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Analytical Investigation of the Dynamics

of Tethered Constellations in Earth Orbit (Phase II)

Contract NAS8-36606

Quarterly Report #5

For the period 22 March 1986 through 21 June 1986

Principal Investigator

Dr. Enrico C. Lorenzini

July 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

N86-32513 -CB- 178901), . ANALYTICAL IB VESTIGA1ION \ . .

O F TH E BYKAHICS CFTETBEEE D CONSTELLATIONS

 :

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IN EAET H OR:BIT, EHASE 7 ( Smithsonian „: . . V; . ,

Astrophysical VoJbservatory) 53 p •-. >CSCL :22A •:.. ,
Analytical Investigation of the Dynamics

of Tethered Constellations in Earth Orbit (Phase II)

Contract NAS8-36606

Quarterly Report #5

For the period 22 March 1986 through 21 June 1986

Principal InvestigatoiDr. Enrico C. Lorenzini

Co-Investigators

Mr. David A. Arnold

Mr. Mario Cosmo

Dr. Mario D. Grossi

July 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

Abstract 3

Figure Captions 4

SECTION 1.0 INTRODUCTION 5

2.0 TECHNICAL ACTIVITY DURING REPORTING PERIOD AND PROGRAM

STATUS 5

2.1 3-Dimensional Model Of The 4-Mass Tethered System ... 5

2.2 Software Implementation Of The 3-Dimensional Model And

New Integration Subroutine 12

2.3 Simulation Runs Of The Station Keeping Phase Of The 4-

Mass Tethered System With/Without Longitudinal Dampers 14

2.4 Damping Algorithms For In-Plane Librational Arid Lateral

Oscillatory Modes 32

2.5 Station-Keeping Simulation Runs With/Without Librational

And Lateral Damping Algorithms 34

2.6 Concluding Remarks 57

3.0 PROBLEMS ENCOUNTERED DURING REPORTING PERIOD 58

4.0 ACTIVITY PLANNED FOR THE NEXT REPORTING PERIOD 58
Page 3

Abstract

This quarterly report deals with the analytical model and software implementation of n-mass, three dimensional tethered systems. The newly developed

computer code is then used for simulating the 3-dimensional dynamics of a 4-mass

tethered system. The 4-mass tethered system is designed in order to provide a

micro-g environment on board the Space Station while variable-g experiments are

performed on a laboratory moving along the upward tether segment. Extensive

simulation runs illustrate both the undamped, free response and the damped

dynamics response of the system. Passive longitudinal dampers and subsequently

active damping algorithms, effective against the in-plane dynamics, are added to

the system. Their separate effects upon the various vibrational modes are

clearly shown by relevant simulation runs.

PRECEDING PAGE 81WIC i*>t
Figure Captions

Page 4

Figure 2.1.1

Figure 2.1.2

Figure 2.3.1

Figure 2.3.2a-v

Figure 2.3.3a-m

Figure 2.5.1a-za

Figure 2.5.2a-za

Geometry for the n-mass, 3-dimensional tethered system.

Schematic of the relationships between generalized coordinates

and Cartesian coordinates.

Schematic of the 4-mass tethered system for micro-g/variable-g

applications.

Three-dimensional, free dynamic response of the 4-mass

tethered system without dampers for a 7200 sec station-keeping

phase. Figures 2.3.2t and u show the side-view and front-view

of the system at 20 sec intervals for 3600 sec. Figure 2.3.2v

is the equal scale, side-view at 20 sec intervals for 1000 sec

(simulation run no. 1).

Three-dimensional, free dynamic response of the 4-mass

tethered system with 3 longitudinal dampers in series with

the associated tether segments. Same initial conditions of

simulation run no. 1. Station-keeping phase of 7200 sec

(simulation run no. 2).

Three-dimensional, free dynamic response of the 4-mass

tethered system with 3 longitudinal dampers for initial

condition 90° out-of-phase with respect to simulation

runs no. 1 and 2. Station-keeping phase of 7200 sec.

Figures 2.5.Iz and za show the side-view and front-view

of the system at 40 sec intervals for 7200 sec (simulation

run no. 3) .

