IAC-06-D STABILITY OF THE SPACE CABLE. John Knapman Knapman Research, United Kingdom ABSTRACT

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1 IC-6-D4.3.4 STILITY OF THE SPCE CLE John Knapman Knapman Research, United Kingdom STRCT The space cale consists of evacuated tues supported y fast-moving ojects, called olts, travelling inside them. The olts are connected to the tues via permanent magnets stailized electronically for minimal power consumption. Version 1 was designed to replace the role of a first-stage rocket y lifting space vehicles from the ground to 5 km at 58 km/hour. Version rises to 14 km has wider applications. Comparale proposals include the space fountain the launch loop. In the long term, such fixed infrastructure is more economical than rockets. key issue with these dynamically supported structures is lateral staility, particularly in the presence of varying cross winds in the stratosphere. The relevant partial differential equations are derived elow with a set of solutions. proposal is presented for providing staility y means of tethers, pipes, a support structure at each end. Each support is a miniature space cale rising to 15 km, it ears the load of the tethers pipes that would otherwise have to e carried y the main structure. INTRODUCTION To escape the earth s gravity well, there are advantages in uilding a fixed infrastructure rather than using rockets. mong these advantages are: 1. Low cost per journey ecause there is a much reduced need to carry fuel reaction mass. Low energy consumption pollution 3. Enalement of gentle rides that allow wide sections of the pulic to visit space without the need for special training or high fitness The space cale is smaller than the launch loop should therefore cost less. 5 The first version was designed to replace the role of first-stage rockets y accelerating a rocket s upper stages to 58 km/h at a height of 5 km. nother version of the space cale is descried here that reaches to near space (14 km) is suitale for astronomy other science also for space tourism. Version is aout three times as expensive as the first, ut it has many more uses. Figure 1 shows a size comparison. The most famous form of fixed infrastructure proposed is the space elevator. 1 The challenge with that idea is to find a strong enough material capale of forming long threads. n alternative form of fixed infrastructure was proposed in 198 to overcome this prolem. Called the space fountain, it relies on pellets, which are fast projectiles travelling inside evacuated tues. Coils in the tues decelerate the rising pellets accelerate the falling ones, thus causing an upward force that supports the weight of the tues of any vehicles. Further work (renamed Starridge 3 ) showed that the energy consumption of the space fountain is unfortunately very high. oth the space elevator the space fountain involve a vertical structure reaching to geostationary orit. Vehicles reach orit when they attain that height, a process that could take hours or days. Lofstrom 4 proposed the launch loop, which consists of a elt travelling at 14 km/sec, reaching a height of aout 8 km extending over km of the earth s surface. This is much more efficient than the space fountain at getting vehicles out of the earth s atmosphere into orit. Version Version 1 Figure 1 Size comparison etween versions In the space fountain, the launch loop the space cale, energy momentum are transferred electromagnetically from the travelling pellets, elt or olts to the accelerating vehicle. This overcomes the prolem in the space elevator of transmitting power to the vehicle. One of the questions with these proposals is how to achieve staility. The troposphere stratosphere can e very turulent, the winds are strong. The launch loop requires tethers to e attached at certain points anchored to the earth s surface to stailize it. One prolem with this approach is that the tethers cause sustantial extra tension in the space cale, this leads to greatly increased forces costs. This 1

