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On the deployment dynamics of tether connected two-body systems
Vol.6, pp. 1183-1197 Pergamon Press Ltd., 1979, Printedin Great Britain Acta Astronautica
On the deployment dynamics of tether connected two-body systemsi" V. J. MODI* Department of Mechanical Engineering, The University of British Columbia, Vancouver, B.C., Canada V6T IW5 AND
A. K. M I S R A § Department of Mechanical Engineering, McGill University, Montreal, Quebec, Canada H3A 2K6 Abstract--A general formulation of the deployment dynamics of tether connected two-body systems taking into account the three dimensional librational motion, longitudinal and transverse vibrations of the continuous tether, eccentricity of the orbit and aerodynamic drag in a rotating, oblate atmosphere, is presented. Three simple deployment procedures (uniform, exponential and their combination) are considered. In the beginning analytical solutions are obtained for the degenerate case of negligible vibrations in a circular orbit, which help in establishing trends for the more general situation. This is followed by the general dynamics of the system investigated through integration of the linearized autonomous equations of motion numerically. Typical plots are presented which describe the effects of various system parameters on the tether dynamics during deployment of a subsatellite from the Space Shuttle.
Introduction TETHERED satellite systems have been proposed for various applications: to provide artificial gravity to a space station, measurement of long wavelength solar and planetary radiations, generation of high resolution gravity and magnetic field maps, carrying out low altitude scientific experiments, etc. Some of these proposals will become realities with the advent of the Space Shuttle. NASA is developing a Shuttle/Tethered Satellite System which will be operational in late 1981 while the concept will be demonstrated in an earlier test flight. Furthermore, ESA has recommended a 10 km tether system to be an integral part of the Shuttle/AMPS Laboratory. The vast potential of tether connected systems has led to numerous investigations of their dynamics, mostly dealing with the post-deployment behaviour. Relatively less attention has been focused on the dynamics and control, of the system during deployment and retrieval although large librational motions may result under certain conditions. Ebner (1970) has carried out a simple planar analysis of a deploying cable-connected space system. The pitch oscillations were evaluated approximately using the tTbe investigation reported here was supported by the National Research Council of Canada, Grant No. A-2181. ~Professor. §Assistant Professor. 1183
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V.J. Modi and A. K. Misra
principle of conservation of angular momentum and this information was used to analyze the transverse vibrations of the cable. On the other hand, Stuiver and Bainum (1973) studied the planar deployment control of the Tethered Orbiting Interferometer satellite ignoring the vibrations of the tether. An algorithm to find the control forces for a given set of initial and terminal conditions was described. A passive deployment procedure requiring no actuators to control the motion of the subsatellite was studied by Kane and Levinson (1977). However, the vibrations of the tetered were not included in the analysis. A tension control law for the deployment and retrieval of tethered systems to be used in conjunction with the Space Shuttle was formulated by Rupp (1975). Librational motion in the orbital plane was analyzed and growth of pitch oscillations during retrieval was noted. The tether, however, was assumed to be straight (no transverse vibrations). The above system was subsequently studied in detail by Baker et al. (1976) taking into consideration the three dimensional character of the dynamics and the aerodynamic drag in a rotating atmosphere. A simulation showing the shape of the tether during deployment and retrieval was carried out by Kulla (1976). Although the transverse vibrations of the tether were included, the longitudinal oscillations and the out-of-plane motion of the system were ignored. The deployment problem was considered briefly by Buckens (1977) while studying the three dimensional dymanics of tethered two-body systems. However, the longitudinal strain in the tether and the aerodynamic forces were not included. Recently, Kalaghan et al. (1978) analyzed the "Skyhook" system in detail, taking various environmental effects into account. However, the tether was not treated as a continuum. The objective of this paper is to attempt a somewhat general formulation of the deployment and retrieval dynamics of tether connected two-body systems taking into account the three dimensional librational motion, lognitudinal and transverse vibrations of the continuous tether, eccentricity of the orbit and the aerodynamic drag in a rotating oblate atmosphere. The equations describing the motion of the system are derived using the classical Lagrangian procedure. The transverse vibrations are expanded in terms of a set of admissible functions. Small oscillations from a reference state are considered and the equations of motion are linearized. The generalized forces arising due to the aerodynamic drag are evaluated. Analytical solutions are obtained for the degenerate case of negligible vibrations in a circular orbit, which help in establishing trends for more complex situations. With this background, the general dynamics of the system is investigated by integrating the linearized, coupled, non-autonomous equations of motion numerically. Effects of various system parameters on the librations and vibrations of the tether are determined. Equations of motion Figure 1 shows a system of two bodies having masses mo and mb, respectively and connected by a tether with a mass density pc. At any instant t, let S be the centre of mass of the system, which is located with respect to the centre of the Earth E by the radial distance R0, inclination i of the orbit to the equatorial plane, argument of the perigee 0p and true anomaly 0. Let A and B be the
On the deployment dynamics of tether connected two-body systems
1185
orbit
x6
0
x
perigee
equatorial plane N Fig. 1. Geometry of motion.
centres of mass of the end-bodies while Po and Pb are the points of attachment of the tether having an instantaneous unstretched length L. A set of co-ordinate axes x0, Y0, z0 located at S is so oriented that y0-axis coincides with the local vertical, z0-axis represents the local horizontal, while x0-axis is the orbit normal. The orientation of the principal axes xo, yo, Zo of body A is described by a set of modified Eulerian rotations as follows: //o about yo-axis giving x'., y~, z~; ~/o about z~-axis resulting in x~, y~, z~ and finally ao about x~-axis yielding xo, yo, zo. The principal axes Xb, Yb, and Zb of body B are obtained from x0, Y0, z0 axes through a similar set of rotations ab, 13b and 3'b. The orientation of the axes Xc, yc, Zc (y~ coinciding with the nominal tetherline) can be defined by only two analogous rotations a~ and ,/~, since the rotation about the axis of the tether can be ignored. The axial strain E is assumed to be uniform along the length of the tether or in other words only one term is retained in the expansion of the lognitudinal vibrations. The out-of-plane and inplane transverse vibrations of the tether, denoted by u and v, respectively, are superposed on the resultant tetherline. In general, PoPb does not pass through A , B, or S. Let R~, Rb and R~
1186
V.J. Modi and A. K. Misra
measured from S be the position vectors of the centres of mass A and B and a point on the deformed tether, respectively. Clearly, maRa + mbRo + f0 L pcRc dyc = O.
