e.
D--mot + 2 -+2 -+3" By using (3), (4) and when tl = 0, we obtain ' " Z (sin213~j] B = - A , - I 1 2 + O 0 + ~3A ,-1 m o2 p 2 gkS[(sln2[3)-
(24)
Since Y1 = sin[3, ]'2 = 0, ")'3 = cos[3, then we have
3
2
4*
"
[2 = 2 m v ( A - C ) s i n 2 ~ 3 - 9 m v 9 2
ekS(
1
sin 213- Z D cos213) cos2l)
and from (24) we obtain ,-1
~ + 9 0 } 8 9 2 z~;kSA
,
,
~cos2 213.t_~m0(A3 2 -C)A
,-I
sin2[3 = 0 ( 2 5 )
From (25) it is obvious when the motion is stationary the angle I3 takes the values (rtk')k: ----oo =+m O! {2 + /tk'tk=+°° ~k' = - m 'and k' is an integer. It is cleat that the first set of points, the motion is stable (i. e. axis ofsymmett3, of the satellite coincides with the unit vector 0-1R0), but at the second set of points, tile motion is unstable.
STABILITY OF ORBITING BODY
However in the second case 11 = 12,A*
4" ~Q) N/ A FIGURE OF THE SYSTEM WI'IH "IHE AX ;I.LS NOTE: (I]TIIE RJ(~ID BOOY (1) tHE ROTOK (~) I~lE VISCOELA~'I IC PLA I"E (4) 111EClRCIJLAR ORBI'r ($) ~ ' E R
OF GRAvrrY
(61CJ~.NTEIt OF MASS 4?)~I~OJ~'~I(~N OF G ONCXY PLANE (I) CN-NODIELLINE
137