Space Webs
Final Report
Authors: Gunnar Tibert, Mattias Gärdsback
Affiliation: KTH Engineering Sciences, Royal Institute of Technology,
SE-10044 Stockholm, Sweden
ESA Research Fellow/Technical Officer: Dario Izzo
Contacts:
Gunnar Tibert
Tel:
+46(0)8 790 7961
Fax:
+46(0)8 796 9850
e-mail: tibert@kth.se
Dario Izzo
Tel:
+31(0)71 565 3511
Fax:
+31(0)71 565 8018
e-mail: act@esa.int
Available on the ACT website
http://www.esa.int/act
Ariadna ID: 05/4109
Study Duration: 4 months
Contract Number: 19702/06/NL/HE
Abstract
This report presents a study of the controlled deployment of a large web in space. The
study is composed of three major parts:
1. The design of the space web starts with a study of different geometries, i.e. three
or four corners. For each geometry, three different mesh topologies are investigated,
i.e. triangular, square or hexagonal meshes. An analysis by the force method showed
that only the web with a square mesh could be prestressed by a centrifugal force
field. The triangular mesh had a too high degree of static indeterminacy, which
resulted in compressed elements. The hexagonal mesh had a too high degree of
kinematic indeterminacy and became too distorted under the centrifugal force field.
The square mesh webs were subsequently analysed in terms of out-of-plane flexibility
and vibrational characteristics. Preliminary investigations on the choice of material
for the web and the probability of web failure due to micro-meteoroid impact were
also performed. MATLAB routines that automatically generates the web with an
arbitrary size, mesh width and sag-to-span ratio have been developed.
2. A successful deployment requires an adequate folding pattern. A literature review
identified the star-like folding pattern as a promising candidate. The folding is performed in two distinctive stages. First, the web is folded towards the central hub in
a way so that three or four radial arms are formed, depending on the chosen geometry. Then, the radial arms can be either coiled around the central hub or folded in
a zig-zag manner towards the hub. The MATLAB-generated web from part 1 is fed
into new MATLAB routines, which folds the web according to the various pattern
described above.
3. The dynamic deployment of the space web is analysed by a two-dimensional analytical model in MATLAB and a full three-dimensional model by the commercial
finite element software LS-DYNA. The developed analytical models can simulate
the deployment of the arms from a position coiled around the hub or reeled up on
spools. A simple control strategy was found in literature and implemented in the analytical model with successful results. The MATLAB-generated model of the folded
web was inserted into the software LS-DYNA. For an uncontrolled deployment, the
finite element model yields the expected coiling off-coiling on oscillating behaviour.
The control law with the drooping characteristics is not implemented in the finite
i
element model, but analyses with a simplified control law shows good agreement
between the analytical and the finite element results for the deployment of the star
arms.
ii
Contents
Abstract
i
List of Symbols and Abbreviations
viii
1 Introduction
1
1.1
The Japanese Furoshiki experiment . . . . . . . . . . . . . . . . . . . . . .
1
1.2
Aims and scope of project . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
2 Literature review
4
2.1
The Heliogyro solar sail
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
2.2
The LOFT radio astronomy facility . . . . . . . . . . . . . . . . . . . . . .
4
2.3
The Znamya-2 experiment . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
2.4
Other studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
3 Web design
12
3.1
Web material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.2
Web geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.3
Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.4
Stresses in a non-spinning web . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.5
3.4.1
Forces in interior cables and equivalent in-plane stresses . . . . . . . 19
3.4.2
Force in edge tether . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Stresses in a spinning web . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.5.1
Approximate analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.5.2
Analysis by the force method . . . . . . . . . . . . . . . . . . . . . 24
3.6
Out-of-plane stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.7
Eigenfrequencies and eigenmodes . . . . . . . . . . . . . . . . . . . . . . . 34
iii
3.8
Cable fracture due to micro-meteoroid impact . . . . . . . . . . . . . . . . 35
3.9
Choice of web architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4 Folding patterns for centrifugal deployment
4.1
4.2
39
Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.1.1
Wrapping around a hub . . . . . . . . . . . . . . . . . . . . . . . . 39
4.1.2
Star pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
Proposed folding pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.2.1
Incomplete star pattern
. . . . . . . . . . . . . . . . . . . . . . . . 45
4.2.2
Zig-zag folding of star arms . . . . . . . . . . . . . . . . . . . . . . 46
4.2.3
Wrap-around folding of star arms . . . . . . . . . . . . . . . . . . . 48
5 Deployment
50
5.1
Deployment strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5.2
Analytical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.2.1
Variation of the angular momentum . . . . . . . . . . . . . . . . . . 54
5.2.2
Model 1: deployment of point mass . . . . . . . . . . . . . . . . . . 54
5.2.3
Model 2: deployment of distributed masses . . . . . . . . . . . . . . 55
5.2.4
Model 2(a): deployment of a tether . . . . . . . . . . . . . . . . . . 55
5.2.5
Model 2(b): deployment of a circular space web . . . . . . . . . . . 55
5.2.6
Model 2(c): deployment of the space web using the star pattern . . 56
5.2.7
Deployment of coiled up mass or arm . . . . . . . . . . . . . . . . . 58
5.2.8
Numerical solution of the analytical model . . . . . . . . . . . . . . 58
5.3
Finite element model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.4
Control strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5.5
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.5.1
Validation of the analytical model . . . . . . . . . . . . . . . . . . . 62
5.5.2
Deployment of coiled up circular net . . . . . . . . . . . . . . . . . 62
5.5.3
Deployment of coiled up star folded web . . . . . . . . . . . . . . . 63
5.5.4
Free deployment of star folded web . . . . . . . . . . . . . . . . . . 65
5.5.5
Free deployment of star arms . . . . . . . . . . . . . . . . . . . . . 65
5.5.6
Controlled deployment of star arms . . . . . . . . . . . . . . . . . . 65
5.5.7
Rotational velocity according to Salama et al. . . . . . . . . . . . . 72
5.5.8
Differences between the analytical and the FE model . . . . . . . . 72
iv
6 Conclusions
78
6.1
Web design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
6.2
Folding pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
6.3
Deployment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
v
List of Symbols and Abbreviations
Roman Symbols
Aca
Cable area, p. 63
Ad
Cross-sectional area of strand, p. 24
Asw
Area of space web, p. 15
b
Number of elements, p. 17
c
Number of kinematic constraints, p. 17
dc
Distance between parallel web elements, p. 25
Eca
Young’s modulus for cable material, p. 63
E
Kinetic energy, p. 52
F
Applied force, p. 55
g
Gravitational acceleration (9.80665 m/s2 ), p. 12
H
Total length of star arm, p. 56
h
Membrane thickness, p. 23
j
Number of nodes, p. 17
Jz
Moment of inertia of the centre hub, p. 54
Length of interior web members, p. 18
L
Deployed length, p. 55
l
Length to mass dm, p. 55
L
Angular momentum, p. 50
M
Applied momentum, p. 54
vi
m
Number of mechanisms, p. 17
mc
Mass at web corner, p. 22
mh
Mass of the hub, p. 50
mp
Particle mass that will just break a cable, p. 36
mw
Total mass of space web, p. 50
N
Tensile force in tether, p. 54
n
Number of corners in space web, p. 15
n
Number of radial tethers, p. 54
Nf
Number of fractures per unit length and unit time, p. 36
Np
Micro-meteoroid flux, p. 35
p
Vector of consistent nodal loads, p. 19
Pf
Failure probability due to micro-meteoroids, p. 37
q
Vector of membrane stresses (N/m), p. 19
R
Radius of edge tether, p. 14
r0
Radius of central hub, p. 22
R̈
Acceleration of point mass, p. 55
ρca
Cable density, p. 63
rm
Radius of circular membrane, p. 22
S
Side length of space web, p. 14
s
Number of states of self-stress, p. 17
Set
Total length of edge tethers, p. 16
Srt
Total length of radial tethers, p. 16
tint
Force in interior web elements, p. 19
x, y, z
Cartesian coordinates, p. 20
Greek Symbols
∆
Half the distance between fold lines, p. 45
vii
µ
Mesh topology parameter, p. 17
ω
Angular velocity, p. 22
ϕ
Deflection angle of tether, p. 55
ψ
Angle between tether and symmetry line, p. 23
ρ
Density, p. 12
ρA
Surface density, p. 56
ρd
Density of strand, p. 24
ρL
Line density, p. 55
Sag-to-span ratio of edge tether, p. 15
max
Maximum sag-to-span ratio of edge tether, p. 16
σu
Ultimate strength, p. 12
Btheta
Angle between folds along the centre line, p. 42
Abbreviations
AO
Atomic oxygen, p. 13
IHRT
Institute for Handling Devices and Robotics, p. 2
ISAS
Institute of Space and Astronautical Science, p. 2
LOFT
Low-Frequency Telescope, p. 4
SVD
Singular Value Decomposition, p. 17
VUT
Vienna University of Technology, p. 2
viii
Chapter 1
Introduction
In the space industry there is an increasing need for larger structures, which require new
ingenious solutions as the launch fleet practically is unchanged since the early eighties.
The prior trend in large deployed structures was to passively obtain the required accuracy
by designing stiff structures that could be reliably tested under the influence of gravity
[12, 14]; a typical example is the AstroMesh reflector antenna [54]. The current trend is
towards larger and very flexible structures, such as solar sails [8], which makes ground
testing absolutely impossible.
ESA Advanced Concepts Team has looked at the possibility of constructing large space
antennas and solar power systems by deploying and stabilising a large web in space. The
web will not contain any hard structures for prestressing, but rely on the stiffening effect
of centrifugal forces by spinning the whole assembly of central satellite, corner satellites
and web. Clever deployment strategies are required to control the deployment this very
flexible structure. If not properly taken into account, the very high flexibility of the large
web during deployment may create chaotic dynamics with high risk of entanglement. Such
dynamic phenomena will inevitably lead to failure and loss of the mission. An example of
such a chaotic behaviour is the deployment sequence of the Inflatable Antenna Experiment
by NASA in 1996. The long inflatable feed booms did not behave as expected due to several
reasons, e.g. residual air. As reliable tests cannot be performed under gravity, numerical
analysis is required.
1.1
The Japanese Furoshiki experiment
The idea for the space webs originates from the Japanese “Furoshiki Satellite” [36, 37].
That is composed of a large membrane or net held in tension by controlled corner satellites
or by spinning the whole assembly. The large aperture of the “Furoshiki” can be used as a
phased antenna or as a solar power satellite. An idea put forward by Kaya et al. [18] is to
build up the antenna or solar power elements by robots that crawls on the web like spiders.
1
Figure 1.1: Crawling robot Roby Space III (Junior) on web (Courtesy of ESA).
The robots are developed by the Institute for Handling Devices and Robotics (IHRT) at
the Vienna University of Technology (VUT) in Austria. The robot has a dimension of
100 × 100 × 50 mm3 and a mass of less than 1 kg, Figure 1.1. This robot can crawl on the
web if the mesh width of the web is between 30 and 50 mm [16].
In early 2006, the Institute of Space and Astronautical Science (ISAS) in Japan performed
an in-space deployment experiment of a triangular space web with side length 17 m, [38].
The satellite containing the web was launched by a S-310-36 sounding rocket. Each of the
three corner satellites was released radially by a spring at an initial velocity of 1.2 m/s.
The satellites reached their maximum distance from the the central satellite (10 m) about
8 s after deployment initiation. Thruster control was installed to prevent the bouncing
back of the corner satellites after the deployment. At present, the post-experiment analysis
is not complete [38].
1.2
Aims and scope of project
The aims of the present project are:
1. Design a web architecture that can be adequately prestressed by centrifugal forces
and study the web behaviour in terms of stiffness and vibration characteristics.
2. Find a suitable folding pattern for the deployment by centrifugal forces.
3. Develop analytical and numerical models for the study of the web deployment.
2
The report is structured as follows:
• The first chapter (this) introduces the concept and presents the latest results from
the Japanese sounding rocket experiment.
• The second chapter contains an extensive literature review of other spin-stabilised
structures.
• The third chapter contains the first part of this study, the design of the web architecture. The web design section includes: material, geometry, topology, stresses
due to spinning, out-of-plane stiffness, eigenfrequencies and eigenmodes, and fracture probability due to micro-meteoroid impact. Numerical routines are written in
MATLAB for the automatic generation of webs with various geometries and topologies. These webs are then analysed by pre-existing MATLAB routines for the static
and dynamic analysis of the webs.
• In the fourth chapter, folding patterns suitable for the developed web architectures
are analysed. Numerical routines are written in MATLAB for automatic folding of
the webs. A literature review on folding patterns precedes the selection of the most
suitable one.
• The fifth chapter contains analysis of the space web deployment. Analytical routines
are written in MATLAB to quickly assess various deployment control strategies. The
folded web from section four is exported from MATLAB to the software LS-DYNA
for a full three-dimensional deployment analysis involving contact between centre
hub and cables and between individual cables. The control strategy identified
by the analytical routine is implemented in the LS-DYNA runs.
3
Chapter 2
Literature review
The use of centrifugal forces to stabilised large space structures is not new, as the literature
review below shows.
2.1
The Heliogyro solar sail
The Heliogyro has been the subject of in-depth analysis since its introduction by MacNeal [23] in 1967. The conceptual basis of the Heliogyro is based on a helicopter’s rotors.
As shown in Figure 2.1, the heliogyro is made up of a centrally located payload and
control structure with long thin blades extending outward. The blades constitute the sail
of the craft, and can be cyclically rotated to obtain attitude control. The overwhelming
advantage of the Heliogyro is its low stowage volume and ease of deployment. This is in
part due to the lack of boom structure required by the blades. The blades are typically
comprised of very long, 1 to 3 metre wide sheets that can be stowed in rolls, obviating the
need for complex folding and packaging. Deployment of the blades is obtained by rotating
the base craft and gradually unrolling the stowed blades. This rotation causes a centrifugal force which acts to rigidize the otherwise thin film blades and must be maintained
throughout the mission. Centrifugal force is selected as the preferred method for rigidising
the long narrow sails on the basis of minimum weight and minimum complexity [28, 62].
2.2
The LOFT radio astronomy facility
Schuerch and Hedgepeth [13,49] present a feasibility study of a large-aperture paraboloidalreflector low-frequency telescope (LOFT). Its central component is a parabolic reflector
surface which is deployed and contour-stabilised by a slow spinning motion around its
axis of symmetry, orbiting at an altitude of 6000 km. The diameter of the reflector is
1500 m, the focal-length-to-diameter ratio 0.5 and the total height is 1020 m, Figure 2.2.
A conductive aluminium web with 0.40 m mesh width is supported by a stainless steel
4
(a)
(b)
Figure 2.1: The Heliogyro solar sail, [52].
support net and back stays. The central deployable mast has a diameter 3.04 m and a
length of 760 m. The estimated total weight of the system is 2640 kg or 1.5 g/m2 .
Schuerch and Hedgepeth [49] concluded that “the burden of technology development becomes primarily one of structural design. Practical and credible methods must be devised
to fabricate, package, deploy in space, and maintain adequate dimensional tolerances in
a structure of truly unprecedented size and performance.”
The angular velocity of the LOFT was one revolution per 11.4 minutes (0.00916 rad/s).
The selected angular velocity was a compromise: fast enough to generate sufficient tensile
stresses in the net and to avoid dynamic coupling with the slower orbital frequency, but
slow enough to keep the demand for orientation control torques at tolerable levels [49]. The
size of the packaged LOFT is 5.5 m in diameter and 5.9 m in height, Figure 2.3(a). During
deployment, the spin propulsion system transfer angular momentum through the front
stays to the reflector. The system is programmed to provide the total angular momentum
to the structure at a time when about 60% of the reflector is deployed. After the propulsion
system has been shut down the radial deployment continues, but Coriolis forces slow
the rotational speed down as the network is deployed into an approximately flat disk.
Calculations performed by Schuerch and Hedgepeth show that complete deployment can
be accomplished in less than two days [49].
2.3
The Znamya-2 experiment
The Russians have twice deployed large, sail-like mirrors in space. The wheel-like mirrors,
named Znamya, were spun on motor-driven axles to keep their shape through centrifugal
5
Figure 2.2: The baseline design of the LOFT concept, [49].
force. A 20-metre-diameter version, Figure 2.4, was successfully tested in space in 1993
using a Progress resupply vehicle that had just undocked from the MIR space station. The
simple deployment process was driven solely by spinning up the stowed reflector using an
on-board electric motor. Observed from the MIR space station, the test demonstrated
that such spin deployment can be controlled by simple means. While the reflectors can
demonstrate technologies for solar sailing, their principal use was to illuminate northern
Russian cities during dark winter months to aid economic development [28, 29].
However, a 25-metre version failed in 1999, when it tangled on an antenna jutting out
from the Progress spacecraft that was deploying it. The antenna had been used in the
docking maneuver, and was supposed to have been retracted before sail deployment. A
mission operations software was to blame [8].
2.4
Other studies
Onoda et al. [39] recently performed a preliminary analytical investigation of a spinstabilised solar sail and verified the concept by a 2.2-m-diameter model experiment under
gravity and normal air pressure. The membrane deployed as expected at a constant angular velocity. Miyazaki and Iwai [32] developed a mass-spring network model for the
simulation of the deployment phase of a spinning solar sail. A comparison was made
between membrane and mass-spring simulations for a 2-m-diameter solar sail model as
shown in Figure 2.5. The angular velocity of the hub is not controlled and thus rotates
so that the total angular momentum is conserved. After full deployment at around 0.25
6
s, the sail starts to repackage itself in the opposite direction. They concluded that the
differences between the membrane and mass-spring models were small and also that a
deployment simulation is difficult using a commercial software [32].
(a)
(b)
Figure 2.3: The LOFT concept: (a) packaged configuration and (b) during deployment, [49].
Figure 2.4: Znamya deployment test, 4 February 1993.
7
t=0s
t = 0.05
t = 0.10
t = 0.15
t = 0.20
t = 0.25
t = 0.30
t = 0.35
t = 0.40
(a)
(b)
Figure 2.5: Deployment simulations by Miyazaki and Iwai using (a) membrane elements and (b)
masses and springs. From [32].
Kanemitsu et al. [17] investigated the self-deployment of a 2-metre-diameter antenna by
the centrifugal force. The antenna is stowed as a polygonal column before deployment
with solar panels on each trapezoidal piece. A series of tests, in which the zero gravity
environment of space was simulated by water, was performed. Although the antenna deployed completely under the test conditions, absolute zero gravity could not be simulated.
8
Therefore, a structural deployment analysis was performed using the multi-body software
ADAMS, which showed that the antenna does not deploy completely under the action a
centrifugal forces. The reasons for this are believed to be inadequate modelling of interference between different segments and the mechanisms which control the deployment [17].
Mori et al. [34, 59] analyse the spinning deployment of clover type solar sail. The clover
type sail has a quadratic main sail with two fan parts in each corner, Figure 2.6. The sail
is folded in a star-like pattern, so that each section is first line-shaped and then coiled
around the central hub. The deployment is performed in two stages. First, the line-shaped
parts are coiled off the central hub to form a cross as they are fixed to the hub at their
roots. Then, the constraints are released and the remainder of the sail is deployed. Mori
et al. [34] perform two deployment experiments: a spinning table in ambient environment
and an in-orbit experiment using a sounding rocket. The diameter of the sail for the
ground experiment was 2.5 m, whereas it for the sounding rocket experiment was 10 m.
The ground experiments showed that the coiling off-coiling on phenomenon did not occur
due to the air resistance, so in-orbit experiments are required. To adjust the deployment
time for the space experiment, tip masses were added. To prevent re-coiling of the sail
around the centre hub, a one way clutch mechanism was used. If the centre hub rotates
faster than the tip of the sail, the clutch is locked, whereas if the tip rotates faster, the
clutch is slipping so that the motions of the sail and centre hub are uncoupled. The stickslip clutch is a simple way to achieve a controlled deployment. A mass-spring model was
later used to simulate the two deployment stages, Figure 2.7. The one way clutch was
incorporated in the model. Simulations using a coarse model show a behaviour close to
that observed in the experiments.
Figure 2.6: The clover type solar sail. From [34].
9
t=0s
t = 0.5
t=1
t=0s
t = 0.5
t=1
t=2
t=3
t=4
t=2
t=3
t=4
(a) First stage deployment
(b) Second stage deployment
Figure 2.7: Deployment of the clover type solar sail. From [34].
Matunaga et al. [25,27] introduce a triangular spinning solar sail that is tether-controlled
during and after deployment. The sail is composed of three corner satellites connected
by tethers and large triangular film surface, Figure 2.8. The sail is folded in way so
that three radial arms are formed. These are then rolled up on special motor-controlled
mechanisms attached to the corner satellites. During deployment, the length between
the corner satellites are controlled by the tethers, which take all the tension. Hence,
the mechanism paying out the sail film does not have to cope with large forces, which
simplifies the deployment control. Matunaga et al. [27] performed ground experiments
with air thrusters on the corner satellites. The corner satellites in the experiment weighed
about 42 kg with dimensions 0.6 × 0.6 × 0.47 m3 . The initial side length was 730 mm,
whereas the final one was 1760 mm. Although the experiment showed that the model
could be made to spin, the corner satellites were too heavy to be controlled by the air
thrusters. Matunaga et al. also performed simulations using a mass-spring model. Two
simulations were run: one without tethers between the corners and one with, in order to
show the advantage of having tethers. Their simulations are shown in Figure 2.9, where
it can be observed that the simulations starts when the tethers are fully deployed, but
where the sail is yet to be deployed. The sail without tethers shows a more uncontrolled
behaviour with bouncing motion and distorted shape. Matunaga et al. [26] previously
analysed the deployment of a 30 × 30 m2 quadratic sail with radial tethers. Two tether
control option were introduced in the mass-spring model: tension or length control. For
the case of no control or tension control, the usual coiling off-coiling on phenomenon
was observed, whereas in the case with length control, stable deployment was achieved.
A combination of length control in the early phased of deployment and tension control
in the latter ones also yielded a stable deployment. Snapshots from a trial simulation is
shown in Figure 2.10.
Other studies that investigate various aspects of spin-stabilised space structures are [30,
35, 39, 43, 61]
10
Figure 2.8: Tether-controlled spinning solar sail. From [27].
(a) Non-tethered
(b) Tethered
Figure 2.9: Deployment of triangular sail: (a) without tethers and (b) with tethers. From [27].
Figure 2.10: Deployment of quadratic solar sail. From [26].
