UZJL^JM

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Analytical Investigation of the Dynamics

of Tethered Constellations in Earth Orbit (Phase II)

Contract NAS8-36606

Quarterly Report # 2

For the period 22 June 1985 through 21 September 1985

Principal Investigator

Dr. Enrico Lorenzini

October 1985

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
Analytical Investigation of the Dynamics

of Tethered Constellations in Earth Orbit (Phase II)

Contract NAS8-36606

Quarterly Report # 2

For the period 22 June 1985 through 21 September 1985

Principal Investigator

Dr. Enrico Lorenzini

Co-Investigatora

Mr. David A. Arnold

Dr. Mario D. Grossi

Dr. Gordon E. Gullahorn

October 1985

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

Abstract 3

Figure Captions 4

SECTION 1.0 INTRODUCTION 5

2.0 TECHNICAL ACTIVITY DURING REPORTING PERIOD AND PROGRAM

STATUS 6

2.1 Introductory Remarks 6

2.2 Updated Mathematical Model With Elastic Tethers And Longitudinal Dampers 7

2.3 General Considerations Regarding The Damping Of Attitude

Librations And Transverse Oscillations 11

2.4 Selection Of The Control Parameters 14

2.5 New Deployment Control Law 17

2.6 Computer Simulation Of The Deployment Phase 19

2.7 Computer Simulation Of The Station-Keeping Phase ... . 28

2.8 Concluding Remarks 34

3.0 PROBLEMS ENCOUNTERED DURING REPORTING PERIOD 34

4.0 ACTIVITY PLANNED FOR THE NEXT REPORTING PERIOD 35
Page 3

Abstract

This Quarterly Report analyzes the deployment maneuver of three-axis vertical constellations with elastic tethers. The deployment strategy devised in

Quarterly Report #1 has been improved. Dampers have been added to the system.

Effective algorithms for damping out the fundamental vibrational modes of the

system have been implemented. This Quarterly Report also shows simulations of a

complete deployment and a subsequent station keeping phase of a three-mass

constellation.

PRECEDING PAGE BLANK NOT FILMED
Page 4

Figure Captions

Figure 2.2.1 - Geometry and Lagrangian Coordinates for a Three-Mass

Constellation

Figure 2.2.2 - Schematic Model of the Longitudinal Damper

Figure 2.3.1 - Schematic Model of the Rotational/Transverse Damper

Figure 2.6.1 - Dynamic Response of a Three-Mass Constellation During

(a) - (n) Deployment

Figure 2.7.1 - Dynamic Response of a Three-Mass Constellation During

(a) - (1) Station-Keeping
Page 5

1.0 INTRODUCTION

This is the second Quarterly Report submitted by SAO under contract NAS8-

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

Earth Orbit (Phase II)," Dr. Enrico Lorenzini, PI, and covers the period from 22

June 1985 through 21 September 1985.
Page 6

2.0 TECHNICAL ACTIVITY DURING REPORTING PERIOD AND PROGRAM STATUS

2.1 Introductory Remarks

The two-dimensional equations of motion for three-mass constellations were

derived in Quarterly Report #1. In that Quarterly Report, however, the two

tethers connecting the three masses were assumed to be unstretchable. The control law used for deployment was a rate control law without angular feedbacks.

The deployment maneuver was fast but the control law was unable to damp out

completely the attitude and transverse oscillations of the constellation. This

time, on the contrary, the two tethers are assumed to be elastic with generic

characteristics. The right hand terms of the equations of motion developed in

Quarterly Report #1 are therefore transformed as shown in the following sections .

Two longitudinal dampers, aligned with the two tethers, have been added to

the system. Angular feedbacks have been implemented in the rate control law in

order to damp out the attitude and transverse vibrations of the constellations.

The following sections deal with the incorporation of the above mentioned additional features into the mathematical model. The appropriate control parameters

are then selected and simulations of a deployment and a station-keeping phase

are performed.
Page 7

2.2 Updated Mathematical Model With Elastic Tethers And Longitudinal Dampers

The two-dimensional equations of motion (2.2.15) in Quarterly Report #1

were derived under the general assumption that the three masses were generically

located in the orbital plane. The only assumption regarding internal forces was

that a force (tension) TI was exchanged between masses mi and ni2 and a similar

force (tension) T3 was exchanged between masses ma and m3 . In the above-mentioned Quarterly Report the tethers were assumed to be unstretchable and the

tether tensions were therefore derived by solving the related equations of motion for given values of the parameters on the left hand side of the equations.