2 prolem can e overcome y erecting a pair of support structures, which are miniature space cales, to a height of 15 km. This is an adaptation of a means of deflecting the space cale at the surface stations, descried in Version 1. That work did not deal with the oscillation modes. Now, the oscillations are examined. We present equations for this motion show how support, tethers pipes can provide the necessary staility. SPCE CLE VERSION 1 The space cale consists of several pairs of evacuated tues in which fast-moving projectiles, known as olts, support the weight. The tues have a diameter of 5 cm, in version 1, reach a height of 5 km while covering a distance on the ground of 15 km. Magnetic levitation is used to maintain a practically frictionless connection etween the olts the tues. t each end there is a surface station that has the jo of initially accelerating the olts then, when they return, of turning them around again. The surface stations may e on the ground or at sea. Tethers Incoming olts mit Main Tunnel ccelerator Tues Gantry Decelerator Outgoing olts Figure Plan view of surface station with amit t each surface station, there is a gantry tunnel (Figure ). They turn the descending olts to the horizontal pass them to a circular arrangement of superconducting magnets called the amit. The radius of the amit is 33 metres, the olts velocity is up to.5 km/sec. The olts proceed ack along the ramp up into a tue at an angle of 56 to the horizontal. The tues are paired: one carries descending one carries ascending olts. Surface Station Levitation Force Tension Surface Station Figure 3 Space cale shape distriution of forces s illustrated in Figure 3, the shape of the space cale resemles that of an inverted catenary; the exact equation is given in reference 5. Magnetic levitation etween a tue the olts inside it causes a force perpendicular to the olts direction of travel. This force supports the tue consequently lowers the olts trajectory as if the weight of each olt had increased. In the lower parts of the tue, the force is at an angle only supports part of the tue s weight. The rest of the weight is sustained y tension in the tue, which transfers the force to the top. t the top, the olts are travelling nearly horizontally support the whole weight of the tue there. They also support part of the weight of the lower parts of the tue. Figure 3 shows the relative forces at different points. This arrangement, wherey the forces are perpendicular to the olts motion, is different from that of the space fountain, where olts have to e retarded or accelerated to support the tues. This is how the space cale avoids the heavy losses caused y heating of the coils in the space fountain. The space cale makes further sustantial energy savings y using a novel application of permanent magnets to magnetic levitation. Permanent magnets have een demonstrated successfully in magnetic earings 6 in a prototype train technology known as Inductrack. 7 The space cale takes this method further to minimize the losses due to eddy currents, as descried in reference 5. SPCE CLE VERSION Version 1 was aimed at reducing the cost of launching space vehicles. Version reaches almost three times as high to 14 km which is near space rather than in the upper atmosphere. Version can still e used for launching space vehicles, ut it is high enough to enale other applications. Many earth satellites are placed in orit for scientific work, the most famous eing the Hule space telescope. However, at 14 km the space cale can provide a platform for science with many advantages over satellites. The iggest advantage is that instruments can e raised or lowered easily for upgrade maintenance; they can even e accessed in place. Consider how eneficial that would have een when errors were discovered in the Hule s mirror. The cost of producing such instruments can e drastically lowered if they can e accessed readily after deployment. nother potential revenue earner is tourism. Like the space elevator, the space cale can transport people to near space at a gentle 15 km/hour, making the experience accessile to a wide pulic. For those customers wanting the thrill of weightlessness reentry, it is quite feasile to uild vehicles that can drop ack into the atmosphere glide or fly home. The reentry velocity into the atmosphere from eing stationary at 14km is aout 3 km/h much less violent than reentry from orit.

3 Version achieves the greater height y douling the olts mass to 1 kg increasing their velocity to 3.4 km/sec. The amit radius ecomes 6 metres. The angle of inclination at the surface stations is 74. The greatest change is the tension in the tues. The preferred material is Kevlar *, ecause it is widely used reasonaly priced. In version 1, up to 4.5 kg of Kevlar per metre is needed to sustain the tension, ut this increases to 19 kg per metre at the top in version. In oth cases, a factor of four is included as a safety margin, as is customary engineering practice. The height of 14 km is carefully chosen; aove this height, the exponential factor in equation (3) (see section Calculations ) kicks in more severely. t km, for example, 46 kg per metre would e needed with a olt velocity of 1 km/sec, leading to sustantial increases in amit size other costs. nother advantage of the altitude of 14 km is that it is elow the level at which space deris is known to orit. Ojects oriting at this height fall to earth rapidly. Calculations The equations are similar to those previously reported except for two changes. In version 1, a constant tue weight per metre was assumed that was sufficient to sustain the maximum tension. In version, the tue weight varies y a factor of 1 must e calculated explicitly. Furthermore, the additional height makes it worth taking into account the reduced gravity, which was neglected in version 1. Define α=4ρ/s, where ρ = kg/m 3 is the density of Kevlar, S=345 MegaPascal (MPa) is its strength, 4 is the safety factor. The weight w of the tue per metre is given y w = ( α T mt )g (1) Here m t is the mass of the tue due to magnets other materials, g is the acceleration due to gravity, T is the tension, which also satisfies T = wdr T () r Here T is the tension at ground level, r is the distance from the centre of the earth, r is the value of r at ground level. The solution to these equations is mt α ( r r ) g mt T = T e (3) α α ( ) α ( r r )g w = α T m t ge (4) * Kevlar is a Dupont registered trademark. See MatWe Material Property Data at URL If g is the acceleration due to gravity at ground level, then g = g R r. These equations lead to the differential equation r & θ g Cr & θ [( mt αt ) g r & θ V qt ] && r = (5) 1 Cr & θut s in version 1, it has the form & r = f ( r&, r) (6) This can e solved numerically in the same way using the Runga-Kutta method. In equation (5), a dot denotes differentiation with respect to time. The angle θ is that sutended at the centre of the earth in plane polar coordinates. The constant C is given y C = s mv, where s is the spacing etween olts at ground level, m is the mass of a olt, V is the velocity of a olt at ground level. The symols q u are given y: 1 g q = 3 V rθ & (7) 1 = & r& u rθ 3 V & rθ (8) We also have & V r& θ = r (9) V = V g( r R ) (1) STILITY wind lowing across the space cale will deflect it. If the winds were predictale steady, the surface stations could compensate y deflecting the cale in the opposite direction. However, winds are often gusty or turulent, this leads to oscillations. nalysis of the equations (section Derivation of the Staility Equations ) shows that there are solutions that involve waves that travel along the tues at the velocity of the olts, ut there are also solutions with positive negative exponential terms. The positive exponential terms often dominate, which means that, once the space cale has ent eyond a certain curvature, its rate of ending increases indefinitely. t heights aove aout 1 km, the air is still. The prolems occur elow that height, particularly when there are strong jet-stream winds. In the launch loop, the proposed solution is to fix tethers from the cale down to the ground. However, this places considerale extra weight on the overall structure, requiring sustantially thicker cales ecause of the exponential law in equation (3). In version of the space cale, the effect of this extra weight is to doule the thickness of Kevlar needed at the top, sustantially increasing the olt velocity hence the amit radius, requiring over four times the permanent-magnet strength in the upper parts of each tue. 3

