{NASA-CR-17875U) ANALYTICA L INVESTIGATIO N

OF THE DYNAMIC S OF TETHEBE D CCNSTELLATION S

IN EART H DEBIT, PHAS E 2 Quarterly Report,

22 Sep. - 21 Dec. 1985 (Smithsonian

Astrophysical Observatory) 53 p

Analytical Investigation of the Dynamics

of Tethered Constellations in Earth Orbit (Phase II)

N36-23629

Unclas

G3/18 16979

Contract NAS8-36606

Quarterly Report #3

For the period 22 September 1985 through 21 December 1985

Principal Investigator

Dr. Enrico Lorenzini

February 1986

Prepared for

National Aeronautics and Space Administration

Marshall Space Flight Center, Alabama 35812

Smithsonian Institution

Astrophysical Observatory

Cambridge, Massachusetts 02138

APR198 6

RECEIVED

HASASn FACILITY ,;

, ACCESS DEFT

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 Two-Dimensional Equations Of Motion For The N-Mass

System 5

2.2 Revised Two-Dimensional Equations Of Motion For The 3-

Mass System 12

2.3 Another Model For The N-Mass System. Cartesian Coordinate Equations Of Motion 15

2.4 The 3-Mass System. Cartesian Coordinate Equations Of

Motion 21

2.5 Generating Sinusoidal Acceleration Profiles 23

2.6 G-Tuning 32

2.7 Concluding Remarks 45

3.0 PROBLEMS ENCOUNTERED DURING THE REPORTING PERIOD 52

4.0 ACTIVITY PLANNED FOR THE NEXT REPORTING PERIOD 52
Analytical Investigation of the Dynamics

of Tethered Constellations in Earth Orbit (Phase II)

Contract NAS8-36606

Quarterly Report #3

For the period 22 September 1985 through 21 December 1985

Principal Investigator

Dr. Enrico Lorenzini

Co-Investigators

Mr. David A. Arnold

Dr. Mario D. Grossi

Dr. Gordon E. Gullahorn

February 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
Page 3

Abstract

This quarterly report deals with the development of a two-dimensional

analytical model that describes the dynamics of an n-mass vertical tethered

system. Two different approaches are described: in the first one the control

quantities are the independent variables while in the second one the Cartesian

coordinates of each mass expressed in the orbiting reference frame are the

independent variables. The latter model has been used in the 3-mass version to

simulate the dynamics of the tethered system in applications involving the

displacement of the middle mass along the tether. In particular the issues

related to reproducing predetermined acceleration profiles and "g-tuning" are

reported.

PRECEDING PAGE BLANK NOT FUMED
Figure Captions

Page 4

Figure 2.1.1

Figure 2.3.1

Figure 2.3.2

Figure 2.3.3

Figure 2.5.1 a-1

Figure 2.5.2 a-1

Figure 2.6.1 a-m

Figure 2.6.2 a-m

Geometry and generalized coordinates for the n-mass

tethered system.

Geometry and Cartesian coordinates for the n-mass

tethered system.

Schematic of the tension forces acting on a single

mass of an n-mass tethered system.

Geometrical relations between generalized coordinates

and Cartesian coordinates.

Dynamic response of the 3-mass tethered system for a

sinusoidal motion of the middle mass with a peak-to-peak

amplitude of 2000 m and an angular frequency of 10"3

rad/sec.

Same as in Figure 2.5.1 with a peak-to-peak amplitude of

the motion of the middle mass equal to 8000 m.

Dynamic response of the 3-mass tethered system during a

transfer maneuver of the middle mass designed for achieving

a predetermined, steady state g-level ("g-tuning"). The

control law is a half cycle sinusoid with a peak-to-peak

amplitude of 2000 m.

Same as in Figure 2.6.1 with a control law of different type.

A modified hyperbolic tangent is used to perform the transfer

maneuver instead of the half cycle sinusoid.
Page 5

1.0 INTRODUCTION

This is the third 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. It covers the period from 22

September 1985 through 21 December 1985.

2.0 TECHNICAL ACTIVITY DURING REPORTING PERIOD AND PROGRAM STATUS

2.1 Two-Dimensional Equations Of Motion For The N-Mass System

The Lagrangian formulation has been used to derive the two-dimensional

equations of motion for an n-mass tethered system. The equations of motion are

expressed in the orbiting reference frame formed by the local vertical (z) and

the local horizon (x) centered at the system C.M. The Lagrangian coordinates,

shown in Figure 2.1.1, are the tether lengths (£i) , the deflections («i) of each

mass from the line connecting the first mass to the last mass and the angle (6)

between that line and the local vertical. The assumptions are like those used

for the 3-mass system, namely: point masses, circular orbit of the C.M., second

order gravity potential expansion, elastic tethers. Following a procedure similar to the one in Quarterly Report #1 and referring to Figure 2.1.1, the kinetic

energy of the system is given by:

