The simplest mathematical model used for the simulation of TSS motion is considered in order to review the dynamics of tethers. This model is discussed. Also reviewed are elasticity effects, perturbation sources, orbit eccentricity and natural frequencies. Tethered systems provide a unique opportunity for skilled measurement activities in space. Some of the experiments envisaged require the measurement of very small mechanical quantities. This implies that the level of dynamical noise on instruments output be low or that system response to excitation, either external or internal, be sufficiently known. In this respect, the first TSS flights will be very useful, but much work will be needed in order to have reliable estimates of structural damping in different future systems

Bergamaschi, S.

English

NASA Technical Reports Server

N86-28411
A REVIEW OF TETHER INDUCED DYNAMICAL FEATURES
Silvio Bergamaschi
Institute of Applied Mechanics
PRECEDING PAGE BUNK WOT'FILMED
103
https://ntrs.nasa.gov/search.jsp?R=19860018939 2020-03-20T14:14:39+00:00Z
INTRODUCTION
A Calk 'in the general field of tether fundamen-
tals cannot be started without a mention to the S.A.0.report of
September 74 (Owhere the "Skyhook" (this was the first name g^_
ven to the TSS) was presented. In fact, as pointed out in {2}
and (3), it is true that tethers in space have been conceived
since the last years of the 19-th century, but it is also true
that until the 70-s they had been part of more or less visiona-
ry concepts. On the contrary, in {1 } the idea was put on sound
engineering grounds and the compatibility .with the Space Shut-
tle was clearly shown. Consequently, investigations of possible
uses of tethers in space were undertaken and in the succeeding
years ^ the peculiarities of TSS motion were investigated extensi
vely, so that at present it can be said that, if elastic effects
are ignored,' TSS dynamics is sufficiently well known. Further,
it can also be said that the experience acquired in past and
present investigations is sufficient to allow the simulation of
the motion of more complex tethered systems with a reasonable d£
gree of accuracy. <
SOME FEATURES OF TETHER DYNAMICS
In order to make a review of the most peculiar features
of tether dynamics let us consider the simplest mathematical mo
del having been used for the simulation of TSS motion. However,
as it will appear from the assumptions below, the sa'me model is
also useful to investigate a larger class of tethered platforms.
104
iln fact, most of the systems proposed so far for future appli-i'
cations have mechanical features (mass, inertia moments, or-
bit) and operational requirements which are largely different
from those of the first.satellite, but the relevant environmeii
tal forces (gravity gradient, Coriolis during manoeuvres,eccen
tricity excitation,etc.) will be the same.
Let us assume that:
the system is composed by a massive main body (space station
or other) in circular orbit and by a smaller platform conne£
ted to it by means of a variable length tether; moreoverjthe
mass of the tether is negligible;
the Earth is spherical and homogenous, so that oblateness
and higher order gravitational perturbations are ignored;
- non gravitational forces, as aerodynamic drag or other ela-
sticity effects are very small so that they can be neglected;
- 'the platform is a point mass.
. In this case, it is well known that the system
has two stable equilibrium configurations, aligned with the l£
cal vertical, so that the classical methods of mechanical vibra_
tions can be used to investigate the motion of the platform in
the neighbourhood of them. Therefore, if the gravitational e-
nergy is reduced to a quadratic form with respect to the ratio
I/a of the tether length to the semimajor axis of the station
orbit arid if the small amplitude approximation is also made,the
dynamical equations can be written as:
105
0 + 2 - 6 + Sn'e£.
SL ,
+ 2 — <J> + 4n <]
Jt
(1)
(2)
where:
- 9 is the offset angle from local vertical in the orbit
plane
- $ is the out of plane offset angle
- n is the orbit mean motion
- the dots mean differentiation with respect to time.
First, let us consider station keeping condi-
tions, where tether length is constant. From (1) and (2)it is im
mediate to verify that motion is stable, consisting of two un-
coupled librations with constant amplitudes. It can be seen
from fig. 1 that tether periods are slowly increasing functions
of the altitude h.
T(m)'
60-
56-
52-
48-
44-
200 4o"b"~
FIG. 1.
106
h(Km.)
If tether length is not constant in time, as
during deployment and retrieval, the terms proportional to £
in eqs.(1) and (2) are different from zero. First, let us con_
sider the last (forcing) term in the 9 equation; its structu
re suggests that it is originated by-the Coriolis force which pu_
shes the platform away from the local vertical. In plane in-
stability can occur during the first phase of deployment and
the last of retrieval; if the same control law for S. is used
in the two phases, they are equally critical. The asymmetry be_t
ween deployment and retrieval is apparent from the velocity de^
pendent terms in both the equations: it is -seen that when the