Three-dimensional, free dynamic response of the 4-mass

tethered system with longitudinal dampers and with in-plane

librational/lateral damping algorithms. Same initial

conditions of simulation run no. 3. Station-keeping phase

of 10,000 sec. Figures 2.5.2z and az show the side-view and

the front-view of the system at 40 sec intervals for 7200 sec

only. (Simulation run no. 4).
Page 5

1.0 INTRODUCTION

This is the fifth quarterly report submitted by SAO under contract NAS8-

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

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

period from 22 March 1986 through 21 June 1986.

2.0 TECHNICAL ACTIVITY DURING REPORTING PERIOD AND PROGRAM STATUS

2.1 3-Dimensional Model Of The 4-Mass Tethered System

The 2-dimensional equations of motion (Cartesian coordinates) derived for

the n-mass system in Section 2.3 of Quarterly Report No. 3 have been extended to

the third dimension .

The derivation of the 3-dimensional equations of motion follows the line

shown in Section 2.3 of Quarterly Report No. 3 with the addition of the out-ofplane coordinate y. With reference to Figure 2.1.1 the radius vector Pi from

the origin of the orbiting reference frame to the ith-raass is given by:

Pi = Xil + yij + Zik (2.1.1)

The inertial acceleration of the ith-mass is given by:

- . -

+ 2n x pi + n (n-?i) - rtln l (2.1.2 )

where:

Page 7

?! = a + Pi .(2.1.3)

After substitution of equations (2.1.2) and (2.1.3) into equation (2.1.1) we

finally get:

rii = [xi - 20 ii - n2Xi]l + [yi]j + [5i + 2flxi - n2(Zi-a)]k (2.1.4)

The equation of motion for the ith-mass in vector form is given by:

= Fgl + FT1 + Fpl (2.1.5)

where Fgi is the gravity force on the ith-mass , FTI is the net force due to

tether tensions acting upon the same mass and Fpl is the external perturbation

force .

As in Quarterly Report No. 3 the gravitational force is obtained from the potential energy as follows:

Fgi = - Wt (2.1.6)

where VA is, in this case, given by:

V± = - - min

2

[2a 2 + 2azi-xi

2

-yi

2

 + 2zi

2

] (2.1.7)

Substituting in equation (2.1.6) and performing the derivatives we get:

- m10

a

[-x1!-yi5+ (a + 2zt)k ] (2.1.8)
Page 8

After substitution of equations (2.1.8) and (2.1.4) into equations (2.1.5) and

by assuming (temporarily) Fpl = 0, we finally obtain the three scalar equations

of motion for the ith-mass as follows:

(2.1.9)

i -t- 2fixj. =

The expressions of the tensional forces are as follows:

Fixi = Txi — Tx,i-i

Fry! = Tyl - Ty.t-i (2.1.10)

FTzi = Tzi — TZ i i-i

where the index i is associated with the tether connecting the ith-mass to the

i+lth-mass . In equations (2.1.10) the tension components are as follows:

> Txt = Ti (Xi.! - X

Tyi = Ti (yi,i - yO//,. (2.1.11)

Tzi = Ti (2i<1 - Zi)/£i

The tension modulus Ti is given by equation (2.3.13) of Section 2.3 in Quarterly

Report No . 3 .

In the case of a 4-mass tethered system the 3-dimensional equations of

motion, inclusive of three longitudinal dampers placed in series with each

tether segment, are finally as follows:
Page 9

xj. = 2nZl + T! (x

x2 = 20z2 + T2 (x3-x2)/(m2£2)

x3 = 2flz3 - T2 (x3-x2)/(m3£2)

yi = -n2yi + ^ (y2-yi)/(m1£1)

9a = -n2y2 + T2 (y3-y2)/(m2£2) - T! (y2-y1)/(m2£1)

y3 = -n2y3 - T2 (y3-y2)/(m3£2)

zj. = 3fl2zi - 20*1. + Tx (z2-z1)/(m1£1) (2.1.12)

z2 = 3fl2z2 - 2nx2 + T2(z3-z2)/(m2£2) - T! (z2-Zl)/ (m2£1)

z3 = 3Q2z3 - 2Qx3 - T2(z3-z2)/(m3£2)

Ti = ktl (£i-£di-£d) = kti/ti

T2 = kt2 (£2-£d2-£c2) = kt2£t2

T3 = kt3 (£3-£d3-^c3) = kt3£t3

£d2 = kt2£t2/b2 - k2£d2/b2

£d3 = kt3£t3/b3 - k3£d3/b3

In equations (2.1.12) Ktl = EiAi/£cl is the stiffness of the ith-tether , £tl is the

tether stretch, bi is the damping coefficient and kj. is the stiffness of the ith-

longitudinal damper (bi and kA were called respectively kdi and kdi in Quarterly

Report No. 3) . The meaning of the other symbols is the same as in Quarterly

Report No . 3 .

