a 

Analytical Investigation of the Dynamics 

of Tethered Constellations in Earth Orbit (Phase I I) 

, Contract NAS8-36606 

c. 

986 through 21 September 1986 

~ ~ 

noipal fnvsst 

Enrico C. Lor 

ber 1986 

~ 

~ ~ ~ ~~ ~ 

~~ ~~ 

~~ ~~ ~~ - 

Smithsonian Institution ~ 

Astro-physical Observatory 

Cambridge, Massachusetts 03138 

I The Smithsonian Astrophysical Observatory 1 

I is a member of the I 

OF TPE LYNAHICS CF TE'IHEEEC CCKSIELLATPONS 

124 EAhlH ORbf'l, PBASE 2 Cuarterly lieport, 

22 Jan. - 21 Sep, 1986 lsrnithscnian 

AstroFhysical ChsErvatory) 59 F CSCZ 22A H1/13 

J .- 

N87- 1 la 15 

Unclas 

44E63 
Analytical Investigation of the Dynamics 

of Tethered Constellations in Earth Orbit (Phase 11) 

Contract NAS8-36606 

Quarterly Report #6 

For the period 22 June 1986 through 21 September 1986 

PrincipalDr. Enrico C. Lorenzini 

Mr. David A. Arnold 

Mr. Mario Cosmo 

Dr. Mario D. Grossi 

October 1986 

Prepared for 

National Aeronautics and Space Administration 

Marshall Space Flight Center, Alabama 35812 

Smithsonian Institution 

Astrophysical Observatory 

Cambridge, Massachusetts 02138 

The Smithsonian Astrophysical Observatory 

is a member of the 

Harvard-Smithsonian Center for Astrophysics 
CONTENTS 

Page 

Summary ................................. 3 

Figure Captions ............................. 4 

SECTION 1.0 INTRODUCTION. ...................... 6 

2.0 TECHNICAL ACTIVITY DURING REPORTING PERIOD AND PROGRAM 

STATUS ......................... 6 

2.1 

2.2 

2.3 

2.3.1 

2.3.2 

2.3.3 

2.4 

3.0 

4.0 

Algorithms For Damping Out-Of-Plane Librations ..... 6 

Simulation Runs With/Without Out-Of-Plane Libration 

Damping Algorithms .................. 14 

Environmental Models .................. 38 

Drag Model 39 

J2 Model ........................ 46 

ThermalModel ..................... 50 

Concluding Remarks 56 

PROBLEMS ENCOUNTERED DURING REPORTING PERIOD ....... 57 

ACTIVITY PLANNED FOR THE NEXT REPORTING PERIOD ...... 57 

....................... 

................... 

i 
Page 3 

Summary 

This Quarterly Report addresses the following topics related to the dynamics of the 4-mass tethered system: 

The development of damping algorithms for damping the out-of-plane libration of the system and the interaction of the out-of-plane control with the 

other degrees of freedom. 

The development of environmental models to be added to the dynamics simulation computer code. The environmental models are specifically a new drag routine based on the Jacchia's 1979 model, a Jz model and an accurate thermal model 

of the wire. 
Page 4 

Figure Captions 

Figure 1. 

Figure 2a-b. 

Figure 3a-b. 

Figure 4a-i. 

Figure Sa-b. 

Figure 6a-r. 

Figure 7a-b. 

Figure 8. 

Figure 9a-d. 

Figure 10. 

Figure 11. 

In-plane and out-of-plane libration angles vs. time for a case 

in which the out-of-plane damping system is disactivated. The 

gains of the control laws are: Kp2, K,=O. 

Dynamic response vs. time for a case in which a modified 

($ = constant) yo-yo control law is adopted for out-ofplane libration control while the in-plane control system 

is disactivated. The gains of the control laws are: 

Ks=O, K,=4. 

The same as in Figure 2 except that this time the in-plane 

control system is activated. The gains of the control laws 

are: Ke=l, K,=4. 

The same as in Figure 2 (the in-plane control system is 

disactivated). The gains of the control laws are: 

Ke=O, K,=10. 

The same as in Figure 3. The gains of the control laws 

are: KezO.1, K,=10. 

Dynamic response vs. time for a case in which the 

standard yo-yo control law (variable $) is adopted 

for out-of-plane libration control. The in-plane control 

system is also activated. The gains of the control laws 

are: Ke=0.2, K,=10. 

Front-view and side-view respectively of the system under 

the conditions of Figure 6. "he snapshots are every 100 

sec for 36,000 sec. The x-axis scale is largely expanded. 

Atmospheric density vs. altitude for different exospheric 

temperatures. The curves are obtained by means of an 

analytical fit of the Jacchia's 1977 model. 

Atmospheric density vs. time and exospheric temperature 

vs. time over one orbit at 450 km altitude and 2845 

inclination. Figure 9a-b show the results for the sun at 

the Spring Equinox and Figure c-d for the sun at the Summer 

Solstice. 