By assuming the tethers to be elastic, the tether tension is computed by the

elastic stretch times the tether stiffnesses as follows:

—

— — ElA —l

T3 = (t, - £03) (2.2.1)

£3

In equations (2.2.1), referring to tether #1 and tether #2 as shown in

Figure 2.2.1, E is the elastic modulus of the tether material, A is the tether

cross section, £ is the actual tether length and t0 is the natural tether

length. Note that all the quantities referred to tether #2 have the subscript 3

for consistency of notation.

Two longitudinal oscillation dampers (one per tether) have been also added

to the system in order to damp out the elastic longitudinal oscillations of the

two tethers. In this model the longitudinal dampers are assumed to be two

classical spring-dashpot systems with a length gain K<j and a velocity gain Kj.

We can therefore write:
Page 8

3 (bUlut)

(Space Station)

Lagrangian coordinates:

6 * in-plane angle

e * lateral deflection

1] - tether length of tether '1

(3 * tether length of tether »2

/c.m.

nass)

z (local vertical)

t

a (orbit seni-aajor axis)

to the center of the Earth

Figure 2.2.1

Figure 2.2.2
Page 9

= Kdl/dl

T3 = Kd3£d3 + Kd3£d3 (2.2.2 )

The tethers' tensions, according to equations (2.2.1), and taking into account

the lengths of the longitudinal dampers, are also given by:

rp £l"l to a a \ V f

(£3 -*d3 -£c 3 ) = Kt3£t3 (2.2.3)

where EiA1/£1 = Ktx and £^^£5 = Kt3 are the tethers' stiffnesses and £tl, £t3

are the elastic stretches of the respective tethers.

By combining equations (2.2.2) and (2.2.3) we finally get the expressions for

£dl and cd3 as follows:

^dl = IT" (£l - £dl - ^cl) - ^— £dl

(2.2.4)

Equations (2.2.3) and equations (2.2.4) should be added to equations (2.2.15) of

Quarterly Report #1 in order to get the complete set of equations of motion for

the system with elastic tethers and longitudinal dampers. The second order

derivatives £1 and 13 must be expressed explicitly in the last two equations of

motion in order to make them suitable for numerical integration. If we define 7

= R3£3 - RI^I so that 7 = R3/3 - RI^I (in order to shorten the notation) the

final equations are as follows
Page 10

TI = Kti (£1 - £di - £ci)

T3 = Kt3 (13 - £d3 - £C3)

I _ Kti . Kdi , £di = - (£1 - £dl

 - £cl) - £21 £dl

Kdi Kdi

/!d3 = £^ (£ 3 -£d3-£=3 ) - ^I<d3 (2.2.5)

(£ +ili) + m3 (l3 - 7^3) + RZ (mi

-O) (£i + 7) + 3nz£xcos 5 sin 0 (£x + 27) - 3n2£1R2e cos (20)]

m3 [2£3 (^ -0) (£3 + 7) + 3n2£3cos 9 sin 6 (£3 + 27) + 3n2£3R2e cos (26)]

Ra (mi +m3 ) [2ce (6 - fl) - 3Q2e

2

 cos ^ sin

[e (^-O)

 2 + 3n2

esin20] + mtot/

e (T1/£1+T3/£3) /[R2

£1 = (biajj - b2a1j)/(a11a22 -

£3

 =

where :

miR3

m3Ri

m3(l-R3) (2.2.6)

(^-n) 2 (£1+7) +

m3[n2(3cos25-l) (£3-27) + (0-n)* (£3-1-7)

] + T3
Page 11

Equations (2.2.5) together with (2.2.6), and with a suitable expression for the

tether commanded lengths £ci and £cs> are the complete mathematical model of the

system under investigation. Suitable control algorithms will be developed in

the next subsections.