coefficients are positive (i.e.during deployment) librations
are damped by length increase, on the contrary, during retri^
val self excited librations can occur, so that the most critjL
cal situation is encountered in the last phase of retrieval.
ELASTICITY EFFECTS
The considerations made so far have ignored £
lasticity effects. Unfortunately, (from the point of view of
simulation problems), the tether is an elastic continuing so
that it can undergo a variety of vibrations: longitudinal, to_r
sional, lateral (both in plane and out of plane). A prelimin£
ry evaluation of the frequencies involved in the TSS case has
been made in {4} and {5} and further investigation is curren-
tly in progress.The same analysis can be extended to different
tethered systems, but since-now it can be expected that the
107
frequencies of elastic modes will be much higher than those
of pendulum-like librations. This is a well known feature of
TSS dynamics; its occurence is the cause of serious difficul_
ties in simulating the motion with purely numerical methods.
In fact the integration step must be small enough to allow
correct simulation of the short period component of the mo-
tion, so that the time needed for a physically meaningful
numerical simulation can easily be excessive.
A major problem is the evaluation of the dy_
namical noise acting on the platform. In fact, one appealing
feature of tethered platforms is the possibility of attaining
high pointing accuracies by isolating them from the noise origi
nated in the primary. To make an example, Aeritalia has inve-
stigated the possibility of actively controlling the motion
of the point of attachment of the tether to the SATP (Science
and Application Tethered Platform) in order to achieve a poin_
ting accuracy of the order of 1 arcsec in attitude control.In
this frame the tether itself can be viewed as a passive damper
the efficiency of.which must be tested carefully.
This is because at present very scarce infomia
tion is available even about the properties (in particular a-
bout structural damping) of the materials to be possibly used
in the first TSS flights. It is expected that post flight an£
lysis of the accelerations at the satellite will provide some
informations on tether damping, but the tuning (if possible)
of a tether to a given.system,in order to maximize energy di^
sipation, will certainly require further experiments.
108
One additional problem is the possibility of
j
coupling between the attitude motion of the platform and so-
me of the higher modes of tether lateral vibrations. While
coupling can be expected on the basis of approximate modal a_
nalysis, a reasonable estimate of vibration amplitudes requi^
res the knowledge of excitation sources and system damping.^
gain, at present experience is lacking , so that the analysis
on TSS will be useful as a first step to understand more com
plex systems and, in particular, to discriminate between what is
really important from what is negligible (at different levels of
accuracy).
Perturbation sources
In spite of the problems, mentioned above, a pre_
liminary knowledge of the response of a tethered system to
most likely perturbations is fundamental, in order to make an
evaluation of the order of magnitude of the dynamical noise
to which it is expected to be subject.
What follows is a tentative list of the best
known mechanical perturbing actions; from the comparison of
their dynamical features with system natural frequencies it
is possible to have a feeling of their impact on the motion
and, consequently, on experiments requirements.
Orbit .eccentricity
Nominally, the orbit of the Space Station (S.S.)
will be circular, but the actual orbit will be allowed a resi-
109
dual eccentricity e. In a pendulum-like system e affects the
in plane motion originating a forced libration with amplitu-
de equal to e and period equal to the orbital period. With
3f 1
-7
-4
e = 10 and a tether length o  0 Km, the acceleration am
plitude would amount to ~ 1.2*10
Earth oblateness
The well known secular regression of the li-
ne of nodes and advance of the apsidal line should not cause
major dynamical problems to a tethered system. However, per-
haps it is less widely known {6} that Earth oblateness causes
both semimajor axis a and inclination i of a circular or-
bit to undergo variations with periods equal to half the orbit
period, i.e. with frequency equal to the out of plane libra-
tion frequency. Therefore resonance can occur and vibration am
plitudes can increase in time.
The mathematical modelization of this dynami-
cal feature is not simple. At present, it is believed that,
due to the relatively short time span of the mission.TSS li-
bration amplitudes cannot grow to undesired levels; however
further investigation is needed if longer missions, in con-
nection to the S.S., are envisaged.
Temperature changes
Tether length in the stressed equilibrium cori
figuration parallel to the local vertical depends on tempera-
ture. In low altitude, low inclination orbits, space tempera-
110
ture changes by some 10 deg. twice per orbit, so that the pojs_
sibility exists that longitudinal vibrations be excited du-
ring the transition from sunlight to Earth shadow and vice-
versa.