The relations between system quantities (xi-.^Xnjyi, ---- ,yn;zi, ----- ,Zn) and output/control quantities are derived as in the two-dimensional case. The major

difference is that in the 3-dimensional case the lateral displacements of the

masses with respect to the line connecting the two end masses have an in-plane

and out-of-plane components, respectively called tn and £0il and that the outof-plane libration angle (<p) must be added to the output/control variables. If

we call £* the radius vector from mi to mn we have (see Figure 2.1.2) :

£* = £* (cosv?sin0! -f sinpj + cospcostfk) (2.1.13)

We can therefore easily obtain the in-plane (0) and out-of-plane libration angle

(<f>) as follows:
Page 10

X x

n-1

Figure 2.1. 2
Page 11

= tg-1 [(Xl-xn)/(Zl-zn)]

(2.1.14)

<p = sin-1 [(yi-yn)/£*]

where:

£* = [(Xl-xn)

2

 + (Yi-Yn)2 + (Zl-zn)

2

]

V2

 (2.1.15)

The lateral deflections £1 = [en2 + £oi2J can be obtained by computing the

coordinates of the point of intersection between the line through the end masses

and its perpendicular through the ith-mass; that is to say by solving the following system of equations:

z-Zi = - (y-yi) = - (x-Xi)

m n

(2.1.16)

z-Z! = m(y-y!) = n (x - Xj.)

where :

(2.1.17)

n = tg0 = (x!-xn)/(zi-zn)

By solving equations (2.1.16) we get the coordinates of the intersection point

as follows:
Page 12

+ (z1+1-Zl)tg0]/(l-tg20)

yci = [yi - yi+i m2 + (zi<1 -z 1 )m]/(l-m2) (2.1.18)

2ci = [xci - xi<:l]tg0 + Zi,i

The lateral deflections «i are therefore given by:

(zi+1-zci)

2

 + (yi*i-yci)2]

1/2

 i = l,..,n- 2 (2.1.19)

and the in-plane (en) and out-of-plane components (£oi) are given by:

/ \i/2

fii = ^ix2 + en3) sign (xi-a-Xd)

= [(Xi.i-xci)

2 + (z^i-Zci)2] sign (x^! - xcl) (2.1.20)

Finally the distance between two contiguous mass points (£1) is simply given by:

+ (yi.i-yi)2 + (Zi.i-Zi)2] (2.1.21)

2.2 Software Implementation Of The 3-Dimensional Model And New Integration Subroutine

A new software code has been implemented for the numerical integration of

the 3-dimensional model. This software code can handle any tethered system of n

masses and m < n-1 longitudinal dampers (in series with the tether segments).

The location of the longitudinal dampers may be anywhere and must be specified

by the user. Additional subroutines provide the control laws for reeling in and

out the tethers connecting the various platforms.
Page 13

The output of the computer code has been modified in order to accommodate

the greater number of variables to be plotted. Moreover a subroutine has been

written for formatting one of the output files in the format suitable for the

graphic software of SKYHOOK. The computer code can therefore produce the familiar side-view and front-view of the constellation shape by plotting sequential

shots at a specified interval of time. A new integration subroutine has also

been added as an option to the variable step, IV-order Runge-Kutta. The RungeKutta subroutine is a very stable subroutine the only drawback of which is the

introduction of an artificial numerical damping1 when long integration step size

are adopted. The new integrator is a variable step (constant relative accuracy)

predictor-corrector, initialized by a Runge-Kutta subroutine. The predictorcorrector is recommended in reference [1] over the Runge-Kutta, the Gear and

other integration subroutines as the one that provides the best compromise among

stability, speed and accuracy, especially for non-linear equations. We have

compared the predictor-corrector subroutine to the IV-order Runge-Kutta on a

test case simulation run of the 4-mass tethered system. When we adopted a

relative accuracy of 10~6 (which produces good results with the Runge-Kutta subroutine) the two integration subroutines produced exactly the same results. The

predictor-corrector subroutine, however, has been 40% faster than the RungeKutta, proving its advantage over the latter. The Runge-Kutta did not show any

artificial numerical damping for relative accuracy better or equal to 10"6.