Acceleration components in the inertial reference frame 

due to the Ja gravity term vs. time over one orbit 

at 450 km altitude and 2845 inclination. 

Acceleration components in the orbiting reference frame 

due to the Ja gravity term vs. time for the same 

conditions of Figure 10. 
Page 5 

Figure 12. 

Figure 13. 

Figure 14. 

Schematic of the geometry of solar illumination for a 

tethered system. 

Temperature vs. time over one equatorial orbit at 450 km 

altitude with illumination angle B=O for a stainless 

steel tether of 2 mm diameter. 

Same as in Figure 13 for a kevlar tether of 2 mm diameter. 
Page 6 

1 . 0 INTRODUCTION 

This is the sixth Quarterly Report submitted by SA0 under contract NAS8- 

36606, "Analytical Investigation of the Dynamics of Tethered Constellations in 

Earth Orbit (Phase 11) ,It Dr. Enrico C. Lorenzini, PI. This report covers the 

period from 22 June 1986 through 21 September 1986. 

. 

2.0 TECHNICAL ACTIVITY DURING REPORTING PERIOD AND PROGRAM STATUS 

2.1 Algorithms For Damping Out-Of-Plane Librations 

Damping of out-of-plane librations is achieved by controlling the lengths 

of the three tether segments connecting the four platforms. The out-of-plane 

librational dynamics of a multi-mass system can be modeled, with good approximation, by reducing the system to a two-mass system if the bowing of the former is 

small. The simplified equation of motion that describes the unperturbed, small 

out-of-plane librations of a two-mass tethered system is as follows: 

For a two-mass tethered system the energy dissipated per cycle per un 

therefore given by (L N Lo for 6L << Lo): 

(2.1.1) 

t mass is 

where rv is the period of the out-of-plane libration equal to x/n. From simple 

physical considerations and also from equation (2.1 .If) it is immediately evi- 
Page 7 

dent that the energy dissipated during an out-of-plane damping cycle does not 

depend upon the orbital rate n. The out-of-plane libration is in fact very 

weakly coupled with the longitudinal degrees of freedom, consequently tension 

control laws, which rely on such coupling, are poorly effective in damping outof-plane librations. A simple mechanical equivalent of the out-of-plane dynamics of a tethered system is a pendulum of variable length on the ground (the 

pendulum has negligible Coriolis forces due to the Earth rotational rate and the 

librational frequency is approximately constant if the length variations are 

small). The yo-yo technique developed in reference [l] and [Z] are perfectly 

suitable for damping the out-of-plane librations of a tethered orbital system. 

The same technique is on the contrary surpassed by the S-type [3] control law 

for damping the in-plane libration because the latter control law exploits the 

Coriolis forces generated by the orbital rate n. In order to clarify this point 

we can use equation (2.1.11) for computing the energy dissipated by substituting 

into that equation various control laws. In the case of small damping we can 

assume that the libration over one cycle is approximately sinusoidal. We have 

therefore 

p N psin (2nt) 

+ N- 2nY,cos (2nt) 

(2.1.2) 

(2.1.3) 

An ideal yo-yo control law (bang-bang type) can be expressed in formulae as 

f 01 lows : 

!/e, = (l+K,(o) if sign(p) = sign (+) 

e/e, = (l-K,Y,) if sign(cp) # sign (+) 

(2.1.4) 
Page 8 

where p is the amplitude of the out-of-plane libration. We can rewrite equation (2.1.11) as follows: 

(2.1.5) 

where M, is the out-of-plane component of the damping torque per unit mass and 

L, is the out-of-plane component of the angular momentum. By making use of 

equation (2.1.4) we finally have: 

(2.1.6) 1 1 1 ED,, N -2(L,, -!=in = -32O1@’K,L.’ 

The fractional damping of the p degree of freedom can be obtained by equating 

the kinetic energy variation for that degree of freedom to the energy dissipated 

per cycle (per unit mass) given by equation (2.1.6). We have: 

or 

After defining: 

we obtain: 

eN4- “ (ideal yo -yo, out -of -plane) 9 CO 

(2.1.7) 

(2.1.8) 

(2.1.9) 

(2.1.10) 
Page 9 

An ideal yo-yo control law, as expressed by equation (2.1.4) is unfeasible because it requires an infinite tether velocity. In actuality the ideal yo-yo 

control law is approximated by a sinusoidal control law (we call it smooth yoyo) that can be defined as: 

By substituting equation (2.1.11), its derivative and equation (2.1.3) into 

equation (2.1.1 t) we obtain : 

ED,, N - (2 .l. 12) 

After making use of equation (2.1.7) and of equation (2.1.9) we finally obtain: 

ZNr- (smooth yo -yo, out - of -plane) 6L 

P LO 

(2 . 1 .13) 

By using the smooth yo-yo instead of the ideal yo-yo we have therefore a decrease in damping effectiveness of roughly 20%. 