2.3 General Considerations Regarding The Damping Of Attitude Librations And

Transverse Oscillations

The overall attitude motion of a three-mass system with the middle mass at

the system c.m. can be well approximated by the two-mass system's equations of

motion. For the purpose of deriving a tether control algorithm for damping out

the attitude librations, the above mentioned approximation is more than satisfactory. The simplified and linearized equation of motion for in-plane attitude

oscillations is as follows:

61* - 2(6-(l)li + 3nz£20 = 0 (2.3.1)

In equation (2.3.1) the second term is the dissipative one. Therefore the

energy dissipated per each libration cycle is given by:

Ed = 2 I it (6 -fl)0 dt (2.3.2)

J o

where r is the period of the in-plane libration. Our goal is to implement a

tether control law that makes Ed » 0.
Page 12

If we consider that 6 is cosinusoidal (with the assumption of t = 0 when 0=0 )

and that fl is constant, a good tether control law appears to be:

t = L0 (1 -K00) ' (2.3.3)

where the gain Kg must be greater than zero and t0 is the tether length for 6 =

0. It is important to note that the implementation of the control law (2.3.3)

assumes that the value of the angle 8 is provided to the reeling systems of the

constellation. If we now substitute (2.3.3) into (2.3.2), omitting the second

order terms, we get:

^ 2i'0Ke ( A0 (2.3.4)

2Odt - A0

In equation (2.3.4) the dominating term is the first integral under parenthesis

while the second integral is negligible. The orbital rate, therefore, strongly

influences the damping of the in-plane librations. It is interesting to note

that the trajectory followed by the end masses in a damping cycle is not an

eight-shaped yo-yo cycle but an S-shaped cycle. In other words the tether is

shortened during the retrograde part of the libration and is lengthened during

the prograde part as shown in Figure 2.3.1. The control law expressed by equation (2.3.3) can also effectively damp the transverse vibrations of the middle

mass (coordinate e in Figure 2.3.1) by modifying the previous control law for

the two tethers as follows:

3 = to3 (1 - K06 + K£ e/£3) (2.3.5)
Page 13

Figure 2.3.1
Page 14

The tether control law, therefore, is sensitive both to the overall attitude of

the constellation (0) and to the transverse displacement of the middle mass (e) .

A simplified version of (2.3.5) can be obtained by assuming Ke = K0 so that,

with reference to Figure 2.3.1, we get:

[1-M0 + «/£i)] = 101 (1 -

£3 = tot [1 -Kfl(0 - e/£3)] = loi (1 - Ke03) (2.3.6)

This simplified version, where knowledge of the values of angles 6^ and 83 is

required, is the one adopted in the following simulations.

2.4 Selection Of The Control Parameters

In subsection 2.2 we derived the analytical expression of a control law

potentially effective in damping out the in-plane oscillations and the transverse oscillations of the system. In subsection 2.3 we also dealt with the

addition of passive longitudinal dampers to the system. In this subsection we

shall preliminarily compute the values of the gains (control parameters) in the

damping active control laws and in the passive longitudinal dampers.

In order to maximize the energy transfer between the tethers' longitudinal

vibrations and the respective longitudinal passive dampers' oscillations the

natural frequency of each damper should be equal to the natural frequency of the

associated tether. Unfortunately the tethers' frequencies change during deployment because the tethers' lengths change. Since a passive damper is not capable

of variable tuning, it must be designed for specific tuning. By actively damp-
Page 15

ing longitudinal oscillations by means of reel -control this problem is overcome

at the expense of further sophistication of the system. This possibility has

not been eliminated and will be explored in the follow-up to this study.

Each passive damper has been tuned to the frequency of the associated

tether at the tether length pertaining to the station-keeping phase. In this

way the most effective damping of longitudinal vibrations is obtained during

station-keeping while the damping performance is degraded during deployment.