11
Chapter 3
Web design
The following part of the report studies various design aspects not related to the subsequent folding and deployment of the space webs. Prior studies, [18, 38], have not looked
into the various choices of web geometry and mesh topology. The prestress distribution
in the web is not uniform due to the centrifugal force field and it is required that the web
is in tension everywhere. The prestress in the web provides the out-of-plane stiffness and
that stiffness may depend on the choice of mesh topology. The spinning of the web and
the low out-of-plane stiffness create undesirable dynamic phenomena, such as travelling
waves and excess out-of-plane deformations, which affect the performance of the web. Due
to all these aspects, a thorough analysis is required to obtain an adequate design of the
web.
3.1
Web material
The web should be manufactured from a very light, but strong and stiff material. A
large modulus is not necessarily an advantage since that requires a higher manufacturing
accuracy. A fundamental property is that the material should be very flexible in bending
so that the web members easily can be folded. Table 3.1 lists properties of fibres for
candidate materials. The factor governing the mass of the web is the breaking length,
σu /ρg, whereas, in a micro-gravity environment, the most important factor is the tensile
strength, σu . As the Zylon fibre has superior properties compared to the other fibres,
Zylon is chosen as the fibre for manufacturing the space web.
Most high performance fibres suffers from strength degradation when exposed to light [11].
The strength of Zylon decreases with exposure to sunlight and must be protected not
only from ultraviolet (UV) light but also from visible light. Experiments by the manufacturer show that the residual strength of Zylon after six months exposure to daylight is
about 35% [58]. Tests performed by Seely et al. [50] show that unshielded Zylon fibres
lost 55% of their strength in only 12 months, whereas fibres that has been shielded from
light (stored in heavy black polyethylene film) lost only 13% in the same time period
12
Table 3.1: Properties of some candidate material fibres [11, 50].
Trade
name
Zylon
Dyneema
Kevlar
Vectran
Generic Type
name
PBO
AS
HPPE
SK76
Aramid 49
TLCP
HS
σu
E
εu
ρ
(GPa) (GPa) (%) (g/cm3 )
5.8
180
3.5
1.54
3.7
116
3.8
0.97
3.6
130
2.8
1.44
2.9
65
3.3
1.40
σu /ρg
(km)
384
389
255
211
due to unidentified causes. Gittemeier et al. [7] recently analysed the effects of various
coatings on the strength degradation rate of the fibres Zylon and Spectra 2000 (similar
to Dyneema) when exposed to UV light and atomic oxygen (AO). Both fibre types show
severe strength degradation and slight mass loss, when exposed to the space environment.
The general conclusion of Gittemeier et al. [7] is that “further coating work is needed
to improve the performance of Spectra and Zylon” for long-term space missions. Gittemeier et al. [7] also measured the mass loss and found it to be at most 7%, which can be
disregarded in the subsequent simulations.
The strength and stiffness of Zylon are also temperature dependent and at 200◦ C, the
relative strength and stiffness are 75% and 90%, respectively [58]. The knot and loop
strength of Zylon is 30% of its tensile strength, [58], which must be taken into account
since the web presumably will be knotted. The abrasion resistance of Zylon is higher
than Kevlar , but much lower than high molecular weight polyethylene fibres, such as
Dyneema and Spectra [58].
The company Phillystran manufactures Zylon AS strands in a variety of dimensions,
which are shown in Table 3.2. The modulus of the strands varies from 68 to 80% of the
modulus of the fibre, which is due to the braiding [50].
Table 3.2: Mechanical properties of low modulus Zylon strands from Phillystran [42].
Part
number
PSAS Z15
PSAS Z30
PSAS Z45
PSAS Z60
PSAS Z75
PSAS Z90
PSAS Z105
PSAS Z150
PSAS Z180
dmin × dmax
(mm2 )
0.43 × 0.66
0.61 × 0.86
0.74 × 1.22
0.79 × 1.50
0.82 × 1.83
0.86 × 1.90
0.86 × 2.03
0.91 × 3.17
1.02 × 3.43
13
Tu
E
Aρ
(N) (GPa) (kg/km)
534
145
0.19
1068
145
0.39
1601
145
0.61
2135
138
0.79
2669
138
0.98
3025
131
1.22
3559
131
1.41
4804
124
2.04
5783
124
2.44
Considering the cumulative effects of light and temperature degradation, knots and loops,
and braiding, the following safety factors are suggested for the strength and modulus. If
the strength and modulus values are taken from Table 3.2, the effects of the braiding is
already accounted for. The reduction factors due to temperature are 0.75 and 0.90 for the
strength and modulus, respectively. For a knotted web, only the strength is affected and
the reduction factor is 0.30. It is clear from the published tests, that the Zylon fibres
must be shielded from light. The choice of shielding material and its effectiveness is still
unclear, so it may be assumed that the strength reduction due to light and atomic oxygen
is 50% throughout the mission life (if less than one year). The environmental effects affects
the ductility of the material, which means that the stiffness do not necessarily diminish
with the same rate as the strength. Due to lack of information, it is assumed that the
stiffness reduction due to light and atomic oxygen is 25%. This reasoning leads to the
following safety factors for the strength
fs,σ =
1
≈ 8.89
0.75 · 0.30 · 0.50
temp.
knot
⇒
fs,σ = 9
(3.1)
light
and for the modulus
fs,E =
1
≈ 1.48
0.90 · 0.75
temp.
⇒
fs,σ = 1.5
(3.2)
light
Thus, a safety factor a high as 9 is needed for the strength, but one of only 1.5 is required
for the modulus. This brief material excursion shows that a careful choice of material
is required so that the web does become unnecessarily heavy. In the remainder of the
this report we are using the characteristic strengths and stiffnesses of the web material
(Zylon ) and thus assuming that improvements in material technology will take place
before the space web will be launched.
3.2
Web geometry
Two different web geometries will be analysed:
• a central hub and three corner masses, and
• a central hub and four corner masses.
The hub and the corner masses are all connected by tethers. The tether connecting a
corner mass with the hub is called radial tether and that connecting two corner masses
is called edge tether. The radial tethers are straight whereas the edge tethers are circular
arches with a radius R. For a given side length S of the web, the radius of the edge tether
is
1 + 42
S
(3.3)
R=
8
14
where is the sag-to-span ratio. For a given sag-to-span ratio, the total area of the space
web is computed as
2
2 2
1
π 1 − 4
1 + 4
4
(3.4)
cot +
−
arcsin
Asw = nS 2
4
n
16
8
1 + 42
where n is the number of corners. For perfectly straight edge cables (which is physically
impossible for a non-spinning web since the force in the edge tether would be infinitely
large), the area is simply
π
nS 2
cot
(3.5)
lim Asw =
→0
4
n
For a quick comparison of the total surface area of various web geometries, the relative
surface area as a function of the sag-to-span ratio is shown in Figure 3.1. The relative
area decreases almost linearly with increasing sag-to-span ratio.
1
0.9
0.8
0.7
0.6
Asw
0.5
S2
0.3
0.2
0.1
0
0
0.05
0.10
max = 20.710%
max = 13.397%
0.4
0.15
0.20
0.25
Figure 3.1: The relative surface area of the web as a function of the sag-to-span ratio of the edge
cables.
The maximum permissible sag-to-span ratio is found when a radial tether is tangent to
the edge circle, which yields
max =
1
2
1
sin
π
2
−
π
n
−
15
1
sin
2
π
2
−
π
n
−1
(3.6)
(a) max = 20.710%
(b) max = 13.397%
Figure 3.2: Configurations with maximum sag-to-span ratios.
For n = 3, max = 13.397% and for n = 4, max = 20.710%. Configurations with the
maximum sag-to-span ratios are shown in Figure 3.2. It is clear that a large sag-to-span
ratio produces very pointy vertices, which is undesirable. The total length of the radial
tethers is
√
3S
for n = 3
1
√
(3.7)
Srt = nS
π =
2 sin n
2 2S for n = 4
whereas the total length of the circular edge tethers is
1 + 42
4
Set = nS
arcsin
4
1 + 42
3.3
(3.8)
Topology
Once the geometry has been defined, the triangular areas defined by the tethers are to
be filled by a mesh. There are several choice for the topology of that mesh. Schuerch
and Hedgepeth [49] chose a quadrangular mesh for the LOFT concept, although previous
reports on the LOFT, [44, 45], used a triangular mesh. The reasons for finally choosing
the square mesh over the triangular one were, [49]:
• the shearing stiffness provided by the diagonal elements were not significantly greater
than the stiffening effect derived from centrifugal forces; deleterious out-of-plane
motions of the surface were reduced by deleting the diagonals, as vibrational energy
then goes into the less harmful in-plane mode of deformation.
• the square mesh has the ability to undergo large shearing deformations, without
requiring in-plane strains or creases in the material; characteristics important for
packaging purposes.
16
Figure 3.3: A uniform stress spinning web, [19].
Kyser [19,20] derives analytical relationships for mesh geometry of a circular spinning web
with uniform prestress. He shows that the elements of the web must be oriented as spirals
in order to obtain a uniform prestress, Figure 3.3. A spiral-like web will carry a radiallydirected loading in such a way that the resulting tension decreases towards the centre. A
uniformly stressed web is obviously interesting structurally, but the non-constant mesh
width will create problems for the crawling robots as the mesh width becomes very large
far away from the centre of the web. Thus, this web topology is not of interest for the
present project.
The web topology is a crucial aspect of the web design as this dictates the static and
kinematic properties of the web through the generalised Maxwell’s rule [1]
3j − b − c = m − s
(3.9)
where j is the number of joints, b the number of elements, c the number of kinematic
constraints, m is the number of mechanisms and s the number of states of self-stress.
A simple comparison between different web topologies can be done by considering a triangle, a square and a hexagon, which are constrained from rigid body motions, i.e. c = 6.
The triangle has j = b = 3 and s = 0, which yields m = 0. For the square and hexagon,
m = 2 and m = 6, respectively. Thus, the triangle is shear-stiff, whereas the square and
the hexagon have several internal mechanisms. The number of mechanisms for a specific
web topology must be found by a Singular Value Decomposition (SVD) of the equilibrium
matrix [40, 55].
The mesh topology is defined by the parameter µ, which is equal to 3 for a triangular mesh,
4 for a square mesh and 6 for a hexagonal mesh. The six possible space web configurations
with the above topologies and three and four corners are shown in Figure 3.4. A MATLAB
routine has been written to generate the configurations shown in Figure 3.4. The routine
17
can handle arbitrary values of the element length and the sag-to-span ratio. In terms of
manufacturing, the most regular web configurations are:
• Triangular web and triangular mesh (TriTri), n = 3, µ = 3,
• Triangular web and hexagonal mesh (TriHex), n = 3, µ = 6, and
• Quadratic web and quadratic mesh (QuadQuad), n = 4, µ = 4.
The advantage with these configurations is the irregularities are restricted to the edges,
whereas for the other three configurations, i.e. Figs. 3.4(b), (d) and (f), irregularities
appear also along the radial tethers. Thus, the subsequent analyses was initially limited
to these three regular configurations, but, as will be shown in section 3.5, the TriQuad
configuration (n = 3, µ = 4) is also of interest and will therefore be included.
(a) n = 3, µ = 3
(b) n = 3, µ = 4
(c) n = 3, µ = 6
(d) n = 4, µ = 3
(e) n = 4, µ = 4
(f) n = 4, µ = 6
Figure 3.4: Possible web and mesh configurations for n = 3, 4 and µ = 3, 4, 6.
The factor governing the mass of the web is the total length of interior elements, i.e.
excluding radial tethers and element forming the circular boundaries. Preliminary analyses
show that the total length increases exponentially with diminishing ratio /S, where is
the length of interior web members, but increases almost linearly with the decrease of
the sag-to-span ratio, . However, for an upper bound estimate of the total length of web
18
elements, it is assumed that = 0. This yields the following equation for the total length
of interior elements:
(3.10)
= An,µ log10
+ Bn,µ
log10
S
S
Least squares solutions for the four interesting combinations yield:
A3,3 = −1.0118, B3,3 =
0.1469 (R2 = 1.0000)
(3.11a)
2
A3,6 = −1.0073, B3,6 = −0.3197 (R = 0.9997)
(3.11b)
A4,4 = −1.0015, B4,4 =
0.2973 (R2 = 0.9997)
(3.11c)
A3,4 = −0.9903, B3,4 = −0.0391 (R = 0.9996)
(3.11d)
2
−4
Hence, a 100×100 m2 web with 30 mm mesh width requires 100·10(−1.0015·log10 (3·10 )+0.2973) ≈
669 km of cable (excluding radial and edge cables). Using the thinnest Zylon strand in
Table 3.2, the total weight of such a web will be at least 127 kg.
3.4
Stresses in a non-spinning web
For a web with several internal mechanisms, m > 0, its equilibrium geometry is governed
by the force distribution [48,57]. Since three different topologies are to be investigated, i.e.
triangular, square and hexagonal, an easy way of comparing them structurally is desired.
In addition, an simple analytical relationship between the forces in the web and the forces
in the tether is of importance in order to simplify the calculations. In this section, a way
comparing the webs in terms of stresses is proposed.
3.4.1
Forces in interior cables and equivalent in-plane stresses
The factor governing the out-of-plane stiffness of the web is the force tint in the interior
elements. However, since the mesh size will be very small, the web can be seen as a
membrane, with in-plane stresses q T = (qx qy qxy ) (N/m). Following the approach by Lai
et al. [21], the interior forces tint can be transformed to equivalent membrane stresses q
via natural stresses. The consistent nodal forces p due to initial stresses σ in a Cartesian
coordinate system is written as [2]
T
(3.12)
p = B σdV = BT qdA
For a constant strain triangle (CST), [2],
⎤
⎡
y23 0 y31 0 y12 0
1 ⎣
0 x32 0 x13 0 x21 ⎦
B=
2A
x32 y23 x13 y31 x21 y12
19
(3.13)
where xij = xi −xj and yij = yi −yj and A is the triangle area. Transforming the consistent
Cartesian force components p into skew components, or ‘natural forces’, parallel to the
edges of the element, yields the triangle side forces, [21]:
⎡
⎤⎛ ⎞
⎛ ⎞
12 y13 y23 12 x13 x23 12 (x32 y13 + x13 y32 )
qx
t12
1
⎣23 y12 y13 23 x12 x13 23 (x31 y12 + x12 y31 )⎦ ⎝ qy ⎠
⎝t23 ⎠ =
(3.14)
4A
t31
qxy
31 y12 y32 31 x12 x32 31 (x23 y12 + x12 y23 )
Solving Eq. (3.14) for q for an equilateral triangle with side length and side 12 parallel
to the x-axis, the side force t12 = t23 = t31 = tint /2 yields
√ t
int √ tint
=
(3.15)
qT
3
3
0
3
which means√that a web with a triangular mesh can be seen as a membrane under uniform
tension q = 3tint /.
For a square mesh, the procedure is simpler as the sides are parallel to the axes of the
Cartesian coordinate system. For a rectangle with side lengths 2a and 2b in the x and y
directions, respectively,
⎡
⎤
−(b − y)
0
(b − y)
0
(b + y)
0
−(b + y)
0
1 ⎣
0
−(a − x)
0
−(a + x)
0
(a + x)
0
(a − x) ⎦
B=
4ab
−(a − x) −(b − y) −(a + x) (b − y) (a + x) (b + y) (a − x) −(b + y)
(3.16)
The consistent nodal loads are
⎛ ⎞
⎛
⎞
p1x
−bqx − aqxy
⎜p1y ⎟
⎜−aqy − bqxy ⎟
⎜ ⎟
⎜
⎟
⎜p2x ⎟
⎜ bqx − aqxy ⎟
⎜ ⎟ b a
⎜
⎟
⎜p2y ⎟
⎜−aqy + bqxy ⎟
T
⎜ ⎟=
⎟
B qdxdy = ⎜
⎜p3x ⎟
⎜ bqx + aqxy ⎟
−b −a
⎜ ⎟
⎜
⎟
⎜p3y ⎟
⎜ aqy + bqxy ⎟
⎜ ⎟
⎜
⎟
⎝p4x ⎠
⎝−bqx + aqxy ⎠
p4y
aqy − bqxy
(3.17)
For uniform element forces, tij = tint , in the square-meshed web: t12 = tint /2 = p2x = −p1x ,
t23 = tint /2 = p1y = −p4y , t34 = tint /2 = p3x = −p4x and t41 = tint /2 = p4y = −p1y . For a
square mesh, ij = , these conditions yield the consistent nodal loads for a single square
element
t
tint int
T
(3.18)
q4 =
0
√
which is lower than q T
3.
3 by
For the hexagonal mesh the procedure is not as straightforward as for the triangular or
the square meshes. One way is to assemble six equilateral triangles into a hexagon and
compute the six consistent nodal loads at the vertices. The extra, seventh, node in the
20
middle of the hexagon should not have a resulting nodal load due to symmetry. Computing
the nodal forces at each vertex of the hexagon and then transforming those to element
side forces yield the following conditions
√
√
3qx + qxy =
3qy − qxy = tint /2
2
2
√
√
=
3qy + qxy =
3qx − qxy = tint /2
2
2
√
√
√ √
=
3qx − 4qxy + 3ny =
3qx + 4qxy + 3qy = tint /2
4
4
t12 = t45 =
(3.19a)
t23 = t56
(3.19b)
t34 = t61
Solving Eq. (3.19) for the equivalent membrane stresses yields
1 tint 1 tint
T
√
0
q6 = √
3 3 (3.19c)
(3.20)
Thus, the equivalent uniform membranes stresses can be written in the general form
1
tint k tint
nT
=
3
(3.21)
with k =
µ2 − 13µ + 36
0
µ
12
where µ = 3, 4 and 6 for the triangular, square and hexagonal mesh, respectively. As the
shear stress qxy = 0, the normal membrane stress in each direction is constant, just like a
soap film.
3.4.2
Force in edge tether
The transformation from element forces to equivalent membrane stresses makes the determination of the force in the circular edge tether straightforward. The radius R of the
edge tether as a function of the side length S and the sag-to-span ratio is given by (3.3).
The axial force in a circular ring subjected to a uniformly distribution radial line load n0
is [63]
R
(3.22)
tet = Rq0 = 3k tint
where k = (µ2 − 13µ + 36)/12.
3.5
Stresses in a spinning web
A non-spinning web is prestressed by pulling the circular edge tethers, which induce a
uniform tension in the web as shown in section 3.4. In such a case there is a direct relation
between the radius and force of the edge tether, Eq. (3.22). However, if the web is spinning,
the stress will be non-uniform due to the centrifugal force field acting on the whole web.
In this case, it becomes possible to have straight edge tethers. It still might be beneficial
21
to have a circular edge tether as the outer parts of the web may not obtain a sufficiently
large stress by the centrifugal field alone. In those cases, the non-uniform stress by the
centrifugal forces on the web is increased by the uniform stress provided by the circular
edge tether.
First, an approximate analysis was performed to investigate the ratio of the web stresses
from spinning and those from the circular edge tether. Then, a more thorough analysis
by the force method, [40, 55], was done.
3.5.1
Approximate analysis
For a circular membrane of radius rm that is fully clamped to a hub of radius r0 , the
radial stress is [43, 51]
3+ν 2 2
r02
2
(3.23)
σr (r) =
ρω (rm − r ) 1 + 2
8
r
where
1 − ν 3 + ν − λ2 (1 + ν)
=
3 + ν 1 + ν + λ2 (1 − ν)
λ = r0 /rm
(3.24)
(3.25)
and ω is the angular velocity. For the special case r0 = r = 0, Eq. (3.23) simplifies to
max(σr )r0 =0 =
3+ν 2 2
ρω rm
8
(3.26)
For the Heliogyro solar sail, the radial stress at the root of the blade is [23]
1
2
max(σr )HG = ρω 2 rm
2
(3.27)
where rm is the length of the blade. As this stress is slightly higher than that for a circular
membranes and independent of the Poisson’s ratio, it will be used in the comparison below.
The centrifugal force on the corner mass mc , at distance rm from the centre of the web, is
Fc = mc rm ω 2
(3.28)
This force is equilibrated by the circular edge cables according to Figure 3.5. The distance
to the corner mass is
S
rm =
(3.29)
2 cos πn
and the angle between the edge tethers and the symmetry line in Figure 3.5 is
4
π
− arcsin
ψ=
2(6 − n)
1 + 42
22
(3.30)
Fc = mc rm ω 2
mc
ψ
ψ
tet
tet
Figure 3.5: Equilibrium between the centrifugal force on the corner mass and the edge tethers.
For a straight edge cable, = 0, the angle ψ is π/6 for a triangular geometry and π/4 for
a square one. An equilibrium equation in the radial direction yields the edge cable force
due to the spinning of the corner mass
tet,mc =
Fc cos ψ
2
(3.31)
¿From Eq. (3.22), this cable force yields the following uniform membrane stress
2mc ω 2
4
tet,mc
π
=
− arcsin
cos
σω,mc =
Rh
h(1 + 42 ) cos πn
2(6 − n)
1 + 42
(3.32)
where h is the thickness of the membrane. Comparing the stress due to the density of the
web, (3.27), with that due to the corner masses, (3.32), yields
ρtS 2
σω,ρ
(3.33)
= kσ
σω,mc
mc
where
kσ =
π
n
16 cos cos
(1 + 42 )
π
2(6−n)
23
− arcsin
4
1+42
(3.34)
The factor kσ approaches infinity as the sag-to-span ratio approaches zero. Assuming that
the minimum sag-to-span ratio is 1%, the factor kσ is approximately 15 for n = 3, which
yields
ρhS 2
σω,ρ
≈ 15
(3.35)
max
σω,mc
mc
The product ρt is the surface density of the equivalent membrane. The surface density of
the triangular, square and hexagonal mesh can be written as
√
7
67
4 Ad ρ d
2
√ − 1 µ + 9 − √ µ + 16 3 − 18 + √
(ρt)µ =
(3.36)
3 3
3 3
3
where Ad is the cross-sectional area of the web wire and ρd is the density of the wire
material. The surface density is largest for the triangular mesh:
√
2 3Ad ρd
(3.37)
(ρt)µ=3 =
Inserting (3.37) into (3.35) and dropping the “max” notation yields
S Ad ρ d S
σω,ρ
≈ 52
σω,mc
mc
(3.38)
Assuming that the web is manufactured from Phillystran PSAS Z15 strands, which has
Ad ρd = 0.19 g/m by Table 3.2. If S = 100 m, = 0.03 m and mc = 10 kg, the stress ratio
σω,ρ /σω,mc = 329. Thus, the stress due to the centrifugal force on the web is more than
300 times larger than that created by the corner masses and edge cables close to the hub.
Increasing the corner masses to 50 kg only decrease the ratio five times. However, for a
smaller web with S = 10 m the stress ratio is 100 times smaller at 3.29. Thus, the effects
of curving the edge tethers may be negligible for most elements of a large web, but not for
a smaller web.