A modification of the smooth yo-yo control law can be obtained by assuming 

that @ in equation (2.1.11) is constant and equal to the amplitude of the first 

out-of-plane libration cycle &. Consequently equation (2.1.13) is modified to 

(2 .l. 131) 

The fractional damping is therefore increasing with the decrease of @. "his 

result seems theoretically remarkable- In practice; however, the control nf 

out-of-plane librations excites in-plane librations because the latter are 
Page 10 

strongly coupled with tether length variations. This spill-over effect pumps 

the in-plane libration amplitude. By adopting the modified smooth yo-yo control 

law the spill-over does not decrease with 8; the in-plane libration builds up. 

The in-plane control consequently tries to damp out the in-plane libration and 

by doing so impairs the effectiveness of the out-of-plane libration damping. In 

actuality therefore the result shown by equation (2.1.131) is not achievable. 

We want now to compare the in-plane libration damping with the out-of-plane 

libration damping. The simplified equation of motion that describes the unperturbed, small in-plane librations of a two-mass tethered system is as follows: 

The energy dissipated over a cycle per unit mass is therefore given by (L N L, 

for 6L << L,): 

+e 

ED,@ = -2 l ti(d-n)ddt N -2L0 lT(8-n)ddt (2.1.15) 

where re is the in-plane libration period equal to 2?r/ (fin) . The equation, 

similar to equation (2.1.7) , applicable to in-plane librations can be obtained 

by multiplying equation (2.1.14) by 4 and intergrating from o to re. Since 

dd2/dt = 286 and N Lo we obtain: 

Equation (2.1.16) can therefore be written as follows: 

(2.1.17) 
Page 11 

where ATy' stands for the kinetic energy of the 6 degree of freedom relative to 

the LH-LV reference frame. 

The smooth yo-yo control law for damping the in-plane libration is expressed as follows: 

t/.& = [ 1 +KeJsin (2 flClt)] (2.1.18) 

By following the same procedure adopted for computing the out-of-plane fractional damping but using equation (2.1.17) instead of (2.1.7) we arrive at the 

following result: 

(smooth yo - yo, in - plane) (2.1.19) 

This result stems from the fact that by using a yo-yo type control law we do not 

exploit the Coriolis forces generated by the orbital rate fl. The fractional 

damping turns therefore to be the same for in-plane librations and out-of-plane 

librations; it would be completely different had we used a tension control law. 

The conclusion just reached prompted us to design the S-type control law for 

damping the in-plane libration. The S-type control law is formulated as follows : 

&/Lo = (1-KeO) (2.1.20) 

The associated fractional damping, in the case of light damping, is given by: 

(S - type, in -plane) AT 

- x 6t ~ 'J=L,B (2 .l. 21) 
Page 12 

The most important feature is that the S-type control law provides a fractional 

damping that is independent of the amplitude J. Heavy damping can be easily 

achieved with this type of control law even at small amplitude e. In the case 

of heavy damping, however, equation (2.1.21) provides only a first order approximation of the fractional damping because the in-plane libration is no longer a 

pure sinusoid over a cycle; equation (2.1.21) gives an overestimation of the 

fractional damping in that case. 

We have also computed the fractional damping for out-of-plane librations 

achieved by using other non linear control laws. Table 1 summarizes the fractional damping achievable vs. the various control laws shown in column one for 

the out-of-plane dynamics. Table 2 shows similar results for the in-plane dynamics. 

Table 1 

Out-of-Plane Libration Damping by Means of Tether-Length Control Law 

Ideal Yo-Yo 

Smooth Yo-Yo 

Modified 

Smooth 

Yo -Yo 

96 Control 

Law 
Page 13 

Table 2 

In-Plane Libration Damping by Means of Tether-Length Control Laws 

Controlw Mathematical- -DamDina 

Smooth Yo-Yo L/t. = [l+Keesin (2 6nt) 3 

S -Type L/L. = (1-KeB) 

se !24- AT 

e LO 

- 

References to Section 2.1 

1. Baker W .P . et a1 . "Tethered Subsatel lite Study , '' NASA 

'I'M X-73314, March, 1976. 

2. Spenser T.M. "Atmospheric Perturbation and Control of 

a Shuttlenethered Satellite," Proceedings of the 8th IFAC 

Symposium, Automatic Control in Space, 1980. 
Page 14 

2.2 Simulation Runs Withmithout Out-Of-Plane Libration Damping Algorithms 

The previous section dealt with the yo-yo damping technique for a two-mass 

tethered system. In this section we generalize it to a four-mass tethered system. As already shown for the in-plane control system in Quarterly Report No. 5 

Section 2.4 the generalization is readily implemented by controlling the three 

tether segments proportionally to their unstretched lengths. In formulae 

(2.1.22) 

Equations (2.1.22) are similar to equations (2.4.3) of Quarterly Report No. 5 

except for the term &OL that is the tether length variation per unit length for 

the out-of-plane libration control. The variable LDLOL depends on the particular 

control law adopted. 