(la)
Rb = Ro + do + L~ - db,
(Ib)
Furthermore,
and R~ = Ra + d . + r .
(lc)
where do and db are the position vectors of the points of attachment relative to the corresponding centres of mass Ls = Lsjc = L(1 + ~)j~,
(ld)
r~ = Uic + yc(1 + ¢)j~ + vkc.
(le)
and
Here i , jc and kc are the unit vectors along x , y~ and z~ axes, respectively. Substituting eqns (lb)-(le) in (la), Ro and Rb can easily be expressed in terms of L,, do, db, ui~ and vkc. The librational motion of the system and the vibrations of the tether are likely to have very little effect on the orbital motion. Hence, the orbit can be computed separately, and will be assumed here to be Keplerian, i.e. R0 = h2/[/~E(1 + ecO)],
(2a)
Ro20 = h,
(2b)
and
where h is the angular momentum per unit mass of the system, e the eccentricity of the orbit and ~n the gravitational constant of the Earth. It is convenient to expand the transverse vibrations in series forms in terms of a set of admissible functions: u = ~'~ O,(y~)A,(t),
(3a)
v = ~ O,(y~)B,(t).
(3b)
n~l
and
n=l
On the deployment dynamics of tether connected two-body systems
1187
The functions O.(y~) can be somewhat arbitrary, as long as they satisfy at least the geometric boundary conditions. In the present analysis, ~,(Yc) = (2) tp sin ( n ~ ' y d L ) ,
n = 1, 2 . . . . .
oo.
(4)
The expansions transform the integrals with time-varying limits to products of A,, B,, functions of L and their time derivatives. The equations of motion are obtained using the classical Lagrangian formulation
A(ar _a + au=O, ' d t \Oq~/
aqi
(5)
cgqi
where qi represents the generalized co-ordinates ac, ~'c, aa,/3a, ~/o, Orb,/3b, 3~b,~, A,, B,, (n = 1, 2 . . . . . o0) and Qi is the corresponding generalized force arising due to the environmental and control forces. The expressions for kinetic and potential energies are too lengthy and hence omitted for brevity. They can be found in an investigation by Modi and Misra (1978). The resulting non-linear, non-autonomous and coupled ordinary differential equations are rather lengthy and difficult to handle even with the help of a computer. Hence, as a first step to extract some information, small oscillations from a reference state are considered and the equations linearized. Usually, the non-trivial equilibrium configuration is taken as the reference state. However, in the present case this configuration changes continuously with time. Therefore, an average value of ac0, which will be determined later, is used and it is assumed that the displacements from this reference state are not too large. Non-trivial equilibrium values of the other generalized coordinates are very close to the trivial ones and hence, the latter may be taken as the reference. Furthermore, for most practical applications, the length of the tetherline is much larger than the distance of the points of attachment from the respective centres of mass, i.e. do, db ~ L,. Considerable simplification of the equations of motion can be achieved without affecting the overall dynamics of the system substantially, provided do and db are assumed to be zero. It may be pointed out that the linearization causes the motion in the orbital plane to be uncoupled from the out-of-plane motion. Hence, the two motions can be studied separately.
Aerodynamic forces Many tethered satellite systems are likely to have low altitude orbits where the aerodynamic drag is significant compared to the gravity gradient and affects the overall dynamics substantially. The aerodynamic force on an element of the system can be expressed as A F = - I [ 2 C d p & A VIVI,
(6a)
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V.J. Modi and A. K. Misra
where Ca and p are the drag coefficient and density of the air, respectively, while AA is the projected area of the element normal to its velocity V relative to the atmosphere. The coefficient Cd has a small variation with the cross-section of the element and the atmospheric conditions, however, in the present analysis it is assumed to be constant. The density of the air can be written in the form, p = po exp ( - h/ho),
(6b)
where h is the altitude of the element in discussion. The reference height h0 varies slightly with h, however, the variation is ignored here and h0 is so chosen as to match the lower portion of the tether accurately. The relative velocity V depends on the inertial velocity of the centre of mass of the system, the velocity of the atmosphere due to its rotation about the Earth's axis and the velocity of the element relative to the centre of mass of the system (resulting due to the librations and vibrations). The last one is comparatively small and will be ignored here. With this approximation, it may be shown that (7a)
V = RoocOsiio + Rojo + R d O - trci)ko,
where or is the speed of rotation of the Earth about its axis and
~=0+0p.
(7b)
It is convenient to resolve V along the tether axes Xo yc and zc, i.e. V = Vxci~ + Vycjc + Vzckc - (RotrcOsicTc + RosT~)ic + [ - RoTcOsica~sTc + Roca~CTc + Ro(O - trci)sac]jc + [RoorcOsisa~ Syc - l~osacCT~ + Ro(O - trci)ca~]kc.
(7c)
Clearly, IVI = [Ro2{(~rc~si) 2 + ( 0 - trci) 2} + Ro2]v2.
(7d)
The projected areas of the end-bodies normal to the velocity V will depend on their shapes• For the tether, however, the shape is cylindrical and the projected area of an element is ALdc(V~c + vec)!/2/IV[. Hence from eqn (6a), AF = - l l 2 C d p A L d d V ~
(8a)
+ vZ Pt2v --ZCJ v .