The maximum acceleration at the platform d£
pends on the coefficient of thermal expansion a of the tether.
Testing of candidate materials should take into account also
this aspect of"the problem; in fact it does not seem impossi-
ble, in line of principle, to use a tether with a very small
a value.
Internal sources of perturbation
One of the features of platforms tethered to
the S.S. is the possibility to act as almost independent sub_
systems with minimum interference with other S.S.activities.
The tether, however can transmit disturbances to the plat-
form; in this way noise can be originated by Shuttle docking,
station keeping manoeuvres, crew motion, etc.
At present, no reliable estimate is possible
of the dynamical noise at the platform, because of lack of
information on tether damping properties. In this concern,as
mentioned before, the results of a study about the possibiljL
ty of using the tether as a structural damper could pay for
the effort.
Ill
JNatural frequencies
From the review above it can be seen that
perturbing forces can be categorized into two groups with
respect to the frequency of the excitation. Long period
forces are those with frequencies comparable to the mean
motion (eccentricity and Earth oblateness effects) while
short period forces are .those with much higher frequencies
(at least one order of magnitude). The former ones are not
likely to excite tether elastic modes,while the latter, on
the contrary, can do that. This is the reason why a numeri-
cal example of the frequencies possibly involved is preseii
ted below in a study case.
Let us assume that the orbit is circular
4
at 500 Km height and that the platform mass is m:5*10 Kg.
Also, the tether parameters are:
p = 1.5 Kg/m2 mass density
y = 0.5 Kg/m mass per unit length
E = 7*1O10 N/m2Young modulus
Longitudinal vibrations
If it is assumed that the tether end at the
S.S. is fixed and that, at the platform, the inertia force
must be balanced by the elastic stress, approximate values
of the periods of the first longitudinal vibration modes
can easily be found {7}.
The first f-ive periods are reported in Tab.1
for two different tether lengths.
112
T(sec)y(Km)
T 1
T 2
T 3 • ' • ' - •
T 4
T 5
1
9.2
0.29
0.15
0.10
0.07
10
29.5
2.9
1.5
0.98
' 0.7
TAB. 1. -
Lateral vibrations
The same approach can be adopted for lateral
vibrations. The main difference is that in the fundamental mo
des elasticity is not relevant and that in plane and out of
plane librations have different periods. On the contary, ela-
stic effects are dominant in upper modes, so that higher fre-
quencies are almost coincident. For this reason, only the pe-
riods from T 2 to T 5 are reported in Tab.2.
T( seem (Km)
T 2
T 3
T 4
. T 5 .
- 1 .
141
70
47
,35
, 10 .
306
156
105
79
TAB. 2. -
113
Torsional vibrations
Torsional vibrations are believed to have a
minor impact on the overall noise, because of the very large
ratio between platform yaw moment of inertia and tether iner-
tia. Only the torsional spring mass mode could be excited by
disturbances originated in the platform itself. No evaluation
is made of the corresponding period because of the uncertain-
ty of the parameters involved.
The period of the upper modes is, however,
in the range of 1 sec or less for both the lengths.
Platform attitude motion
The evaluations of the periods is quite un_
certain, because of the large variability range of the para-
meters involved. However, let us assume that the distance of
the platform c.o.g. from the tether attachment point be equal
to 5 m and that the radius of gyration both in pitch and in
roll be 3 m. The periods are shown in Fig.2 vs.tether length.
A
Ta(sec)
120-
80-
40-
8
FIG. 2. -
/(Km.)
4J&.
114
(CONCLUSIONS
Tethered systems provide a very interesting
and, in some sense, unique opportunity for scientic activity
in space. Some of the experiments envisaged so far, however re_
quire the measurement of .very small mechanical quantities (ac-
celerations etc.). This implies that the level of dynamical
se on instruments output be low or that system response to
tations, either external or internal, be sufficiently known.
In this respect, the first TSS flights will
be very useful, but much work will be needed in order to have
reliable estimates of structural damping in different future
systems.
REFERENCES
.{1,} Colombo,G.et.al. "Shuttle-borne "Skyhook": A New Tool for
Low-Orbital-Altitude Research". S.A.O.Re_
port,Sept. 1974.
{2} Tiesenhausen (Von),G. "Tethers in Space.Birth and Growth
of a New Avenue to Space Utilization".
NASA TR-82571 MSFC.Feb.1984
{3} Bekey.I. "Applications of Space Tethers".I.A.F.P£
per 84-436.
{4} Bergamaschi,S. "A Theoretical and Experimental Investi-
gation on TSS Dynamics". Proposal in re-
sponse to NASA-CNR/PSN AO.n.OSSA 1 -84 ,
July 1984.
{5} Gullahorn,G. "Investigation and Measurement of Dynamic
Noise in Tethered Satellite Systems".Pro-
posaL.-in response to NASA-CNR/PSN AO. n.
OSSA 1-84, July 1984.
115
i{6} Sterne,I.E. "An Introduction to Celestial Mechanics";
Interscience Tracts on Physics and Astro-
nomy., 1960, Ch.5.2.
{7} Volterra,E.,Zachmanoglou,E.C. "Dynamics of Vibrations".
Merrill Books,1965, Ch.4.3 .
116


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