Several simulation runs have been done in order to derive a reasonable 

technique for damping the out-of-plane oscillations. The smooth yo-yo control 

law has been extensively tested either in the standard (variable @) or in the 

modified version (constant F). The in-plane libration angle 6 and the out-ofplane libration angle 'p are shown in Figure 1 for a case in which the in-plane 

libration control is activated (Ke=2) while the out-of -plane libration control 

is off (Kp=O). Notice that the in-plane libration is very well damped and that 

the in-plane control does not spill over the out-of-plane libration. The viceversa is not true. The second simulation was run by adopting Ke=O (in-plane 

control off) and Kp=4 in the modified smooth yo-yo control law. Figure 2a 

shows the out-of-plane angle vs. time. The fractional damping (for light damp- 
Page 15 

A 

2n 

ing: fractional damping N logarithmic decrement = A; damping ratio N -) of 

the angle p is approximately 11%. Figure 2b shows the in-plane angle and the 

out-of-plane angle vs. time for the same case. This figure clearly shows the 

spill of the out-of-plane control over the in-plane libration. The amplitude of 

the in-plane angle is building up. A third simulation was run in order to 

address the issue of the back action of the in-plane control over the out-ofplane damping. The control law adopted for the out-of-plane libration is the 

same of the previous run with the same gain k, = 4. The in-plane control is 

activated with a gain ke = 1 in the S-type control law. Figure 3a shows the 

out-of-plane angle vs. time for a time duration of 5000 sec. Because of the 

activation of the in-plane control the out-of-plane damping is reduced from 11% 

to 6.5%. The in-plane amplitude growth due to the spill-over is, on the contrary, reduced from 2O to 1" as shown in Figure 3b. In the fourth simulation 

run Ke=O and K,=10. This simulation has been run for 10,000 sec to show better 

the build-up of in-plane librations. Figure 4a depicts the out-of-plane libration angle vs. time. A fractional damping N 26% has been achieved. The outof-plane libration phase-plane, depicted in Figure 4b, stresses this result. 

Figure 4c shows the in-plane libration angle build-up (together with the out-ofplane angle vs. time) while Figure 4d depicts the phase-plane of the in-plane 

libration. The fractional tether length variation 6L/L, for out-of-plane libration control is shown in Figure 4e (in the case of the modified smooth yo-yo, 

6f/fo has a constant amplitude). By substituting the values of 6C/L0 = 0.12 

(remember that 6L is a peak-to-peak variation) into equation (2.1.131) for the 

first cycle we obtain a fractional damping AJ/J = 38% that is 

greater than the measured one (26%). The difference can be attributed to two 

different effects: 1) Equation (2.1.131) and all the other similar equations 

are valid for light damping and they overestimate the damping if otherwise. 2) 

The tether elasticity deforms the eight-shaped damping cycle invoked by equation 

(when pl/p = 1) 
Page 16 

(2.1.11). This latter effect is readily noticeable in Figure 4f that shows the 

trajectory of mass no. 1 with respect to the orbiting reference frame (located 

approximately at the system C.M.). For completeness Figure 49 and 4h show the 

in-plane and out-of-plane components respectively of the distances of the middle-masses from the straight line through the end-masses (lateral deflections) . 

Figure 41 depicts the moduli of such distances. In the fifth simulation the 

same gain k, = 10 of the previous simulation was adopted for the out-of-plane 

libration control while Kg was set equal to 0.1 in an attempt of limiting the 

in-plane oscillation growth without impairing the damping of out-of-plane librations. Figure 5a shows the out-of-plane libration angle vs. time. The fractional damping reduction is almost negligible: from 26% to 25%. Figure 5b 

shows the in-plane libration angle vs. time and the out-of-plane angle vs. time. 

The in-plane amplitude growth is reduced compared to the previous run. All the 

other features of this run are similar to the previous one. Because of the 

results obtained by adopting the modified yo-yo control law we decided to switch 

to the standard yo-yo (variable p) for the following simulations in order to 

reduce the spill-over due to the out-of-plane libration control. 

The sixth simulation run adopts K,=10 in the standard yo-yo control law 

while KO is set equal to 0.2. This simulation has been run for 36,000 sec. 