T = i> nu vxi (2.1.1)

2Z. I ii|

In equation (2.1.1) the velocity of the ith-mass is given by:
Page 6

Figure 2.1.1
Page 7

= v± + (Ox ?i) (2.1.2)

and since we have:

= VBI + ( x PBI) (2.1.3)

= 3 + Pi (2.1.4)

we can write equation (2.1.2) as follows:

= VB1 + (0 x a) + (0 + (I) x pBi (2.1.5)

so that:

|vn|a = [XBI + Oacostf + (^-0)zBi] -I- [zBi + Oa sin 9 - (6 - fl) xB1] (2 .1 .6)

Since the body reference frame is centered at the system C.M. we have:

= 0 (2.1.7)

Moreover, since the C.M. remains, through time, at the center of the orbiting

reference frame we also have:

(2.1.8)

By manipulating equation (2.1.6) we finally get:
Page 8

n

2Z-,m*ixBi

2(6 -fl) (xBiZBi - xB1zB1)} + | mtotn2a2 (2.1.9)

where the last term is the kinetic energy of the system concentrated at the C.M.

The potential energy of the system is given by:

Since:

where:

|n| = [xi2 + (a - Zi) 2]

V2

 (2.1.11)

We can therefore write the potential energy as:

ft

where:

(2.1.13)

= O2a

3

= ZBI cos 0 — xB1 sin 8 (2.1.14)

after algebraic manipulations and because of equations (2.1.7) we finally get:
Page 9

V = - ^ min 2(3co s

 20 - l)zBi2

 + Q2 (3sin*0 -l)xBi

2

-6 fi« sin 6 cos 0 xB12B1)| - mtotn

2

a

2

 (2.1.15)

In equation (2.1.15) the last term is the potential energy of the system when

the total mass is concentrated at the C.M. ; its value is constant for a circular orbit.

Assuming that eA « li and defining RI = mi/mtot

 we can

 write:

i

—/ ^

J=i

(2.1.16)

n-l

=/ ^ ^j ~ ZBCMI

and:

XDJ. = -

Xfl2

 ==

 Ci ~

(2.1.17)

XBCMI

xBn —
Page 10

where ZBCMI and XBCMI

 are t^ie co°i~dinates, in a body reference frame with the

origin at mi, of the system C.M. They can be expressed as follows:

(2.1.18)

n-2

=/_ Rj+i «j

where:

R<i n) = 2^ Rj (2.1.19)

Equations (2.1.16) and (2.1.17) relate the coordinates of the masses in body

axis to the Lagrangian coordinates.

The Lagrangian function (L) is readily obtained by subtracting the potential

energy from the kinetic energy so that the equations of motion are given by:

5L ,

= fi i = 1 ..... n

..... (2.1.20)

dt

 x

d_

dt

After many algebraic manipulations the final structure of the equations of motion is as follows:
Page 11

b/] + 2(0 -O) B « + (§ - d) C e = Q£ /mtot

D[e + ge] + 2 (6 - to) E t + (6 + d) F / = Q£ /mtot (2.1.21)

2(0-0), 3n2sin0cos0, 3Q2 (sin

 20 - cos *6) ]* + tBt + /Le = Q0/mtot

In equations (2.1.21) t, i, t are column vectors of dimension n-1 and e, «, «

are also column vectors of dimension n-1 with the last row equal to zero. The

matrices A, B, C, D, E, E, H, L are non dimensional matrices and G is on the

contrary dimensional, being a function of e, e, t, i. Their complex expressions

are not reported in this quarterly report for the sake of brevity. The coefficients in equations (2.1.21) are given by:

b = 0« (1-3 cos20) - (0'-0)2

d = 302sin0cos0 (2.1.22)

g = O2 (1-3 sin20) - (0-fi)2

The equations of motion (2.1.21) together with (2.1.22) will be fully developed

in the next section for the case n=3 . Note that equations (2.1.22) are a set of

2 (n-1) equations while in general the degrees of freedom of n masses in a twodimensional space are 2n . In our case, however, the motion of the system C.M.

is constrained so that the total degrees of freedom of the system are 2n-2.
Page 12

2.2 Revised Two-Dimensional Equations Of Motion For The 3-Mass System

From the generalized case of the previous section we derive in this section

the equations of motion for a 3 -mass system. The tedious computation of the

matrix components in equation (2.1.21) is not reported. The equations of motion

finally result in:

-Ri) [£\ + b£i] + RXR3 [£2 ••• b£2] - 2 (0-O)RiR2e- (0-d)RiR2e = Q£ /mtot

R3(1-R3)[£*2 + b£2] + RiR3 [£\ + b£x] + 2(^-0)R2R3e +(^-d)R2R3c = Q£ /mtot

R2(1-R3) [€ + ge] - 2(^-fl)R2v - (S+d)R2v = Qe/mtot (2.1.23)

C/i+v] +R3£2 C/a-v]

2

+R3£2 [£2-v] -R2

ot

where b, d and g are given by equations (2.1.22) while v and v are given by:

v = R3£2 - Ri*!

v = R3^2 - Rx^ (2.1.24)

The generalized forces had already been derived in Quarterly Report #1; they

are given by:

Q£i = -T.