The stiffnesses of tether #1 and tether #2 for two millimeter diameter kevlar

tethers and fully deployed tether lengths are:

Kti = E!Ai/£isk = 67.81 N/m

= 6.781 N/m (2.4.1)

In order to tune the longitudinal dampers we must select the dampers' stif

fnesses in equations (2.2.2) as follows:

Kd3 = Kt3 (2.4.2)

The spring-mass mode longitudinal oscillation frequencies for the constellation

under investigation in the fully deployed configuration are:

icini) = 2.735 x 10'* ra d/sec

= V/E3A3/(£3.kin3) = 2.735 x 10'2 ra d/sec (2.4.3)
Page 16

Some test simulations have been performed in order to identify good damping

coefficients in equations (2.2.2). By increasing the damping coefficients the

dampers proved to be more effective. An adimensional coefficient £ = 0.9 was

adopted for both longitudinal dampers although the dynamic response for supet —

critical damping coefficients has not been explored yet. Consequently the dimensional damping coefficients are as follows:

= 4460.24 N/m/ sec

Kd3 = 2£w«m3 = 446.024 N/m/ sec (2.4.4)

The selection of the gains (equations 2.3.6) for the rotational/transverse oscillations dampers (active damping system) has been made on a trial and error

basis. The major constraint is the tether length variation over a damping

cycle. The maximum tether length variation was finally selected as 1% of the

fully deployed tether length per degree of system's libration. The gain in

equations (2.3.6) is therefore:

Ke = 0.55 (2.4.5)

By adopting the above mentioned value the damping of the constellation's librations is very good. A more than satisfactory damping of the system's transverse

oscillations is also achieved. The transverse oscillations' damping can be

further improved by using equations (2.3.5) instead of (2.3.6), namely by adopting different values for the gains Kg and Kc. This option, however, has not

been explored yet. Results of the simulation runs performed will be illustrated

in subsections 2.6 and 2.7.
Page 17

2.5 New Deployment Control Law

The tether rate control law adopted in the deployment simulations of Quarterly Report #1 has been modified as follows. The initial part of the deployment is still an exponentially accelerated phase followed by an exponentially

decelerated phase with continuous tether speed at the transition between the two

phases. In formulae we can write:

Phase I (acceleration)

*ic = iu e« if £XI < £lc < £1T (2.5.1)

£30 = £31 e0* simultaneous transition for the two tethers

Phase II (deceleration)

*lc = (*1T - *lf)e-* + £lf if £ic > /IT

(2.5.2)

where /9 = a/(£lf/£1T - 1.) and a = nsin(2^c).

All the characteristic tether lengths pertaining to tether #1 and tether #2, in

formulae (2.5.1) and (2.5.2), are in the ratio £i.k//a«k- The angle 0C is the

constant in-plane angle assumed by the constellation during the acceleration

phase. In the deployment control law the deceleration phase does not make the

tether velocity approach zero. The tethers' controllers simultaneously activate

the rotational damper when the tether speed in each tether, during the deceler-
Page 18

ating phase, is equal (or very close) to the tether speed required by the rotational damper for that tether length and that libration angle. The equality of

the tether speeds, however, does not guarantee that the tether lengths, required

by the two different control laws, match. For this reason provisions must be

made for providing a smooth transition. Appropriate values of 6C and of the

transition length £u must be adopted in order to reduce the tether length

mismatch at the transition. In addition a transition control law must be devised in order to compensate for the residual tether length mismatch. The

transition control law must be devised in such a way that its initial and final

velocity are zero. Therefore a semi-cycle of a cosinusoidal law has been adopted. All the above mentioned considerations can be translated into formulae as

follows (similar formulae apply to tether #2) :

If: [^ic].cc.Ph..e ta [^lelrot.d-p.on that implies -0(tlc-llt) ~

then:

- ftr ~ KS0) (2.5.3)

ftr = f,tr COS(| ±) (2.5.4)

and:

fotr — (*lc rot dup.on ~ ^lc »cc.ph««») / *l»k

In equation (2.5.4) AT is the transition time.
Page 19

The control parameters which proved to provide a good dynamic response for

the deployment maneuver are as follows:

£1T = 500 m

0C = 30° (2.5.5)

AT = 2000 sec

A detailed description of the deployment maneuver obtained by using the above

mentioned control law and control parameters is given in the next subsection.