Figure 6a shows the angle p vs. time. The fractional damping is, at the beginning, around 17%. The fractional damping starts decreasing past 15000 sec. The 

variation is, however, non-monotonic: the fractional damping goes almost to 

zero around 30000 sec and goes up again to 6% at 32000 sec. Various sources can 

contribute to the deviation of the fractional damping behavior from equation 

(2.1.13) : 1) the tether length variation for yo-yo control, as expressed in 

equazion (2.i.iij. has no frequency feed'iack; the controller goes slowly out of 

phase with the p oscillation because of the frequency variation due to damping. 
Page 17 

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Page 26 

This phenomenon, however, is not very important in the present simulation as it 

clearly appears from Figure 6b that shows the tether length variation for outof-plane libration damping (&wL) vs. the out-of-plane angle. The phase difference is fairly constant as it can be inferred by the crossing points of the 

eight-shaped curves close to the p=O vertical line. 2) The system is probably 

settling down to a state in which energy, through exchange among the various 

degrees of freedom, is flowing into the least damped degrees of freedom. This 

conclusion can be preliminarily inferred from Figure 6c that shows the angular 

foot-print of the system (out-of-plane libration vs. in-plane libration) . At 

approximately the same time when the damping starts decreasing the system gets 

trapped in an eight-shaped angular foot-print. 

Figure 6d shows the in-plane angle vs. time along with the out-of-plane 

angle vs. time. The in-plane angle is clearly forced by the tether length 

variation for the control of out-of-plane librations. By adopting the standard 

smooth yo-yo control law, however, the spill-over decreases with time. Figures 

6e and 6f are the phase-planes for the (p and 19 oscillations respectively. Figures 6g and 6h show the tether length variation per unit tether length for the 

out-of-plane and in-plane oscillation control respectively. Figure 6i and 6j 

show the in-plane and out-of-plane components respectively of the lateral 

deflections of the Space Station (solid line) and the G-laboratory (dotted 

line). The in-plane component of the lateral deflection of the G-laboratory is 

excited at the beginning of the simulation by the out-of-plane control spillover but it decreases as the spill-over decreases. The out-of-plane components 

of lateral deflections of both the Space Station and the G-laboratory increase 

slnwly with tine. Since thero is no c~ntrnl ~ser EUC~ csm,;=snents iz a?? these 

simulation runs the energy is probably accumulating in these degrees of freedom. 

Figures 6k, 6L and 6m depict the tether length variations for tether no. 1, no. 
Page 27 

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Page 30 

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a9 

two. 

IW z 

PO 0 S 

uo 

E' 

-8 

0 

a 

o 

d 

2. 

0 

0 

2 

MASS SYMBOL 

'~00.00 IbO.00 liO.00 Bb.00 Ub.00 O'.OO -bO.OO -'eO.OO 2120.00 -'160.00 -iOO.OO -kO.OO -iBO.W -hO.OO 

10 

2A 

3+ 

4x 

7 

-360.00 -b30.00 -hO.OO -beO.OO -kZO.OO -b60.00 -kOO.OO -bUO.OO -kBO.OO -'lZO.OO -560.00 -'eOO.OO -bUO.OO -bw.w OUT-OF-PLONE COtIPONENT IMETERSI 

- -7 

-920.00 -b60.00 -'~ooo.oo -~~ouo.oo -'1080.00 -'l120.00 -'l160.00 -'1200.00 -'lZuO.OO -'lzBO.oO -i320.00 -i360.00 -mlUOO.OO -'lUUO.W -iUM.WFigure 7a. Front-view. Snapshots every 100 sec for 36,000 sec. 
Page 37 

MASS SYMBOL 

10 

2A 

3+ 

rlx 

0 

0 

'D 

'100.00 8b.00 6b.00 4b.00 2b.00 0.00 -20.00 -'uO.OO -60.00 -bO.OO -'lOO.OO -120.00 -140.00 -160.00 -180.00 -200.00 

-220.00 -240.00 -260.00 -210.00 -300.00 -310.00 -340.00 -360.00 -380.00 -UOO.?O -420.00 -YYO.OO -460.00 -U80.00 -500.00 

IN-PLRNE COMPONENT IMETERSI -10 

-k20.00 -b40.00 -b60.00 -bBO.OO -800.00 -b20.00 -b40.00 -b60.00 -bBO.OO -iOO.OO -720.00 -740.00 -760.00 -780.00 -100. 

Figure 7b. Side-view. Snapshots every 100 sec for 36,000 sec. 
Page 38 

2 and no. 3 respectively. In order to give an idea of the acceleration fiuctuations during the out-of-plane damping process Figure 6t? depicts the three components, in the orbiting reference frame, of the acceleration on board the Space 

Station. The residual acceleration is due to the fact that all the initial 

oscillations have not been damped out completely. The in-plane libration however can be damped out very effectively by disactivating the out-of-plane yo-yo 

control and by increasing the gain of the in-plane control system. The following figures, therefore, should not be interpreted as the limit performance of 

the system. Figures 6p, 6q and 6r show the flight direction (ACH) , transverse 

(ACL) and vertical (ACH) component of the acceleration on board the G-laboratory 

vs. time. Because of the relatively large residual libration amplitudes the 

residual acceleration fluctuations on board the G-laboratory are larger than the 

acceleration fluctuations on board the Space Station. Finally Figures 7a and 7b 

show the expanded-scale (the expansion coefficient is different for the two 

figures) front-view and side-view respectively of the system with snapshots every 100 sec up to 36,000 sec. The residual oscillations of the system are 

clearly evident in these two figures. 