References to Section 2.2

1. D'Souza, A.F. and V.K. Garg, "Advanced Dynamics, Modeling and Analysis."

Prentice Hall, 1984.
Page 14

2.3 Simulation Runs Of The Station Keeping Phase Of The 4-Mass Tethered System

With/Without Longitudinal Dampers

The new computer code that models the dynamics of n-mass, 3-dimensional

tethered systems has been used to simulate the dynamic response of a 4-mass

tethered system. The system under investigation, depicted in Figure 2.3.1, consists of : the Space Station with a mass ma = 306.752 metric tons located at

the C.M. of the system, an end mass, 1114 = 10 metric tons, deployed upward on a

10 km long, 2 mm diameter kevlar tether, a variable-g platform (ma) of 5 metric

tons that can crawl along the upward tether segment and a balancing mass, m! =

10 metric tons deployed downward on a 10-15 km tether of the same material and

shape as the upward tether. The tether length (£x) of tether no. 1 can be adjusted in such a way as to maintain the C.M. of the entire system at the desired

vertical location on board the Space Station. In the following simulation runs

of this quarterly report the variable-g platform (ma) is arbitrarily placed at a

distance of 1 km from the Space Station on the upper tether segment. The dynamic response of the system has been simulated during the station-keeping

phase. The first simulation has been run, without any passive and/or active

dampers, starting from a perturbed initial state of the system. The initial inplane libration angle of the system is 0.51° while the initial out-of-plane

angle is 0.42°. The initial lateral deflections of mass m2 (Space Station) and

ma (variable-g laboratory) with respect to the line through the end masses are

respectively as follows: en = -89.m, £j2 = 93.m (in plane components) ; CQI

 = ~

73.m, €02

 = -56.m (out-of-plane components); tt = 10500.m, £2 = 1000.m, £3 =

9000.m. The reason for having some numbers which are not perfectly round in the

initial conditions is that the initial conditions are supplied to the computer

code in terms of the masses' Cartesian coordinates. The Cartesian coordinates,

however, do not provide an immediate picture of the system configuration;
Page 15

Q

 m4 (End Mass)

m3 (G-Lab.)

m9 (SS)

O mi (Bal. Mass)

To Earth

Figure 2.3.1
Page 16

/

which is better represented by coordinates like 6, ip,e's and £'s.

The following set of plots, shown in Figure 2.3.2a-x, has been obtained by

simulating for 7200.sec the undamped response of the system due to. the perturbed

initial conditions previously described. The in-plane and out-of-plane libration angles are plotted in Figure 2.3.2a. The oscillation periods are Tp ~

3240.sec and Tv ~ 2810.sec respectively, which correspond to the undamped, free

oscillation angular frequency of \/3 fl and 2fl respectively, where fl = 1.119 x

10~3 rad/sec at 450 km of altitude. Figures 2.3.2b and c show the phase-planes

6—6 and <p-<f> respectively. These two plots clearly show the undamped response

of the in-plane and out-of-plane libration and the small effects due to the

lateral vibrations of the masses that are depicted by the small wiggles superimposed to the elliptical graphs. Figure 2.3.2d,e and f depict the tether lengths

of tethers no. 1, 2 and 3 respectively. The shape of the plots is due to the

superimposition of the longitudinal, lateral and librational frequencies that

all affect the tether lengths plus beating phenomena. The highest oscillation

frequency is the longitudinal frequency in tether no.2, the shortest tether and

therefore the stiffest. The in-plane components of the lateral displacements

€n and £12, respectively of mass ir>2 and ma, are shown in Figure 2.3.2g; while

the out-of-plane components «oi

 anc* €oa are shown in Figure 2.3.2h. Figure

2.3.21 shows the moduli of the lateral displacements (et) and (e2) ; the decrease in amplitude is due only to a beating phenomenon between the components

and is not due to damping. Figures 2.3.2J and k are the phase-planes ei-£i and

Cj-ea respectively. The decrease in amplitude is again due to a beating phenomenon. The tether tensions of tethers no. 1, 2 and 3 are . depicted in Figures