2.6 Computer Simulation Of The Deployment Phase

Several simulation runs of the deployment maneuver have been performed in

order to select the appropriate control parameters previously described in subsections 2.4 and 2.5. In this subsection the dynamic response of the system

during deployment is shown for that "best" selection of parameters. The deployment maneuver was started from initial tethers' lengths of £u = 20 m and £31 =

200 m. The system with elastic tether shows a strong tendency to go slack in

the very beginning of the deployment maneuver. Any small initial mismatch may

cause a temporary slack tether condition. An in-line thruster and an appropriately tuned longitudinal damper can help considerably in relieving the slack

tether tendency. A detailed analysis of the initial deployment phase should

follow the present study of the deployment maneuver. This type of analysis has

been postponed, however, for follow-up studies of the three-mass constellations.

Figure 2.6.1 from (a) to (n) shows the dynamic response during deployment with

longitudinal dampers tuned to the station-keeping tether lengths and rotational/transverse dampers activated as described in the previous subsection.
Page 20

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 11000 12000

TIME (Sec) Rot -f Longit Dampers, Deployment

(b)

1000 2000 3000 4000 SOOO 6000 7000 BOOO 9000 10000 11000 12000

Time (Sec) Deployment Maneuver

Figure 2.6.1 (a) and (b)
Page 21

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 11000 12000

TIME (Sec) Longit Damper. Csi= 9. Deployment

(d)

0 ) 000 2000 3000 4000 5000 6000 7000 8000 9000 10000 11000 12000

TIME (Sec) Rot + Longit Dampers, Deployment

Figure 2.6.1 (c) and (d)
Page 22

0 1000 2000 3000 4000 5000 6000 7000 BOOO 9000 10000 11000 13000

TIME (Sec) Rotat -t- Longit Dampers, Deployment

005 -

(f)

0 5 10 IS 20 25

THETTA (Deg) Rot + Longit Dampers, Deployment

30

Figure 2.6.1 (e) and (f)
Page 23

350^

300 —

U

I

§

 25°r

e ZOO Iisol

50 r

r

h

oh

I I ! I > [ l._t _ r i j ] i | j.. I I I I 1 I I i I ..1 -1-

^

J

j

4

H (g)

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 11000 12000

Time (Sec) Deployment

00

-100 -50 0 SO 100 150 200

XI (M) Ml Trajectory Side View. Deployment

250 300

Figure 2.6.1 (g) and (h)
Page 24

C'

05 r04 ,—

03 —

r

t~

02 —

F

§ -01 •

-03

-

0 1000 3000 3000 4000 5000 6000 7000 8000 9000 10000 11000 12000

TIME (SEC) Rot -t- Longit Dampers, Deployment

-.0005 t—1

-05 -04 -03 -02 -01 0 01 02

EPSI (M) With Epsi Feed-back. Deployment

04

Figure 2.6.1 (i) and (1)
Page 25

14E-5

]

 (m)

0 1000 2000 3000 4000 5000 6000 7000 BOOO 9000 10000 11000 12000

TIME (Sec) Longit Damper CSI= 9, Deployment

3E-5.L

2E-5

flE-5 ,

-2E-5 •-

r

f

f"i 1 1 1 t • L 1 1 ! 1 > 1 1 i 1 • ! , i i i 1 i i i i ! . , , 1 1 . 1 1 . 1 1 1 tl I 1 1 1 1 1 1 1 1 1 1 1 1 1 > 1 1 1 > 1 1 1 •z. (n)

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 11000 12000

TIME (Sec) Longit Damper CS\~ 9. Deployment

Figure 2.6.1 (m) and (n)
Page 26

The initial conditions of this deployment maneuver are an in-plane attitude of

30° with respect to the local vertical and an initial alignment error of the

three masses (e) of 0.05 m. The dynamic response, however, does not differ

significantly from a qualitative standpoint for varying values of the above

mentioned parameters. Figure (a) and Figure (b) show the tether length and

tether speed respectively of tether #1 vs. time. The same quantities for tether

#2 can be obtained from those of tether #1 by scaling them up by a factor of 10.