The contribution to the generalized force Q~ is given by an,,
Q, = Fa • -~-q/+ F,
. a R , + [.L,XF,(y~) • ~a R ~ aq, Jo
d
y~,
(8b)
On the deploymentdynamics of tetherconnected two-body systems
1189
where Fa and Fb are the total aerodynamic forces on the end-bodies. For the Space Shuttle/Tethered Subsatellite system, /~b, ~tc ~/~o and the generalized forces can be expressed as:
Q~c=-ll2Cd[d~(V2~+ V2¢)t/2f ~ y~o(y~)dyc + pbAblV[L]V~,
(9a)
Q~=-l/2Cd[dc(V2~+ V2c)lrZf ~ ycp(yc)dyc + pbAb[V[L]V~cca~,
(9b)
Q.=-l/2Cd[d~(V2c +
V2~)'/2 foLycp(yc) dy.+pbAb[V[L]Vy~,
Qan = --1/2Cdd~(V2~ + V~c)l/2Vxc
~Pn(Yc)P(Yc)dye,
fo
Qbn = - 1/2 Cad~(V2c + V2c)ll2Vz¢
(9c) (9d)
~.(Yc)P(Y¢) dye,
(9e)
n = 1,2,...oo,
where Pb is the density of air at the subsatellite end. state For a deploying system, the nontrivial equilibrium configuration changes continuously with time. It is convenient to determine the steady state attitude of the non-deploying and non-vibrating system and use it as the reference state. Furthermore, the eccentricity of the orbit.is likely to be very small and the reference state of a corresponding circular orbit can serve as the reference state of the elliptic orbit. With this as background, it may be shown that ac0 and Too are the solutions of the following equations: Reference
ml~eL Z[~2SacoCaco(3 C27co -- S2yco)= -- 112 CdRo2[orcOsi sacoSyco + ([~ -- orci)caco] × f(0, i, ac0, T~0),
(10a)
4ml~eL2f~2SycoCTcoC2acO= 112 CaRo2(o'cOsicyco)Cacof(O,i, aco, ~/co),
(lOb)
where f ( O , i, acO , ~cO) ~" [(orcO$i)2( 1 -- C2acOS2~IcO) "~ (~'~ -- orCi)2C201cO
r",,,y.,{,._ +
+ 2 ( £ t - orci)orcOsis coCa~oSy~oj [Jo -
d,.ld. (lOt)
p.o = mJm,
IZb = mb/m, I~ = p~L/m, P.* = I~JI~,.L.
Note that, in general, or is non-zero and ac0 as well as ~/c0obtained from eqn (10) vary with O. However, or ~ [1 and a suitable reference state can be obtained
1190
V.J. Modi and A. K. Misra
by solving eqn (10) after putting tr = O. This results in Yco = O,
(1 la)
while a~0 is the root of the transcendental equation
3m~.Lec.~oS.co=(-1/2)C~Ro2[dAc~Ol foLp(yA{Yc-(~ +'~)LI dyc For the Shuttle/Subsatellite system, ~o >> ~b,/~c and eqn (1 lb) simplifies to
cao[3(mb +'~lmc)Saco+(1/2)CdRo2{(wldcacol+pbAgL}]=O,
(llc)
where:
= 2p,(hdLca~o)2[1 -
(1 +
Lca~olho) exp (-Lcacolho)],
(1 ld)
p, = density of air at the altitude o f the centre of mass of the system = p0 exp ( - hJho), (1 le) and pb = P, exp ( -
Lca~dho).
(1 If)
It may be noted that for downward deployment a~0 is close to ~r. An examination of eqn ( l l c ) shows that the quantity inside the square bracket cannot be zero for real values of aco, if
(3mb + mc) < (I/2)CdR~p~A~JL,
(1 lg)
and hence
Caco=O,
i.e.
ac0=Ir/2
or
3~r/2.
Deployment p r o ¢ ~ u r e s D e p l o y m e n t or retrieval of the subsatellite can be carried out by controlling the tension in the tether through a reel mechanism as proposed by Rupp (1975). T w o simplest d e p l o y m e n t rates are: (i) Uniform d e p l o y m e n t rate, i.e. /_~= cL~, where L~ is the initial length. Clearly L = L~(1 +
ct).
(12a)
On the deployment dynamics of tether connected two-body systems
1191
(ii) Exponential deployment where the rate at any instant is proportional to the deployed length, i.e./~ = cL. Integrating, L = L~ exp (ct).
(12b)
The constant c is positive for deployment and negative for retrieval. For elliptic orbits, it is more convenient to express L in terms of the true anomaly 0 and eqn (12) transforms to L = LiD + (c/fl){H(O) - H(0i)}],
(13a)
L = Li exp [(c/fl){H(O) - H(0i)}],
(13b)
and
where
H(O) =
- (1/2)(1 -
e2)tfzF(O)+
2 t a n - i [{(1 - e ) / ( l + e ) } Irz t a n ( 0 / 2 ) ] ,
(13c)
and 0i = i n i t i a l t r u e a n o m a l y .
It may be noted that, for circular orbits, H(O) = O. Let us assume that the length of the tether increases from Li to L I during n orbits. Hence, from eqn (13), (c/l)) corresponding to the two deployment procedures are given by c/l) = ( L f l L i - 1)/2wn,
(14a)
clfl = In ( Ltl L~)/2,rn.
(14b)
and
In practice, the deployment is likely to involve a combination of the two procedures discussed above. For example, a deployment strategy may be as follows: cL,
L = ]cLm, I c ( L t + L2 - L),
Li <- L <- Lm, Lz<--L<--L2,
05)
L2 <--L <--L/.
The first and third correspond to exponential rate while the second rate is
1192
V.J. Modi and A. K. Misra
uniform. The length history can be written as
[
Li exp
L =1 L~[1 +
[(c/[l){H(O)
(c/fl){H(O)
-
-
H(0i)}], H(0~)}],
1 [ (L2 + L I ) - Lm exp [ - ( c l I l ) { H ( O ) - H(0z)}],
Oi--~ 0--~ Or, OI --~ 0 -- 02,
(16)
02--~ 0--~ 0t.
The average tension in the tether can be calculated from eqn (5c), since L is known as a function of 0. Results and discussion For the special case where the transverse and longitudinal vibrations are negligible, the orbit is circular and the reference is along the local vertical, exact analytical solutions to the librational equations can be obtained for uniform or exponential deployment (Modi and Misra, 1978). The set of equations describing the general dynamics, however, is rather complex and must be solved numerically with the help of a computer. The in-plane and out-of-plane motions were analyzed separately. There is a small coupling between the two motions through the aerodynamic drag if the rotation of the atmosphere is taken into account; but this was ignored. In spite of this, the computation was rather expensive, since the minimum step size required to handle the high frequency transverse vibrations (several hundred cycles/orbit for small tether lengths) was quite small (of the order of 10-5 orbit). Although extensive calculations were carried out, only a few representative plots characterizing the general trends are given here for brevity. The deployment and retrieval of a 150 kg subsatellite from the Space Shuttle having a mass of 2 x l0 s kg was considered. The tether was assumed to have a mass density of 1.5 kg/km and a stiffness of (5 x 104/L) Him. In-plane motion during the deployment of the subsatellite from 1 km to ---100 km is presented in Fig. 2. The deployment is governed by eqn (15) and the parameters c, L1 and L2 are chosen to be 5 x 10-4sec -1, 10km and 90km, respectively. It may be noted that the pitch angle attains a maximum of about 23°, but becomes quite small towards the terminal phase of deployment. It is interesting that although there are no initial transverse vibrations, B~ builds up to about 100 m due to the coriolis excitation. Subsequently, Bm decreases when the rate of deployment is reduced. It may be observed in Fig. 2(b) that non-zero initial transverse vibrations lead to the superposition of high frequency vibrations on the previous results. There are significant librations and transverse deformations of the tether for large L if the aerodynamic drag is taken into account (Fig. 2(c); note that the transverse vibrations are plotted to a different scale). The out-of-plane motion during deployment of the subsatellite using the combination strategy is shown in Fig. 3. Although the initial distrubances decay, the motion subsequently builds up at the terminal phase due to the aerodynamic forces. Figure 4 is a representative plot for the out-of-plane motion during retrieval using an exponential law. It is clear that small initial libration and vibrations are likely to increase during retrieval.