2.3 Environmental Models 

We have developed an entirely new package of subroutines which model the 

environment at the altitudes of interest to the Space Station. 

We have modeled the major perturbations acting upon a tethered system in 

the range of altitudes from 400 km to 600 km. The major perturbations are: 

drag, J2 gravity term and thermal as dealt with in detail in the next subsections. 
Page 39 

2.3.1 Drag Model - 

As a density model we have adopted the version by C.E. Roberts Jr., reference [l], who developed an analytical model of upper atmosphere which provides 

atmospheric density as a function of exospheric temperature and altitude. The 

model is based on Jacchia's 1970, taking also into account the '77 corrections 

[2]. The temperature equation has been modified in order to have a simple 

analytical expression of the temperature as a function of the height h. 

The atmospheric density is given by: 

where : 

i=1 

(2.3.1) 

constituents = N2, Ar, He, 02, 0, H 

molecular mass (gr/mole) 

number of particles of each constituent per cm3/Avogadro's 

number 

computed data at h = 125 km 

exospheric temperature 

thermal diffusion coefficients 

f (exospheric temperature) = A1 + A~TEx + A3exp(AJ"I'x) 

(2.3.2) 

(2.3.3) 

125 km 

90 km 

to be chosen in order to produce a minimum in the absolute 

value of the percentage difference with the Jacchia's 

model for altitudes between 125 and 1000 km and for TEX 

between 600°K and 2000OK 
Page 40 

Ra - effective earth's radius = 6356.766 km 

(2.3.4) 

m go - gravity acceleration on the ground = 9.80665 - 

s= 

R - universal gas constant = 8.31432 J/OK-mole 

The number of particles of atomic hydrogen dH is equal to zero for h < 500 km. 

As a result of the above mentioned computations Figure 8 shows the atmospheric 

density vs. altitude, in the range of interest to our system, for different 

exospheric temperatures. 

In order to obtain a more accurate density model we have improved the 

"static model" by taking into account the phenomena which affect the values of 

the global exospheric temperature [3]. We have modeled the effects of solar 

activity and diurnal variations but we have neglected the geomagnetic activity. 

Following these assumptions we can express the local exospheric temperature TL 

as follows: 

where : 

RT 

m 

n 

e 

'1 

7 

0 

night minimum of the global exospheric temperature, 

function of solar activity 

cost = .3 = amplitude of temperature variation 

cost = 2.2 

cost = 3.0 

I 0+b 1/2 

14-b 1/2 

IH + p + p sin (H+7) I 

latitude of the satellite 

(2.3.5) 
Page 41 

60 declination of the sun 

H hour angle of the sun 

B lag of the temperature maximum with respect to the sun's 

culmination = cost = -37O 

7 - asymmetry of temperature distribution = cost = +6O 

7 - location of temperature distribution asymmetry = cost = +430 

Figures 9a, 9b, 9c and 9d show the density and the local exospheric temperature 

variations vs. time (1 orbit) in two different seasonal situations: Spring 

equinox and Summer solstice. 

The deceleration of the i*-mass mi due to drag is given by: 

where : 

ballistic coefficient of the ith-mass 

cross section of the ith-mass 

drag coefficient of the ith-mass 

= - CD , iAi 

2% 

p(hi) = atmospheric density at height hi 

height of the ith-mass = Ri-Q (l-fsin2Xi) 

flattening factor of the Earth 

velocity of the i'"-mass relative to the atmosphere 

latitude of the ith-mass 

earth's equatorial radius 

(2.3.6) 

We assume that the atmosphere rotates rigidly with the Earth and, differently 
Page 42 

from other authors, we take into account the relative motion between the orbiting frame and the ith-mass mi. The relative velocity is therefore given by: 

(2.3.7) 

where & is the Earth's angular velocity while Ri is the radius vector from the 

Earth's center to the ith-mass. We have 

(2.3.8) 

where R, is the radius vector of the origin of the orbiting reference frame, ri 

is the radius vector from that origin to the i*-mass and iY is the orbital rate. 