Q = -T2 (2.1.25)

«-2

Qe = -eCTi/ti + T2/£2)

Qe = 0
Page 13

A comparison of equations (2.1.23) with equations (2.1.15) of Quarterly Report

#1, taking into account the different notation, shows that some inaccuracies

were present in the latter equations. However those inaccuracies involved terms

which multiplied v or v. v and v represent the distance of m2 from the system

C.M. and its rate of change respectively. In all the simulations reported in

Quarterly Reports #1 and #2 the values of v and v were zero or close to it so

that the results shown in those reports are correct. As a confirmation of this

we ran with the new model one of the case reported in Quarterly Report #2

obtaining the same results. When simulating the middle mass travelling along

the tether the incorrect terms are, on the contrary, important and the use of

the new model is therefore mandatory.

In equations (2.1.23) the independent variables are the controlled or the

observed parameters. To associate control laws to the system equations is

therefore a straight forward process. Equations (2.1.23), however, have the

disadvantage of being non-normalized. In matrix form the equations (2.1.23) can

be written:

A x = b (2.1.26)

where x = [0, €, li, Ij]"1, A is a symmetric 4x4 matrix and b = [bj., bj, t>3,

b4]

T. The numerical integration of equations (2.1.26) implies the inversion of

matrix A. Even if it is not a problem for a 4x4 matrix, the inversion becomes

more and more time consuming for bigger matrices (greater number of masses).

For this reason a different model in Cartesian coordinates has been derived, as

shown in the next section. For the sake of completeness the components of

matrices A and b are given below:
Page 14

an = Ri

a22 = R2(1-R2)

a33 = Ri(l-Ri)

a44 = R3(1-R3)

au = a21 = -R2v (2.1.27)

ai4

 = a4i = R2R3£

a2a = a32 = 0

aJ4 = a42 = 0

+R3£2[£2-v] -R2(1-R2) e

- Ra(l-R2 )g e + 2(^-fl)R2 v + dR2v (2.1.28)

b4 = Q£ /mtot - bCRaJl-RaJfj+RiRjfi] - 2(0-n)R2R3 e + dR2R3€

Equations (2.1.27) and (2.1.28) are complemented by equations (2.1.25), (2.1.24)

and (2.1.22) .
Page 15

2.3 Another Model For The N-Mass System. Cartesian Coordinate Equations Of

Motion

The model derived in the previous section is very convenient from the control viewpoint since the variables of the system equations are the control parameters. However, because of the difficulty of inverting large matrices, this

model's numerical integration is not efficient. In this section, we will therefore develop the equations of motion for a two-dimensional, n-mass tethered

system in Cartesian coordinates relative to the orbiting reference frame. The

equations in this form can be directly integrated without matrix inversion; on

the other end the relations between the system variables and the controlled/observed variables are much more complex.

Referring to Figure 2.3.1 let us define the radius vector of the generic

i^-mass as follows:

rt = a + pt (2.3.1)

where:

Pi = Xi I + zi k

(2.3.2)

a = — ak

In the case of 0 = constant we can write the inertial acceleration of the 1thmass as follows:

*ii = ?i + 20 x pi + n x (n x ?±) (2.3.3)
Page 16

Figure 2.3.1
Page 17

This equation can also be expressed as:

= Pi + 20 x ?i + 5 (n-?i) - rjfl l (2.3.4)

Since in the two dimensional case fi is always perpendicular to ?i the third term

in equation (2.3.4) is equal to zero. By using equations (2.3.2) we finally

have:

rn = [xi - 2flzi - fl2Xi]l + [zj. + 2Qxi - fl2 (zi-a)]k (2.3.5)

The equation of motion for the ith-mass is therefore:

= F,i -f FTi + Fpi (2.3.6)

In equation (2.3.6) Fgl denotes the gravity force acting upon the i^-mass while

FTI is the net force due to tether tension acting upon the same mass and Fpl is

the perturbation force.

Without limiting the generality of the model we assume Fpi = 0. The gravitational force is obtained from the potential energy as follows:

Fgl = -V Vi (2.3.7)

where Vi can be readily derived from equation (2.1.12) obtaining:

(2.3.8)

Substituting in equation (2.3.7) and performing the derivatives we get:
Page 18

(2.3.9)

The two scalar equations of motion for the i^-mass can be finally written as

follows :

Xi - 2fl ii = Fixi/mi

(2.3.10)

Zi - 302 Zi + 2O Xi =

These equations are usually called the Hill's equations.