2.3.21, m and n respectively. The shapes of these plots are exactly like those

of the associated tether lengths since the tension is proportional to the tether

stretch, the tether lengths at rest are constant and there are no longitudinal
Page 17

dampers. Figure 2.3.2p shows the flight-direction (ACH), transverse (ACL) and

vertical component (ACV) respectively of the acceleration, at the C.M. of the

Space Station, induced by the transient dynamics of the system. The level of

acceleration, shown in the figure, is only due to the initial conditions and may

be theoretically reduced to zero by a perfect damping system. A large section

of the study of the 4-mass system will be devoted to the damping of the various

vibrational modes of the system and the effect of the external perturbations on

the acceleration level. Figure 2.3.2p clearly shows the contribution of the

various vibrational modes to the acceleration components. The acceleration components at the variable-g laboratory (mj) are shown in Figures 2.3.2q, r and s

respectively. It is clear from the figures that the flight-direction and transverse components are primarily due to the lateral oscillations of mass m3 while

the vertical component is affected by various vibrational modes. The vertical

component also shows the bias due to the 1 km distance from the system C.M.

Figure 2.3.2s t, u and v are the familiar SKYHOOK-type plots. In these plots

the shape of the system, at time intervals of 20 sec for 3600 sec, is depicted

as seen from the side (radial/in-plane) in Figure 2.3. 2t and from the front

(radial/out-of-plane) in Figure 2.3.2u. The scale of the abscissa is largely

expanded in order to show the otherwise undetectable oscillations. The overall

length of the system is also normalized. The mass numbering is the same adopted

in Figure 2.3.1 so that the Space Station is mass no. 2. . The Earth is therefore

on the side of mass no. 1 in the direction of the positive ordinates. Without

the expansion of the scale of the abscissa the former plots look like Figure

2.3.2v, that is the equal-scale side-view of the system plotted every 20 sec for

1000 sec only.

The first stage of damping is implemented into the system by adding three

longitudinal dampers in series with the three tether segments. Each damper is
Page 18

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tuned to the frequency of the associated, single degree of freedom, oscillating

system (associated tether plus the two masses at that tether's tips) and the

damping coefficient is selected in order to achieve a damping ratio of approximately 0.9. The dramatic positive effect of the longitudinal dampers upon the

longitudinal oscillatory modes is shown in Figure 2.3.3a-m. The time histories

of the libration angles 9 and <p and .the lateral displacements ei and £2 are

negligibly influenced by the longitudinal dampers and they are, therefore, very

similar to the respective plots of the previous simulation run. The tether

lengths of tethers no. 1, 2 and 3, for the system with longitudinal dampers, are

shown in Figure 2.3. 3a, b and c. These figures must be compared with Figures

2.3.2d, e and f respectively of the previous run. The longitudinal dampers

abate very effectively the higher frequency longitudinal oscillation modes but

are not effective against the longitudinal oscillation forced by lateral and/or

t

librational oscillations. Figures 2.3.3d, e and f, depicting the tension in

tethers no. 1, 2 and 3 respectively, must be compared to Figures 2.3.21, m and

n. Similar considerations to those pointed out for the tether lengths apply to

these tension plots. Figures 2.3.3g, h and i show the lengths of the longitudinal dampers associated to tethers no. 1, 2 and 3 respectively. Since each

longitudinal damper is tuned to the frequency of the associated tether, the

length of each longitudinal damper is equal to the elastic stretch of the associated tether. The acceleration components ACH (flight-direction), ACL (transverse) and ACV (vertical) on board the Space Station are shown in Figure 2.3.3J.

By comparing Figure 2 . 3 . 3 j with Figure 2.3.2p it is clear that the vertical

component of the acceleration is the most affected by the longitudinal dampers;

its amplitude is reduced and it becomes comparable to the other two components.

Figures 2.3.3k, t and m show the acceleration components on board the variable-g

laboratory. The flight-direction component and the transverse component are

almost identical to those shown in Figures 2.3.2q and r of the previous Simula-
Page 28

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tion. The vertical component is on the contrary positively affected by the

«

longitudinal dampers (compare with Figure 2.3.2s); the higher frequency components have been abated and the acceleration level is closer to the steady state

value. Since the side-view and the front-view of the system (SKYHOOK-type

plots) are negligibly affected by the longitudinal dampers, they are not shown

in this simulation run.

2.4 Damping Algorithms For In-Plane Librational And Lateral Oscillatory Modes

The damping algorithms for the in-plane librational and lateral oscillatory

modes have been devised by generalizing and modifying the damping algorithms

derived in Quarterly Report No. 2 for the 3-mass, 2-dimensional model.