Hence 

(2.3.9) 

The aerodynamic forces (per unit mass) are readily computed by substituting 

equation (2.3.9) into equation (2.3.6). 
Page 43 

1 m! VI Y L 

0 

t- / / 

2 

I 

w 

(D 

2 

I 

W 

(D 

2 2 

ti 

I 

2 

I 

w 

N 

0 

d 

(D 

0 

02 

a 

0 

0 

(D 

0 

OD 

ua 

0 

(D 

ua 

0 

d 

ua 

0 

N 

u) 

0 

10 

d 

0 

d 

d 

0 

N 

d 

0 

0 

d 

0 

OD 

m 

0 

(0 

I? 
Page 44 

1150 

1100 

u 

950 

BOO 
Page 45 

2.6E- 12 

2.4E- 12 

2.2E- 12 

2E-12 

mTl.OE- 12 

. M 

1, 

1.4E-12 

1.2E- 12 

1E-12 

8E-13 

HEIGHT AT THE EQUATUR = 450 Km 

INC = 28.5 deg 

SWER SOLSTICE 

- 

- 

- 

- 

- 

- 

- 

- 

- 

- 

- 

- 

- 

- 

- 

IIIIIIII 111111111111l1 l111111111lIlIIIIIIIIIII 1111111111111~ 

Figure 9c+ Figure 9d+ 

- 

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 

TIME (sec) 

1100 c 

h 1050 

0, 

x 

000 1 

I1 1 I I I 1 I I I I I I I I I I I I I I 1 I 1 I I I I I I I I 1 I I 1 I I I I I I I I I I I I I 1 1 I I I I I I I I I I 14 

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 

TIME (SEC) 
Page 46 

2.3.2 J1 Model - 

For satellites other than the geosynchronous ones the zonal harmonics of 

the Earth’s gravity field are the major source of orbital perturbations. The 

expression of the potential of the Earth’s gravity field for the ith-mass of the 

system, limited to the zonal harmonic terms, is given by: 

(2.3 .lo) 

where Q is the earth equatorial radius, p the earth gravitational constant, J,, 

are the zonal harmonic coefficients, Pno are Legendre polinomials of argument 

sin6i, 61 is the latitude of the i*-mass and Ri is the radius vector of the same 

mass. If we limit the expansion of equation (2.3.10) to the J1 term (Jl = 0 in 

the geocentric reference frame) we obtain the expression of the J1 perturbation 

potential as follows: 

The gravity perturbation force due to J1 is therefore given by: 

J1 c31 = -grad (Vi ) 

Since in functional form Vi Ja = V:’ [Ri, sindi (Ri) ] we have: 

(2.3.11) 

(2.3.12) 

where xil = Xi, xil = Yi, xi3 = Zi are the components of the radius vector Ri of 

the ith-mass in the inertial reference frame. By substituting equation (2.3.11) 
Page 47 

into equation (2.3.12) we finally get: 

FJz = - - 3 p J2 & (1-5sina6i) - xi 

xi 2 Ri5 

(2.3.14) 

Figure 10 shows the components in inertial reference frame of the J3 gravity 

force over one orbit with an inclination of 280.5 and an orbital altitude of 450 

km. The components of the Ja gravity force in inertial reference frame, given 

by equations (2.3.14), must be transformed into orbiting reference frame components in order to be introduced into our dynamics simulation computer code. 

This transformation is per formed according to : 

(2.3.15) 

where the subscript (0) stands for orbiting reference fame and the subscript (I) 

stands for inertial reference frame. The transformation matrix [TI is given by: 

(2.3.16) 

where f, 3 and 

while i, 7 and G are the unit vectors of the orbiting reference frame given by: 

are the unit vectors of the geocentric inertial reference frame 
Page 48 

(2.3.17) 

where go is the radius vector of the origin of the orbiting reference frame and 

G, is the velocity vector. The orbiting reference frame orbits the Earth at 

constant angular rate CA = Vo/Ro. Figure 11 shows the components of the Jz gravity force in orbiting reference frame for the same case shown in the previous 

figure. 

2.3.3 Thermal Model - 

The thermal environment produces a periodical variation of the tether 

length (contractions and expansions) with consequent variations of the acceleration levels. The thermal inputs into the tether are: 

A) Solar radiation 

B) Earth albedo 

C) Earth infrared radiation 

D) Aerodynamic heating 

E) Internal heating (i.e., Joule effect ...) 

The only cooling process is the radiation emitted by the wire. We have developed a new routine, neglecting the thermal inputs D and E, which calculates the 

temperature gradient of the tether per unit length as follows: 

(2.3.18) 
Page 49 

s 

0 

u) 

u) 

0 

0 

0 In 

0 

0 

u) 

d 

0 

0 

0 

f 

0 

0 In 

c) 

h 

0 

0 

In 

N 

0 

0 

0 

N 

0 

0 

2 

0 

0 

2 

0 

0 

In 

0 
Page 50 

0 

0 

In 

In 

0 

0 

0 

VJ 

0 

0 

In 

d 

0 

0 

0 

d 

0 

0 

In 

c) 