On the deployment dynamics of tether connected two-body systems
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1193
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1194
V . J . Modi and A. K. Misra
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1195
On the deployment dynamics of tether connected two-body systems
Li= 100 kin, Lf = 4 k m , i = 9 0 °, e =0.001, a = 220km+Earth's rad.
4(]
~'c 20
-20 20O
,oo
Al,m 0
-100
-200 5O A2'mo ~
.
~
i
i
l
t
t
i
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-50 0
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|
1
2
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3 orbits
I
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4
5
6
Fig. 4. Out-of-plane dynamics during retrieval of a 100km tether, using an exponential
law, including aerodynamic effects. Concluding remarks Some of the salient features of the analysis may be summarized as follows: (i) A general formulation of the deployment dynamics of tether connected two-body systems is presented. (ii) Numerical solutions for the general motion are presented for uniform and exponential deployment and their practical combinations. (iii) Transverse vibrations can build up due to coriolis excitation even though there are no initial deformations. (iv) There are large librations and transverse oscillations of the tether due to the aerodynamic drag if the subsatellite is close to the Earth (100-120 km). (v) Although the initial out-of-plane motion decays during deployment, it subsequently builds up in the terminal phase if the rotation of the atmosphere and the eccentricity of the orbit are taken into account in evaluating the aerodynamic drag. (vi) The results presented here should prove useful in selecting a suitable control system for deploying tether connected two-body systems.
1196
V.J. Modi and A. K. Misra
References Baker W. P., Dunkin J. A., Galaboff Z. J., Johnston K. D., Kissel R. R., Rheinfurth M. H. and Siebel M. P. L. (1976) Tethered Subsatellite Study. NASA TMX-73314, Marshall Space Flight Center, Alabama. Buckens F. (1977) On the motion stability of tethered satellites configurations. In Proc. of the 12th Int. Syrup. on Space Technology and Science, (Edited by H. Nagasu) pp. 351-358. AGNE Publishing Inc., Tokyo. Ebner S. G. (1970) Deployment of rotating cable-connected space stations. J. Spacecraft and Rockets 7, 1274--1275. Kalaghan P. N., Arnold D. A., Colombo G., Grossi M. D., Kirschner L. R. and Orringer O. (1978) Study of the Dynamics of a Tethered Satellite System (Skyhook). Final Report, Contract NAS8-32199, Smithsonian Institution, Astrophysical Observatory, Cambridge, Mass. Kane T. R. and Levinson D. (1977) Deployment of a cable-supported payload from an orbiting spacecraft. J. Spacecraft and Rockets 14, 409--413. Kulla P. (1976) Dynamics of tethered satellites. In Proc. of the Syrup. on Dynamics and Control of Non-Ridid Spacecraft, pp. 349-354. Frascati, Italy. Modi V. J. and Misra A. K. (1978) Deployment dynamics of tethered satellite systems. AIAA/AAS Astrodynamics Conference, Palo Alto, California, Paper No. 78-1398. Rupp C. C. (1975) A tether tension control law for tethered subsatellites deployed along local vertical. NASA TMX-64963, Marshall Space Hight Center, Alabama. Stuiver W. and Bainum P. M. (1973) A study of planar deployment control and iihration damping of a tethered orbiting interferometer satellite. J. Astro. Sci. 20, 321-346.
Appendix Nomenclature A A., BN c da, db
e E G(0), H(O) h i i0,lo, k0
ia, ia, ko ; ] ib, jb, kb; / i . | . kc !, J, K L, Ls
area of cross-section of the tether coefficients of O~ in the expansion of u and v, respectively coefficient describing deployment rate position vectors of the points of attachment relative to the respective centres of mass eccentricity of the orbit modulus of elasticity of the tether material functions of 0, eqn (13c) angular momentum per unit mass of the system or altitude inclination of the orbit to the equatorial plane unit vectors along the orbit normal, local vertical and local horizontal, respectively unit vectors along xo, y°, za ; xb, Yb, Z6; and xc, Yc, zc axes, respectively unit vectors along inertial coordinate axes unstretched and stretched
L. Lf, Llk
m ma, rob, m¢
n, ns
q~ Q~ rc R0
1~, Rb, 1~
t T u, v
U, U., U~ V
lengths of the tether, respectively initial length, final length and initial length of the kth stage, respectively total mass of the system mass of the end-bodies and tether, respectively number of orbits, total and per stage, respectively generalized co-ordinates: ao,/3o, %; ab, /3b, Yb; ac, 3'c; At, Bj and • generalized forces vector defined by eqn (le) radial distance of the centre of mass of the satellite system from the centre of the Earth position vectors of the centres of mass of the end-bodies and a point on the tether, respectively time kinetic energy out-of-plane and in-plane transverse vibrations of the tether, respectively total, elastic and gravitational potential energy, respectively velocity of the centre of mass
On the deployment dynamics of tether connected two-body systems
of the system relative to air V~c, V,~, Vzc components of V along x . yc and zc axes, respectively x0, Y0, z0 co-ordinate axes along the orbit normal, local vertical and local horizontal, respectively xo, y., z. ; / xb, Yb, Zb; / body coordinate axes xc, Yc, Zc ao,/3o, ~,o; ] ab,/3b, Ys; modified Eulerian rotations of ac, Yc the end-bodies ac0 equilibrium value of ac • axial strain in the tether O true anomaly Op argument of the perigee
O+0p ~o, P.b, ~ , ~,
mass ratios, eqn (lOc)
1197
/ze gravitational constant of the Earth p density of air ~ , L constants Pb,P,,P densities, eqn (11) Pc mass per unit length of the tether ~r angular velocity of the Earth's atmosphere r tension in the tether • , nth admissible function X, ~ Variables defined by o~JL and yelL, respectively o,. (nw/LX%Jpc) m fl mean orbital rate Dots and primes represent differentiation with respect to t and O, respectively.