3.5.2
Analysis by the force method
The analysis by the force method uses the routines described in [40, 55]. For this analysis
it is assumed that the side length S = 100 m and that the distance between the web
strands dc = 30 mm. The interior web elements are assumed to be made from lightest
strand in Table 3.2 (PSAS Z15), which has a length density of 0.19 g/m. The radial and
edge tethers are assumed to be made from the strand with a length density of 1.22 g/m.
As the number of nodes and elements of a web with 30 mm mesh width is too large
for our MATLAB routines, a larger mesh width is used in the computations. However,
the cross-sectional area of the web strands are scale up so that the total weight of the
web remains constant even though the mesh width changes. The total length for different
element lengths are given by Eq. (3.10). As a certain distance between individual elements
24
is required by the crawling robots, the comparison will be made for a constant distance,
dc . The relationship between the element length and mesh width is simply
⎧√
⎪
⎨ 3/2 if µ = 3
(3.39)
dc = if µ = 4
⎪
⎩√
3
if µ = 6
For a web with dc = 30 mm, the web strands have a cross-sectional area Aint = 0.19 ·
10−3 /1540 = 1.23 · 10−7 m2 , whereas the cross-sectional area of the tethers is Atether =
1.22 · 10−3 /1540 = 7.92 · 10−7 m2 . For example, assume that a square web with a square
mesh have a mesh width of 2 m. The total length of the web strands for the 30 mm
mesh width is 669 km by (3.10), whereas the total length is 9.97 km for the web with 2
m mesh width. The cross-sectional area for the web strands in the model thus becomes
Aint = 669/9.97 · 1.23 · 10−7 = 8.25 · 10−6 m2 . Note that the cross-sectional area of the
tethers does not change. The mass in each corner is 10 kg and the angular velocity is 1
rad/s. The exact value of the angular velocity is not relevant as the ratio of the stresses
only depends on the masses and dimensions, Eq. (3.38).
Web with four corners and square mesh (QuadQuad)
The element forces of the QuadQuad web is shown in Figure 3.6(a). It is clear that the
large force from the corner masses are transferred through the radial tether and to the
last row of interior web elements. Thus, the edge tether is subjected to a smaller load
that the web strands, which is unsatisfactory. By decreasing the cross-sectional area of
the radial tether by a factor 106 , it is virtually removed from the calculations. The corner
loads are now transferred to the edge tether as intended, 3.6(b).
The magnification of cross-sectional area of the web strand for larger mesh widths to
keep the web mass constant should produce mesh-independent stresses in the web. The
element forces for four different mesh widths (10, 8, 6 and 4 m) are shown in Figure 3.7.
The equivalent radial force intensities (N/m) at the centre elements for the four cases are:
(a) 144/10 = 14.4, (b) 133/8 = 16.6, (c) 98/6 = 16.3 and (d) 64/4 = 16 N/m. Equation
(3.27) yields
1
1 0.19 · 10−3 · 2 · 0.030 2
· 1 · 502 = 15.8 N/m
nr = σr t = ρtω 2 b2 ≈ ·
2
2
2
0.030
(3.40)
Hence, the equivalent root membrane stresses in the square web is similar to the analytical
value. An interesting observation is that the circumferential forces do not change very
much with the distance from the centre along the centre line for each segment. However,
they decrease in the circumferential direction from the centre line towards the radial
tether.
The effect of curving the edge cables are shown in Figure 3.8. The forces in the outer
element increase as the sag-to-span ratio increases, whereas the root stress does not change
much. Note that in Figure 3.8(d), some elements are compressed, which obviously cannot
be accepted. For this case, the stress decreases in a significant portion of the web. This
25
may be explained by the fact that the distance from the centre to the edge along the
centre lines is shorter than for the other cases ( = 0–5%).
The results of the analysis of the QuadQuad web suggest that a straight edge cable
produces an adequate force distribution. Radial tethers are not necessary and if they are
used, they should be integrated with the web in non-structural way, i.e. in a way so that
they cannot share the load with the web elements, as they would change the load paths
in a detrimental way.
64
64 617 4
68
71
74
77
79
81
82
83
83
83
83
82
81
79
77
74
71
68
64
500
500 0 4
4 61764
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
381
393
405
414
422
429
433
436
438
438
436
433
429
422
414
405
393
381
4
68
381 106 19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19 106 381
68
4
19
94
105
115
123
129
134
137
139
139
137
134
129
123
115
105
94
19
4
71
393
94 18 31
31
31
31
31
31
31
31
31
31
31
31
31
31
31 18 94
393
71
4
19
31
53
62
70
77
81
85
86
86
85
81
77
70
62
53
31
19
4
74
405
105
53 4 43
43
43
43
43
43
43
43
43
43
43
43
43 4 53
105
405
74
4
4
77
4
79
4
81
4
82
4
83
4
83
4
83
4
83
4
82
4
81
4
79
4
77
504
4
507
4
510
19
31
43
54
62
68
73
76
78
78
76
73
68
62
54
43
31
19
4
414
115
62
54 2 52
52
52
52
52
52
52
52
52
52
52 2 54
62
115
414
77
19
31
43
52
60
67
71
75
76
76
75
71
67
60
52
43
31
19
4
422
123
70
62
60 2 60
60
60
60
60
60
60
60
60 2 60
62
70
123
422
79
19
31
43
52
60
67
71
75
76
76
75
71
67
60
52
43
31
19
4
429
129
77
68
67
67 3 67
67
67
67
67
67
67 3 67
67
68
77
129
429
81
19
31
43
52
60
67
71
74
76
76
74
71
67
60
52
43
31
19
4
433
134
81
73
71
71
71 3 71
71
71
71
71 3 71
71
71
73
81
134
433
82
19
31
43
52
60
67
71
74
76
76
74
71
67
60
52
43
436
137
85
76
75
75
74
74 3 74
74
74 3 74
74
75
75
76
85
19
31
43
52
60
67
71
74
76
76
74
71
67
60
52
43
438
139
86
78
76
76
76
76
76 3 76 3 76
76
76
76
76
78
86
19
31
43
52
60
67
71
74
76
76
74
71
67
60
52
43
438
139
86
78
76
76
76
76
76 3 76 3 76
76
76
76
76
78
86
19
31
43
52
60
67
71
74
76
76
74
71
67
60
52
43
436
137
85
76
75
75
74
74 3 74
74
74 3 74
74
75
75
76
85
31
19
31
43
52
60
67
71
74
76
76
74
433
134
81
73
71
71
71 3 71
71
71
71
71
19
31
43
52
60
67
71
74
76
76
74
429
129
77
68
67
67 3 67
67
67
67
67
67
19
31
43
52
60
67
71
75
76
76
75
422
123
70
62
60 2 60
60
60
60
60
60
60
75
76
76
75
19
31
43
52
60
67
71
414
115
62
54 2 52
52
52
52
52
52
52
52
31
4
19
31
43
54
62
68
73
74
405
105
53 4 43
43
43
43
43
4
19
31
53
62
70
77
81
71
393
94 18 31
31
31
31
31
31
4
19
94
105
115
123
129
134
68
381 106 19
19
19
19
19
19
19
4
381
393
405
414
422
429
433
64 617 4
4
4
4
4
4
4
4
64
4
68
71
74
77
79
81
82
76
78
43
85
43
86
31
137
19
43
19
438
31
438
19
137
519
83
4
4
83
519
83
519
4
436
19
4
4
60
60 2 60
62
70
123
422
79
60
52
43
31
19
4
52
52 2 54
62
115
414
77
54
43
31
19
4
513
43
43 4 53
105
405
74
53
31
19
4
510
31
31 18 94
393
71
94
19
4
507
19
504
67
52
68
43
43
77
31
19
19
19
115
19
422
4
81
62
31
123
429
4
82
70
31
129
433
4
62
19
414
4
79
105
405
4
77
393
4
74
4
68
510
513
515
517
518
519
519
519
519
518
517
515
513
510
507
504
500
52
60
67
71
75
76
76
75
71
67
60
52
43
31
19
4
52 0 52
52
52
52
52
52
52
52
52
52
52 0 52
52
52
52
513
52
60
67
71
75
76
76
75
71
67
60
52
43
31
19
4
60
60
60
60
60 0 60
60
60
60
60
60
60
60
60 0 60
60
60
60
60
515
19
31
43
52
60
67
71
75
76
76
75
71
67
60
52
43
31
19
4
67
67
67
67
67
67 0 67
67
67
67
67
67
67 0 67
67
67
67
67
67
517
19
31
43
52
60
67
71
75
76
76
75
71
67
60
52
43
31
19
4
71
71
71
71
71
71
71 0 71
71
71
71
71 0 71
71
71
71
71
71
71
518
43
52
31
19
43
31
75
76
76
75
71
67
60
52
43
75 0 75
75
75 0 75
75
75
75
75
75
75
76
76
75
71
67
60
52
43
76
76
76
76
76
76
76
76
76 0 76 0 76
76
76
76
76
76
76
19
31
43
52
60
67
71
75
76
76
75
71
67
60
52
43
76
76
76
76
76
76
76
76
76 0 76 0 76
76
76
76
76
76
76
19
31
43
52
60
67
71
75
76
76
75
71
67
60
52
43
74
75
75
75
75
75
75
75 0 75
75
75 0 75
75
75
75
75
75
75
19
71
19
31
4
19
52
4
4
60
31
71
75
76
76
75
71
67
60
52
43
19
75
76
31
31
519
4
76
19
76
519
4
76
19
75
31
4
74
19
519
4
74
19
519
4
71 0 71
71
71
71
71 0 71
71
71
71
71
71
71
518
67
71
75
76
76
75
71
67
60
52
43
31
19
4
67
67 0 67
67
67
67
67
67
67 0 67
67
67
67
67
67
517
60
67
71
75
76
76
75
71
67
60
52
43
31
19
4
60 0 60
60
60
60
60
60
60
60
60 0 60
60
60
60
60
515
75
76
76
75
71
67
60
52
43
31
19
4
52
60
67
71
52
52 0 52
52
52
52
52
52
52
52
52
52
52 0 52
52
52
52
513
43
52
60
67
71
75
76
76
75
71
67
60
52
43
31
19
4
43 0 43
43
43
43
43
43
43
43
43
43
43
43
43 0 43
43
43
510
31
43
52
60
67
71
75
76
76
75
71
67
60
52
43
31
19
4
31
43
19
67
71
31
71
52
60
43
71
71
75
67
60
52
67
43
31
52
19
43
43
60
67
75
60
52
71
67
31
60
60
75
52
43
71
67
19
60
52
75
43
31
71
67
4
43
75
31
19
4
31
4
19 106 381
68
381
4
4
71
31
52
19
4
515
71
19
74
4
517
134
83
43
83
4
19
139
4
519
67 3 67
67
68
77
129
429
81
67
60
52
43
31
19
4
71
81
436
4
52
436
139
4
518
4
19
31
4
517
518
31
19
83
60
137
31
515
507
4 0 500
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
19
31
43
52
60
67
71
74
76
76
74
71
67
60
52
43
31
19
4
19 0 19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19 0 19
504
19
31
43
52
60
67
71
75
76
76
75
71
67
60
52
43
31
19
4
31
31 0 31
31
31
31
31
31
31
31
31
31
31
31
31
31
31 0 31
31
507
19
31
43
52
60
67
71
75
76
76
75
71
67
60
52
43
31
19
4
43
43
43 0 43
43
43
43
43
43
43
43
43
43
43
43
43 0 43
43
43
510
52
4
3 71
71
71
73
81
134
433
82
71
67
60
52
43
31
19
4
43
137
438
4
83
67
73
85
31
139
438
4
76
86
31
139
436
83
78
71
19
4
513
504
4
31 0 31
31
31
31
31
31
31
31
31
31
31
31
31
31
31 0 31
31
507
19
31
43
52
60
67
71
75
76
76
75
71
67
60
52
43
31
19
4
19 0 19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19 0 19
504
19
31
43
52
60
67
71
74
76
76
74
71
67
60
52
43
31
19
4
4
500 0 4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4 0 500
500
504
507
510
513
515
517
518
519
519
519
519
518
517
515
513
510
507
504
500
61764
64
(a) s = 10
(b) s = 10
Figure 3.6: Element forces (N) in a spinning web with 5 m mesh width: (a) Aradial = Aedge and
(b) Aradial = Aedge · 10−6 (green denotes tension and red compression).
.
26
500
500
0
514
524
17
17
17
530
17
68
534
534
17
106
17
132
530
17
144
524
17
144
132
514
500
17
17
106
68
0
500
500 0 3
3
503 27
500
68
0
17
68
68
68
524
68
106
106
106
0
68
132
106
68
144
68
144
106
106
132
106
68
0
106
106
68
68
505 54
514
106
3
17
106
68
106
132
132
132
17
68
132
106
0
144
144
132
132
132
132
132
144
144
0
106
132
132
68
132
17
68
3
530
144
144
17
68
534
144
106
144
144
132
144
144
0
144
0
144
144
144
144
0
144
0
144
132
144
144
144
106
144
68
144
534
68
106
132
132
132
17
68
524
132
106
106
106
0
144
144
132
132
132
132
144
132
144
0
17
3
144
106
132
68
132
132
68
106
514
68
0
17
500
106
106
68
68
0
132
106
68
106
106
500
514
17
524
144
68
68
132
17
17
144
106
144
68
68
0
106
530
17
534
68
132
17
68
106
3
530
17
3
68
0
127
68
113
94
94
133
68
127
94
68
35
509
3
102
35
508
3
91
35
133
68
133
94
94
68
127
68
507 3
75
35
113
68
133
94
35
113
94
127
94
505 3
54
35
94
68
113
94
68
126
94
133
133
35
113
133
68
0
500
503
3 0 500
27
3
0
126
133
113
126
133
133
126
127
113
35
68
94
3
3
107 510
35
133
68
3
102 509
133
133
3
91 508
35
68
94
133
35
127
133
3
75 507
113
68
27 503
54 505
35
68
94
113
133
68
94
113
133
35
68
94
126
133
0
94
94
113
113
126
133
133
133 0
113
126
68
94
94
113 0
126
126 0
133 0
133
133 0
126
113
133
126
133 0
126
113
133
113
133
126
133
133
133
113
126
126 0
133
133
126
113
113
126
133
133
113
113 0
94
127
133
127
35
514
127
68
113
35
126
94
113
68
0
500
500
94
35
505 54
514
17
17
524
510
3
107
35
3
107 510
126 0
113
113 0
94
126
126
113
113
126
133
113
126
126 0
133
113
133
126
126
113
133
126
113
113 0
126
127
35
94
113
113
127
3
68
113
94
102 509
35
113
68
3
91 508
35
3
524
17
68
17
534
0
68
113
68
35
508 91
17
106
106
68
144
17
530
132
510
3
107
35
17
3
0
509
3
102
35
113
94
68
127
507 75
17
68
94
113
35
510 107
534
132
106
508
3
91
35
94
68
35
509 102
530
0
94
510 107
3
17
35
509 102
17
3
534
68
68
508 91
132
106
507
3
75
35
524
3
17
530
35
507 75
0
505 3
54
35
17
3
514
503 3
27
0
68
3
35
503 27
0
3
500 0 3
500
94
68
0
27
503 3
94
94
68
68
35
0
68
94
35
54
505 3
94
113
68
113
35
75
3
507
94
127
68
127
35
91
3
508
94
133
68
133
35
102
3
509
94
133
68
133
35
107
3
510
94
127
68
127
35
107
3
510
94
113
68
113
35
102
3
509
0
94
94
68
94
35
91
3
508
94
68
0
68
68
35
75
507 3
75 507
35
35
35
54
505 3
3
54 505
3
27 503
3
27
503 3 0 500
500
0
(a) s = 5
(b) s = 7
500 502
504
506
507
508
509
509
510
510
509
509
508
507
506
504
502 500
500 0 2
2
2
2
2
2 0 500
2
2
2
2
2
2
2
2
2
2
2
17
33
47
58
67
74
79
81
81
79
74
67
58
47
33
17 2
2
502 17 0 21
21
21
21
21
21
21
21
21
21
21
21
21
21
21 0 17 502
502 17 0 21
21
21
21
21
21
21
21
21
21
21
21
21
21
21 0 17 502
2
17
33
47
58
67
74
79
81
81
79
74
67
58
47
33
17 2
500 0 2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2 0 500
502
504
506
507
508
509
509
510
510
509
509
508
507
506
504
502
500
500
500 501 503 504 505 506 507 508 508 509 509 509 509 509 509 509 509 508 508 507 506 505 504 503 501 500
500 0 1
2
2
2
2
2
2
2
2
1 0 500
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
9
17 24 30 36 41 45 49 52 54 55 56 56 55 54 52 49 45 41 36 30 24 17 9 1
1
501 9 0 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 0 9 501
2 10 19 27 35 41 47 52 56 59 62 63 64 64 63 62 59 56 52 47 41 35 27 19 10 2
503 17 19 0 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 0 19 17 503
2 10 19 27 35 41 47 52 56 59 62 63 64 64 63 62 59 56 52 47 41 35 27 19 10 2
504 24 27 27 0 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 0 27 27 24 504
2 10 19 27 35 41 47 52 56 59 62 63 64 64 63 62 59 56 52 47 41 35 27 19 10 2
505 30 35 35 35 0 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 0 35 35 35 30 505
2 10 19 27 35 41 47 52 56 59 62 63 64 64 63 62 59 56 52 47 41 35 27 19 10 2
506 36 41 41 41 41 0 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 0 41 41 41 41 36 506
2 10 19 27 35 41 47 52 56 59 62 63 64 64 63 62 59 56 52 47 41 35 27 19 10 2
507 41 47 47 47 47 47 0 47 47 47 47 47 47 47 47 47 47 47 47 47 0 47 47 47 47 47 41 507
2 10 19 27 35 41 47 52 56 59 62 63 64 64 63 62 59 56 52 47 41 35 27 19 10 2
508 45 52 52 52 52 52 52 0 52 52 52 52 52 52 52 52 52 52 52 0 52 52 52 52 52 52 45 508
2 10 19 27 35 41 47 52 56 59 62 63 64 64 63 62 59 56 52 47 41 35 27 19 10 2
508 49 56 56 56 56 56 56 56 0 56 56 56 56 56 56 56 56 56 0 56 56 56 56 56 56 56 49 508
2 10 19 27 35 41 47 52 56 59 62 63 64 64 63 62 59 56 52 47 41 35 27 19 10 2
509 52 59 59 59 59 59 59 59 59 0 59 59 59 59 59 59 59 0 59 59 59 59 59 59 59 59 52 509
2 10 19 27 35 41 47 52 56 59 62 63 64 64 63 62 59 56 52 47 41 35 27 19 10 2
509 54 62 62 62 62 62 62 62 62 62 0 62 62 62 62 62 0 62 62 62 62 62 62 62 62 62 54 509
2 10 19 27 35 41 47 52 56 59 62 63 64 64 63 62 59 56 52 47 41 35 27 19 10 2
509 55 63 63 63 63 63 63 63 63 63 63 0 63 63 63 0 63 63 63 63 63 63 63 63 63 63 55 509
2 10 19 27 35 41 47 52 56 59 62 63 64 64 63 62 59 56 52 47 41 35 27 19 10 2
509 56 64 64 64 64 64 64 64 64 64 64 64 0 64 0 64 64 64 64 64 64 64 64 64 64 64 56 509
2 10 19 27 35 41 47 52 56 59 62 63 64 64 63 62 59 56 52 47 41 35 27 19 10 2
509 56 64 64 64 64 64 64 64 64 64 64 640 64 0 64 64 64 64 64 64 64 64 64 64 64 56 509
2 10 19 27 35 41 47 52 56 59 62 63 64 64 63 62 59 56 52 47 41 35 27 19 10 2
509 55 63 63 63 63 63 63 63 63 63 630 63 63 63 0 63 63 63 63 63 63 63 63 63 63 55 509
2 10 19 27 35 41 47 52 56 59 62 63 64 64 63 62 59 56 52 47 41 35 27 19 10 2
509 54 62 62 62 62 62 62 62 62 620 62 62 62 62 62 0 62 62 62 62 62 62 62 62 62 54 509
2 10 19 27 35 41 47 52 56 59 62 63 64 64 63 62 59 56 52 47 41 35 27 19 10 2
509 52 59 59 59 59 59 59 59 590 59 59 59 59 59 59 59 0 59 59 59 59 59 59 59 59 52 509
2 10 19 27 35 41 47 52 56 59 62 63 64 64 63 62 59 56 52 47 41 35 27 19 10 2
508 49 56 56 56 56 56 56 560 56 56 56 56 56 56 56 56 56 0 56 56 56 56 56 56 56 49 508
2 10 19 27 35 41 47 52 56 59 62 63 64 64 63 62 59 56 52 47 41 35 27 19 10 2
508 45 52 52 52 52 52 520 52 52 52 52 52 52 52 52 52 52 52 0 52 52 52 52 52 52 45 508
2 10 19 27 35 41 47 52 56 59 62 63 64 64 63 62 59 56 52 47 41 35 27 19 10 2
507 41 47 47 47 47 470 47 47 47 47 47 47 47 47 47 47 47 47 47 0 47 47 47 47 47 41 507
2 10 19 27 35 41 47 52 56 59 62 63 64 64 63 62 59 56 52 47 41 35 27 19 10 2
506 36 41 41 41 410 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 0 41 41 41 41 36 506
2 10 19 27 35 41 47 52 56 59 62 63 64 64 63 62 59 56 52 47 41 35 27 19 10 2
505 30 35 35 350 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 0 35 35 35 30 505
2 10 19 27 35 41 47 52 56 59 62 63 64 64 63 62 59 56 52 47 41 35 27 19 10 2
504 24 27 270 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 27 0 27 27 24 504
2 10 19 27 35 41 47 52 56 59 62 63 64 64 63 62 59 56 52 47 41 35 27 19 10 2
503 17 190 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 0 19 17 503
2 10 19 27 35 41 47 52 56 59 62 63 64 64 63 62 59 56 52 47 41 35 27 19 10 2
501 9 0 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 0 9 501
1 9
17 24 30 36 41 45 49 52 54 55 56 56 55 54 52 49 45 41 36 30 24 17 9 1
500 0 1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
1 0 500
500 501 503 504 505 506 507 508 508 509 509 509 509 509 509 509 509 508 508 507 506 505 504 503 501 500
(c) s = 9
(d) s = 13
2
21
504 33
2
506 47
2
57
2
81
95
21
510 81
2
98
510 81
2
509 79
2
95
504 33
2
57 0
40
81 0
70 0
57
57
40
70
40
70
70
57
81
70
57
57
95
40
90
70
98
40
95
70
57
98
57
57
40
95
57
57
40
70
81
70
70
57
57
40 0
57
40
2
58 507
21
40
2
67 508
21
40
2
74 509
21
40
57
57 0
40
81
70
70
90
81
2
79 509
21
40
57
70
81
40
90
81
70 0
95
90
2
81 510
21
40
57
70
81
90
90
81
98
95
2
81 510
21
40
57
70
81
70
57
95
90
98
98
2
79 509
21
40
57
70
81
90
95
40
98
90
81 0
70
98
95
95
98
2
74 509
21
40
57
70
81
90
95
98
40
98
81
95
98
95
90
98
98
90
95
2
67 508
21
40
57
70
81
90
90 0
81
98
98
95
95
98
81
90
95
90
81
98
95
90
40
81
81
98
95 0
98
98
95
70
81
90
57
2
58 507
21
40
2
47 506
21
70
81
57
90
95
57
40
2
33 504
21
40
70
81
70
81
57
57
70
90
95
90
95
81
81
90
98
98
98
90
95
95
98
98
95
90
98 0
95
90
98
90
70
21
40
40
57
70
81
90
95
98
98
95
81
70
57
90
90
81
70
40
57
81
95
95 0
98 0
81 0
90 0
57 0
70 0
40
40 0
57
70
81
90
95
98
98
95
90
81
70
95 0
90 0
81
70
57
40
40 0
21
81
70
57
21
90
98 0
95
90
95
98
81
90
57
70
57
40
70
81
90
95
98
98
98 0
98
95
95
95
90
81
70
57
40
95
90
81
98
81
70
57
40
70
506 47
2
90
81
21
95
98
81
90
57
70
70
40
81
90
95
98
98
95
90
81
90
57
70
81
40
90
95
98
98
95
95 0
98
98
81
90
57
70
90
40
95
98
98
95
90
81
70
57
40
21
507 58
2
95
90
98
98
57
40
21
508 67
2
98
95
81
70
81
90
57
70
95
40
98
98
95
90
90 0
95
98
81
81
70
57
40
21
509 74
2
98
98
21
95
57
40
90
57
70
98
40
98
95
90
81
70
57
70
98
40
95
90
81
81 0
90
95
57
70
95
40
90
81
70
57
40
21
81
90
40
57
70
57
90
40
81
70
70 0
81
90
57
57
40
21
509 79
2
70
81
40
70
57
40
21
509 74
2
57 0
70
70
40
57
40
21
508 67
57
40
40
21
507 58
2
40
40 0
21
2
47 506
21
40
2
33 504
21
2
Figure 3.7: Element forces (N) in a spinning web with different mesh widths: (a) 10, (b) 8, (c)
6 and (d) 4 m (green denotes tension and red compression).