The deployment is performed in approximately 3 hours. This value however is

affected by the initial tether length and is therefore eventually affected by

the position of the reeling system on the Space Station. More important is the

initial tether speed which must be as close as possible to the initial design

speed, according to the law £x = a £j, in order to avoid tether slackening. In

Figure (b) the different phases of the deployment control law are evident: the

rotational damper-on results in the ripple on the plot at approximately 5500

sec, while the transition law-off results in the second ripple at 7500 sec.

Figure (c) and Figure (d) show the tether #1 elastic stretch and longitudinal

damper's length respectively vs. time. Here again it must be pointed out that

the system should start the deployment with its natural tether stretch: an

overstretch or an understretch induces longitudinal oscillations and reduces the

margin for positive tension in the tether. It is also interesting to note the

strong coupling between the elastic stretch and the longitudinal damper's

length, as well as the capability of the system to damp out longitudinal oscillations as shown by the ripple around 8000 sec. This capability will be shown

more clearly in the next subsection. Figure (e) shows the in-plane angle (8)

vs. time. Note that the acceleration phase of deployment is designed for a

constant in-plane angle. Figure (f) is the phase plane 8-6 and clearly shows

the effectiveness of the rotational damper in damping out the attitude libration

of the constellation. Figure (g) shows the tether tension for tether #1 which
Page 27

is approximately equal to that for tether #2. The tension is proportional to

the tether elastic stretch (Figure c) . Figure (h) shows the side-view of the

trajectory of mass #1 (Space Station) with respect to the system center of mass.

Mass #3 (the ballast) follows a specular trajectory scaled up by a factor 10.

Mass #2 (the micro-g facility) remains very close to the system c.m. throughout

the deployment maneuver. Figure (i) is the lateral displacement (e) of mass #2

vs. time. When the rotational/transverse damper goes on at 5500 sec this oscillation begins to be damped out. Figure (£) is the phase plane e-e. The damping

of e is less effective than that of the in-plane angle 6 because the rotational/transverse damper is tuned to the in-plane oscillation. By using a control law like (2.3.5) instead of the simplified (2.3.6), that is, by using a

multi-frequency damping technique, the damping of the transverse oscillation can

be further improved. Finally Figure (m) and (n) give the horizontal (along the

local horizon) and vertical components respectively of the acceleration at the

micro-g facility vs. time. It is clear from these figures that the initial

values of the acceleration due to the transverse oscillations of the system and

the transient oscillations due to snapping of the tethers during transitions in

the deployment control law are damped out very effectively by the various dampers. We shall return to this topic again in the next subsection which deals

with modal vibrations damping during station-keeping.
Page 28

2.7 Computer Simulation Of The Station-Keeping Phase

The effectiveness of the dampers during station-keeping is shown in the

following set of plots. These plots have been obtained by simulating the dynamic response of the constellation during a station-keeping phase under the

following initial conditions. The simulation begins with a tether length for

tether #1 equal to 909 m and for tether #2 equal to 9090 m; the initial inplane angle is 1° while the initial lateral deflection of mass #2 is 0.10 m.

Figure 2.7.1(a) shows the tether length for tether #1 vs. time while Figure (b)

is the side view of mass #1 trajectory. Figure (c) shows the tether's elastic

stretch vs. time. Both these quantities are referred to tether #1: tether #2

has a qualitatively similar behavior. Figure (e) shows the in-plane angle vs.

time and Figure (f) is the phase plane 6-6. Figure (g) shows the lateral

deflection (e) of mass #2 (the micro-g platform) vs. time and Figure (h) is the

phase plane e-e. The saw-teeth-like shape of this plot, near the beginning of

the simulation, is due to an unsufficiently small plotting step and therefore it

has no dynamic meaning. It is evident from these plots that any initial perturbations of the constellation are abated very effectively by the dampers. The

final result is shown in Figure (i) and (1) which give the horizontal and vertical component respectively of the acceleration at mass #2 vs. time. The maximum

accelerations at the end of this simulation are much smaller than 10"8g. This

last point must not be misinterpreted: it means only that the system can effectively abate transient vibrations while the steady state acceleration level at

mass #2 will depend upon the external steady state perturbations. These perturbations are not simulated in this analysis which deals with the transient response of the three-mass constellation.
Page 29

(a)