Referring to Figure 2.3.2 the force FTi is given by:

FTI = -[Txi + Tx.i-i]! - [Txi - T^i-ijk (2.3.11)

where Txi, Txi are the components of the tension TI in the tether connecting

the i^-mass to the i+l^-mass; they are respectively given by:

(2.3.12)

Tml = Ti sin0i = Ti — ^ 5^_ ^

Referring to Quarterly Report #1 the tension TI is given by:

(2.3.13)

In equation (2.3.13), £1 is the geometric distance between the i^-mass and the

i + l^-mass, tci is the controlled length of that tether segment, t^t the length
Page 19

of the associated longitudinal damper and £tl the tether stretch. The length

£ct can follow a prescribed control law if that particular tether segment is

actually controlled or it is the tether length at rest if that tether segment is

uncontrolled. The length £dl is obtained by solving the following equation:

r^ <*i - *di - /d) - ,— *di (2.3.14)

kdi kdl

Equation (2.3.14) is the generalization of equation (2.2.5) in Quarterly Report

#2. Kdi and Kdi are the stiffness and damping coefficient of the ith-longitudinal damper respectively.

We must now derive the expressions that relate the system quantities

(xi xn, Zj. zn, Xi,...xn, Zj. zn) to the output quantities (9,

£ii---.£n-i i «!,... en-j, ii t..., tn-i, «i «n-2) • In this report we will

derive the expressions that relate the variables while the first derivatives,

when necessary, are numerically computed in the computer program. Referring to

Figure 2.3.3 we have:

-1 fXj - Xn)

JZ! - Zn|

(2.3.15)

The lateral displacement £1 is derived by computing the coordinates of the point

of intersection

 ;between the straight line through mi and mn and the straight

line perpendicular to it passing through mi. The equations of the above mentioned lines are respectively given by:

z — Zi = — (x — Xi)

 : i .

— ^n ^1 / \ I** o -i f-\

^ij-fl ~~ ^~^~^~~~~~~ \ ^ "i"*1!/ I ^ . o . -Lo I
Page 20

Figure 2.3.2

Figure 2.3.3
Page 21

The solution of equations (2.3.16) are the coordinates of the point of intersection xci, zci. After some manipulations we get:

Xoi = [Xl - Xt+l tg*0 + (Zt+1 -Zi)tg 0)]/(l-t,«0)

(2.3.17)

So that finally ei is given by:

- xcl)

2

 + (zi+i - zci)*]1/2'sign (xi+1 - xci) (2.3.18)

Equation (2.3.18) and equations (2.3.17) provide the relations between system

variables and output variables.

2.4 The 3-Mass System. Cartesian Coordinate Equations Of Motion

The equations derived in the previous section for an n-mass system are

reported, in this section, for a 3 -mass system. The equations of motion become

as follows:

Xl = 20 Zl + Ti (xaXj = 20 z2 + T3 (x3-x2)/(mj£j) + TI (xi-xa)/(ma£i)

x3 = 2nz3 + Ta (xa-x3)/(m3£a)

zx = 3n2z1-20x1 + ^(za-zO/Cm^O (2.4.1)

za = 3O2za-2nx2 + T2(z3-za)/(mJ£2)+T1(zi-za)/(ma£1)

z3 = 3O2z3-2Ox3 + T3(zj-z3)/(m3£j)
Page 22

where:

(2.4.2)

The relations between system variables and output variables become:

,-xa)« + (z3-Z3)*] (2.4.3)

2-xci)2 + (zj-zcl)

2

] 'sign(x2-xcl)

where:

+ z2 , (2.4.4)

Equations (2.4.1) through (2.4.4) have been numerically integrated in a newly

developed computer code to obtain the results illustrated in the following sections .
Page 23

2.5 Generating Sinusoidal Acceleration Profiles

A unique feature of a 3-mass, vertical, tethered system is the capability

of providing a variable-g level in the middle platform by simply moving the

platform along the tether. The middle platform experiences a vertical acceleration, linearly dependent upon the distance from the system C.M., given in g-unit

by the following formula:

av = 9.81 x 3O2£C.M. (2.5.1)

By controlling the distance £C.M. it is possible, in principle, to produce a

desired vertical acceleration profile. Actually the scenario is more complex:

any displacement of the middle mass from a rest condition generates a transient

response of the tethered system. Namely, the motion of the middle mass induces

longitudinal oscillations of the elastic tethers and its non-zero velocity interacts with the orbital rate producing a Coriolis force that deflects, away

from the vertical, the acceleration measured at the middle platform. During the

acceleration or deceleration phase, moreover, the non-zero acceleration of the

platform is added to the gravity gradient acceleration. The acceleration profile obtained is therefore a result of many parameters, its departure from the

nominal profile depending primarily upon the control laws adopted and the effectiveness of the dampers.