On the deployment dynamics of tether connected two-body systemsi" V. J. MODI* Department of Mechanical Engineering, The University of British Columbia, Vancouver, B.C., Canada V6T IW5 AND
A. K. M I S R A § Department of Mechanical Engineering, McGill University, Montreal, Quebec, Canada H3A 2K6 Abstract--A general formulation of the deployment dynamics of tether connected two-body systems taking into account the three dimensional librational motion, longitudinal and transverse vibrations of the continuous tether, eccentricity of the orbit and aerodynamic drag in a rotating, oblate atmosphere, is presented. Three simple deployment procedures (uniform, exponential and their combination) are considered. In the beginning analytical solutions are obtained for the degenerate case of negligible vibrations in a circular orbit, which help in establishing trends for the more general situation. This is followed by the general dynamics of the system investigated through integration of the linearized autonomous equations of motion numerically. Typical plots are presented which describe the effects of various system parameters on the tether dynamics during deployment of a subsatellite from the Space Shuttle.
Introduction TETHERED satellite systems have been proposed for various applications: to provide artificial gravity to a space station, measurement of long wavelength solar and planetary radiations, generation of high resolution gravity and magnetic field maps, carrying out low altitude scientific experiments, etc. Some of these proposals will become realities with the advent of the Space Shuttle. NASA is developing a Shuttle/Tethered Satellite System which will be operational in late 1981 while the concept will be demonstrated in an earlier test flight. Furthermore, ESA has recommended a 10 km tether system to be an integral part of the Shuttle/AMPS Laboratory. The vast potential of tether connected systems has led to numerous investigations of their dynamics, mostly dealing with the post-deployment behaviour. Relatively less attention has been focused on the dynamics and control, of the system during deployment and retrieval although large librational motions may result under certain conditions. Ebner (1970) has carried out a simple planar analysis of a deploying cable-connected space system. The pitch oscillations were evaluated approximately using the tTbe investigation reported here was supported by the National Research Council of Canada, Grant No. A-2181. ~Professor. §Assistant Professor. 1183
1184
V.J. Modi and A. K. Misra
principle of conservation of angular momentum and this information was used to analyze the transverse vibrations of the cable. On the other hand, Stuiver and Bainum (1973) studied the planar deployment control of the Tethered Orbiting Interferometer satellite ignoring the vibrations of the tether. An algorithm to find the control forces for a given set of initial and terminal conditions was described. A passive deployment procedure requiring no actuators to control the motion of the subsatellite was studied by Kane and Levinson (1977). However, the vibrations of the tetered were not included in the analysis. A tension control law for the deployment and retrieval of tethered systems to be used in conjunction with the Space Shuttle was formulated by Rupp (1975). Librational motion in the orbital plane was analyzed and growth of pitch oscillations during retrieval was noted. The tether, however, was assumed to be straight (no transverse vibrations). The above system was subsequently studied in detail by Baker et al. (1976) taking into consideration the three dimensional character of the dynamics and the aerodynamic drag in a rotating atmosphere. A simulation showing the shape of the tether during deployment and retrieval was carried out by Kulla (1976). Although the transverse vibrations of the tether were included, the longitudinal oscillations and the out-of-plane motion of the system were ignored. The deployment problem was considered briefly by Buckens (1977) while studying the three dimensional dymanics of tethered two-body systems. However, the longitudinal strain in the tether and the aerodynamic forces were not included. Recently, Kalaghan et al. (1978) analyzed the "Skyhook" system in detail, taking various environmental effects into account. However, the tether was not treated as a continuum. The objective of this paper is to attempt a somewhat general formulation of the deployment and retrieval dynamics of tether connected two-body systems taking into account the three dimensional librational motion, lognitudinal and transverse vibrations of the continuous tether, eccentricity of the orbit and the aerodynamic drag in a rotating oblate atmosphere. The equations describing the motion of the system are derived using the classical Lagrangian procedure. The transverse vibrations are expanded in terms of a set of admissible functions. Small oscillations from a reference state are considered and the equations of motion are linearized. The generalized forces arising due to the aerodynamic drag are evaluated. Analytical solutions are obtained for the degenerate case of negligible vibrations in a circular orbit, which help in establishing trends for more complex situations. With this background, the general dynamics of the system is investigated by integrating the linearized, coupled, non-autonomous equations of motion numerically. Effects of various system parameters on the librations and vibrations of the tether are determined. Equations of motion Figure 1 shows a system of two bodies having masses mo and mb, respectively and connected by a tether with a mass density pc. At any instant t, let S be the centre of mass of the system, which is located with respect to the centre of the Earth E by the radial distance R0, inclination i of the orbit to the equatorial plane, argument of the perigee 0p and true anomaly 0. Let A and B be the
On the deployment dynamics of tether connected two-body systems
1185
orbit
x6
0
x
perigee
equatorial plane N Fig. 1. Geometry of motion.
centres of mass of the end-bodies while Po and Pb are the points of attachment of the tether having an instantaneous unstretched length L. A set of co-ordinate axes x0, Y0, z0 located at S is so oriented that y0-axis coincides with the local vertical, z0-axis represents the local horizontal, while x0-axis is the orbit normal. The orientation of the principal axes xo, yo, Zo of body A is described by a set of modified Eulerian rotations as follows: //o about yo-axis giving x'., y~, z~; ~/o about z~-axis resulting in x~, y~, z~ and finally ao about x~-axis yielding xo, yo, zo. The principal axes Xb, Yb, and Zb of body B are obtained from x0, Y0, z0 axes through a similar set of rotations ab, 13b and 3'b. The orientation of the axes Xc, yc, Zc (y~ coinciding with the nominal tetherline) can be defined by only two analogous rotations a~ and ,/~, since the rotation about the axis of the tether can be ignored. The axial strain E is assumed to be uniform along the length of the tether or in other words only one term is retained in the expansion of the lognitudinal vibrations. The out-of-plane and inplane transverse vibrations of the tether, denoted by u and v, respectively, are superposed on the resultant tetherline. In general, PoPb does not pass through A , B, or S. Let R~, Rb and R~
1186
V.J. Modi and A. K. Misra
measured from S be the position vectors of the centres of mass A and B and a point on the deformed tether, respectively. Clearly, maRa + mbRo + f0 L pcRc dyc = O.
