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 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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 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 4 513 52 4 515 60 4 504 4 507 19 31 4 518 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 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 4 519 74 4 519 76 52 19 4 517 19 4 519 4 518 19 71 4 517 71 67 60 4 513 52 71 67 67 60 19 19 43 4 507 31 4 504 19 0 67 60 34 46 55 63 70 74 77 79 79 77 74 70 63 55 46 34 22 8 34 0 34 33 33 33 33 32 32 32 32 32 33 33 33 33 34 0 34 34 465 34 45 55 63 69 74 77 79 79 77 74 69 63 55 45 34 21 8 46 45 0 45 44 44 44 44 43 43 43 44 44 44 44 45 0 45 46 45 466 7 21 33 45 54 62 69 73 77 78 78 77 73 69 62 54 45 33 21 7 468 53 55 55 54 0 54 54 53 53 53 53 53 53 53 54 54 0 54 55 55 53 468 67 71 60 67 52 31 71 71 0 67 67 0 60 31 71 75 71 71 67 67 76 71 75 67 71 76 71 76 67 75 75 71 76 67 76 75 67 76 71 67 60 52 43 19 75 76 7 62 68 73 76 78 78 76 73 68 62 54 44 33 20 7 62 0 61 61 61 61 61 61 61 61 61 0 62 62 63 63 61 469 61 68 73 76 77 77 76 73 68 61 54 44 33 20 7 68 68 0 68 67 67 67 67 67 68 0 68 68 69 69 70 66 470 7 20 33 44 53 61 68 72 76 77 77 76 72 68 61 53 44 33 20 7 471 71 74 74 73 73 73 72 0 72 72 72 72 72 0 72 73 73 73 74 74 71 471 7 20 32 44 53 61 67 72 75 77 77 75 72 67 61 53 44 32 20 7 471 73 77 77 77 76 76 76 75 0 75 75 75 0 75 76 76 76 77 77 77 73 471 31 19 76 31 76 75 519 4 71 0 71 71 71 71 71 71 71 518 71 67 60 52 43 31 19 4 67 75 71 43 470 67 0 67 67 67 67 67 67 517 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 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 19 4 4 500 19 31 504 19 43 4 507 19 52 4 510 19 60 4 513 19 67 4 515 19 71 4 517 19 74 4 518 19 76 4 519 19 76 4 519 19 74 4 519 19 71 4 519 19 67 4 518 19 60 4 517 19 52 4 515 19 43 4 513 31 4 510 507 4 504 54 69 45 8 34 8 462 68 68 68 0 61 428 0 12 429 428 429 429 430 431 430 431 431 430 431 430 430 430 429 12 12 0 428 11 11 11 11 429 429 429 430 430 9 10 10 25 36 46 54 10 60 9 66 9 69 9 72 9 73 73 9 72 9 69 9 66 60 10 54 46 36 25 12 25 0 24 23 22 21 21 20 20 20 20 20 20 20 21 21 22 23 24 0 25 429 68 471 75 72 68 45 76 67 73 77 67 76 77 67 77 76 67 77 72 67 76 73 68 61 53 44 33 20 471 471 471 7 68 0 68 68 69 69 70 66 470 68 61 54 44 33 20 7 62 0 61 61 61 61 61 61 61 61 61 0 62 62 63 63 61 469 76 78 78 76 73 68 62 54 44 33 20 7 54 62 68 73 54 0 54 54 53 53 53 53 53 53 53 54 54 0 54 55 55 53 468 54 62 69 73 77 78 78 77 73 69 62 54 45 33 21 7 45 0 45 44 44 44 44 43 43 43 44 44 44 44 45 0 45 46 45 466 34 45 55 63 69 74 77 79 79 77 74 69 63 55 45 34 21 8 34 0 34 33 33 33 33 32 32 32 32 32 33 33 33 33 34 0 34 34 465 22 34 46 55 63 70 74 77 79 79 77 74 70 63 55 46 34 22 8 22 22 462 459 21 34 21 45 465 20 53 466 20 61 7 7 8 8 8 0 468 469 7 20 66 470 (a) s = 10 428 75 77 72 62 44 55 33 61 54 72 77 67 46 22 0 8 459 69 63 33 55 53 44 67 77 61 21 465 0 500 500 63 21 466 69 61 78 53 44 33 20 53 53 78 43 33 70 61 7 7 43 79 32 20 469 19 0 19 504 19 4 4 66 468 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 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 32 20 7 43 19 54 62 44 69 79 19 7 19 4 5000 19 75 7 519 74 44 63 33 70 77 77 75 72 67 61 53 43 32 19 7 77 0 77 0 77 77 77 77 78 78 79 79 75 77 77 75 72 67 61 53 43 32 19 7 471 75 79 79 78 78 77 77 77 77 0 77 0 77 77 77 77 78 78 79 79 75 7 19 32 43 53 61 67 72 75 77 77 75 72 67 61 53 43 32 19 7 471 73 77 