The first case that we analyze is the generation of sinusoidal acceleration

profiles with the middle mass starting from the system C.M. and moving in between the upper end mass and the initial C.M. position. The Space Station mass

adopted in the following simulations is consistent with the post-IOC phase of

the Space Station program. The other parameters of the micro-g/variable-g labo-
Page 24

ratory are not intended to be definitive: they provide a possible configuration

of the system that is not based upon optimization criteria. The system parameters are as follows:

Space Station Mass (mi): 3.06752 x 10s kg

Middle Mass (m2) : 5 x 103 kg

End Mass (m3) : 104 kg (2.5.2)

Tether Length: 10 km

Orbital Height at the

System C.M.: 450 km

The tether is a 2. mm diameter kevlar tether. In these particular simulations

the longitudinal dampers are non-adaptative; they are tuned respectively to the

longitudinal frequency of the associated tether segments at their initial tether

length. The angular frequenices are given by:

(2.5.3)

where £clo and £C20 are the initial, controlled tether lengths for tether #1 and

tether #2 respectively. The damping coefficient in both the longitudinal dampers is assumed equal to .9. The rotational (0) and lateral (e) active damping

algorithms are operated according to the description given in section 2.3 and

2.4 of Quarterly Report #2.

The control laws adopted for the sinusoidal acceleration variation are as

follows:
Page 25

£d = £cio + A£c 1 + sin (wct + - ff)

£CJ = £cao - A£c 1 + sin (wct + - JT) (2.5.4)

In equations (2.5.4) A£c is the amplitude of the sinusoidal displacement so that

the peak-to-peak displacement is 2A£C and wc is the angular frequency. Neglecting effects such as tether elasticity and lateral displacement of the mass mj,

the resulting vertical acceleration measured at the middle platform, in zero

order approximation, is given by:

av = 3n»£c.M. - A£cwc sin (wct + — TT) (2.5.5)

Assuming that the position of the C.M. does not move appreciably during the

motion of the middle platform (as is the case in zero order approximation) we

have:

£C.M. * A£c |l + sin (w0t + 4 JT) 1 (2.5.6)

Finally we get:

av = 3n«A£c + A£c (3fJ2 - u>l) sin (wct + - ir) (2.5.7)

£i

The resulting vertical acceleration variation is therefore dependent upon both Q

and wc. The point of zero acceleration is no longer the system C.M. but occurs

according to the following formula:

wct = sin'1 [l/(wc/3Q2 - 1)] - - TT + k2ir k = 1,2,... (2.5.8)
Page 26

I f we < O then the second term in equation (2.5.5) is negligible. Since the

effects of tether elasticity and Coriolis acceleration are also smaller for low

values of wc, the resulting acceleration variations of ma are closer to nominal.

In the following set of simulations we assume wc = 10"3 rad/sec while the

effect of a different amplitude A£c is explored. The first group of plots shown

in Figure 2.5.1 a-1 is relevant to a peak-to-peak displacement 2A£C = 2000 m.

All the dampers are active. The simulation has been run for 11000 sec which is

approximately 2 orbits. Figure 2.5.la shows the sinusoidal length variation of

tether #1 vs. time. Figure 2.5.1b represents the longitudinal damper length

variation for the same tether. Figure 2.5.1c is the in-plane (6) angle vs. time

while Figure 2.5.Id is the phase plane 6-6. Figure 2.5.1e is the phase plane ei where e is the lateral displacement of mj from the line through mi and ms.

These last two figures show that the tethered system, after dissipating the

initial transient motion, follows a steady state limit cycle sustained by the

motion of nij. Figure 2.5.If shows the components of the acceleration measured

at the middle platform. The vertical component (av) is sinusoidal in shape but,

as mentioned before, the zero-g acceleration point is no longer at the crossing

of the system C.M. Because of the middle mass self acceleration the vertical

component (av) overshoots towards the positive values (acceleration directed

downwards). Note also that a much smaller, but not negligible, horizontal component (ah) appears. It is related to the Coriolis force. Figure 2.5.1g is an

isometric, polar diagram of the acceleration. The vector connecting the 0,0

point to any point along the curve gives the actual orientation and modulus of

the acceleration. After the transient phase is over the acceleration vector

rotates steadily around the C.M. of the middle platform at the angular frequency

we. Figure 2.5.In is the isometric side view trajectory of the middle mass with

respect to the system C.M. Negative numbers in the z axis meaning upwards. For
Page 27

3400

2200 -

400 -

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

TIME (Sec)

Figure 2.5.1af

Figure 2.5.1b

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

TIME (Sec)
Page 28

I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I ! I I ! I I I I I I I I I ! I I I I I I ! I I I0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 11000

TIME (Sec) RoVat. + Longit. Dampers

Figure 2.5.let

Figure 2.5.Id

.0003 -

.0002 -

0004 M-J i i i I [ i I I I | I I I I | I I I I [ I I I I [ I I I I | I I I I | I I I I | i I I I j I I I I [

-.0004 -

-.35 -.3 -.35 -.2 -.15 -.1 -.05 0 .05

THETA (Deg) Phase Plane

.2 .25
Page 29

5. .oi -

tn

a.