(la)
Rb = Ro + do + L~ - db,
(Ib)
Furthermore,
and R~ = Ra + d . + r .
(lc)
where do and db are the position vectors of the points of attachment relative to the corresponding centres of mass Ls = Lsjc = L(1 + ~)j~,
(ld)
r~ = Uic + yc(1 + ¢)j~ + vkc.
(le)
and
Here i , jc and kc are the unit vectors along x , y~ and z~ axes, respectively. Substituting eqns (lb)-(le) in (la), Ro and Rb can easily be expressed in terms of L,, do, db, ui~ and vkc. The librational motion of the system and the vibrations of the tether are likely to have very little effect on the orbital motion. Hence, the orbit can be computed separately, and will be assumed here to be Keplerian, i.e. R0 = h2/[/~E(1 + ecO)],
(2a)
Ro20 = h,
(2b)
and
where h is the angular momentum per unit mass of the system, e the eccentricity of the orbit and ~n the gravitational constant of the Earth. It is convenient to expand the transverse vibrations in series forms in terms of a set of admissible functions: u = ~'~ O,(y~)A,(t),
(3a)
v = ~ O,(y~)B,(t).
(3b)
n~l
and
n=l
On the deployment dynamics of tether connected two-body systems
1187
The functions O.(y~) can be somewhat arbitrary, as long as they satisfy at least the geometric boundary conditions. In the present analysis, ~,(Yc) = (2) tp sin ( n ~ ' y d L ) ,
n = 1, 2 . . . . .
oo.
(4)
The expansions transform the integrals with time-varying limits to products of A,, B,, functions of L and their time derivatives. The equations of motion are obtained using the classical Lagrangian formulation
A(ar _a + au=O, ' d t \Oq~/
aqi
(5)
cgqi
where qi represents the generalized co-ordinates ac, ~'c, aa,/3a, ~/o, Orb,/3b, 3~b,~, A,, B,, (n = 1, 2 . . . . . o0) and Qi is the corresponding generalized force arising due to the environmental and control forces. The expressions for kinetic and potential energies are too lengthy and hence omitted for brevity. They can be found in an investigation by Modi and Misra (1978). The resulting non-linear, non-autonomous and coupled ordinary differential equations are rather lengthy and difficult to handle even with the help of a computer. Hence, as a first step to extract some information, small oscillations from a reference state are considered and the equations linearized. Usually, the non-trivial equilibrium configuration is taken as the reference state. However, in the present case this configuration changes continuously with time. Therefore, an average value of ac0, which will be determined later, is used and it is assumed that the displacements from this reference state are not too large. Non-trivial equilibrium values of the other generalized coordinates are very close to the trivial ones and hence, the latter may be taken as the reference. Furthermore, for most practical applications, the length of the tetherline is much larger than the distance of the points of attachment from the respective centres of mass, i.e. do, db ~ L,. Considerable simplification of the equations of motion can be achieved without affecting the overall dynamics of the system substantially, provided do and db are assumed to be zero. It may be pointed out that the linearization causes the motion in the orbital plane to be uncoupled from the out-of-plane motion. Hence, the two motions can be studied separately.
Aerodynamic forces Many tethered satellite systems are likely to have low altitude orbits where the aerodynamic drag is significant compared to the gravity gradient and affects the overall dynamics substantially. The aerodynamic force on an element of the system can be expressed as A F = - I [ 2 C d p & A VIVI,
(6a)
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V.J. Modi and A. K. Misra
where Ca and p are the drag coefficient and density of the air, respectively, while AA is the projected area of the element normal to its velocity V relative to the atmosphere. The coefficient Cd has a small variation with the cross-section of the element and the atmospheric conditions, however, in the present analysis it is assumed to be constant. The density of the air can be written in the form, p = po exp ( - h/ho),
(6b)
where h is the altitude of the element in discussion. The reference height h0 varies slightly with h, however, the variation is ignored here and h0 is so chosen as to match the lower portion of the tether accurately. The relative velocity V depends on the inertial velocity of the centre of mass of the system, the velocity of the atmosphere due to its rotation about the Earth's axis and the velocity of the element relative to the centre of mass of the system (resulting due to the librations and vibrations). The last one is comparatively small and will be ignored here. With this approximation, it may be shown that (7a)
V = RoocOsiio + Rojo + R d O - trci)ko,
where or is the speed of rotation of the Earth about its axis and
~=0+0p.
(7b)
It is convenient to resolve V along the tether axes Xo yc and zc, i.e. V = Vxci~ + Vycjc + Vzckc - (RotrcOsicTc + RosT~)ic + [ - RoTcOsica~sTc + Roca~CTc + Ro(O - trci)sac]jc + [RoorcOsisa~ Syc - l~osacCT~ + Ro(O - trci)ca~]kc.
(7c)
Clearly, IVI = [Ro2{(~rc~si) 2 + ( 0 - trci) 2} + Ro2]v2.
(7d)
The projected areas of the end-bodies normal to the velocity V will depend on their shapes• For the tether, however, the shape is cylindrical and the projected area of an element is ALdc(V~c + vec)!/2/IV[. Hence from eqn (6a), AF = - l l 2 C d p A L d d V ~
(8a)
+ vZ Pt2v --ZCJ v .