77 77 76 76 76 75 0 75 75 75 0 75 76 76 76 77 77 77 73 7 20 32 44 53 61 67 72 75 77 77 75 72 67 61 53 44 32 20 7 471 71 74 74 73 73 73 72 0 72 72 72 72 72 0 72 73 73 73 74 74 71 471 4 19 33 63 20 66 7 519 4 19 31 519 4 76 20 61 7 470 4 74 19 76 21 45 469 60 43 52 31 71 71 75 67 76 60 52 43 31 52 4 510 71 67 75 60 76 52 43 31 60 75 52 76 43 31 19 52 75 43 76 31 19 4 515 43 75 31 22 34 8 466 43 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 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 4 519 8 465 43 31 76 8 462 60 75 19 462 0 8 52 31 60 465 462 459 465 466 468 469 470 471 471 471 471 471 471 470 469 468 466 8 0 459 8 8 8 8 7 7 7 7 7 7 7 7 7 7 7 7 7 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 2 106 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 2 261 69 2 69 71 73 75 76 77 78 78 78 78 77 76 75 73 71 69 66 64 1 467 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 253 473 1 64 261 271 279 287 292 297 299 301 301 299 297 292 287 279 271 261 7 41 253 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 7 66 51 51 2 66 42 50 58 64 68 71 73 73 71 68 64 58 50 42 66 261 -2 14 9 9 14 69 15 15 15 15 15 15 15 15 15 15 15 -2 42 42 14 2 15 15 14 271 271 26 31 36 39 40 40 39 36 31 26 9 9 50 71 50 71 19 15 26 -3 19 20 20 20 20 20 20 20 -3 26 15 2 2 279 279 19 19 19 19 23 9 9 29 30 30 29 26 -19 -19 26 58 73 58 73 15 31 23 22 15 22 22 2 2 23 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 76 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 78 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. 108 108 108 108 107 107 106 106 106 106 106 106 106 107 107 108 108 108 108 108 238 187 162 147 138 131 127 125 124 125 127 131 138 147 162 187 238 108 115 115 114 114 113 113 112 112 112 111 111 111 112 112 112 113 113 114 114 115 115 115 232 190 167 153 144 138 133 130 128 128 128 130 133 138 144 153 167 190 232 115 108 187 147 128 116 108 103 100 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 72 79 89 103 131 106 112 133 111 98 89 81 75 69 64 61 59 61 64 69 75 81 89 98 111 133 112 112 130 109 96 86 78 71 64 58 52 49 52 58 64 71 78 86 96 109 130 112 106 127 100 86 76 68 61 54 49 46 49 54 61 68 76 86 100 127 106 106 125 98 84 74 65 57 49 40 31 40 49 57 65 74 84 98 125 106 111 128 107 94 84 76 68 61 52 42 33 42 52 61 68 76 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 75 67 59 49 33 0 33 49 59 67 75 84 93 107 128 111 106 125 98 84 74 65 57 49 40 31 40 49 57 65 74 84 98 125 106 111 128 107 94 84 76 68 61 52 42 33 42 52 61 68 76 84 94 107 128 111 112 130 109 96 86 78 71 64 58 52 49 52 58 64 71 78 86 96 109 130 112 106 127 100 86 76 68 61 54 49 46 49 54 61 68 76 86 100 127 106 112 133 111 98 89 81 75 69 64 61 59 61 64 69 75 81 89 98 111 133 112 106 131 103 89 79 72 66 61 57 56 57 61 66 72 79 89 103 131 106 112 138 115 102 93 85 80 75 71 68 67 68 71 75 80 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 98 100 103 108 116 128 147 187 108 114 190 159 141 129 121 115 111 109 107 107 107 109 111 115 121 129 141 159 190 114 115 232 190 167 153 144 138 133 130 128 128 128 130 133 138 144 153 167 190 232 115 108 238 187 162 147 138 131 127 125 124 125 127 131 138 147 162 187 238 108 108 108 108 108 107 107 106 106 106 106 106 106 106 107 107 108 108 108 108 115 115 114 114 113 113 112 112 112 111 111 111 112 112 112 113 113 114 114 115 115 (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. 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