-.01 -

-.03 -

-30 -20 -10 0 10

EPS1 (M) Epsi-EpsiD Phase Plane

20 30 40

Figure 2.5.let

Figure 2.5.If

.002 -

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

TIME (SEC)
Page 30

.003

tn

^-.002

5

o

^-.004

N-.006

S

-.008

-.01

T—T—1—i—i—i—i—i—i—i—I—i—i—i—i—i—r ~i—i—i—i—i—r

-.008 -.006 -.004 -.002 0 .002

M2 HORIZ. ACCEL..ACH (M/Sec"2)

.004 .006 .008

Figure 2.5.1gt

Figure 2.5.1H

200

0

-200

-400

-600

-800

-1000

-1200

-1400

-1600

-1800

-2000

-ppnn

i i | i i i | i i i | i i i | i i i | i i i | i i i | i i i | i i i | i i i | i i i | i i i | iii | ' > ' | ' ' ' | ' ±

-_ START -

- i ^

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r II i

1

-

-

'-

-

^

-

-

-

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i v \l -

 1

r i l i .. I i i • 1 i i i I i i i I i i i 1 i i i 1 i i i I . i . l i i i 1 i i i 1 i i i 1 i i i 1 i i i 1 i i i 1 i T

-1400 -1200 -1000 -800 -600 -400 -200 0 200 400 600 800 1000 1200 1400

X2 (M) M8 Trajectory Side View, Deployment
Page 31

405

400 -

350

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

Time (Sec)

Figure 2.5.1i-tFigure 2.5.11

366 -

359 -

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

Time (Sec)
Page 32

completeness Figures 2.5.1 i and 1 show the tension in tethers #1 and §2 respectively.

A second case with a peak-to-peak tether length equal to 8000m is shown in

Figure 2.5.2 a-1. Comments similar to those for Figure 2.5.1 hold true in this

case. The shape of the trajectory side view of mass mj is less symmetrical with

respect to .the local vertical; in general, however, the behavior is qualitatively very similar to the previous case. Quantitatively the acceleration components show a dependence with A£e so that they are 4 times bigger than in the

previous simulation.

2.6 G-Tuning

The possibility of varying the acceleration level in the middle platform

provides a "g-tuning" capability. This means that the middle platform is moved

from the zero-g point of the tethered system and is placed at the appropriate

distance in order to have the requested g-level according to equation (2.5.1).

Two different control laws are shown here. They are both open loop control laws

in the sense that the acceleration in the middle mass is not fed back into the

control system. The final acceleration value, therefore, depends only upon the

final distance (^C.M.) achieved. The first control law adopted is a sinusoidal

control law, similar to the one illustrated in the previous section, that has

been truncated after half a cycle. In Figure 2.6.1a-m the dynamic response of

the system is shown in a case with a peak-to-peak displacement of the middle

platform equal to 2000 m. Figure 2.6.la depicts the tether length for tether #1

vs. time: the sinusoidal control law is clearly visible. Figure 2.6.Ib is the

length variation of the longitudinal damper. Further comments on the figure
Page 33

8000 -

0 h- i I t i i i I i i i i I i i i i I

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

TIME (Sec)

Figure 2.5.2a+ „. _ _ .,

Figure 2.5.2b

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

TIME (Sec)
Page 34

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

TIME (Sec) Rotat. + Longit. Dampers

Figure 2.5.2C+

Figure 2.5.2d .

Y

.004

-3 - Z -1 0 1

THETA (Deg) Phase Plane
Page 35

3 H-l I I | I I I I | I I I I |

-.4 r-i i i I i i i i I i i i i I i i i i I i i i i I i i i i I i i i i I i i i i I i i i i I i i i i I i i i i I i i i i I250 -200 -150 -100 -50 0 50 100 150 200 250 300

EPSI (M) Epsi-EpsiD Phase Plane

Figure 2.5.2et

Figure 2.5.2f

0 1000 2000 3000 4000 5000 6000 7000 8000

TIME (SEC)

9000 10000 11000
Page 36

01 I l | l l l l | I l I l | I I l I | l I l l | I l l l | i i i i | i i i i | l l l l | I I l I | I I l l | l i i i | i i i i | i

.005

TM-.005

S -.01

5

S -.03

"-.025

-.03

-.035 -

-.04 i i i -I i i i i i i

-.03 -.035 -.02 -.015 -.01 -.005 0 .005 .01

M2 HORIZ. ACCEL..ACH (M/Sec"2)