In-Plane Libration Damping Algorithms

In Section 2.3 of Quarterly Report No. 2 it is shown that a 3-mass tethered

system librates about the system C.M. Since the libration frequency is independent of the mass and its distance from the system C.M., the statement above can

be generalized to an n-mass system. In order to maintain the Space Station (ma)

at the system C.M. while the in-plane libration damping algorithm is active, the

tether lengths must be controlled with a control law similar to equation (2.3.3)

of Quarterly Report No. 2. The in-plane libration damping algorithm for the 4-

mass tethered system is therefore formulated as follows:
Page 33

£ic = £01 (1-M)

£2= = £02 (l-ke0) (2.4.1)

where £01, £02 and £03 are the steady state values of the tether lengths.

In-Plane Lateral Oscillation Damping Algorithms

The damping of the in-plane lateral oscillatory modes is obtained with a

;

technique similar to that adopted in Section 2.3 of Quarterly Report No. 2.

Note that in the above mentioned report the signs in front of the terms in € in

equations (2.3.5) and (2.3.6) are misprinted and should be inverted. The damping of lateral oscillations is obtained by moving the two middle masses, m2 and

m3 (by controlling the tether lengths) in such a way as to . produce Coriolis

forces opposed to the lateral displacements of the middle masses (negative

work) . The tether control laws for the damping of the in-plane lateral oscillations are, when expressed in terms of tether length variations, as follows:

i

A£lc = keiicn

A£2c = ke i 2 ei2£o2/£o3-l<£iieii (2.4.2 )

where ktn and kei2 are the gains for the lateral oscillation in-plane components

€n and €12 respectively.

By assuming a linear superimposition of effects, equations (2.4.2) must be

added to equations (2.4.1) in order to obtain the tether length control laws

which provide in -plane libration and lateral oscillation damping. The final

result is as follows:
Page 34

£ic = £01 (l-k<,0 + kell€n/£0i)

£2c = £02 (1-M + kei2ei2/*03-keii«n//oa) (2.4.3)

£3= = £03 (1-M - keI2eI2£02/£03

2

)

Equations (2.4.3) are adopted in the next section as damping algorithms for the

in-plane libratory modes of the system.

2.5 Station-Keeping Simulation Runs With/Without Librational And Lateral Damping Algorithms

A third simulation run, similar to the second simulation of this report,

but with different initial conditions, has been made. In the third run the 4-

mass system is modeled like in simulation no. 2 (with longitudinal dampers

only). The third run, however, differs from the second because the initial

conditions of the third run are 90° out-of-phase with respect to run no. 2 (the

initial velocities are different from zero in run no. 3) . The initial conditions of simulation no. 3 are therefore as follows: 6 = 4.7 x 10~4

 deg/sec, <f> =

3.9 x 10'* deg/sec, £x = 10500.m, £2 = 1000.m, £3 = 9000.m, £di = 66.33 m, £d2 =

6.32 m, £d3 = 54.18 m (where £.&'s are the lengths of the longitudinal dampers) ,

e.n = -8.33 x 10~2 m/sec, eI2 = -3.38 x 10'2 m/sec, £0i = -6.83 x 10'2 m/sec, 102 =

-1.91 x 10"1 m/sec, all the other variables are initially set equal to zero.

Figures 2.5.1a-za show the dynamic response of the system over a time of 7200

sec. These plots are intended to be used for comparison with the next simulation run. Figure 2.5.la shows the in-plane and out-of-plane angles which are

almost unaffected by the damping provided by the longitudinal dampers, as can be

seen more clearly in Figures 2.5.1b and c. Figures 2.5. Id, e and f show the

tether tensions of tethers no. 1, 2 and 3 respectively, while Figures 2.5.1g, h
Page 35

and i show the length variations of the associated longitudinal dampers. Figures 2.5.1J and k depict the in-plane (ei) and out-of-plane (CQ) components of

the lateral deflections respectively, while Figure 2.5.l£ shows the moduli of

the lateral deflections. Figures 2.5.1m and n show the phase-plane e^-e^ and

ej-Cj. The inward spiralling shown in Figure 2.5.1m is not due to damping but

is the result of a beating phenomenon.. The tether tensions are shown in Figures

2.5. lp, q and r for tethers no. 1, 2 and 3 respectively. Figure 2.5.1s shows

the components of the acceleration on board the Space Station: ACH is the

flight-direction, ACL the transverse and ACV the vertical component. The acceleration components at the variable-g laboratory are shown in Figures 2.5.It, u