Web with three corners and triangular mesh (TriTri)
The dimensions and properties of the TriTri web are identical to the QuadQuad web
above. However, the radial tethers cannot be removed here as they form one side of the
triangles; above they could be removed as they formed the diagonals of the squares. As
a triangular web has a higher degree of redundancy (static indeterminacy) than a square
one and it is thus more difficult to prestress [56]. The results for various TriTri webs are
shown in Figure 3.9. From the large number of compressed elements (in red), it is clear
that the triangular web cannot be satisfactorily stressed by centrifugal forces. The cause
27
of the prestressing problems is the high degree of static indeterminacy: s = 89 for the
TriTri web compared to s = 10 for the QuadQuad web for the same mesh width. Curving
the edge tethers decreases the static indeterminacy slightly, but does not produce a better
force distribution. The only conclusion is that a web with a triangular mesh is very difficult
to prestress by spinning and it should thus be avoided in the design of space webs.
500
5000 4
504
507
510
513
515
517
518
519
519
519
519
518
517
515
513
510
507
504
500
4
510
4 0 500
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
19
31
43
52
60
67
71
74
76
76
74
71
67
60
52
43
31
19
4
19 0 19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19
19 0 19
504
19
31
43
52
60
67
71
75
76
76
75
71
67
60
52
43
31
19
4
31
31 0 31
31
31
31
31
31
31
31
31
31
31
31
31
31
31 0 31
31
507
19
31
43
52
60
67
71
75
76
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67
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31
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43
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43 0 43
43
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43
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43 0 43
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4
513
52
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60
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52
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52 0 52
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52
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52 0 52
52
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513
52
60
67
71
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76
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52
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4
60
60 0 60
60
60
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60
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60 0 60
60
60
60
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515
19
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43
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67 0 67
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67 0 67
67
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71 0 71
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19 0
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79
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55
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22
8
34 0 34
33
33
33
33
32
32
32
32
32
33
33
33
33
34 0 34
34 465
34
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45 0 45
44
44
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45 0 45
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53
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54 0 54
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53
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53
53
53
53
53
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54 0 54
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53 468
67
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31
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62 0 61
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61 0 62
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61
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67 0 67
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60 0 60
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60 0 60
60
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430 430 430 429
12
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11
11
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429 429 429 430 430 9
10
10
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36
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54 10 60 9 66 9 69 9 72 9 73
73 9 72 9 69 9 66
60 10 54
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25
12
25 0 24
23
22
21
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20
20
20
20
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24 0 25
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68
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68 0 68
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62 0 61
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61 0 62
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54 0 54
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(a) s = 10
428
75
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61
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22 0
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69
63
33
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19 0 19
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43 0 43
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31
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31 0 31
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31
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31
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31
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31
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31
31
31 0 31
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5000
19
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74
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77 0 77 0 77
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75 0 75
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8
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465 462 459
465 466 468
469 470 471 471 471 471 471 471 470 469 468 466
8 0 459
8
8
8
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7
7
7
7
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7
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7
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7
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22
34
45
53
61
66
71
73
75
75
73
71
66
61
53
45
34
22
8
22 0 22
21
21
20
20
20
20
19
19
19
20
20
20
20
21
21
22 0 22
462
459
459
7
20
71
471
7
19
73
471
7
19
75
471
7
19
75
7
471
20
73
20
71
7
471
7
471
20
66
470
20
61
7
21
53
7
469
21
45
7
468
466
22 0 22
462
22
8
8
8 0 459
465 462 459
34
8
(b) s = 10
387
384 -2
384
-2 19
387
380
380419
19
420419 37 16 27 19
27 16 37 420 418
19
418
416
416 414
27 2 27 384
13
13
384 27 2 27
19
19
412 411 410 410 411 412 414
18
18
16 27
42
53
61
68 18 74 17 77 16 80 16 81
81 16 80 16 77 17 74 18 68
61
53
42
27 16
42 0 25
31
29
28
27
26
26
26
26
26
27
28
29
31
25 0 42 37 380
380 37
419 13
25
36
46
54
60
65
68
70
70
68
65
60
54
46
36
25
13 419
53
36 0 42
41
39
38
37
37
37
37
37
38
39
41
42 0 36
420
53
420
387
387
11 24
37
48
58
66
72
77
80
82
82
80
77
72
66
58
48
37
24 11
36
37 0 36
35
34
34
33
33
32
32
32
33
33
34
34
35
36 0 37 36
430
11 23
36
47
57
65
71
76
79
81
81
79
76
71
65
57
47
36
23 11
46
48
47 0 46
45
45
44
44
44
43
44
44
44
45
45
46 0 47
48 46 431
10 22
35
46
56
64
70
75
78
80
80
78
75
70
64
56
46
35
22 10
431 54
58
57
56 0 55
54
54
53
53
53
53
53
54
54
55 0 56
57
58 54 431
10 21
34
45
55
63
69
74
77
79
79
77
74
69
63
55
45
34
21 10
430 60
66
65
64
63 0 62
62
61
61
61
61
61
62
62 0 63
64
65
66 60 430
9 21
34
45
54
62
69
73
77
78
78
77
73
69
62
54
45
34
21 9
430 66
72
71
70
69
69 0 68
68
67
67
67
68
68 0 69
69
70
71
72 66 430
9 20
33
44
54
62
68
73
76
78
78
76
73
68
62
54
44
33
20 9
430 69
77
76
75
74
73
73 0 73
72
72
72
73 0 73
73
74
75
76
77 69 430
9 20
33
44
53
61
68
73
76
77
77
76
73
68
61
53
44
33
20 9
429 72
80
79
78
77
77
76
76 0 75
75
75 0 76
76
77
77
78
79
80 72 429
9 20
32
44
53
61
67
72
75
77
77
75
72
67
61
53
44
32
20 9
429 73
82
81
80
79
78
78
77
77 0 77 0 77
77
78
78
79
80
81
82 73 429
20
32
43
53
61
67
72
75
77
77
75
72
67
61
53
43
32
20
429 73
82
81
80
79
78
78
77
77 0 77 0 77
77
78
78
79
80
81
82 73 429
9 20
32
44
53
61
67
72
75
77
77
75
72
67
61
53
44
32
20 9
429 72
80
79
78
77
77
76
76 0 75
75
75 0 76
76
77
77
78
79
80 72 429
9 20
33
44
53
61
68
73
76
77
77
76
73
68
61
53
44
33
20 9
430 69
77
76
75
74
73
73 0 73
72
72
72
73 0 73
73
74
75
76
77 69 430
9 20
33
44
54
62
68
73
76
78
78
76
73
68
62
54
44
33
20 9
430 66
72
71
70
69
69 0 68
68
67
67
67
68
68 0 69
69
70
71
72 66 430
9 21
34
45
54
62
69
73
77
78
78
77
73
69
62
54
45
34
21 9
430 60
66
65
64
63 0 62
62
61
61
61
61
61
62
62 0 63
64
65
66 60 430
77
79
79
77
74
69
63
55
45
34
21 10
10 21
34
45
55
63
69
74
431 54
58
57
56 0 55
54
54
53
53
53
53
53
54
54
55 0 56
57
58 54 431
10 22
35
46
56
64
70
75
78
80
80
78
75
70
64
56
46
35
22 10
431
46
48
47 0 46
45
45
44
44
44
43
44
44
44
45
45
46 0 47
48 46 431
11 23
36
47
57
65
71
76
79
81
81
79
76
71
65
57
47
36
23 11
430
36
37 0 36
430
35
34
34
33
33
32
32
32
33
33
34
34
35
36 0 37 36
11 24
37
48
58
66
72
77
80
82
82
80
77
72
66
58
48
37
24 11
429
25 0 24
429
23
22
21
21
20
20
20
20
20
20
20
21
21
22
23
24 0 25
12
25
36
46
54 10 60 9 66 9 69 9 72 9 73
73 9 72 9 69 9 66 9 60 10 54
46
36
25
12
10
10
11
11
429 429 429 429 430 430 430
11
11 12
428 0 12
430 430 430
0 428
431 431
431 431
430 429
430
429
428
428
430
431
19
31
42
52
60
66
71
74
76
76
74
71
66
60
52
42
31
19
61
46
52 0 50
49
48
47
47
46
47
47
48
49
50 0 52
46
61 418
18 29
41
50
58
65
69
73
74
74
73
69
65
58
50
41
29 18
54
60
58 0 57
56
55
54
54
54
55
56
57 0 58
416 68
60
54
68 416
18 28
39
49
57
63
68
71
73
73
71
68
63
57
49
39
28 18
60
66
65
63 0 62
61
61
61
61
61
62 0 63
414 74
65
66
60
74 414
17 27
38
48
56
62
67
70
72
72
70
67
62
56
48
38
27 17
65
71
69
68
67 0 66
66
65
66
66 0 67
412 77
68
69
71
65 77 412
16 26
37
47
55
61
66
69
71
71
69
66
61
55
47
37
26 16
68
74
73
71
70
69 0 69
69
69 0 69
411 80
70
71
73
74
68 80 411
418
16 26
37
47
54
61
66
69
70
70
69
70
76
74
73
72
71
70 0 70 0 70
410 81
71
26
37
46
54
61
65
69
70
70
69
70
76
74
73
72
71
70 0 70 0 70
410 81
71
16 26
37
47
54
61
66
69
70
70
69
68
74
73
71
70
69 0 69
411 80
69
69 0 69
16 26
37
47
55
61
66
69
71
71
69
65
71
69
68
67 0 66
412 77
66
65
66
66
17 27
38
48
56
62
67
70
72
72
60
66
65
63 0 62
414 74
61
61
61
18 28
39
49
57
63
68
71
73
73
54
60
58 0 57
416 68
56
55
54
54
73
74
74
18 29
41
50
58
65
69
61
46
52 0 50
418
49
48
47
47
46
19 31
42
52
60
66
71
74
76
76
53
36 0 42
420
41
39
38
37
37
37
70
61
72
66
74
54
71
61
55
37
70 81 410
26 16
37
68 80 411
26 16
74
47
56
65
48
71
57 0 58
60
54
68 416
58
50
41
29 18
49
66
38
60
39
50 0 52
46
61 418
52
42
31 19
41
13
25
36
46
54
60
65
68
70
70
68
65
60
54
46
419
42 0 25
31
29
28
27
26
26
26
26
26
27
28
29
380 37
16 27
42
53
61
68 18 74 17 77 16 80 16 81
81 16 80 16 77 17 74 18 68
61
18
411 410 410 411 412
18
412
19
27
414
414
384
13
2 27
416
416
418
418
19 27 16 37
420
419
19
387 -2 384 380
387
(c) s = 13
70 81 410
26
76
47
73
26 16
37
62 0 63
65
66
60 74 414
63
57
49
39
28 18
69
37
73
37
76
46
0 67
68
69
71
65 77 412
67
62
56
48
38
27 17
47
74
47
74
54
61
70
68
73
54
73
61
66
55
47
61
72
65
61
71
54
37
66
42 0 36
53 420
36
25
13 419
31
25 0 42 37 380
42
27 16
13
27 2 27 384
420 419 37 16 27 19
380
19
384 -2 387
387
53
19
(d) s = 12
Figure 3.8: Effect of curving the edge cables (mesh width 5 m): (a) = 0%, (b) = 2.5%, (c)
= 5% and (d) = 10% (green denotes tension and red compression).
28
130
354
130
138
6
174
23
23
178
6
6
190
23
23
11
392
383
433
2
6
433
174
23
190
242
23
257
124
6
77
6
433
237
124
2
23
-202
6
237
257
11
138
383
-17
124
178 23
54
54
56
65
66
72
73
76
76
76
76
73
72
66
65
56
54
54
52
2 363 377 2 6
6 2 6
6 2 6
6 2 6
6 2 6
6 2 6
6 2 6
6 2 377 363 2 489
52
3
3
3
3
3
3
3
-361
-361
110
142
230
125
78
87
69
73
73
69
87
78
125 110
230 -162
142 2
-162 173
-96
-50
-50
-62
-96
173
-62
363
54 363
-38
131
271
105 165
95
120
103 120
95
165 105
271 131
103
?361 54
-38
-361
-17
-13
-14
-13
-17
-14
-14
-162
-162
100
20
10
193
53
133
79
100 79
133 53
193 10
20
377
173
173
54 377
54
2
-17
-17
-12
2
-12
131
131
2
230
10
17
171
66
126
94
94
126 66
171 17
10
230 2
-14
-14
22
7
2
7
22
6 56
10
10
56 6
271
5
30
145
67
102
102 67
145 30
5
271
3
2
2
3
27
17
17
27
110
17
17
110
-96
193
6
35
120
82
82
120 35
6
193
6
22
22
65 6
-96
65
42
29
42
105
30
30
105
2
125
171
5
47
89
89
47
5
171 -17
125 2
27
27
-17
55
55
6 66
53
35
35
66 6
-12 53 165
165
145
4
59
59
4
145
3
42
42
3
-12
62
78
66
47
47
66
78
133
120
2
2
120
133 -62
6
7
55
55
7
72 6
-62
72
95
67
59
59
67
-13 95
2
87
126
89
89
126
87 2
17
62
62
17
-13
6 73
79
82
2
82
-17 79
73 6
120
102
59
59
102
120
3
29
29
3
-17
69
94
89
89
94
69
100
82
82
100 -50
6
2
55
55
2
766
-50
76
103
102
4
102
-14 103 73 2
2
73
94
47
47
94
17
17
-14
6 76
100
120
120
76 6
-17 100 103
103
67
67
3
42
42
3
-17
126
5
126
-50 73
6 73
79
35
35
79
6
7
7
76
-50
76
120
145
145
-13 120 69 2
2
69
66
66
27
27
-13
6 76
133
6
-12 133
766
95
30
30
95
3
3
-12
87
171
171
87
-62
53
53
6
22
22
736
-62
73
165
5
-17 165 78 2
2
78
17
17
-17
193
193
6 72
726
105
105
3
2
2
3
125
10
-96 125 6
6
10
10
66
-96
66
271
-14 271 110 2
2
110
-14
20
6 65
656
131
131
3
3
230
230
6
6
173
173
56
56
2 -162 -38
2
-162
54377
-361377
54
-361
363 142 363
54
54
2 2
489
52 52
52 489
2
354
130
6
23
11
52
130
6
392
74
-330
237
6
242
383
242
237
-202
433
23
-330
-202
257
242
-17
-330
138
23
11
2
257
-202
383
174
6
23
77
74
11
178
23
11
-330
138 23
23
174
190
6
11
11
178
2
237
237
242
242
23
77
190
257
6
2
257
6
2
-17
190 23
23
-202
11
11
-202
433
-330
23
178
23
-330
178
74
6
190
433
383
6
383
174 23
23
11
11
174
392
23
138
23
138
6
6
130 354
130
(a) s = 11
88
(b) s = 89
88
90 96 100 98 99
96 90 88
104 106
106 104 99 98 100
107 109
109 107
109 110
110 110
4 2 418 88
110 109
5 13
418 2 4
31 5
2
2
2
428
320 339 2 21
33 3 62
69 3 59
61 3 61
59 3 69
62 3 33
21 2 339
320
428 2
4 90
-54
-13
-313
-49
-54
-49
90 4
-13
-137 -313
127
128 111
87
103
91
91
103 87
111 128
127 2 96
2
201
-137 164201
96
-16
-14
-16
-57
164
-57
320
-313 320 100
-34
100
126 215
73
107
70
85
85
70
107 73
215 126
5
5
-313
-34 -137
-19
-33
-7
-33
-19
-7
-137
1 98
17
14
170
63
107
82
82
107 63
170 14
17
3393
164
3 339
164
99
5
-3
-1
-3
5
126
-7 126 201 2 104
2
201
7
16
132
59
88
88
59
132 16
7
104
-7
18
11
11
18
21
14
14
21 106
106
215
6
26
106
71
71
106 26
6
215 -13
5
5
-13
34
23
34
128
16
16
128
33
170
4
39
77
77
39
4
170
33
18
18
-57
-57
46
46
73
26
26
73
107
3
111
132
4
50
50
4
132 -33
111 3 107
34
34
-33
53
62
63
39
39
62 109
-3 63 107
107
106
2
2
106
109
46
46
-54
-3
-54
87
59
50
50
59
87
69
107
77
77
107 -16
69
11
53
53
11
-16
70
71
2
71
109
-19 70 103 3 109
3
103
88
50
50
88
23
23
-19
59
82
77
77
82
59
-1
-49
110
110
85
71
71
85
46
46
-1
-49
91
88
4
88
-14 91
61
82
39
39
82
61
11
11
-14
110
85
106
106
-19 85 91 3 110
3
91
59
59
34
34
-19
107
4
-49 61 110
-3 107 70
110 61
70
26
26
-3
-49
103
132
132
-16 103 59
59
63
63
18
18
-16
110
110
107
6
-33 107 87
3
3
87
16
16
-33
69
170
170
-54 69 109
109
73
73
5
5
-54
111
7
111
-57
62
14
14
62
-57
215
109
-7 215 128 3 109
3
128
-7
33
17
33 107
107
126
126
-13
-13
201
201
21
21
164
164
-137 -34
106
2
2 106
-137
339
104
-313339 104
3
3
99 1 -313 127
1 99
320
3205
98 5
98
2
2 100
100
428 96
96
4
4
90 2 2 90
418
88
88
(c) s = 86, = 2.5%
91
91 93
93 91
98 100
100 98 3 2 406
100 101
406 2 3
100
107
107 109 105
2 413 3
110 111 108 111 111 108 111 111 108 111 110 106 105 109
3 413 2 91
325 3 5
3
3 93
311
5 3 325 311
42 6
93
-306
-306
3
3
-137
3
3
3
121
201
94
106
66
72 3 61
61 3 72
66
106 94
201
121 3 98
98
-137 147
-83
-55
-49
-55
-83
147
311
-306 311 100
-32 112
235
91
140
87
100
87
140 91
112
100
100
235
-306
-32 -137
-12
-15
-13
-13
-15
-12
-137
325
17
9
166
48
108
79
79
108 48
166 9
17
147 325
147
2
-10
-17
-10
2
112
-12 112 201 3 100
100
3
201
8
15
143
65
98
98
65
143 15
8
-12
5
5 101
22
8
8
22
9
9
101
235
5
28
113
71
71
113 28
5
235 3
3
2
2
107
30
16
30
107
94
15
15
-83 94 109
166
5
36
82
82
36
5
166
109 6
22
22
-83
46
46
105
91
28
28
91
2
-15
143
4
50
50
4
143
106 106
4 106
30
30
-15
51
48
36
36
110
-10 48 140 3 110
3
140
113
2
2
113
46
46
-10
66
65
50
50
65
-55 66 111
108
82
82
108
111
8
51
51
8
-55
4
14
87
71
2
71
-13 87 72 108
72
98
50
50
98
16
16
108
-13
79
82
82
-17 79 100 3 111
111
3
100
71
71
46
46
-17
61
98
4
98
-49 61 111
111
79
36
36
79
8
8
41 -49 100
108
113
113
100
-13
4 61
65
65
61 108
30
30
-13
108
5
-10 108 87
3
3 111
111
87
28
28
-10
72
143
143
-55 72 111
48
48
22
22
111
-55
140
5
-15 140 66 108
66
15
15
108
-15
166
166
111
3
3 111
91
91
2
2
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8
-83 106 110
110
9
9
4
-83
106
235
235
2
-12
94
94
-12
105
105 6
17
112
112 3 109
3
109
201
201 107
107
147
147 5
5
101
101 3 -137 -32
3
-137
100
100 325
-306 325
-306
121
311
311
100
100
3
3
413 98
98
3 3
93 2 2 93
406
91
91
(d) s = 68, = 5%
Figure 3.9: Element forces (N) in a spinning web with different mesh widths: (a) 10 and (b)–(d)
5 m (green denotes tension and red compression).