Figure 2.5.2g+

.015 .02 .025 .03

Figure 2.5.2H

,START

-1000

-2000

-3000

-5000 -

-6000 -

-7000 -

-8000 Hi I i l i i i i I l I ~

. I I I I I I I I I I I I I I I I I I I I I 1 I I I I I I I I I I I ! I I I I I I I I I I I I I I I I I I I I I

-5000 -4000 -3000 -2000 -1000 0 1000 2000 3000 4000 5000

X3 (M) M2 Trajectory Side View. Deployment
Page 37

540 -

530 -

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

Time (Sec)

Figure 2.5.21-tFigure 2.5.2£^

390 -

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

Time (Sec)
Page 38

will be provided later on. Figures 2.6.1c and d are the in-plane angle vs. time

and the phase plane 0-6 respectively. These two figures show that after 7200

sec there is still a small residual in-plane oscillation decreasing to zero.

Similar comments are applicable to Figure 2.6.1e and f which represent the lateral deflection e and the phase plane e-e respectively. Figure 2.6.1g shows the

acceleration components, along the orbiting reference frame, measured at the

middle mass. In this "g-tuning" simulation the horizontal component disappears

after the transient phase is over. The vertical component is however experiencing a steep variation at the time when the final distance from the C.M. has been

reached because of the non-zero acceleration provided by the control law at that

point. This effect accounts for the similar steep variation in the longitudinal

damper's length. The system, therefore, needs additional time to reach a steady

state acceleration condition. Figure 2.6.1h shows the isometric, polar diagram

of the acceleration; it stresses again the negative effect of using a control

law (sinusoidal) that has non-zero acceleration at the beginning and at the end.

Figure 2.6.11 is an isometric plot of the trajectory side view of mass ma with

the zero point located at the initial position of the system C.M. Figure 2.6.1&

and m show the tension in tether #1 and tether #2 respectively. The larger

oscillations in tether #2 tension are due to the expanded plotting scale. The

results obtained by using the sinusoidal control law for "g-tuning" prompted us

to adopt a control law with zero acceleration at the end of the maneuver. The

new control law is a modified hyperbolic tangent; the tether lengths are controlled according to:

£x = £10 + A£c (1 - e-*) tanh (at)

(2.6.1)

£2 = £JO - A£c (1-e-^) tanh (at)
Page 39

2200

3000

1800

^1600

o

z

1400

5-

U

2

U

-? 1200

1000

BOO

600 -

400 -

-J—I—I—I—I—I—I—I—I I i i

0 1000 2000 3000 4000 5000 6000 7000

TIME (Sec)

Figure 2.6.1at

Figure 2.6.Ib

1000 2000 3000 4000

TIME (Sec)

5000 6000 7000
Page 40

.25 -

1000 2000 3000 4000 5000

TIME (Sec) Rotat. + Longit. Dampers

Figure 2.6.let

6000 7000

Figure 2.6.Id.

.0002

.0001

^ o

(/}

oo

S.oooi

u

E-

-.0002

-.0003

-.1 -.05 .05 .1 .15

THETA (Deg) Phase Plane

.2 .25
Page 41

1000 2000 3000 4000 5000 6000 7000

-40

Figure 2.6.let

Figure 2.6.If,

-35 -30 -25 -20 -15 -10 -5

EPSI (M) Epsi-EpsiD Phase Plane
Page 42

.002

o

a;

if}

-.002

EZ

t:

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S-.004

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a

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N-.006

S

-.008 -

-.01

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VERT

J I I I I I L_J 1 I I I I I I I I

0 1000 2000

Figure 2.6.1gt

3000 4000

TIME (SEC)

5000 6000 7000

Figure 2.6.1H,

.002

CM

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>-.002

>

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_j

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u

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u

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~i—i—i—I—i—i—i—I—i—i—i—I—i—r

START

END

_L I I I I I I I i i i I 1 1 L.

-.008 -.006 -.004 -.002 0 .002

M2 HORIZ. ACCEL..ACH (M/Sec"2)

.004 .006 .008
Page 43

-200

-400

-600

. -BOO

s

^1000

N

N

-1200

-1400

-1600

-1800

-2000 -

START

-2200 II I I I I ' I ' ' I I ' I ' I I I I I I ' ' I ' ' ' I ' ' ' I ' ' ' I ' ' ' I ' ' ' I ' ' ' I ' ' ' I ' ' ' I ' ' ' I ' l[

-1400 -1200 -1000 -800 -600 -400 -200 0 200 400 600 800 1000 1200 1400

X2 (M) M2 Trajectory Side View, Deployment

Figure 2.6.lit

Figure 2.6.U

400 -

355 -

350

1000 2000 3000 4000

Time (Sec)

5000 6000 7000
Page 44

n r r i i i \ i i i i i i i i i i i i i i i i i i i i i i r

o

o

o

o

o

o

CO

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o

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in

o

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I I I I I I I I I I I I I I I I I I I I I I