The contribution to the generalized force Q~ is given by an,,
Q, = Fa • -~-q/+ F,
. a R , + [.L,XF,(y~) • ~a R ~ aq, Jo
d
y~,
(8b)
On the deploymentdynamics of tetherconnected two-body systems
1189
where Fa and Fb are the total aerodynamic forces on the end-bodies. For the Space Shuttle/Tethered Subsatellite system, /~b, ~tc ~/~o and the generalized forces can be expressed as:
Q~c=-ll2Cd[d~(V2~+ V2¢)t/2f ~ y~o(y~)dyc + pbAblV[L]V~,
(9a)
Q~=-l/2Cd[dc(V2~+ V2c)lrZf ~ ycp(yc)dyc + pbAb[V[L]V~cca~,
(9b)
Q.=-l/2Cd[d~(V2c +
V2~)'/2 foLycp(yc) dy.+pbAb[V[L]Vy~,
Qan = --1/2Cdd~(V2~ + V~c)l/2Vxc
~Pn(Yc)P(Yc)dye,
fo
Qbn = - 1/2 Cad~(V2c + V2c)ll2Vz¢
(9c) (9d)
~.(Yc)P(Y¢) dye,
(9e)
n = 1,2,...oo,
where Pb is the density of air at the subsatellite end. state For a deploying system, the nontrivial equilibrium configuration changes continuously with time. It is convenient to determine the steady state attitude of the non-deploying and non-vibrating system and use it as the reference state. Furthermore, the eccentricity of the orbit.is likely to be very small and the reference state of a corresponding circular orbit can serve as the reference state of the elliptic orbit. With this as background, it may be shown that ac0 and Too are the solutions of the following equations: Reference
ml~eL Z[~2SacoCaco(3 C27co -- S2yco)= -- 112 CdRo2[orcOsi sacoSyco + ([~ -- orci)caco] × f(0, i, ac0, T~0),
(10a)
4ml~eL2f~2SycoCTcoC2acO= 112 CaRo2(o'cOsicyco)Cacof(O,i, aco, ~/co),
(lOb)
where f ( O , i, acO , ~cO) ~" [(orcO$i)2( 1 -- C2acOS2~IcO) "~ (~'~ -- orCi)2C201cO
r",,,y.,{,._ +
+ 2 ( £ t - orci)orcOsis coCa~oSy~oj [Jo -
d,.ld. (lOt)
p.o = mJm,
IZb = mb/m, I~ = p~L/m, P.* = I~JI~,.L.
Note that, in general, or is non-zero and ac0 as well as ~/c0obtained from eqn (10) vary with O. However, or ~ [1 and a suitable reference state can be obtained
1190
V.J. Modi and A. K. Misra
by solving eqn (10) after putting tr = O. This results in Yco = O,
(1 la)
while a~0 is the root of the transcendental equation
3m~.Lec.~oS.co=(-1/2)C~Ro2[dAc~Ol foLp(yA{Yc-(~ +'~)LI dyc For the Shuttle/Subsatellite system, ~o >> ~b,/~c and eqn (1 lb) simplifies to
cao[3(mb +'~lmc)Saco+(1/2)CdRo2{(wldcacol+pbAgL}]=O,
(llc)
where:
= 2p,(hdLca~o)2[1 -
(1 +
Lca~olho) exp (-Lcacolho)],
(1 ld)
p, = density of air at the altitude o f the centre of mass of the system = p0 exp ( - hJho), (1 le) and pb = P, exp ( -
Lca~dho).
(1 If)
It may be noted that for downward deployment a~0 is close to ~r. An examination of eqn ( l l c ) shows that the quantity inside the square bracket cannot be zero for real values of aco, if
(3mb + mc) < (I/2)CdR~p~A~JL,
(1 lg)
and hence
Caco=O,
i.e.
ac0=Ir/2
or
3~r/2.
Deployment p r o ¢ ~ u r e s D e p l o y m e n t or retrieval of the subsatellite can be carried out by controlling the tension in the tether through a reel mechanism as proposed by Rupp (1975). T w o simplest d e p l o y m e n t rates are: (i) Uniform d e p l o y m e n t rate, i.e. /_~= cL~, where L~ is the initial length. Clearly L = L~(1 +
ct).
(12a)
On the deployment dynamics of tether connected two-body systems
1191
(ii) Exponential deployment where the rate at any instant is proportional to the deployed length, i.e./~ = cL. Integrating, L = L~ exp (ct).
(12b)
The constant c is positive for deployment and negative for retrieval. For elliptic orbits, it is more convenient to express L in terms of the true anomaly 0 and eqn (12) transforms to L = LiD + (c/fl){H(O) - H(0i)}],
(13a)
L = Li exp [(c/fl){H(O) - H(0i)}],
(13b)
and
where
H(O) =
- (1/2)(1 -
e2)tfzF(O)+
2 t a n - i [{(1 - e ) / ( l + e ) } Irz t a n ( 0 / 2 ) ] ,
(13c)
and 0i = i n i t i a l t r u e a n o m a l y .
It may be noted that, for circular orbits, H(O) = O. Let us assume that the length of the tether increases from Li to L I during n orbits. Hence, from eqn (13), (c/l)) corresponding to the two deployment procedures are given by c/l) = ( L f l L i - 1)/2wn,
(14a)
clfl = In ( Ltl L~)/2,rn.
(14b)
and
In practice, the deployment is likely to involve a combination of the two procedures discussed above. For example, a deployment strategy may be as follows: cL,
L = ]cLm, I c ( L t + L2 - L),
Li <- L <- Lm, Lz<--L<--L2,
05)
L2 <--L <--L/.
The first and third correspond to exponential rate while the second rate is
1192
V.J. Modi and A. K. Misra
uniform. The length history can be written as
[
Li exp
L =1 L~[1 +
[(c/[l){H(O)
(c/fl){H(O)
-
-
H(0i)}], H(0~)}],
1 [ (L2 + L I ) - Lm exp [ - ( c l I l ) { H ( O ) - H(0z)}],
Oi--~ 0--~ Or, OI --~ 0 -- 02,
(16)
02--~ 0--~ 0t.
The average tension in the tether can be calculated from eqn (5c), since L is known as a function of 0. Results and discussion For the special case where the transverse and longitudinal vibrations are negligible, the orbit is circular and the reference is along the local vertical, exact analytical solutions to the librational equations can be obtained for uniform or exponential deployment (Modi and Misra, 1978). The set of equations describing the general dynamics, however, is rather complex and must be solved numerically with the help of a computer. The in-plane and out-of-plane motions were analyzed separately. There is a small coupling between the two motions through the aerodynamic drag if the rotation of the atmosphere is taken into account; but this was ignored. In spite of this, the computation was rather expensive, since the minimum step size required to handle the high frequency transverse vibrations (several hundred cycles/orbit for small tether lengths) was quite small (of the order of 10-5 orbit). Although extensive calculations were carried out, only a few representative plots characterizing the general trends are given here for brevity. The deployment and retrieval of a 150 kg subsatellite from the Space Shuttle having a mass of 2 x l0 s kg was considered. The tether was assumed to have a mass density of 1.5 kg/km and a stiffness of (5 x 104/L) Him. In-plane motion during the deployment of the subsatellite from 1 km to ---100 km is presented in Fig. 2. The deployment is governed by eqn (15) and the parameters c, L1 and L2 are chosen to be 5 x 10-4sec -1, 10km and 90km, respectively. It may be noted that the pitch angle attains a maximum of about 23°, but becomes quite small towards the terminal phase of deployment. It is interesting that although there are no initial transverse vibrations, B~ builds up to about 100 m due to the coriolis excitation. Subsequently, Bm decreases when the rate of deployment is reduced. It may be observed in Fig. 2(b) that non-zero initial transverse vibrations lead to the superposition of high frequency vibrations on the previous results. There are significant librations and transverse deformations of the tether for large L if the aerodynamic drag is taken into account (Fig. 2(c); note that the transverse vibrations are plotted to a different scale). The out-of-plane motion during deployment of the subsatellite using the combination strategy is shown in Fig. 3. Although the initial distrubances decay, the motion subsequently builds up at the terminal phase due to the aerodynamic forces. Figure 4 is a representative plot for the out-of-plane motion during retrieval using an exponential law. It is clear that small initial libration and vibrations are likely to increase during retrieval.