Web with three corners and hexagonal mesh (TriHex)
The hexagonal mesh ought to have greater potential of being prestressed by the centrifugal
forces as the degree of static indeterminacy is smaller. However, the mechanisms are no
longer orthogonal to the external loads, i.e. the centrifugal loads, which means that the
linear force method is not valid for the computation of the element forces. Switching to
another method more suited for the analysis of singular systems, e.g. Dynamic Relaxation
[60] or Generalised Inverse [53], may yield an accurate solution. This is, however, not
required here since the presence of a significant number in-plane mechanisms indicates
that the web will undergo severe distortions when subjected to the centrifugal force field.
Figure 3.10 shows the 14 in-plane mechanisms (42 mechanisms in total) of a TriHex web
29
with a mesh width of 20.78 m. TriHex webs with mesh widths of 5 and 4 m contain
212 and 350 in-plane mechanisms, respectively. Thus, the problem gets worse quickly as
the mesh width decreases. The conclusion from the analysis of the TriHex web is that it
is unsuitable for the space web as it will undergo too severe distortions during spinning
due to the presence of a large number of in-plane mechanisms that are activated by the
centrifugal forces.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
(l)
(m)
(n)
Figure 3.10: The 14 in-plane mechanisms of a TriHex web with 20.78 m mesh width and sag-tospan ratio of 2.5%.
Web with three corners and square mesh (TriQuad)
As both the triangular webs with manufacturing advantages, i.e. TriTri and TriHex, were
30
discarded due to prestressing and instability problems, it has become necessary to study
the triangular web with a square mesh, i.e. TriQuad. Using the same dimensions and
material properties as above, the results are shown in Figure 3.11. Figure 3.11(a) shows
that some of the radial tether element must be removed as they prevent the formation of
tensile forces and create large forces in the web and not in the edge tethers as for the other
configuration. Once removed, the forces are in tension everywhere and well distributed,
Figure 3.11(b). Curving the edge tether increases the force along the periphery, Figure
3.11(c). Since the vector of centrifugal loads is orthogonal to the subspace of mechanisms,
no mechanism is activated and the displacement are kept small, Figure 3.11(d).
64
66
467 1
2
64
473 253
1
41 7
253
66
7
2
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1 467
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2
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2
2
2
253 473 1 64
261
271
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7 41
253
9
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2
66 42
50
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64
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50 42 66
261
-2 14
9
9
14
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-2 42
42 14
2
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15 14
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31
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31
26
9
9
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15 26 -3 19
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-3 26 15
2
2
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19 19
19 19 23
9
9
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26 -19
-19 26
58
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58
73
15 31 23 22
15
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22
2
2
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23
23
31
4
287
287
20 26 4
20
24
9
9
27 24
64
75
25 -13 27 22 26
64
75
22
-13 24 25
15 36
15
2
2
24
36
292
292
20 29 24 24 11 24 11 24 24
20
9
9
29
68
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68
76
23
23
25
15 39
15
25
2
2
39
24
24
297
297
20 30
20
9
9
30
71
77
71
77
23 25 11
23
15 40
15
25
2
2
40
299
24 24
299
20 30
20
9
9
-13
30
73
78
73
78
23 24 24 23
15 40
15
2
2
40
301
301
20 29
20
27
9
9
29
73
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73
78
22
22
15 39
15
2
2
39
301
301
20 26 4
20
9
9
26
71
78
71
78
22
22
15 36
15
2
2
-19
36
299
299
23 20
20
9
9
23
68
78
68
78
19
15 31
15
2
2
297
19
19 31
297
9
9
64
77
64
77
15 26 -3
15
2
2
26
292
292
9
9
58
76
58
76
15
15
2
2
287
287
15
9
9
50
75
50
75
2
2
14 14
279
279
-2
9
9
73
42
42
73
2
2
66
271
271
9
9
71
71
51
2
2
261
261
69
7 7
69
2 41
2
253 253
66
473 66
1 1
334
336
338
341
343
344
345
346
347
347
347
347
346
345
344
343
341
338
336
334
1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
334 1
334
1 7 8
17
26
35
42
48
52
55
56
56
55
52
48
42
35
26
17
8 7 1
7 8
8 7
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
336
7
7
336
14
14
2
2
17
32 22
30
37
43
48
51
52
52
51
48
43
37
30
22 32
17
9
9
14
338
22 14
338
22
15
15
15
15
15
15
15
15
15
15
15
2
2
14 19
19 14
26
26
17
23
27
30
32
32
30
27
23
17
9
9
30
341
30
341
19 21
15 17 21 19
15
20
20
20
20
20
20
20
2
2
17
35
35
45 19
19 45 27
9
9
31
33
35
35
33
3122 27
37
343
37
343
27 22
27
15 23
15
22
22
22 24 22
2
2
23
23
23
23
42
42
20 31 24
20
9
9
49 29
43
344
30
30
29 49 22 31
43
344
22
15 27
15
24
24
2
2
29
29
27
48
24
48
24
24
20
20
33
9
9
25
25
33
48
345
48
345
23 30
23
15 30
15
30
2
2
30
24
24
52
52
20 35
20
9
9
35
51
346
51
346
23 30 25
23
15 32
15
30
2
2
32
55
55
24 24
20 35
20
9
9
35
52
347
52
347
23 29 29 23
15 32
15
2
2
32
56
56
20 33
20
49
9
9
33
52
347
52
347
22
22
15
15
2
2
30
30
56
24
56
20
20
31
9
9
31
51
347
51
347
22 22
15 27
15
2
2
27
55
55
20 27 27 20
9
9
48
347
48
347
45
15 23
15
2
2
23
52
52
19 19
9
9
43
346
43
346
15 17 21
15
2
2
17
48
48
9
9
37
345
37
345
15
15
2
2
42
42
19
9
9
30
344
30
344
2
2
14 14
35
35
9
22 22 9
343
343
2
2
32
26
26
9
9
341
341
14
2
2
17
17
338
7 7
338
2 8 8 2
336
7
1
64467
64
1
336
334 334
(a) s = 6, m = 285
(b) s = 0, m = 297
318
318
319
319
321
321
322
322
4
323
323
318 4
324
324
4
4
318
4
4
4
4
4
4
4 3254 3254 3254 3254 3254 3254 3254 3254
4 10 9
17
26
34
41
46
49
52
53
53
52
49
46
41
34
26
17
9 10 4
9 9
9 9
319
11
11
11
11
10
10
10
10
10
10
10
10
10
11
11
9
9
319
16
16
4
4
17
37 25
32
40
46
50
53
55
55
53
50
46
40
32
25 37
17
11 25 16
11
321
16
321
25
17
16
16
16
16
16
16
16
16
16
17
4
4
16 21
21 16
26
26
18
24
28
31
33
33
31
28
24
18
322
11 32
11
322
32
21
21
22
22
4
4
17 18
17
21
21
21
21
21
21
21
18
34
34
48 21
21 48 29
323
11 40
11
323
35
37
37
35
322329
23 32
40
29
4
4
16 24 29 23
16
24
24 25 23
24
24
24
24
41
41
21 32 25
21
324
10 46
10
324
51 30
32
32
30 51 24 32
46
24
4
4
16 28
16
25
25
30
30
28
46
46
25 26 25 26 25
21 35
21
325
10 50
10
325
35
50
24 32
24
4
4
16 31
16
32
31
25
25
49
49325
21 37
21
325
10 53
10
37
26
53
24
24
4
4
32
16 33
16
32
33
52
52325
25 25
21 37
21
325 10
10
37
55
55
24 30 30 24
4
4
16 33
16
33
53
53325
21 35
21
325
51
10 55
10
35
55
24
24
4
4
16 31
16
31
53
53325
21 32 25
21
325
10 53
10
32
53
4
4
23 23
16 28
16
28
52
52325
21 29 29 21
325
10 50
10
50
4
4
48
16 24
16
24
49
49325
325
21 21
10 46
10
46
4
4
16 18 22
16
18
46
46
325
10 40
10
325
40
4
4
17
17
41
41
324
11 32 21
11
324
32
4
4
16 16
34
34
323
11 25
11
323
25
4
4
37
26
26
322
11
11
322
4
4
16
17
17
321 9 9
321
4 9 9 4
319 10 319
4 4
318 318
(c) s = 0, m = 297, = 2.5%
(d) s = 0, m = 297
Figure 3.11: A TriQuad spinning web with mesh width 5 m: (a)–(c) element forces and (d)
displacements magnified 10 times.
31
The general conclusion from the prestressing analysis presented above must be that only
square meshes will produce a prestressed spinning space web, both for quadratic and triangular geometries.
3.6
Out-of-plane stiffness
A preliminary comparison will be made of the out-of-plane stiffness for the various mesh
topologies. The linearised local out-of-plane stiffness for b in-plane bars is [9, 55]:
Kµ = b
t
(3.41)
Note that the elastic stiffness can be disregarded. For a constant element distance dc and
uniform stress q0 , the out-of-plane stiffness for the various mesh topologies is
K3 =
q0 √2
√
dc
6 32 3
√ dc
3
≈ 3.464q0
q0 dc
= 4q0
d
√c d
3q0 √c
K6 = 3 dc 3 ≈ 5.196q0
K4 = 4
√
(3.42a)
(3.42b)
(3.42c)
3
Thus, for identical uniform in-plane stresses, the hexagonal web will be stiffer out-ofplane than both the triangular and the square ones. Nevertheless, as shown in section
3.5.2, triangular and hexagonal meshes are not adequately stress under the influence of
centrifugal forces.
A global out-of-plane stiffness analysis is performed by inverting the geometric tangent
stiffness matrix to obtain the geometric flexibility matrix. Only the centre node is fixed
in the analysis. First, an analysis was done with a 100 × 100 m2 web and then with a
10 × 10 m2 web. The material properties are as in the previous section, the corner masses
are 10 kg each and the angular velocity is 1 rad/s. The out-of-plane flexibilities for the
100×100 m2 web are shown in Figure 3.12. The general trends is that the flexibilities tend
to increase as the mesh width decreases, i.e. the web will behave more like a membrane.
An initial analysis of the 10 × 10 m2 web showed that the combination of very light web
material and small web dimensions yielded too small element forces for a straight edge
tether configuration. To increase the forces, the angular velocity was set to 3 rad/s and
the edge tether was curved to = 2.5%. The resulting force distribution is shown in
Figure 3.13(a). The uniform force field generated by the edge cable dominates over the
centrifugal forces (which can be neglected for small web dimensions). The resulting outof-plane flexibilities are shown in Figure 3.13(a). By the changes in the angular velocity
and edge curvature, the flexibilities are now reasonably low. Exactly what “reasonably
32
low” means is presently not clear; no requirements for the maximum out-of-plane stiffness
of a space web has been stated as no mission involving this type of structure has been
formulated.
This analysis confirms that for a large web, the forces generated by the curved edge tether
can be neglected compared to the forces by the centrifugal force field. In a smaller web, it
is the curved edges that prestress the web to an approximately uniform prestress.
108
108
108
108
107
107
106
106
106
106
106
106
106
107
107
108
108
108 108
108
238
187
162
147
138
131
127
125
124
125
127
131
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147
162
187
238 108
115
115
114
114
113
113
112
112
112
111
111
111
112
112
112
113
113
114
114
115
115
115
232
190
167
153
144
138
133
130
128
128
128
130
133
138
144
153
167
190
232
115
108
187
147
128
116
108
103
100
98
97
98
100
103
108
116
128
147
187 108
114
190
159
141
129
121
115
111
109
107
107
107
109
111
115
121
129
141
159
190
114
108
162
128
111
101
94
89
86
84
83
84
86
89
94
101
111
128
162 108
114
167
141
125
115
108
102
98
96
94
93
94
96
98
102
108
115
125
141
167
114
107
147
116
101
91
85
79
76
74
73
74
76
79
85
91
101
116
147 107
113
153
129
115
105
98
93
89
86
84
84
84
86
89
93
98
105
115
129
153
113
113
144
121
108
98
91
85
81
78
76
75
76
78
81
85
91
98
108
121
144
113
107
138
108
94
85
77
72
68
65
64
65
68
72
77
85
94
108
138 107
112
138
115
102
93
85
80
75
71
68
67
68
71
75
80
85
93
102
115
138
112
106
131
103
89
79
72
66
61
57
56
57
61
66
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89
103
131 106
112
133
111
98
89
81
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64
61
59
61
64
69
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81
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109
96
86
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71
64
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52
49
52
58
64
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96
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130
112
106
127
100
86
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61
54
49
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49
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68
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86
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127 106
106
125
98
84
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49
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31
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49
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125 106
111
128
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94
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61
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42
33
42
52
61
68
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84
94
107
128
111
106
124
97
83
73
64
56
46
31
0
31
46
56
64
73
83
97
124 106
111
128
107
93
84
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59
49
33
0
33
49
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84
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125 106
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106
127
100
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68
61
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49
54
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68
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100
127 106
112
133
111
98
89
81
75
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64
61
59
61
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81
89
98
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133
112
106
131
103
89
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66
61
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103
131 106
112
138
115
102
93
85
80
75
71
68
67
68
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85
93
102
115
138
112
113
144
121
108
98
91
85
81
78
76
75
76
78
81
85
91
98
108
121
144
113
107
138
108
94
85
77
72
68
65
64
65
68
72
77
85
94
108
138 107
107
147
116
101
91
85
79
76
74
73
74
76
79
85
91
101
116
147 107
113
153
129
115
105
98
93
89
86
84
84
84
86
89
93
98
105
115
129
153
113
108
162
128
111
101
94
89
86
84
83
84
86
89
94
101
111
128
162 108
114
167
141
125
115
108
102
98
96
94
93
94
96
98
102
108
115
125
141
167
114
108
187
147
128
116
108
103
100
98
97
98
100
103
108
116
128
147
187 108
114
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159
141
129
121
115
111
109
107
107
107
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111
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121
129
141
159
190
114
115
232
190
167
153
144
138
133
130
128
128
128
130
133
138
144
153
167
190
232
115
108
238
187
162
147
138
131
127
125
124
125
127
131
138
147
162
187
238 108
108
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108 108
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112
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111
111
112
112
112
113
113
114
114
115
115
(a) s=9, m=366, max flex = 238 mm/N
115
(b) s=10, m=447, max flex = 232 mm/N
115 115 115 114 114 114 113 113 113 113 113 113 113 113 113 113 113 113 113 114 114 114 115 115 115 115
115
312 248 214 194 180 171 163 158 154 151 149 148 148 148 149 151 154 158 163 171 180 194 214 248 312 115
115
248 199 174 158 147 139 133 129 125 123 121 120 120 120 121 123 125 129 133 139 147 158 174 199 248 115
120 120 120 119 119 119 119 118 118 118 118 118 117 117 117 117 117 117 117 117 117 117 117 118 118 118 118 118 119 119 119 119 120 120 120
120
373 299 259 234 217 205 195 188 182 177 174 171 168 167 165 165 164 165 165 167 168 171 174 177 182 188 195 205 217 234 259 299 373 120
120
299 244 214 195 181 171 164 157 152 148 145 142 140 139 138 137 137 137 138 139 140 142 145 148 152 157 164 171 181 195 214 244 299 120
119
259 214 190 173 162 153 146 141 136 133 130 127 125 124 123 122 122 122 123 124 125 127 130 133 136 141 146 153 162 173 190 214 259 119
115
214 174 153 140 130 123 118 114 111 108 107 106 105 106 107 108 111 114 118 123 130 140 153 174 214 115
119
234 195 173 159 149 141 134 129 125 122 119 117 115 114 113 112 112 112 113 114 115 117 119 122 125 129 134 141 149 159 173 195 234 119
114
194 158 140 128 119 113 108 104 101 98
97
96
95
96
97
98
101 104 108 113 119 128 140 158 194 114
119
217 181 162 149 139 132 126 121 117 114 111 109 107 106 105 104 104 104 105 106 107 109 111 114 117 121 126 132 139 149 162 181 217 119
114
180 147 130 119 111 105 100 96
89
88
88
88
89
91
93
119
205 171 153 141 132 125 119 114 110 107 104 102 100 99 98 98 97 98 98 99 100 102 104 107 110 114 119 125 132 141 153 171 205 119
118
195 164 146 134 126 119 113 109 105 102 99 97 95 93 92 92 92 92 92 93 95 97 99 102 105 109 113 119 126 134 146 164 195 118
93
91
96
100 105 111 119 130 147 180 114
114
171 139 123 113 105 99
94
90
87
84
82
81
81
81
82
84
87
90
94
99
105 113 123 139 171 114
118
188 157 141 129 121 114 109 104 100 97 94 92 90 89 87 87 87 87 87 89 90 92 94 97 100 104 109 114 121 129 141 157 188 118
113
163 133 118 108 100 94
89
85
81
78
76
75
75
75
76
78
81
85
89
94
100 108 118 133 163 113
118
182 152 136 125 117 110 105 100 96 93 90 88 86 84 83 82 82 82 83 84 86 88 90 93 96 100 105 110 117 125 136 152 182 118
113
158 129 114 104 96
90
85
80
76
73
71
69
69
69
71
73
76
80
85
90
96
104 114 129 158 113
118
177 148 133 122 114 107 102 97 93 90 87 84 82 80 78 77 77 77 78 80 82 84 87 90 93 97 102 107 114 122 133 148 177 118
118
174 145 130 119 111 104 99 94 90 87 83 80 78 75 74 73 72 73 74 75 78 80 83 87 90 94 99 104 111 119 130 145 174 118
113
154 125 111 101 93
87
81
76
72
68
65
63
62
63
65
68
72
76
81
87
93
101 111 125 154 113
117
171 142 127 117 109 102 97 92 88 84 80 77 74 71 69 68 67 68 69 71 74 77 80 84 88 92 97 102 109 117 127 142 171 117
113
151 123 108 98
84
78
73
68
64
60
57
55
57
60
64
68
73
78
84
91
98
117
168 140 125 115 107 100 95 90 86 82 78 74 71 67 65 63 62 63 65 67 71 74 78 82 86 90 95 100 107 115 125 140 168 117
117
167 139 124 114 106 99 93 89 84 80 75 71 67 63 60 57 55 57 60 63 67 71 75 80 84 89 93 99 106 114 124 139 167 117
117
165 138 123 113 105 98 92 87 83 78 74 69 65 60 54 49 46 49 54 60 65 69 74 78 83 87 92 98 105 113 123 138 165 117
117
165 137 122 112 104 98 92 87 82 77 73 68 63 57 49 40 32 40 49 57 63 68 73 77 82 87 92 98 104 112 122 137 165 117
117
164 137 122 112 104 97 92 87 82 77 72 67 62 55 46 32 0
117
165 137 122 112 104 98 92 87 82 77 73 68 63 57 49 40 32 40 49 57 63 68 73 77 82 87 92 98 104 112 122 137 165 117
117
165 138 123 113 105 98 92 87 83 78 74 69 65 60 54 49 46 49 54 60 65 69 74 78 83 87 92 98 105 113 123 138 165 117
117
167 139 124 114 106 99 93 89 84 80 75 71 67 63 60 57 55 57 60 63 67 71 75 80 84 89 93 99 106 114 124 139 167 117
117
168 140 125 115 107 100 95 90 86 82 78 74 71 67 65 63 62 63 65 67 71 74 78 82 86 90 95 100 107 115 125 140 168 117
117
171 142 127 117 109 102 97 92 88 84 80 77 74 71 69 68 67 68 69 71 74 77 80 84 88 92 97 102 109 117 127 142 171 117
118
174 145 130 119 111 104 99 94 90 87 83 80 78 75 74 73 72 73 74 75 78 80 83 87 90 94 99 104 111 119 130 145 174 118
118
177 148 133 122 114 107 102 97 93 90 87 84 82 80 78 77 77 77 78 80 82 84 87 90 93 97 102 107 114 122 133 148 177 118
118
182 152 136 125 117 110 105 100 96 93 90 88 86 84 83 82 82 82 83 84 86 88 90 93 96 100 105 110 117 125 136 152 182 118
118
188 157 141 129 121 114 109 104 100 97 94 92 90 89 87 87 87 87 87 89 90 92 94 97 100 104 109 114 121 129 141 157 188 118
118
195 164 146 134 126 119 113 109 105 102 99 97 95 93 92 92 92 92 92 93 95 97 99 102 105 109 113 119 126 134 146 164 195 118
119
205 171 153 141 132 125 119 114 110 107 104 102 100 99 98 98 97 98 98 99 100 102 104 107 110 114 119 125 132 141 153 171 205 119
119
217 181 162 149 139 132 126 121 117 114 111 109 107 106 105 104 104 104 105 106 107 109 111 114 117 121 126 132 139 149 162 181 217 119
119
234 195 173 159 149 141 134 129 125 122 119 117 115 114 113 112 112 112 113 114 115 117 119 122 125 129 134 141 149 159 173 195 234 119
91
108 123 151 113
113
149 121 107 97
89
82
76
71
65
60
54
49
46
49
54
60
65
71
76
82
89
97
107 121 149 113
113
148 120 106 96
88
81
75
69
63
57
49
40
31
40
49
57
63
69
75
81
88
96
106 120 148 113
113
148 120 105 95
88
81
75
69
62
55
46
31
0
31
46
55
62
69
75
81
88
95
105 120 148 113
113
148 120 106 96
88
81
75
69
63
57
49
40
31
40
49
57
63
69
75
81
88
96
106 120 148 113
113
149 121 107 97
89
82
76
71
65
60
54
49
46
49
54
60
65
71
76
82
89
97
107 121 149 113
113
151 123 108 98
91
84
78
73
68
64
60
57
55
57
60
64
68
73
78
84
91
98
108 123 151 113
113
154 125 111 101 93
87
81
76
72
68
65
63
62
63
65
68
72
76
81
87
93
101 111 125 154 113
113
158 129 114 104 96
90
85
80
76
73
71
69
69
69
71
73
76
80
85
90
96
113
163 133 118 108 100 94
89
85
81
78
76
75
75
75
76
78
81
85
89
94
100 108 118 133 163 113
114
171 139 123 113 105 99
94
90
87
84
82
81
81
81
82
84
87
90
94
99
105 113 123 139 171 114
114
180 147 130 119 111 105 100 96
93
91
89
88
88
88
89
91
93
96
100 105 111 119 130 147 180 114
114
194 158 140 128 119 113 108 104 101 98
97
96
95
96
97
98
101 104 108 113 119 128 140 158 194 114
115
214 174 153 140 130 123 118 114 111 108 107 106 105 106 107 108 111 114 118 123 130 140 153 174 214 115
104 114 129 158 113
119
115
248 199 174 158 147 139 133 129 125 123 121 120 120 120 121 123 125 129 133 139 147 158 174 199 248 115
115
115
32 46 55 62 67 72 77 82 87 92 97 104 112 122 137 164 117
259 214 190 173 162 153 146 141 136 133 130 127 125 124 123 122 122 122 123 124 125 127 130 133 136 141 146 153 162 173 190 214 259 119
120
299 244 214 195 181 171 164 157 152 148 145 142 140 139 138 137 137 137 138 139 140 142 145 148 152 157 164 171 181 195 214 244 299 120
312 248 214 194 180 171 163 158 154 151 149 148 148 148 149 151 154 158 163 171 180 194 214 248 312 115
120
373 299 259 234 217 205 195 188 182 177 174 171 168 167 165 165 164 165 165 167 168 171 174 177 182 188 195 205 217 234 259 299 373 120
115 115 115 114 114 114 113 113 113 113 113 113 113 113 113 113 113 113 113 114 114 114 115 115 115 115
120
120 120 119 119 119 119 118 118 118 118 118 117 117 117 117 117 117 117 117 117 117 117 118 118 118 118 118 119 119 119 119 120 120 120
(c) s=13, m=738, max flex = 312 mm/N
(d) s=17, m=1238, max flex = 373 mm/N
Figure 3.12: Out-of-plane flexibilities (mm/N) for a 100 × 100 m2 spinning web and different
mesh widths: (a) 6 m, (b) 5 m, (c) 4 m and (d) 3 m.