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(N) 2

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Page 45

The two constants a and ft set the slope of the curve, ft having been introduced

in particular to slow down the acceleration phase (initial phase) of the control

law. Results from a simulation with 2A£C = 2000 m, a = 1/2000 sec"1, ft = 1/1000

sec"1 are shown in Figure 2.6.2a-m. The sequence of plots is equivalent to the

one previously illustrated. Note that the transient phase at the end of the

maneuver has disappeared. In the same 7200 sec time frame the system reaches a

quieter state than in the previous simulation. The initial transient phase is

still there; it is partially due to the imperfect initial conditions. Figure

2.6.2g shows the acceleration components in the local orbiting reference frame

at the middle platform. Note that at the end of the maneuver the horizontal

acceleration component has completely disappeared while the vertical component

has reached its steady state value.

2.7 Concluding Remarks

The newly developed Cartesian coordinate model provides a very convenient

tool for analyzing multi-mass tethered systems. The new model of the 3-mass

tethered constellation has been used to perform simulations which show the capabilities and the limitations of the tether system with respect to the reproduction of predetermined acceleration profiles on board the middle platform. Sinusoidal vertical acceleration variations of the desired frequency and amplitude

can be obtained at the expense of a smaller horizontal acceleration component.

This component deflects the total acceleration from the local vertical during

the sinusoidal motion of the middle platform.

The "g-tuning" capability has been also demonstrated. An open loop control

law that provides a satisfactory dynamic response has been devised. The middle
Page 46

2400

2200 -

1000 2000 3000 4000 5000 6000 7000

Figure 2.6.2at

Figure 2.6.2b

202 (Hi—I—i—i—m—I—i—i—i—i—I—i—i—i

1.78 b=l

1000 2000 3000 4000

TIME (Sec)

5000 6000 7000
Page 47

0 1000 2000 3000 4000 5000 6000 7000

TIME (Sec) Rotat. + Longit. Dampers

Figure 2.6.2c

Figure 2.6.2d.

.0001 -

5E-5 -

on

&

-5E-5 -

-.0001 -

.02 .04 .06 .08 .1

THETA (Deg) Phase Plane

.13 .14
Page 48

o -

-25 -

o 1000 2000

Figure 2.6.2e+

3000 4000

TIME (SEC)

5000 6000 7000

Figure 2.6.2f

015 M | i i i | i i i | i i i | i i i | i i i | i i i | i i i | i i i [ i i i | i i i [ i i i | i i i j i

035

 rr I i i i I i i i I

-.03 -

-26 -24 -22 -20 -18 -16 -14 -12 -10 -8

EPSI (M) Epsi-EpsiD Phase Plane

-6 -4 -2
Page 49

.004 |—r

.003

w

§-.002

0-

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u

<-.004

CM

-.006 -

\

HOR

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i i I I i i i I _ I _ I - 1 - 1 _ I _ I I I I I I i i i i I I i i I I _ I i

0 1000 2000

Figure 2.6.2gt

3000 4000 5000 6000 7000

TIME (SEC)

Figure 2.6.2h

I ' ' ' I i—i—i—i—rI ' ' ' I T

.004

.002

CM

*

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u

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H--004

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START

END

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-.008 -.006 -.004 -.002 0 .002

M2 HORIZ. ACCEL..ACH (M/Sec"2)

.004 .006 .008
Page 50

200

0

-200

-400

-600

-BOO

-1200

-1400

-1600 -

-1800 -

-2000 -

-8200

400

i i I I I I I I I M

START

i I i i i I i i i I i i i I i i i I i i i I I i i i I i i i I i i i I i i i I i i

-1400 -1300 -1000 -BOO -600 -400 -200 0 200 400 600 800 1000 1200 1400

X2 (M) M2 Trajectory Side View, Deployment

Figure 2.6.2it

Figure 2.6.2£ ,

395 -

1000 2000 3000 4000

Time (Sec)

5000 6000 7000
Page 51

i i i i r i i r i i i i I i i i i i i r i I i r

I , , , , I

CO CO

co c*o

CO

CO

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co

C\J

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(N) Z
Page 52

mass is moved from the system C.M. to the desired location along the tether

according to a modified hyperbolic tangent control law. Time constants for the

control law have been derived in order to have a satisfactory transient response .

3.0 PROBLEMS ENCOUNTERED DURING THE REPORTING PERIOD

None

4.0 ACTIVITY PLANNED FOR THE NEXT REPORTING PERIOD

During the next reporting period further studies will be performed on the

"g-tuning." We will investigate the case of displacing the middle mass from a

position near the Space Station to the zero-g point in order to achieve a zero

acceleration condition. We will also investigate the effect, upon the overall

dynamics of the constellation, of a fast displacement of the middle mass from

one tether tip to the other.