4000 6000 8000 10000

TIME (SEC) ROTAT-rLONGIT DAMPERS ACTIVE

12000 14000

920 -

914 -

912 -

910 - (b)

0 2 4 6 8 10

XI (M) Ml TRAJECTORY SIDE VIEW

12 14 16

Figure 2.7.1 (a) and (b)
Page 30

49 —

r

48^

E E < 44 —

E r

h

u

42 —

t

39k I

3000 4000 6000 8000 10000 1EOOO

Time (sec) Longit Damper CS1= 9

J

j

H

1

^

1

J

14000

456

454

45

448

£ 446

444

442

_J

2000 4000 6000 6000 10000

TIME (SEC) ROTAT +LONGIT DAMPERS ACTIVE

13000 14000

(d)

Figure 2.7.1 (c) and (d)
Page 31

f

L

L

8 -

6U

[

5 I

E 2 L-

0 r1

1

(e)

zooo 4000 6000 8000 10000

TIME (SEC) ROTAT +LONGIT DAMPERS ACTIVE

12000 14000

0004 '—

024 6

THETA (DEC) ROTAT +LONGIT DAMPERS ACTIVE

Figure 2.7.1 (e) and (f)
Page 32

eg)

2000 4000 6000 8000 10000

TIME (SEC) ROTAT +LONGIT DAMPERS ACTIVE

14000

00

-8 -4-2 0 2 4 6

EPSI (M) TET LEN CONTROL EPS! FEED-BACK

Figure 2.7.1 (g) and (h)
Page 33

, , . 1 . . , i . . j . 1 . 1 I . . 1 I < 1 1 1 1 I J . »

t ft 3

5E-S jIf

r

sr 1 t

: r

i*j r

V)

\ r

S5E-5 j—

5 L

£u 1 r-

« 0001 •

o

z

-=000 15

L

L

- 0002 {-

t , i

' !

i A "i

• A ^

M / ' A '

'1 1 1 (V 1 \ l\ /^ /^- ~ — i )i\lii\/\ / \J \s ^-^ ^~^ ^

1

 i 1 / ~i

1 1 , i • V -1

I

j

i

J

j

~~1

3j

-j

j

0 2000 4000 6000 8000 10000 13000 14000

Time (sec) Longit Damper CS1= 9

008

006

004

| 002

S

>

id

U

LJ

i i r i ' i i 1 r i 1 i i i [ i i i 111 ; r i ^ i i

H -

(i)

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-

- 1

: ^

1

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j- 002 — B —

u

i

>-004

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-006

-ooa

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-

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_

.

-i — 1 i ' - * — L t i t 1 _(_ i t 1 i i_ t 1 > i — i 1 — -J — i — i — 1 — * ' — i — ! — : — ia)

0 2000 4000 6000 6000 10000 12000 14000

Time (sec) Longit Damper CS!= 8

Figure 2.7.1 (i) and (1)
Page 34

2.8 Concluding Remarks

Passive dampers for tethers' longitudinal oscillations have been added to

the system with elastic tethers. Active control of rotational/transverse oscillations have been also added to the constellation. A new deployment strategy

which drives the system to a steady state condition in approximately 3 hours has

been devised. The initial part of deployment with elastic tethers, however,

requires a more detailed investigation; as a result of such investigation the

deployment duration may increase depending upon the minimum separation length of

the end masses at the start of deployment. The dampers have been designed to

provide an effective damping of the longitudinal, libratory and transverse oscillations of the system. The effectiveness of the dampers has been proven by

computer simulations of the deployment and station-keeping phase. The damping

of the modal vibrations can be further improved by multi-frequency active damping.

3.0 PROBLEMS ENCOUNTERED DURING REPORTING PERIOD

None
Page 35

4.0 ACTIVITY PLANNED FOR THE NEXT REPORTING PERIOD

During the next reporting period we will analyze the dynamics of three-mass

constellations when the middle mass travels along the tether. Stability constraints and appropriate control laws will be sought. The investigation of the

dynamics of a five-mass system, vertically oriented, attached to the Shuttle

will also be initiated. This system potentially could be used as a scientific

platform for measuring geophysical gradients.