33
413
413
0
412
412
4
4
4 0 4
4
4
4
412
4
4
412
4
4
412
4
412
411
5
4
411
4
4
411
4
5
411
4
4
4
4
4
412
5
4
412
413
5
4
5
4
4 0
4
5
4
5
5 0
5
4
5
5
5 0
5
5
5
4
5
5 0
5
5
5
5 0
5
5
5
5 0
5
5
5
5
5
5
5
4
5
5
5
5
5
5
5
4
4
5
0 5
5
5
4
5
4
5
4
5
4
5
0 5
5
4
5
0 5
5
4
4
5
4
5
5
4
0 4
4
108
106
75
75
108
116
117
116
115
113
112
111
111
110
111
111
112
113
115
116
117
116
108
75
74
107
117
119
118
117
115
114
113
112
112
112
113
114
115
117
118
119
117
107
74
74
105
116
118
118
117
115
113
111
110
109
110
111
113
115
117
118
118
116
105 74
74
103
115
117
117
115
112
110
107
106
105
106
107
110
112
115
117
117
115
103 74
74
101
113
115
115
112
109
106
102
100
99
100
102
106
109
112
115
115
113
101 74
74
100
112
114
113
110
106
101
96
92
89
92
96
101
106
110
113
114
112
100 74
74
99
111
113
111
107
102
96
88
81
76
81
88
96
102
107
111
113
111
99 74
74
98
111
112
110
106
100
92
81
67
52
67
81
92
100
106
110
112
111
98 74
74
98
110
112
109
105
99
89
76
52
0
52
76
89
99
105
109
112
110
98 74
74
98
111
112
110
106
100
92
81
67
52
67
81
92
100
106
110
112
111
98 74
74
99
111
113
111
107
102
96
88
81
76
81
88
96
102
107
111
113
111
99 74
74
100
112
114
113
110
106
101
96
92
89
92
96
101
106
110
113
114
112
100 74
74
101
113
115
115
112
109
106
102
100
99
100
102
106
109
112
115
115
113
101 74
74
103
115
117
117
115
112
110
107
106
105
106
107
110
112
115
117
117
115
103 74
74
105
116
118
118
117
115
113
111
110
109
110
111
113
115
117
118
118
116
105 74
74
107
117
119
118
117
115
114
113
112
112
112
113
114
115
117
118
119
117
107 74
75
108
116
117
116
115
113
112
111
111
110
111
111
112
113
115
116
117
116
108
75
75
106
108
107
105
103
101
100
99
98
98
98
99
100
101
103
105
107
108
106
75
411
411
412
4
5
412
4
5
412
4
4
75
75
107
4
4
75
105
4
4
5
74
103
411
5
5
4
74
101
4
4
74
100
411
5
5
4
74
99
411
5
4
74
98
4
4
74
98
411
5
5
74
98
4
4
5
74
99
411
5
5
4
5
5
5
5
4
5
74
100
4
4
74
101
411
5
5
4
4
5
5
5
5
4
5
5
4
5
74
103
4
4
74
105
4
5
5
74
107
412
5
4
4
4
5
5
5
5
4
5
4
5
5
5
4
4
74
108
412
5
5
4
4
5
5
5
0 5
5
5
5
5
4
5
4
5
5
5
4
74
106
4
4
5
5
5
5
5
4
5
5
5
5
5
4
5
4
5
5
5
75
75
412
5
4
4
5
4
4
5
4
75
75
412
5
5
4
5
5
5
5
5
5
5
5
5
4
5
4
5
5
5
4
0 5
5
5
5
5
5
5
4
4
4
4
5
5
5
5
5
5
5
5
5
5
5
5
5
0 5
5
5
5
5
4
5
4
5
5
5
4
5
5
5
5
4
5
5
4
5
5
5
5
5
5
5
5
5
5
5
4
413
4 0 413
4
4
0 4
412
4
4
5
4
5
5
5
5
5
5
5
5
5
5
5
0 5
5
5
5
5
5
5
5
5
5
5
4
5
5
5
4
5
4
5
5
5
5
5
0 5
5
5
5
5 0
5
5
5
5
5 0 5
5
5
5
5
5
5 0
5
4
5
5
5
5
5
0 5
5
5
5
5
5
5
5
4
4
5
5
5
5
4
4
4
4
4
5
4
5
5
5
0 5
5
5
5
5
0 5
5
5
5
5
5
5
5
5
0 5
5
4
0 4
4
5
5
5
4
4
5
0 5
5
5
5
5 0 5
4
5
5
5
5
5
4
4
4
5
0 5
5
5
5
4
5
5
5
5
5
5
4
4
5
5
5
5
5
5
5
5
4
5
4
4
4
4
5
412
412
412
412
4
5
4
4
5
5
5
5
5
5
5
4
5
4
4
5
5
5
5
5
5
5
5
4
5
4
4
5
5
4
4
5
5
5
5
5
5
5
5
5
4
4
412
4
5
5
5
5
5
5
4
5
5
5
5
5
5
5
5
5
412
411
411
4
5
4
5
5
5
4
4
5
5
5
5
5
5
4
5
4
5
5
5
5
4
4
5
5
5 0 5
5
5
5
5
4
4
5
5
412
5
5
4
5
5
5
4
4
5
5
5
4
5
4
5
5
5
4
4
5
5
5
5
5
5
411
411
411
4
5
4
5
5
5 0 5
5
5
5
4
4
5
5
4
4
5
5
5 0 5
5
5
5
5
411
4
4
4
4
5
5
411
4
5
4
5
5
5
4
4
5
5
411
4
5
4
5
5
5
4
4
5
5
5
4
4
411
4
5
5
4
5
5
5
4
5
411
4
5
4
5
4
4
5
5
411
412
5
4
5
4
5
5 0 5
4
5
4
5
5
4
5
4
5
4
4
5
5
412
4
5
5
5 0 5
4
4
4
411
4
5
5
4
4
5
4
5
412
4
5
4
4 0 4
4
5
4
412
412
4
412
4
5
5
5
5
5
5
5
5
5
5
5
5
5
5
4
4
4 0 4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4 0 4
412
4
4
4
5
5
5
5
5
5
5
5
5
5
5
5
5
5
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4 0 413
0 4
412 411 411 411 411 411 411 411 411 412 412 412 412
412
412
412 413
413 412 412
(a) s = 10, m = 447, = 2.5%
75
75
75
74
74
74
74
74
74
74
74
74
74
74
74
74
74
74
75
75
75
(b) max flex = 119 mm/N
Figure 3.13: Element forces (N) and out-of-plane flexibilities (mm/N) for a 10 × 10 m2 spinning
web with mesh width 0.5 m.
3.7
Eigenfrequencies and eigenmodes
The eigenfrequencies and free vibration modes of the system are of importance to judge
the effects of travelling waves, [4], out-of-plane damping requirements, and other dynamic
phenomena, e.g. orbital maneuvering. In this analysis, the material properties and dimensions are the same as in the stress and out-of-plane analyses above with the addition of a
mass of 100 kg at the centre node.
The exact tangent stiffness matrix for a stressed three-dimensional truss element with
nodes i and j is, [9]:
⎡ 2
⎤
cx
cx cy
cx cz
−c2x −cx cy −cx cz
⎢ cx cy
c2y
cy cz −cx cy −c2y −cy cz ⎥
⎢
⎥
2
2 ⎥
⎢
−
t
AE
c
c
c
c
c
−c
c
−c
c
−c
ij
ij
x
z
y
z
x
z
y
z
z
z ⎥
⎢
Ke =
2
⎢ −c2x −cx cy −cx cz
⎥
c
c
c
c
c
ij
x
y
x
z ⎥
x
⎢
2
2
⎣−cx cy −cy −cy cz cx cy
cy
cy cz ⎦
2
−cx cz −cy cz −cz
cx cz
cy cz
c2z
⎡
⎤
(3.43)
1
−1
⎢
⎥
1
−1
⎢
⎥
⎥
tij ⎢
1
−1
⎢
⎥
+
⎥
1
ij ⎢
⎢−1
⎥
⎣
⎦
−1
1
−1
1
34
where
xi − xj
ij
yi − y j
cy =
ij
zi − zj
cz =
ij
cx =
(3.44a)
(3.44b)
(3.44c)
The consistent mass matrix is [2]:
⎡
⎤
2
1
⎢ 2
1 ⎥
⎢
⎥
⎥
ρAij ij ⎢
2
1
e
⎢
⎥
MC =
⎥
2
6 ⎢
⎢1
⎥
⎣ 1
2 ⎦
1
2
(3.45)
When computing the eigenfrequencies and eigenmodes of the web it is assumed that the
web is completely free in space, i.e. no node is constrained. Hence, six of the frequencies
belongs to three global rigid body translations (f = 0 Hz) and three rigid body rotations
(f = 0 Hz). The rigid body motions are identified by the eigenmodes and the four lowest
non-rigid body motions are plotted.
The eigenfrequencies and corresponding eigenmodes for a 10 × 10 m2 (3 rad/s) and the
100 × 100 m2 (1 rad/s) QuadQuad webs are shown in Figure 3.14. The element forces
are those generated by the centrifugal forces, Figure 3.13(a) for the small web and Figure
3.8 for the large web. It should be noted that the frequency for the rigid body rotations
was f10m = 0.477 Hz and f100m = 0.159 Hz. The low eigenfrequencies further confirm
the flexible nature of the system. Eigenmodes 3 and 4, Figure 3.14(c) and (d), bear some
similarities with in-plane mechanisms, so it is clear that the system has some weak in-plane
modes.
3.8
Cable fracture due to micro-meteoroid impact
The risk of micro meteoroid damage to web-like structures was estimated by MacNeal
[24]. He observed that design of compression members is usually governed by the elastic
stability, whereas for tension members, which can be very thin and long, fracture due
to meteoroids will be the critical design consideration [24]. The micro-meteoroid flux in
space is
10−17
(particles/m2 /s)
(3.46)
Np =
mp
where mp is the particle mass (kg). The particle mass that will just break the cable is
3
d
4πσu
mp = 3/2 1/2
(3.47)
2.5
vp vo
35
(a) f10m = 0.114 Hz, f100m = 0.100 Hz
(b) f10m = 0.482 Hz, f100m = 0.160 Hz
(c) f10m = 0.573 Hz, f100m = 0.219 Hz
(d) f10m = 0.573 Hz, f100m = 0.219 Hz
Figure 3.14: Four lowest eigenmodes of the QuadQuad web with a sag-to-span ratio of 2.5%: (a)
out-of-plane mode,(b) in-plane mode, (c) and (d) similar in-plane mechanism-like
modes.
where d is the cable diameter, the mean velocity of meteoroids vp = 30 km/s, the empirical velocity parameter vo = 6.5 km/s and the ultimate strength of the cable material
(Zylon) σu = 5.8 GPa. The number of fractures per unit length of cables per unit time
is
πd
Nf =
(3.48)
Np
2
Combing (3.46)–(3.48) yields the number of fractures for a Zylon cable:
1.41 · 10−18
(fractures/m/s)
d2
4.46 · 10−5
(fractures/m/year)
=
d2mm
Nf =
(3.49)
where dmm is the cable diameter in millimetres.
The space web has a certain degree of redundancy which means that it can still function
satisfactorily until a significant portion of the members have been fractured. MacNeal [24]
shows that the probability of failure for the structure (or a member composed of several
individual elements) is
s
(3.50)
Pf = 1 − e−Nf τ
36
where is the element length, τ is the time and s is the degree of statical indeterminacy
(degree of redundancy).
Schuerch and Hedgepeth [49] studied the number of fractures for the LOFT baseline
design. A conductor mesh of 2.5 mm wide and 6 µm thick aluminium will experience
450 fractures per year. This equals one fracture per 20 × 20 m2 in ten years and will not
significantly affect the performance of the LOFT. Fracture of a supporting cable is more
critical and the cables were design to produce a single-fracture probability of less than 1%
in four years. For the space web, the critical elements are the edge cables, which should
be designed using several wires on the local level and as a framework on the global level,
Figure 3.15.
For the present space webs, the diameter of the thinnest strand is 0.43 mm, which leads
to 4.46 · 10−5 /0.432 ≈ 2.4 · 10−4 fractures/m/year. For an web element length of 30
mm and a mission time of 10 years, the failure probability due to micro-meteoroids is
Pf = (7.2 · 10− 5)s , which is very small. Hence, it can be concluded that the failure risk
for a space webs mission is not due to micro-meteoroids.
Figure 3.15: Redundant design of the supporting cable in the LOFT antenna, from [49].
37
3.9
Choice of web architecture
The optimum layout of the web is determined by the following parameters: (i) prestressability, (ii) manufacturability, (iii) mass, (iv) out-of-plane stiffness and (v) eigenfrequencies, but not necessarily in that order. From the analysis above it is clear that only the
square mesh is prestressable by centrifugal forces. A square web with a square mesh is
better from a manufacturing viewpoint and the out-of-plane stiffness and eigenfrequencies
can be adjusted by the angular velocity and corner masses. The remainder of this report
will thus only be concerned with the quadratic web with a quadratic mesh.
38
Chapter 4
Folding patterns for centrifugal
deployment
Fundamental to a successful deployment is a good folding pattern. Several folding patterns
for large space structures have been proposed for various deployment approaches. A good
example is the Miura-Ori [31] for the efficient folding of square solar sails. However, not
all of the proposed patterns will work for a deployment driven by centrifugal forces. The
literature review below studies the available patterns for spin deployment and identifies
the most suitable one for the present application.
4.1
4.1.1
Literature review
Wrapping around a hub
Several studies have dealt with the folding of a circular solar sail by wrapping it around
a cylindrical hub, Figure 4.1. The origin of this idea can be traced back to the early
1960s, [10]. Scheel [47] developed a pattern with straight folding lines, Figure 4.2. In
1992, Temple and Oswald from the company Cambridge Consultants performed a study
of a 276-m-diameter solar sail. The sail had 36 radial spars emanating from the central
hub [10, 28]. Guest and Pellegrino [10] later derived analytical relationships for zerothickness and thin membranes wrapped around polygonal hubs. Inextensional wrapping
around a circular hub is not possible since a curved fold line requires that the membrane is
curved in opposite directions [10]. On the other hand, extensional folding around a cylinder
is possible, but then the membrane is subjected to localised wrinkling and stretching near
the hub [41].
39
Figure 4.1: Hub-wrapping scheme derived by Guest and Pellegrino, [10].
Recently, Japanese researchers have studied the folding scheme by Scheel [47]. Furuya et
al. [5] performed deployment experiment of 300-mm-diameter and 12.5 µm thick circular
sails. They found that for a given angular velocity, the deployment ratio for a segmented
sail folded in a fan-like manner was higher than a single-sheet sail folded according the
Scheel pattern. The reason for the higher deployment ratios is that the segmented fan fold
pattern generates larger deployment forces than the Scheel pattern. Complete deployment
is not possible as the folding of the membrane creates permanent creases which cannot be
fully removed. A two-dimensional folding pattern, e.g. Scheel, has a smaller deployment
ratio than a two-dimensional one, e.g. fan, due to the constraint of creases [6].
Figure 4.2: Hub-wrapping folding pattern by Scheel [47].
For the present project, the hub-wrapping patterns by Scheel [47] and Guest and Pellegrino
[10] have some disadvantages:
• they are developed for circular sheets and not for triangular or square ones, and
40
• the fold lines do not follow the nodal lines of the web, which makes the development
of a folding routine more complicated.
4.1.2
Star pattern
Schuerch and Hedgepeth [49] suggest a folding pattern for the LOFT system where the
supporting structure is sheared to be folded into the hub. This pattern produces a star-like
shape of the structure being folded, Figure 4.3. For the LOFT system the number of star
arms is eight.
Figure 4.3: Star-shaped folding pattern for the LOFT system, [49].
Melnikov and Koshelev [29] proposed a similar pattern for the folding of spin-deployed
circular space structures, Figure 4.4. The arms of the star can then be folded in various
ways as discussed in [29]. Two approaches appear especially interesting due to their simplicity: (i) folding of the arms in a zig-zag manner towards the hub or (ii) coiling the arms
around the hub as in the hub-wrapping concept. If the arm are folded in a zig-zag manner,
the end of the arm must be connected to the hub with a cable so that the deployment
rate of the arm end can be controlled. It is seen from Figure 4.4 that the star pattern is
not exclusively for circular structures, as the folding is linear towards the hub.
41
Figure 4.4: Folding of a circular structure towards the hub to obtain a star-like pattern, [29].
4.2
Proposed folding pattern
It was decided that the star pattern was the most suitable one for the space web as the
web can be deployed in two stages, which presumably makes it easier to control. Another
advantage is the that fold lines coincide with the nodal lines, so that the folding can be
done using simple analytical relationships.
Complete star pattern
The first step of the folding is to fold the web into a ‘star’-like shape. The y-coordinate
of a node on the centre line is described as
y = y0 sin
θ
2
(4.1)
where θ is the fold angle along the centre line (θ = 180◦ for a fully deployed configuration
and 0◦ when completely folded) and y0 is the y-coordinate of the node in the deployed
configuration. Equation (4.1) is the mapping scheme of the nodes along the centre line.
For a node i in the first and second quadrants and lying between side lines the mapping
42
scheme is
xi = x0i cos φ
θ
yi = y0i sin + |x0i | sin φ
2
where
⎛
φ = arccos ⎝
sin 2θ cot πn +
'
tan2
π
n
tan πn 1 + cot2
(4.2)
(4.3)
+ cos2
θ
2
π
n
⎞
⎠
(4.4)
Equations (4.2) and (4.3) describe the movement of a node in the x–y plane during the
‘star’ folding process. Figure 4.5 shows how nodes at positions 1/4, 1/2 and 3/4 of the
fold line moves from the fully deployed to the fully folded configurations; it is evident that
the movement is not linear.
1/4
1/2
3/4
θ = 160◦
x)
θ = 140◦
lin
e
(y
=
θ = 120◦
Centre line (x = 0)
Si
de
θ = 100◦
θ = 80◦
θ = 60◦
θ = 40◦
θ = 20◦
Figure 4.5: Movement of nodes at positions 1/4, 1/2 and 3/4 along the fold line for a square
web. Note that the figure shows only one eighth of the quadratic web.
43
θ = 180◦
θ = 160◦
θ = 140◦
θ = 120◦
θ = 100◦
θ = 80◦
θ = 60◦
θ = 40◦
θ = 20◦
θ = 0◦
Figure 4.6: Complete star folding sequence for a square sheet (note that all lines are not fold
lines).
44
Since the surface in the z-direction has a zig-zag pattern, the mapping for the z-coordinate
is a bit more complex. Assuming that the distance between fold lines is 2∆ in the interior
and ∆ at the centre and along the edges, the relative position of the node between two
fold lines is computed as
(4.5)
χ = y0i /2∆ − y0i /2∆
where x rounds x to the nearest integer towards −∞. The mapping scheme for the
z-coordinate becomes
⎧
⎪
⎨±2χ∆ cos θ
if χ ≤ 0.5
2
zi =
(4.6)
θ
⎪
⎩±2(1 − χ)∆ cos
if χ > 0.5
2
where the ‘−’ sign holds if y0i /2∆ is an even number and the ‘+’ sign otherwise.
In summary, the star pattern is neatly described by analytical relationships which maps
the coordinates from a given position of the deployed configuration to a position of the
folded configuration described by the single variable θ. In this way, the curved edges of
the web can be mapped to the folding pattern even though the edge nodes do not lie on
the fold lines.
4.2.1
Incomplete star pattern
If the central satellite is modelled as a point mass with inertia, the complete star mapping
scheme, which folds the web towards the centre point, can be used. If, however, the central
satellite is modelled with shell elements and its physical dimensions, the complete star
mapping cannot be used. In such a case, the star mapping must be modified to have the
two innermost rings of elements deployed. The starting configuration for this scheme is the
star folding with folding angle θ = 0◦ (all nodes of the arms are positioned along straight
lines). From this position, the two inner rings of elements are deployed. Note that the
incomplete star mapping only works for works for fold angle θ = 0, since the arms have
to be completely folded before the two innermost rings can be deployed. The incomplete
star mapping scheme is written as
x0i cos φ + 2∆sgn(x0i ) tan πn 1 − sin πn
if |x0i | > 2∆ tan πn ; y0i > 2∆
(4.7)
xi =
x0i
if |x0i | ≤ 2∆ tan πn
and
⎧
θ
π
π
⎪
⎨y0i sin 2 + |x0i | sin φ + 2∆ tan n 1 − cos n
yi = 2∆
⎪
⎩
y0i
45
if |x0i | > 2∆ tan πn ; y0i > 2∆
if |x0i | ≤ 2∆ tan πn ; y0i ≥ 2∆
if y0i < 2∆
(4.8)
The z-coordinates are basically unchanged, except for the nodes lying within the first two
element rings defined by the y-coordinate:
Eq. (4.6) if y0i ≥ 2∆
(4.9)
zi =
0
if y0i < 2∆
A comparison between the complete and incomplete star mapping schemes is shown in
Figure 4.7. In order to facilitate a tight wrapping around the hub, the square deployed
part of the web is made circular while preserving the length of the all elements. This last
change is very important since it otherwise will be a significant space between the coiled
web and the surface of the cylindrical hub.