and v respectively. Figure 2.5.Iv clearly shows the steady state, vertical

acceleration bias due to the 1 km offset of the variable-g laboratory from the

system C.M. Figures 2.5.1w, x and y show the x,y and z components of the system

C.M. referred to the orbiting reference frame. Because the initial conditions

have non-zero total angular momentum, the x-component (flight-direction) of the

C.M. drifts, while the y-component (transverse) and the z-component (vertical)

oscillate. The system C.M. is following a slightly elliptic orbit and the orbital plane is also slightly oscillating. Figure 2.5.1z and Figure 2.5.1za show

the side and front-view respectively of the system at intervals, of 40 sec for

7200 sec. The scale of the abscissa is largely expanded in order to show the

various oscillatory modes. Since mass no. 1, the balancing mass, is shown at

the top of the figures, the Earth is on the side of the positive ordinates.

The fourth simulation has been run for 10,000 sec with the same initial

conditions of simulation no. 3. This time the in-plane librational and lateral

damping algorithms are activated in order to damp out the relevant oscillatory

modes. Figures 2.5.2a-za show the results of the simulation. Each figure must

be compared to the equivalent figure from the previous set of plots (Figures
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Page 46

2.5.1a-za). The activation of the in-plane librational and lateral dampers damp

out effectively the in-plane libration and the in-plane components of the lateral oscillatory modes as may be easily seen throughout the set of plots. The

out-of-plane dynamics are, on the contrary, negligibly affected by the dampers.

The out-of-plane dynamics are therefore responsible for the residual oscillations of the system both directly and through couplings. Figures .2.5.2m and n,

for example, show the phase-planes EI-CI and e2 -£2- The limit cycles in these

figures are due to the out-of-plane dynamics. Since the derivatives €1 and ej

change sign when Ci = 0 or £2 = 0 respectively, the two phase-planes have an

unusual shape due to the discontinuities of the derivatives at the zero crossing. The net result of the dampers can be seen in Figure 2.5.2s and Figures

2.5.2t, u and v. The librational/lateral dampers (at least with the current

algorithms) initially produce higher acceleration levels at the Space Station,

but after 10,000 sec the vertical (ACV) and flight-direction (ACH) components

are strongly reduced. The transverse component (ACL) depends upon the out-ofplane dynamics and is therefore negligibly affected by the current dampers. The

initial increase in the acceleration level at the Station will be investigated

more in the next reporting period with the aim of reducing such increase. The

acceleration level at the variable-g laboratory (m3) does not show any initial

amplitude increase with respect to simulation no. 3. The flight-direction (Figure 2.5.2t) and the vertical component (Figure 2.5.2v) are effectively damped

out at the end of the simulation; in particular the vertical component reaches

the steady state value consistent with the 1 km offset from the system C.M.

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longitudinal and librational/lateral dampers switched on. The shape of the constellation is plotted every 40 sec for 7200 sec. These two plots must be compared to Figures 2.5.1z and za respectively; the dramatic effect of the
Page 47

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Page 57

dampers upon the in-plane dynamics is evident from the figures.

2.6 Concluding Remarks

The newly developed computer code, for 3-dimensional, n-mass tethered systems has been utilized for simulating the free dynamics of a 4-mass tethered

system for micro-g/variable-g applications. A new predictor-corrector integrator has been used for the numerical integration of the model. This integrator

is up to 40% faster than the older IV-order Runge-Kutta with comparable accuracy. Simulation runs have been performed to show the contributions of the

various vibrational modes (librational, longitudinal and lateral) to the dynamics of the system and to the acceleration levels on board the Space Station and

the variable-g laboratory. The addition of longitudinal dampers, tuned to the

longitudinal vibrational modes, provides an effective damping of the longitudinal oscillatory modes. More sophisticated damping algorithms have also been

devised for damping the in-plane librational and lateral oscillations of the

system.
Page 58

3.0 PROBLEMS ENCOUNTERED DURING REPORTING PERIOD

None.

4.0 ACTIVITY PLANNED FOR THE NEXT REPORTING PERIOD

Further investigation and simulation efforts will be carried out on the

damping of the out-of-plane oscillatory modes. The modeling of external perturbations will be initiated with the final target of a reliable, accurate model of

the Jj perturbations, atmospheric drag and solar illumination that cause tether

expansions and contractions.