0 

8G 

m2 

og ZF: 

v 

Lv 

0 

0 

0 

Lv 

0 

0 

In 

I 

0 

0 

2 

0 

0 in 

0 

d 

Q) 

k 

& 
n rage 5i 

where : 

r 

U 

tether radius 

Boltzman constant 

tether emissivity 

volume density of the tether 

heat capacity of the tether 

tether mass 

For the expression of thermal inputs we have followed the line adopted by SKYHOOK [4]. We included the earth albedo and we derived a more sophisticated 

model for the partial and the total eclipse of the wire [5]. The expressions of 

the Qi are as follows: 

where : 

a 

a 

A 

f 

7 

OIR 

*e 

tether absorbitivity 

earth albedo (annual average) 

tether area 

view factor = B-sin(2 B)/2 

sun zenith angle 

K 

infrared tether absorbitivity 

earth temperature 

(2.3.19) 

% and sin$ = - 

R 

In the computation of QB and Qc we neglected the solar declination angle, the 

differences between continental and ocean areas and the dependence with the 
Page 52 

latitude and seasonal variations. We also assumed that the reflectance of the 

earth is isotropic and diffuse and that the spectral distribution of reflected 

radiation is equivalent to the spectral distribution of the incident radiation. 

We compared the thermal behavior of two different materials, under the same 

environmental conditions: stainless steel and kevlar. Figure 12 is a sketch of 

the geometry relevant to the thermal effects over the orbit. In Figure 13 and 

14 the tether temperature is plotted vs. time (2 orbits) for a 2 mm diameter 

stainless steel and kevlar tether respectively. The initial temperature is the 

same for the two cases. 

The thermal expansion (or contraction) of a tether segment of length lo is 

given by : 

(2.3.20) 

where aT is the tether thermal expansion coefficient. If we assume a tether 

length of 10 km the expansion/contraction of the stainless steel wire and the 

kevlar wire over one orbit at 450 km altitude are respectively as follows: 

(2.3.21) 

where we have assumed CY^,,^^.^ = 1.2~10-~ OK-l a nd &T,kevlar = -2X10-6 OK-’ while the 

maximum AT have been derived from the previous runs as follows: lAT,,tee~axl N 58OK 

and IATkevlr:*I N 95OK. 
Page 53 
0 

0 

2 

H 

0 

0 

0 

2 

0 

0 

0 

0) 

0 

0 

0 

m 

0 

0 

0 

PO 

0- 

s; v 

W 

TIta 

0 

0 

0 * 

0 

0 

0 

CY 

0 

0 

0 

cu 

0 

0 

2 

0 

Page 54 

M 

r( 
Y 

0 \ M 

x le. 

'rl - urd UVI 

N.... 

BhlQlW.0 0 

00000 

Page 55 
Page 56 

References for Section 2.3 

I 

111 C .E. Roberts. "An Analytical Model for Upper Atmosphere 

Densities Based Upon Jacchia's 1970 Models." Celestial Mechanics, 

Vol . 4, pp. 378"-377, 1971. 

121 L.G. Jacchia. "Thermospheric Temperature, Density and 

Composition: New Models." SA0 Report SR375, March 1977. 

c31 COSPAR Working Group 4, "CIRA. COSPAR International Reference 

Atmosphere 1972." Akademie-Verlag, Berlin, 1972 

141 Kalaghan, P.H., D.A. Arnold, G. Colombo, M.D. Grossi, L.R. 

Kirschner and 0. Orringer. "Study of the Dynamics of a Tethered 

Satellite System (SKYHOOK)." Final Report NAS8-32199, SAO, March 

1978. 

c51 _. "Spacecraft Attitude, Determination and Control." Edited by 

James R. Wertz, Reidel Publishing Company. 

2.4 Concluding Remarks 

A survey of various out-of-plane libration control laws has been carried 

out. Consequently a yo-yo control law with amplitude of the tether length variation proportional to the amplitude of the out-of-plane libration has been selected. This control law provides good damping when applied to a (theoretical) 

two-dimensional system. In the actual 3-dimensional 4-mass tethered system, 

however, energy is transferred to the least damped degrees of freedom (the outof-plane lateral deflections are still undamped in the present simulations) in 

such a way as to decrease the effectiveness of the algorithm for out-of-plane 

iibration control. The addition of damping algorithms for the out-of-plane lateral deflections is therefore necessary (these damping algorithms will be developed in the next reporting period). 

A completely new package of major environmental disturbances relevant to 

the altitude range of the Space Station has been also developed. This package 

consists of: a new drag model based on Jacchia's 1977 model, a thermal model of 
Page 57 

the wire and a model of the J2 gravity perturbation. 

3.0 PROBLEMS ENCOUNTERED DURING REPORTING PERIOD 

None 

4.0 ACTIVITY PLANNED FOR THE NEXT REPORTING PERIOD 

In the next reporting period the algorithms for damping the out-of-plane 

lateral oscillations will be developed. The investigation of the forced dynamics of the 4-mass tethered system during the station-keeping phase will also be 

carried out.