A disadvantage with the proposed incomplete star pattern is that the size of the deployed
inner portion of the web is dependent on the mesh size of the web, which may produces an
unrealistically large radius of the stowed package. However, a general folding routine, where
the fold lines do not have to coincide with the nodal lines, was deemed too complicated
and costly to implement for the present project.
(a)
(b)
Figure 4.7: Visual comparison of the (a) complete and (b) incomplete star patterns.
4.2.2
Zig-zag folding of star arms
The second step is to fold the ‘star’ arms towards the central satellite using zig-zag folds.
This step is somewhat simpler than the previous step since the nodes only move in-plane
and a variant of the zig-zag folding, Eq. (4.6), can be used. The routine outlined below
only works for an equal division between the node along the star arms. The radius from
the centre of the web to node i on the star arm is
'
(4.10)
ri = x2i + yi2
and the angle of the arm is computed as
⎧
yi
⎪
⎨arcsin r
i
β=
⎪
⎩π − arcsin yi
ri
46
if
if
xi
≥0
ri
xi
<0
ri
(4.11)
The position of the folded node measured along the initial arm orientation is, Figure 4.8,
θ
(4.12)
2
where r0 is the radius of the to the first node on the star arm, Figure 4.8. The distance
perpendicular to the initial arm orientation is computed as follows. The relative position
of the node between two fold lines is
ri,1 = r0 + (ri − r0 ) sin
χa = (ri − r0 )/2∆a − (ri − r0 )/2∆a (4.13)
which is similar to Eq. (4.5). The movement of the node perpendicular to the initial
orientation is thus
⎧
⎪
⎨±2χa ∆a cos θ
if χa ≤ 0.5
2
di =
(4.14)
θ
⎪
⎩±2(1 − χa )∆a cos
if χa > 0.5
2
where the ‘−’ sign holds if (ri − r0 )/2∆a is an even number and the ‘+’ sign otherwise.
The radius of the new node position is
'
2
+ d2i
(4.15)
ri,2 = ri,1
and the new angle is
βi,2 = β + arctan
di
ri,1
(4.16)
which yields the new node positions:
xi,2 = ri,2 cos βi,2
yi,2 = ri,2 sin βi,2
zi,2 = zi
(4.17a)
(4.17b)
(4.17c)
(b)
(a)
Figure 4.9: Zig-zag folded star arms: (a) 40◦ and (b) 0◦ folding angle.
47
initial straight position
y
i
(xi , yi )
d+
i (xi,2 , yi,2 )
di
d−
ri,2
zig-zag folded position
r0
β
βi,2
x
Figure 4.8: Zig-zag folding of a star arm.
4.2.3
Wrap-around folding of star arms
Another folding scheme that is attractive due to its simplicity is the wrap-around or coiling
scheme, where the star arms are wrapped around the central satellite. It is important also
for this scheme that the folding is done in a polygonal way and not by a smooth curve
since the fold lines must be positioned where the nodes are. Otherwise, the elements will
become too short for the web to deploy properly. A mapping scheme, which preserves the
lengths of all members can be written using the Denavit-Hartenberg convention, [3]. As
the movement of the nodes during the folding only is in-plane, the Denavit-Hartenberg
transformation matrix can be simplified to
⎤
⎡
cos α − sin α 0 ∆w cos α
⎢ sin α cos α 0 ∆w sin α ⎥
⎥
(4.18)
A(α) = ⎢
⎦
⎣ 0
0
1
0
0
0
0
1
where ∆w is the distance between the fold lines or mesh width, αi is the rotation angle
relative to the previous segment of the star arm. for the first element, the folding angle
is set to β + αi /2 − π/2 as the whole star arm first must be folded 90◦ from its initial
angle β, which found from Eq. (4.11). The folding of the star arms with α = 0 is shown
in Figure 4.10(a). To produce a circle of the star arms that encloses the deployed portion
of the web, the following value for the relative rotation of the star arm segment must be
48
⎡
chosen:
⎞⎤
⎛
∆w
α = − ⎣π − 2 arccos ⎝ '
2 r02 +
∆2w
4
⎠⎦
(4.19)
The x- and y-coordinates of the position of the first node of the star arm (r1 = r0 + ∆w )
after folding is found as elements (1,4) and (2,4), respectively, in A(β − π/2 + α/2).
Similarly, the position of node i is found as positions (1,4) and (2,4) in the resulting
matrix from the product A(β − π/2 + α/2)A(α)i−1 . Applying the Denavit-Hartenberg
transformation for all nodes of the star arms yields the folded configuration in Figure
4.10(b). Since the star arms in this position lie on top of each other, problems might arise
in the finite element simulations. Therefore, a slight change in the relative rotation angle
α is implemented: for node i the relative rotation is set to
αi := α − (i − 1)ϑ
(4.20)
where ϑ is chosen as percentage of α. It is found that ϑ = 0.01α yields enough separation
between the star arms, Figure 4.10(c). Hence, the x- and y-coordinates for node i is now
found at positions (1,4) and (2,4) in the matrix A(β − π/2 + α/2)A(α1 )...A(αi−1 )A(αi ),
where αi is given by (4.20).
(b)
(a)
(c)
Figure 4.10: Hub-wrapping folding of the star arms: (a) initial 90 ◦ folding, (b) most compact
folding with square (aerial and top views), and (c) with separation between the
arms.
49
Chapter 5
Deployment
The most interesting, but also the most complicated, part of the project is the deployment
simulations. This part is divided into three sections: (i) selection of deployment control
strategy, (ii) development of an analytical model for quick assessment of the effect of
changing various parameters and (iii) deployment simulations with the finite element
software LS-DYNA.
5.1
Deployment strategies
The most simple deployment strategy would be to coil the star arms around the centre
satellite and then provide a initial angular velocity large enough to deploy the web. However, as the system conserves the angular momentum, the final spin would be too slow,
as shown by the following calculation. The total angular momentum before release of the
web and the corner masses, Figure 5.1(a), is
m
h
(5.1)
+ mw + 4mc r02 ω0
L0 =
2
where mh is the mass of the hub, mw is the total mass of the web, mc is the mass at each
corner, r0 is the radius of the hub, and ω0 is the initial angular velocity. After deployment,
Figure 5.1(b), the angular momentum is
mh r0 2 mw
+ 2mc S 2 ω1
+
(5.2)
L1 =
2 S
6
The law of the conservation of the angular momentum yields the ratio of the angular
velocities after and before the deployment:
ω1
=
ω0
mh
+ mw +
2
mh r0 2
+ m6w
2
S
50
4mc
+ 2mc
r 2
0
S
(5.3)
L
Jh = 12 mh r02
r0
Jc = 4mc r02
Jw = mw r02
Jh = 12 mh r02
L
Jc = 2mc L2
1
Jw = m w L2
6
(a)
(b)
Figure 5.1: Angular momentum of the system: (a) before deployment and (b) after deployment.
Assume that S = 100 m, r0 = 1 m, mc = 10 kg and mh = 100 kg. With a 30 mm
mesh width, the total weight of the web (including edge tethers) is 122 kg. Inserting these
values into (5.3) yields ω1 /ω0 = 5.26 · 10−4 . Hence, with ω0 = 12.6 rad/s (2 rps), the
final angular velocity will be 6.62 · 10−3 rad/s or 3.8 revolutions per hour, which is way
too slow. Compare this value with that of the 1500-m-diameter LOFT that was designed
to spin with one revolution in about 11 minutes, [49]. Decreasing the size of the web to
L = 10 m and keeping the same hub diameter and masses, except for the mass of the web,
which becomes 1.22 kg, yields ω1 /ω0 = 4.41 · 10−2 . Hence, with an initial spin of 2 rps, the
final spin rate is 5.3 rpm. It appears that some adjustments of masses to obtain a desired
final angular velocity is possible for a smaller web. However, for a larger one, angular
momentum must be added to the system during deployment to obtain a sufficiently high
angular velocity for the deployed web. As mentioned in section 2.2, the spin up system
of the LOFT was programmed to provide angular momentum until 60% of the structure
had been deployed.
Melnikov and Koshelev [29] initially discussed that the angular momentum ideally should
be applied both to the central hub and to the periphery of the structure being deployed,
e.g. by a pair of diametrically opposite thrusters. Such a layout would require that the
thrusters have their own attitude control system to keep the jet motion within the plane of
rotation. Melnikov and Koshelev later disregarded this system on grounds of complexity
and suggested that the angular momentum should be applied to the central hub. They
initially considered several ways of applying this momentum [29]:
• flywheel that initially run at a high speed,
• gas thrusters with different time-dependent control laws for the thrust, and
• electric motors with rigid or drooping characteristics.
51
The system with a flywheel that runs at a very high speed is similar to the case where
the packaged system is first spun and then deployed, as described earlier in this section.
For this case, the initial kinetic energy of the stowed system is
1 mh
+ mw + 4mc r02 ω02
E0 =
2 2
(5.4)
1 100
2
2
=
+ 122 + 4 · 10 · 1 · 12.6 ≈ 16.83 kJ
2
2
whereas the kinetic energy of the deployed system is
1 m h r0 2 m w
+
+ 2mc L2 ω12
E1 =
2
2 L
6
2
1 100
122
1
=
+
+ 2 · 10 · 1002 · (6.62 · 10−3 )2 ≈ 8.84 J
2
2
100
6
(5.5)
Since the energy of the system also is conserved, the excessive kinetic energy creates
undesirable oscillations and eventually failure of the structure. Therefore, the excessive
kinetic energy must be removed from the system during deployment.
When the momentum is provided by thrusters, the angular momentum is accumulated
while the system is being rotated. As there is no requirements to compensate this momentum by a counter-rotating systems, Melnikov and Koshelev [29] state that the theoretical
solution of a constant or time variable momentum suggests that the deployment becomes
oscillatory. Ground testing with gas thrusters is also problematic.
With an electric drive solution, the angular momentum of the system being deployed has
to be compensated by, e.g., a counter-rotating flywheel [29]. The drive characteristics of
the electric rotor connecting the deploying structures and the counter-rotating systems is
fundamental in order to control the deployment. Adjusting the momentum of the electric
drive to keep the angular velocity constant leads to the following drawbacks, [29]:
• the centrifugal forces on the structure will be very large at the end of deployment,
and
• oscillatory, unstable deployment dynamics with first coiling off and then coiling on
again.
Melnikov and Koshelev [29] suggest that a drooping characteristic of the electric drive,
i.e. the momentum is increased as the angular velocity is decreased, and vice versa, would
eliminate the drawbacks above and provide the following advantages:
• The initial angular velocity could be sufficiently high to initiate deployment,
• a high deployment velocity and a short deployment time is possible,
52
• the deployment would be smooth without the coiling on–coiling off dynamic phenomena, and
• a low angular velocity at the end of the deployment produces acceptable centrifugal
forces.
The drooping characteristic of the electric motor produces a stable, self-controlled system:
if the system suddenly was slowed down, the momentum is increased to increase the
angular velocity; on the contrary, if the system was accelerated, the momentum would
decrease to slow down the rotations.
5.2
Analytical model
Simple analytical models can be used to describe the deployment dynamics qualitatively.
First, equations are derived that describe the deployment of an arbitrary number of point
masses symmetrically attached around a cylindrical central hub. If the mass of the space
web is small compared to the point masses at the ends of the cables this would be a good
first model of the deployment. Then, models of the deployment of more complicated space
webs are derived or obtained from literature.
The derivation of the equations follows the same principles used by Melnikov and Koshelev
[29] to describe the deployment of solid reflectors and tether systems from a rotating
central satellite. Hedgepeth [13] also used a similar model for the LOFT system. The
analytical models are based on the following basic assumptions:
• The deployment is symmetric relative to the central axis.
• The radial tethers are straight.
• There is no out-of-plane motion.
• Potential energy effects due to gravity and the elasticity in the cables are neglected.
• The energy dissipation caused by deformation, friction and environmental effects is
neglected.
Three coordinate systems are required for the model, Figure 5.2. All of them have one
(i)
(i)
(i)
axis e3 directed along the axis of rotation and two axis, e1 and e2 , in the plane of
deployment. The first coordinate system is fixed in space with its origin in the centre of
the central satellite. The second coordinate system is rotating with an angular velocity
ω(t) = ϕ˙1 around the same origin as the first. The third system is attached at the periphery
of the central satellite, where the radial arm or tether from the central satellite to the end
(2)
(2)
mass is attached. Axis e1 points in the direction of the tether, e3 is parallel to the axis
(2)
(2)
(2)
of rotation, and e2 is perpendicular to e3 and e1 .
53
mc
(0)
e2
)
(1
2
e
ϕ2
e2(2)
R
e1(2)
L
)r
(1 0
e1
ϕ1
O e(0)
1
Figure 5.2: The analytical model for a point mass. The description of the equation systems is
the same for all the analytical models presented here.
5.2.1
Variation of the angular momentum
Due to the applied momentum and the forces in the tethers, the angular momentum will
change. Therefore, the change of angular momentum for the central cylinder around its
axis of rotation is:
(5.6)
Jz ω̇ = M + nN r0 sin ϕ
where Jz is the moment of inertia of the centre hub around the axis of rotation, M is
the applied momentum, n is the number of radial tethers, N is the tensile force in each
tether. The moment of inertia for the hub is
1
Jz = mh r02
2
5.2.2
(5.7)
Model 1: deployment of point mass
This model can be used if the corner masses are much heavier than the space web and the
tethers. In any case, it is good to start with the simplest case, a point mass. The equation
of motion for a point mass is:
(5.8)
mc R̈ = F
54
where R̈ is the acceleration of the point mass and F is the applied force. Projected along
(2)
(2)
axes e1 and e2 respectively, Eq. (5.8) becomes
(5.9a)
mc r0 ω 2 cos ϕ − ω̇ sin ϕ − L̈ + L (ω + ϕ̇)2 = N
mc r0 ω̇ cos ϕ + ω 2 sin ϕ + 2(ω + ϕ̇)L̇ + (ω̇ + ϕ̈) L = 0
(5.9b)
where ω = ϕ˙1 is the angular velocity of the centre hub, ϕ = ϕ2 is the deflection angle of
the tether relative to the radial direction, L is the length of the tether.
5.2.3
Model 2: deployment of distributed masses
If the mass of the space web or the tethers are of importance, a refined model must be
used. This model will depend on the folding pattern and the geometry of the space web
during the deployment. This requires the introduction of some additional simplifications.
Equation (5.6) is kept unchanged, while Eq. (5.9) is changed. For distributed masses the
equations of motion are replaced with:
(5.10a)
dm r0 ω 2 cos ϕ − ω̇ sin ϕ − ¨l + l (ω + ϕ̇)2 = dN
(5.10b)
dm r0 ω̇ cos ϕ + ω 2 sin ϕ + 2(ω + ϕ̇)l˙ + (ω̇ + ϕ̈) l = 0
where l is the length to mass dm. These equations are then integrated over the area or the
length, but first an expression for dm must be derived. dm is constant along the length
of a tether, whereas it varies linearly along the length of a star arm with the folded web.
The equations of motion for some interesting cases follow.
5.2.4
Model 2(a): deployment of a tether
For a tether, the line density ρL is constant, and dm = ρL dl. The equations of motion
(5.10) describing the deployment of a straight tether, become
L
2
2
(5.11a)
ρL L r0 ω cos ϕ − ω̇ sin ϕ − L̈ + (ω + ϕ̇) = N
2
r0 ω̇ cos ϕ + ω 2 sin ϕ + 2(ω + ϕ̇)L̇ + 0.5 (ω̇ + ϕ̈) L = 0
(5.11b)
As the line density of the tether is constant, the direction from which it is fed out is not
important.
5.2.5
Model 2(b): deployment of a circular space web
Melnikov and Koshelev [29] used this model for a circular split reflector. Hedgepeth [13]
used a similar model for the LOFT system. Here, it is used for a circular space web
55
coiled around the centre hub. As the tension in the tangential direction is small during
the deployment, the model originally used for a split membrane/space web can be used
also for a solid membrane or a space web. If the web is fed out from spools on, or coiled
around, the centre hub, then each sector have the following line density
ρL =
2πρA
(l + LR − L)
n
(0 ≤ l ≤ L)
(5.12)
where ρA is the surface density of the space web, R is the radius of the circular space
web, L is the deployed length of the sector and LR = R − r0 . The resulting equations of
motion thus become
LR L
L 2πρA
2
2
2
L LR −
r0 ω cos ϕ − ω̇ sin ϕ − L̈ + L
−
(ω + ϕ̇) = N
n
2
2
6
(5.13a)
LR L
L
LR −
r0 ω̇ cos ϕ + ω 2 sin ϕ + 2(ω + ϕ̇)L̇ + L2
−
(ω̇ + ϕ̈) = 0
2
2
6
(5.13b)
5.2.6
Model 2(c): deployment of the space web using the star
pattern
The folding into the star pattern was described in the previous section. The deployment
is performed in two steps. The dynamics of the first step, the deployment of the arms,
is most important. When the arms are deployed, the deployment of the rest of the net
should not pose any major problems according to Melnikov and Koshelev [29].
Step 1: deployment of the arms
A space web is first folded into n identical arms positioned symmetrically around the
central hub. The arms can be folded, and then deployed, on spools at the end of the arms,
in a zig-zag pattern, or coiled around the centre hub. The line density ρL of an arm varies
linearly, from zero at the tip of the arm. If the arm is fed out from a spool at the tip of
the arm, the line density becomes
ρL =
2mw
(H − l)
nH 2
(0 ≤ l ≤ L)
(5.14)
where H is the length of the arm, i.e. H = S/2, l is the position on the arm for a small
element of size dl. For a quadratic net, Eq. (5.14) is written as
ρL = 2ρA (H − l)
(5.15)
The mass of the small element of size dl, to be put in Eqs. (5.10) is
dm =
2mw
(H − l)dl
nH 2
56
(5.16)
Attached to the end of each arm is the not yet deployed part of the web with mass
mc =
2mw (H − l)2
nH 2
2
The resulting equations of motion become
⎤
⎡
L2
2
+
−
HL
H
3
mw ⎣
L (ω + ϕ̇)2 ⎦ = N
r0 ω 2 cos ϕ − ω̇ sin ϕ − L̈ +
n
H2
2
H 2 + L3 − HL
r0 ω̇ cos ϕ + ω 2 sin ϕ + 2(ω + ϕ̇)L̇ +
L (ω̇ + ϕ̈) = 0
H2
(5.17)
(5.18a)
(5.18b)
If the arm is initially coiled around the centre hub, the line density becomes
ρL =
2mw
(L − l)
nH 2
(0 ≤ l ≤ L)
The resulting equations of motion become
L3
2mw L2 2
2
r0 ω cos ϕ − ω̇ sin ϕ − L̈ +
(ω + ϕ̇) = N
nH 2 2
6
L
r0 ω̇ cos ϕ + ω 2 sin ϕ + 2(ω + ϕ̇)L̇ + (ω̇ + ϕ̈) = 0
3
(5.19)
(5.20a)
(5.20b)
Step 2: deployment of the web from the arms
The second deployment step can be modelled in different ways, depending on the required
accuracy. One very rough model is suggested in [29]. However, the second step is less
important to control, or preferably, not necessary to control at all [29]. A more accurate,
but still not exact expression, can be achieved by assuming an almost linearly varied line
density from l = L to l = Ltot and integrating the surface density over the surface from
l = 0 to l = L. The resulting equations of motion would then become
C1 r0 ω 2 cos ϕ − ω̇ sin ϕ − L̈ + C2 (ω + ϕ̇) 2 = N
(5.21a)
(5.21b)
C1 r0 (ω̇ cos ϕ + ω2 sin ϕ) + 2(ω + ϕ̇)L̇ + C2 (ω̇ + ϕ̈) = 0
57
where
C1 = 2ρA H 2 tan β
β π
L3 sin β
C2 = ρA
+ ln | tan + |
3 cos β
2
4
⎛
⎞
L3
3
−a ⎟
⎜ 1 − cos2 β a
L
cos3 β
⎟
+ ρL ⎜
+
a
+
+
⎝ 2 cos2 β
⎠
L
2
cos2 β
3 a−
cos β
π
β=
n
Lw
H=
2
L
sin β
γ = arcsin
H
sin(β + γ)
sin β
2(L2tot − L2 ) tan β
ρL = ρ A
L
−L +
2 cos β + a/2
H
Ltot =
tan β
(5.23)
(5.24)
(5.25)
(5.26)
a=H
5.2.7
(5.22)
(5.27)
(5.28)
Deployment of coiled up mass or arm
All the equations of motion derived above can also be used to simulate space webs that
are coiled up around the centre hub if the following changes are made: ω = ω + ϕ̇,
L = L − r0 max 0, |ϕ| − π2 , L̇ = ∓r0 ϕ̇, L̈ = ∓r0 ϕ̈ and ϕ ± max |ϕ|, π2 .
5.2.8
Numerical solution of the analytical model
The change of angular momentum, Eq. (5.6), can be used together with the two equations
of motion, e.g. Eqs. (5.9), to solve the behaviour of the desired deployment. In order to
do this, the three equations must be transformed into a system of nonlinear ordinary
differential equations (ODE) for the three unknowns L(t), ϕ(t) and ω(t). If L is known,
then the equation of motion in the radial direction can be used to determine N . This will
be the case when a control strategy is used for L or when the tether is fully deployed.
Similarly, if ω is controlled the equation of change of angular momentum can be used to
determine the torque. The nonlinear ODE system is on the form
ẋ = f (x)
58
(5.29)
T
where x = L, L̇, ϕ, ϕ̇, ω if there is no control. If a control strategy is used, then the
control parameters are omitted.
Finally, the system is solved by a 4th and 5th order Runge-Kutta scheme in MATLAB
using the routine ode45, an ODE solver that uses error control to determine the integration
step size.
5.3
Finite element model
The FE model was implemented in two steps. First, a model with only a mother satellite,
four daughter satellites and the tethers connecting them was studied to investigate contact and cable behaviour. Then a more realistic model including a space web, including
corner masses, and a central satellite was implemented. The node and element geometry
and connectivity are generated by MATLAB for a desired configuration with suitable dimensions and meshing. The equations of motion are then solved in LS-DYNA [22] using
the explicit central difference integration method.
In the FE model, the cable behaviour is more accurately modelled than in the analytical
model, and perturbations from the ideal symmetric deployment can be investigated. The
cables and tethers are modelled with a great number of cable elements, i.e. truss elements
with a no-compression material. The main differences compared to the analytical model
is that the cables can store elastic energy and that they are not constrained to be straight.
Another important difference is that forces can be distributed in directions other than the
radial. The simulated space web is coarser than the real space web, but the masses of the
cables in the model are determined so that the total masses of the ideal and actual space
webs are identical.