On the deployment dynamics of tether connected two-body systems
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1193
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1194
V . J . Modi and A. K. Misra
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1195
On the deployment dynamics of tether connected two-body systems
Li= 100 kin, Lf = 4 k m , i = 9 0 °, e =0.001, a = 220km+Earth's rad.
4(]
~'c 20
-20 20O
,oo
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-100
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.
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t
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I
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Fig. 4. Out-of-plane dynamics during retrieval of a 100km tether, using an exponential
law, including aerodynamic effects. Concluding remarks Some of the salient features of the analysis may be summarized as follows: (i) A general formulation of the deployment dynamics of tether connected two-body systems is presented. (ii) Numerical solutions for the general motion are presented for uniform and exponential deployment and their practical combinations. (iii) Transverse vibrations can build up due to coriolis excitation even though there are no initial deformations. (iv) There are large librations and transverse oscillations of the tether due to the aerodynamic drag if the subsatellite is close to the Earth (100-120 km). (v) Although the initial out-of-plane motion decays during deployment, it subsequently builds up in the terminal phase if the rotation of the atmosphere and the eccentricity of the orbit are taken into account in evaluating the aerodynamic drag. (vi) The results presented here should prove useful in selecting a suitable control system for deploying tether connected two-body systems.
1196
V.J. Modi and A. K. Misra
References Baker W. P., Dunkin J. A., Galaboff Z. J., Johnston K. D., Kissel R. R., Rheinfurth M. H. and Siebel M. P. L. (1976) Tethered Subsatellite Study. NASA TMX-73314, Marshall Space Flight Center, Alabama. Buckens F. (1977) On the motion stability of tethered satellites configurations. In Proc. of the 12th Int. Syrup. on Space Technology and Science, (Edited by H. Nagasu) pp. 351-358. AGNE Publishing Inc., Tokyo. Ebner S. G. (1970) Deployment of rotating cable-connected space stations. J. Spacecraft and Rockets 7, 1274--1275. Kalaghan P. N., Arnold D. A., Colombo G., Grossi M. D., Kirschner L. R. and Orringer O. (1978) Study of the Dynamics of a Tethered Satellite System (Skyhook). Final Report, Contract NAS8-32199, Smithsonian Institution, Astrophysical Observatory, Cambridge, Mass. Kane T. R. and Levinson D. (1977) Deployment of a cable-supported payload from an orbiting spacecraft. J. Spacecraft and Rockets 14, 409--413. Kulla P. (1976) Dynamics of tethered satellites. In Proc. of the Syrup. on Dynamics and Control of Non-Ridid Spacecraft, pp. 349-354. Frascati, Italy. Modi V. J. and Misra A. K. (1978) Deployment dynamics of tethered satellite systems. AIAA/AAS Astrodynamics Conference, Palo Alto, California, Paper No. 78-1398. Rupp C. C. (1975) A tether tension control law for tethered subsatellites deployed along local vertical. NASA TMX-64963, Marshall Space Hight Center, Alabama. Stuiver W. and Bainum P. M. (1973) A study of planar deployment control and iihration damping of a tethered orbiting interferometer satellite. J. Astro. Sci. 20, 321-346.
Appendix Nomenclature A A., BN c da, db
e E G(0), H(O) h i i0,lo, k0
ia, ia, ko ; ] ib, jb, kb; / i . | . kc !, J, K L, Ls
area of cross-section of the tether coefficients of O~ in the expansion of u and v, respectively coefficient describing deployment rate position vectors of the points of attachment relative to the respective centres of mass eccentricity of the orbit modulus of elasticity of the tether material functions of 0, eqn (13c) angular momentum per unit mass of the system or altitude inclination of the orbit to the equatorial plane unit vectors along the orbit normal, local vertical and local horizontal, respectively unit vectors along xo, y°, za ; xb, Yb, Z6; and xc, Yc, zc axes, respectively unit vectors along inertial coordinate axes unstretched and stretched
L. Lf, Llk
m ma, rob, m¢
n, ns
q~ Q~ rc R0
1~, Rb, 1~
t T u, v
U, U., U~ V
lengths of the tether, respectively initial length, final length and initial length of the kth stage, respectively total mass of the system mass of the end-bodies and tether, respectively number of orbits, total and per stage, respectively generalized co-ordinates: ao,/3o, %; ab, /3b, Yb; ac, 3'c; At, Bj and • generalized forces vector defined by eqn (le) radial distance of the centre of mass of the satellite system from the centre of the Earth position vectors of the centres of mass of the end-bodies and a point on the tether, respectively time kinetic energy out-of-plane and in-plane transverse vibrations of the tether, respectively total, elastic and gravitational potential energy, respectively velocity of the centre of mass
On the deployment dynamics of tether connected two-body systems
of the system relative to air V~c, V,~, Vzc components of V along x . yc and zc axes, respectively x0, Y0, z0 co-ordinate axes along the orbit normal, local vertical and local horizontal, respectively xo, y., z. ; / xb, Yb, Zb; / body coordinate axes xc, Yc, Zc ao,/3o, ~,o; ] ab,/3b, Ys; modified Eulerian rotations of ac, Yc the end-bodies ac0 equilibrium value of ac • axial strain in the tether O true anomaly Op argument of the perigee
O+0p ~o, P.b, ~ , ~,
mass ratios, eqn (lOc)
1197
/ze gravitational constant of the Earth p density of air ~ , L constants Pb,P,,P densities, eqn (11) Pc mass per unit length of the tether ~r angular velocity of the Earth's atmosphere r tension in the tether • , nth admissible function X, ~ Variables defined by o~JL and yelL, respectively o,. (nw/LX%Jpc) m fl mean orbital rate Dots and primes represent differentiation with respect to t and O, respectively.