The coiling up and coiling off phenomena on the centre hub is important to model in
the simulations. Therefore, the centre hub is modelled as a rigid body and not a point
mass. The corner masses, however, can be modelled as point masses without loosing any
important effects.
Contact phenomena will occur because of the large displacements. The contact between
the cables and the rigid bodies, at the centre, is modelled using the kinematic constraint
method [15]. This method enables reeling up of the space web without contact forces
pushing the web away from the rigid bodies, as would be the case with the more common
penalty method. If a cable node is within a small distance from a satellite surface, then
the nodal displacements of the slave nodes (the cables) are transformed, on the global
equations level, so that the degrees of freedom normal to the master surface (the centre
hub) are eliminated. Impact and release conditions are then imposed to ensure conservation of momentum. The contact condition expires when the relative velocity becomes
positive again.
Contact will also occur between the cables in the space web. Here, a nodes to surface
variant of the penalty method is used. In the case of penetration, equal and opposite
59
forces are applied. This implies a more elastic behaviour which is more realistic for the
cable contact.
Damping has not been included in this work. However, damping may be important, especially if it is used as part of the control strategy or to remove excessive energy. LS-DYNA
has routines to deal with damping if this will be required at a later stage.
All the nodes are given initial rotations and initial velocities proportional to the distance from the centre of mass. Torque and external forces can also be applied during the
deployment.
Finally, one of the main problems to solve is how to fold the membrane around the centre
hub in the most realistic way. A folding pattern where the space web is perfectly folded
around a circular hub is impossible to create, since the cables must be modelled straight
and with a certain length determined by the total computation cost. Also, to fold the
cables near the centre hub is not even trivial in reality. Our solution to this problem was
to move the nodes nearest, and inside, the centre hub to the periphery of the hub, see
Figure 4.10. Initial contacts between cable elements in the space web were disregarded,
since higher priority was focused on coiling the space web as near the centre hub as
possible in the initial state.
5.4
Control strategies
The dynamics of the space web deployment using centrifugal forces is dependent on a
suitable folding pattern and a smart control strategy. Free deployment would be the
easiest way to deploy a space web. However, simulations show that this strategy most
likely will lead to the reeling up of the net around the central hub again after the initial
reeling off. Therefore, some kind of control strategy is necessary.
The control law should be selected so that the space web ends up in the desired configuration at the end of the deployment, within a required time period and with no undesirable
oscillations or entanglement of the system. A stable deployment is obtained if the centrifugal force, which is directed radially, is much greater than the Coriolis and inertial
forces [29]:
mω 2 L
mω 2 L
2mL̇ω
mω̇L
(5.30)
(5.31)
or re-written as
γ=
ωL
2L̇
ω2
ω̇
60
1
(5.32)
1
(5.33)
A large ratio between the centrifugal and Coriolis forces is the most difficult to achieve,
and therefore, γ must be observed.
The deployment can be controlled in different ways. The control parameters could be the
torque M , the current length L of the tether, the angular velocity ω of the centre hub or
the force N , that resists the deployment of each segment.
Several, more or less successful, control strategies have been described in literature. Salama
et al. [46] linearly increase the angular velocity ω from 0 to ωmax during a time period of
∆t, and then keep it constant at ωmax , to deploy tethers in the first step of unfolding from
a star folding. They also take advantage of an estimated structural damping of about 5%.
Melnikov and Koshelev [29] use the torque and the velocity of the cable being fed out as
control parameters to deploy the reflector. They examine two different strategies to adjust
the momentum: constant angular velocity and increased momentum as the the angular
velocity decreases. Constant momentum is also mentioned, but not investigated.
Constant angular velocity would require a low angular velocity at the end, and therefore
the angular velocity would have to be low during the entire process. This would imply
increased deployment time, and possibly, difficulties to initiate the deployment because
of small centrifugal forces. They also showed that deployment according to this scheme
gave an oscillatory behaviour, with partial reeling off and reeling up of the reflector.
Instead, they propose increased momentum as the angular velocity decreases. Using this
strategy, they obtain a high initial angular velocity, a low angular velocity in the end,
short deployment time and a stable and smooth deployment without entanglements and
reeling up of the reflector. More exactly, they propose that the momentum should vary
according to the law:
ω
(5.34)
M = M0 1 −
ω0
where M0 is the initial momentum applied to the centre hub and ω0 is the initial angular
velocity of the centre hub. Melnikov and Koshelev [29] found that a higher value of
K = M0 /ω0 produces a more stable deployment. This statement is also confirmed by our
studies.
The deployment velocity L̇ could be constant, or changed smoothly or stepwise. Using
constant deployment velocity Melnikov and Koshelev [29] reported about increased stability with decreased deployment velocity, which is natural since the Coriolis force decreases.
The force N can be obtained using a brake on the spools from where the tethers or space
web arms are uncoiled. Preliminary studies with the analytical model have shown that it
is not necessary to control the deployment velocity when the arms are coiled around the
hub if a momentum according to Eq. (5.34) is applied.
61
5.5
5.5.1
Results
Validation of the analytical model
First, the accuracy of our analytical model is compared with two more complicated models: a mass-spring network model [32] and a finite element model [33]. The folding and
deployment of the simulations by Miyasaki are shown in Figure 2.5. In the analytical
model n identical circle sectors were used, where n cancels out because of symmetry, see
Eqs. (5.6) and (5.13b).
This example should not be favourable for our simple model, as in this example an unrealistically large hub compared with the membrane size makes the effects near the hub
more visible. Furthermore, the folding pattern used here, when the whole web is deployed
simultaneously is more difficult to describe with the analytical model than the more controllable stepwise deployment of a star folded web.
However, good agreement between the results of the different models was found, Figure
5.3, which shows that the present analytical model is reliable. Obviously, the influence of
the centrifugal forces is much greater than the influence of the elastic forces in this case.
This is probably true for most applications where centrifugal forces are used for the deployment. Therefore, the analytical model is sufficiently accurate for initial determination
of the system dimensions and the control characteristics.
5.5.2
Deployment of coiled up circular net
Figure 5.4(a) shows data from a simulation of the free deployment of a circular space web
with radius 10 m. The web is assumed to be folded as in the example above. Again, the
web is first coiled off and then coiled up. The centre hub changes direction during the deployment. Therefore, the web is coiled off not only because of the centripetal acceleration,
but also because the web and the hub move in different directions.
If a torque according to the control law in Eq. (5.34) is applied, then successful deployment
is obtained. The results when M0 = 3ω0 are shown in Figure 5.4(b). The deviation angle
of the web compared to the centre hub, ϕ, is stabilised near 0 at the end of the deployment
in the stable case. It is also interesting to compare the quotient between the centrifugal
and Coriolis forces, γ, in Figures 5.4(a) and 5.4(b). In the stable case, Figure 5.4(b), |γ|
increases fast initially, while γ only increases linearly in the unstable case, Figure 5.4(a).
No additional deployment velocity control is required in this case. Applying a torque is not
trivial, but probably necessary anyway, since conservation of angular momentum implies
a very low angular velocity at the fully deployed state when the web is freely deployed.
62
5.5.3
Deployment of coiled up star folded web
Previously in this study, quadratic space webs folded in the star pattern have been shown
to be of certain interest. The initial configuration of the web coiled around the centre
hub is shown in Figure 4.10. Several different deployment simulations of a large space
web with side 100 m have been performed. The following data has been used in all the
simulations: S = 100 m, mh = 100 kg, r0 = 6.3 m, ρA = 1.267 · 10−2 kg/m2 , mc = 10 kg,
Eca = 180 · 109 Pa, ρca = 1540 kg/m3 , Aca = 2.5/0.030 · 1.23 · 10−7 m2 and t = 2.5 m.
Analytical
Analytical
(a)
(b)
Analytical
(c)
Figure 5.3: Results from a simulation of the deployment of a circular space web using the simple
analytical model, a mass-spring network model [32] and a finite element model [33].
63
10
L
5
0
20
0
ω
-20
100
50
0
- 50
ϕ
γ
50
0
-50
0
1
2
3
4
5
6
7
8
9
10
t
(a) Free
10
L5
0
0
-5
ω-10
-15
-20
150
100
ϕ50
0
6
N4
2
0
0
-10
γ-20
-30
0
5
10
15
20
25
30
t
(b) Controlled
Figure 5.4: Analytical solution of the deployment of a circular net with radius 10 m. The free
deployment is shown in (a). In (b) A torque, M = M0 (1 − ωω0 ), is applied to control
the deployment. L is the deployed length of the net. ϕ is the angle of rotation of
the web from the normal to the centre hub and should stabilise near zero at the end
of the deployment, γ is the quotient between the centrifugal force and the Coriolis
force. R = 10 m, r0 = 0.1 m, mh = 1 kg, ρA = 1.267 · 10−2 kg/m2 , ω0 = 16 rad/s
and M0 = 3ω0 .
64
5.5.4
Free deployment of star folded web
First, the free deployment of the whole space web was simulated. The results are shown
in Figure 5.5 and 5.6. The arms are deployed first even though no restrictions have been
imposed on them to achieve stepwise deployment. But the main problem is the same as
for the deployment of the circular web; the web is deployed and then coiled back on the
hub again. The coiling off can be described analytically, but not coiling on to the hub.
5.5.5
Free deployment of star arms
To be certain that stepwise deployment will occur, some parts of the web can be held
to the centre hub during the deployment of the arms, and then released when the arms
are deployed successfully. This course of events also have the advantage that it is easily
described analytically. Therefore, the free deployment of only the star arms coiled up
around the centre hub was simulated, Figure 5.7. The deployment is initiated by an
initial rotational velocity of 8π rad/s on the hub and the web. The star arms are coiled
off from the hub, but then coiled up on the centre hub again. The centre hub also change
rotational direction during the deployment. The results from the FE simulation can be
compared with the analytical model, Figure 5.8. The agreement between the two models
is very good for the length of the deployed arm, but not for the rotational velocities.
5.5.6
Controlled deployment of star arms
An applied torque, again using Eq. (5.34), can be used also for the deployment of star
arms to stabilise the deployment, see Figure 5.9. Attempts to apply the same torque on
the FE model have been performed, but have not been successful due to the sensitivity
of correct feedback in the required control strategy. Therefore, a torque that is feedback
controlled in the FE model is necessary. This can be performed in the object version of
LS-DYNA [22].
The torque applied in the above example is unrealistically high. Hedgepeth used a torque
approximately equal to 200 Nm in the simulations of the LOFT [13]. To achieve a stable
deployment with applied moments of this order, the deployment velocity must be considerably lower. In reality the hub is smaller, which of course gives lower deployment velocity
for a given rotational velocity. Alternatively, it is possible to decrease the initial rotational
velocity.
Using a hub with radius 1 m and torque equal to 60π ≈ 200 Nm in the analytical model,
the deployment characteristics in Figure 5.10 were obtained for different initial rotational
velocities ω0 . It can be seen that increased ω0 , i.e. increased deployment velocity, requires
higher moment to control the deployment.
65
(a) t=0.0
(b) t=0.1
(c) t=0.2
(d) t=0.3
(e) t=0.4
(f) t=0.5
(g) t=0.6
(h) t=0.7
(i) t=0.8
(j) t=0.9
(k) t=1.0
Figure 5.5: The free deployment of a space web folded according to the star pattern and coiled
up around the centre hub. The rotation is initiated by an initial rotational velocity
ω0 = 8π rad/s.
66
50
L
40
30
20
10
0
0
0.2
0.4
0.6
0.8
1
1.2
0
0.2
0.4
0.6
0.8
1
1.2
0
0.2
0.4
0.6
0.8
1
1.2
0
0.2
0.4
0.6
0.8
1
1.2
ϕ
0
−5
−10
30
center
20
10
ω
0
−10
−20
30
ωtip
20
10
0
t
Figure 5.6: The free deployment of a space web folded according to the star pattern and coiled
up around the centre hub. The results in the graph have been obtained from the
Finite Element model, analytical results cannot be obtained. The four graphs show
the length of the deployed arm L, the angular deviation from the radial direction ϕ,
the angular velocity of the inner hub ωcentre and the angular velocity of the tip of
the arm ωtip .
67
(a) t=0.0
(b) t=0.1
(c) t=0.2
(d) t=0.3
(e) t=0.4
(f) t=0.5
(g) t=0.6
(h) t=0.7
(i) t=0.8
(j) t=0.9
(k) t=1.0
Figure 5.7: The first step of the free deployment, i.e. the deployment of the star arms, of a space
web folded according to the star pattern and coiled up around the centre hub. The
rotation is initiated by an initial rotational velocity ω0 = 8π rad/s.
68
L [m]
40
20
0
0
0.2
0.4
0.6
0.8
1
1.2
0
0.2
0.4
0.6
0.8
1
1.2
0
0.2
0.4
0.6
0.8
1
1.2
10
[rad]
5
ϕ
0
−5
ωcentre [rad/s]
−10
20
0
−20
Analytical
Finite Element
ωtip [rad/s]
40
20
0
0
0.2
0.4
0.6
0.8
1
1.2
t
Figure 5.8: The first step of the free deployment, i.e. the deployment of the star arms, of a space
web folded according to the star pattern and coiled up around the centre hub. The
results obtained with the finite element model can be compared with the results from
the analytical model. The four graphs show the length of the deployed arm L, the
angular deviation from the radial direction ϕ, the angular velocity of the inner hub
ωcentre and the angular velocity of the tip of the arm ωtip . ω0 = 8π rad/s.
69
L [m]
40
20
0
0
0.5
1
1.5
2
2.5
3
0
0.5
1
1.5
2
2.5
3
0
0.5
1
1.5
2
2.5
3
0
0.5
1
1.5
t
2
2.5
3
[rad]
0
ϕ
−5
−10
ωcentre [rad/s]
30
20
10
0
ωtip [rad/s]
50
40
30
20
10
0
Figure 5.9: The first step of the deployment, i.e. the deployment of the star arms, of a space web
folded according to the star pattern and coiled up around the centre hub. A torque
is applied according to the control law in Eq. (5.34) with M0 = ω0 · 105 . The four
graphs show the length of the deployed arm L, the angular deviation from the radial
direction ϕ, the angular velocity of the inner hub ωcentre and the angular velocity of
the tip of the arm ωtip .
70
L [m]
60
ω0 =π/2
40
ω0 =2π
20
ω =8π
0
0
0
50
100
150
200
250
300
0
50
100
150
200
250
300
50
100
150
200
250
300
50
100
150
t
200
250
50
ϕ
[rad]
0
−50
−100
ωcenter [rad/s]
10
5
0
−5
ωtip [rad/s]
4
2
0
−2
−4
Figure 5.10: The first step of the deployment, i.e. the deployment of the star arms, of a space
web folded according to the star pattern and coiled up around the centre hub. A
torque is applied according to the control law in Eq. (5.34) with M0 = 60π Nm and
ω0 = π/2, π, 2π rad/s. The four graphs show the length of the deployed arm L, the
angular deviation from the radial direction ϕ, the angular velocity of the inner hub
ωcentre and the angular velocity of the tip of the arm ωtip . The stability increases
as the initial rotational velocity decreases.
71
5.5.7
Rotational velocity according to Salama et al.
Salama et al. [46] propose linearly increasing angular velocity of the inner hub, from 0
at time t = 0 to ωmax at time t = ∆t, and then constant angular velocity ωmax . There
is no systematic way to choose these parameters, but using the values ωmax = 8π rad/s
and ∆t = 2 s as in the article by Salama et al. [46], results in the deployment in Figure
(5.11). The arms are coiled off the centre hub, and not coiled back on the hub again.
However, there are undesired oscillations in the plane of rotation. The results from the
FE simulation can be compared with the results from the analytical model, Figure (5.12).
The agreement between the two models is very good during the deployment, but the
amplitude and frequency of the above mentioned in-plane oscillations are different.
5.5.8
Differences between the analytical and the FE model
In order to investigate the importance of different effects, simulations were also performed
using a stiff material, with Young’s modulus E = 1000EZylon , in the FE model. In the
case of linearly increased rotational velocity on the centre hub, Figure 5.13, excellent
agreement was obtained between the FE model with stiffer material and the analytical
model, Figure 5.14. Therefore, it can be concluded that the difference between the FE
model and the analytical model, in this case, is due to the effects of the elasticity in the
material when the arms are in tension.
On the other hand, when the same material was used in the case of free deployment
simulations, Figure 5.15, no differences were visible compared to when Zylon was used,
Figure 5.8. The reason is of course that the rotational velocity is too small to create enough
centrifugal forces to keep the arms in tension. Therefore, the effects of the elasticity is
negligible.
Instead, the differences between the two models are explained by the fact that the arms
are not straight here, Figure 5.7. However, this is due to the different directions of the
rotations of the centre hub and the web, since the centre hub changes direction during the
deployment. That is, the coiling off of the arms from the centre hub is not entirely caused
by the centripetal acceleration, but also simply because the arms and the hub move in
opposite directions. This will not occur during a successful centrifugal force dominated
deployment.
The rapid increase, in the analytical model, of the rotational velocity at the tip, from 8π
rad/s initially to the double shortly thereafter, is more likely to cause significant differences
in interesting realistic cases, Figure 5.15.
72
(a) t=0.0
(b) t=0.2
(c) t=0.4
(d) t=0.6
(e) t=0.8
(f) t=1.0
(g) t=1.2
(h) t=1.4
(i) t=1.6
(j) t=1.8
(k) t=2.0
Figure 5.11: The first step of the deployment of a space web folded according to the star pattern
and coiled up around the centre hub. The rotational velocity is constrained and
varies linearly from 0 at t = 0 to 8π rad/s at 2 s.
73
L [m]
60
40
Analytical
Finite Element
20
0
0
0.5
1
1.5
2
2.5
3
0
0.5
1
1.5
2
2.5
3
0
0.5
1
1.5
2
2.5
3
0
0.5
1
1.5
t
2
2.5
3
[rad]
0
ϕ
−5
−10
ωcentre [rad/s]
25
20
15
10
5
0
ωtip [rad/s]
50
40
30
20
10
0
Figure 5.12: The first step of the deployment of a space web folded according to the star pattern
and coiled up around the centre hub. The angular velocity is constrained and varies
linearly from 0 at t = 0 to 8π rad/s at 2 s. The results obtained with the finite
element model can be compared with the results from the analytical model. The
four graphs show the length of the deployed arm L, the angular deviation from the
radial direction ϕ, the angular velocity of the inner hub ωcentre and the angular
velocity of the tip of the arm ωtip .
74
(a) t=0.0
(b) t=0.2
(c) t=0.4
(d) t=0.6
(e) t=0.8
(f) t=1.0
(g) t=1.2
(h) t=1.4
(i) t=1.6
(j) t=1.8
(k) t=2.0
Figure 5.13: The first step of the deployment of a space web folded according to the star pattern
and coiled up around the centre hub. The angular velocity is constrained and varies
linearly from 0 at t = 0 to 8π rad/s at 2 s. The stiffness is 1000 times greater than
for Zylon, see Figure (5.13).
75
L [m]
60
40
20
0
Analytical
Finite Element
0
0.5
1
1.5
2
2.5
3
0
0.5
1
1.5
2
2.5
3
0
0.5
1
1.5
2
2.5
3
0
0.5
1
1.5
t
2
2.5
3
[rad]
0
ϕ
−5
−10
ωcentre [rad/s]
25
20
15
10
5
0
ωtip [rad/s]
50
40
30
20
10
0
Figure 5.14: The first step of the deployment of a space web folded according to the star pattern
and coiled up around the centre hub. The angular velocity is constrained and varies
linearly from 0 at t = 0 to 8π rad/s at 2 s. The results obtained with the finite
element model, with stiff material (Eca = 1000EZylon in Figure (5.14) ), can be
compared with the results from the analytical model. The four graphs show the
length of the deployed arm L, the angular deviation from the radial direction ϕ ,
the angular velocity of the inner hub ωcentre and the angular velocity of the tip of
the arm ωtip .
76
L [m]
40
20
0
0
0.2
0.4
0.6
0.8
1
1.2
0
0.2
0.4
0.6
0.8
1
1.2
0
0.2
0.4
0.6
0.8
1
1.2
10
[rad]
5
ϕ
0
−5
ωcentre [rad/s]
−10
20
0
−20
Analytical
Finite Element
ωtip [rad/s]
40
20
0
0
0.2
0.4
0.6
0.8
1
1.2
t
Figure 5.15: The first step of the free deployment of a space web folded according to the star
pattern and coiled up around the centre hub. The rotation is initiated by an initial
rotational velocity ω0 = 8π rad/s. The results obtained with the finite element
model, with stiff material (Ecable = 1000EZylon ), can be compared with the results
from the analytical model. The four graphs show the length of the deployed arm L,
the angular deviation from the radial direction ϕ, the angular velocity of the inner
hub ωcentre and the angular velocity of the tip of the arm ωtip .
77
Chapter 6
Conclusions
The main conclusions from the respective chapters are as follows.
6.1
Web design
• Only a web with a quadratic mesh topology can be adequately prestressed by spinning. A triangular mesh will contain slack element whereas the hexagonal mesh will
be severely distorted.
• If very low element forces in the regions close to the edges can be accepted, it is not
required to have curved edges.
• If radial tether must be used to control the deployment, they should have no load
bearing function after deployment, since they prevent the web from being fully
prestressed.
6.2
Folding pattern
• The two stage star folding pattern makes it possible to deploy the web in two
distinctive stages to achieve more control.
• The star folding pattern is also of advantage from a modelling point of view since
it can be described with simple equations, mapping the web from the deployed to a
fully folded configuration.
• For the second deployment stage, coiling the arms around the hub was chosen in
favour of a radial zig-zag scheme. The coiling scheme does require additional tethers
that control the deployment rate of the star arms, which is required in the zig-zag
scheme.
78
6.3
Deployment
• An analytical model, originally developed by Melnikov and Koshelev [29] and then
slightly modified, and a finite element model implemented in LS-DYNA have been
developed. The models have then been used to, mainly, simulate the deployment of
space webs folded in star arms and then coiled around a centre hub.
• The analytical and the finite element models produce almost identical results when
the arms are straight, i.e. when energy is transferred from the hub to the arms, and a
stiff material is used, i.e. no elasticity is involved. When material data for Zylon is
used, then the deployment is still similar, but different oscillatory behaviour occurs
after the arms have been fully deployed.
• Both the analytical and finite element models show that free deployment, at least
without damping, is not possible.
• To stabilise the deployment, a torque can be applied to the central hub. The torque
control law implies that the moment increases when the angular velocity of the hub
decreases and vice versa. However, greater maximum moment is required if higher
deployment velocity is desired.
79
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