4 Main Tues Support etween, the displacement of the olt is x δ t δ x δ t δ Hence x δ t δ δt x δ t δ V δx δt (1) Tethers Figure 4 Impression of support tethers more economical solution is to erect a support structure near each surface station (Figure 4). Each support is a space cale in miniature; it sts at right angles to the main space cale. The main space cale is tethered to the supports, not to the ground. Each support s height of 15 km is designed so that the tethers support carry the weight of the space cale up to this height. They also carry the weight of aluminium pipes provided to give extra stiffness. Some tethers are at 45 while others are vertical. The support has to e tethered to the ground, it carries the weight of its own tethers, ut this avoids the cumulative effect of transmitting the weight to the top of the space cale. The details are in the next sections. Derivation of the Staility Equations To underst the effect of external forces (mainly wind) on the space cale, we can derive partial differential equations as follows. Consider two points close together on a curved tue (Figure 5). If a olt reaches at time t, it reaches at time t δt. Then the distance etween is δ x = Vδt (11) z In the limit, this equation ecomes dz z z = V (13) dt Similarly, d z z z z z = V V V (13) dt This leads to the equation for a olt s acceleration d z z z z = V V (14) dt pair of tues is connected y damped elastic struts positioned every u metres. If a olt s mass is m their separation is s, it is convenient to consider an average olt mass m per u metres. Then the force at a strut caused y the olts acceleration in one tue is z v z m V V (15) To this must e added the force due to the tue s acceleration m t z. t Writing m u =m m t, the force F a due to acceleration of the olts the tue is given y z v z Fa = mu m V m V (16) T u R ψ T tδ xδtδ xδ x Figure 5 Displacements as a olt travels from to Figure 6 Contriution of tension alancing F a are three more forces, an external force F generally due to wind, a force F s in the strut the tension T in the tue. The contriution of the tension (Figure 6) over the angle ψ etween points distance u apart is Tsinψ. For small u compared with the radius of curvature R, we have a force T sinψ Tψ Tu R Then the force due to tension at a strut is given y 4

5 F T u z = T = Tu (17) R x The alance of forces at one end of a strut is given y F = F F F (18) a T s strut connects two tues in which the olt velocities are equal opposite. Writing z 1 for the z- displacement in one tue z for the other, we comine equations (16), (17) (18) to otain the following pair of simultaneous partial differential equations for a pair of tues connected y a strut z1 z1 z1 m u mv ( mv Tu) = F F s (19) z z z mu mv ( mv Tu) = F F s () The first term in these equations is the straightforward acceleration of the tues olts, independent of the motion of the olts. The third term represents an effect due to curvature, moderated y tension. Once a tue starts to end, the olts momentum tends to resist turning, thus exerting an outward force that causes the end to increase indefinitely. The second term in equations (19) () represents a propagation effect carried along y the olts at velocity V. If the struts are rigid, then z 1 = z L, there is no propagation. However, with elasticity damping in the struts, a degree of play is possile. The overall effect is that the ending of the tues due to a localized wind is spread along the tues in oth directions rather than eing concentrated in one place. This slows the rate of ending reduces the acuteness of the instaility. If L is the nominal length of a strut, ε is its expansion coefficient, p is the damping coefficient, the force in a strut is given y z1 z F s = p ( z1 z L)ε (1) When F=, the equations have solutions of the form αt γ α V V z e e x γ α e x 1 = () ( ) L αt γ α V V ( e e ) x γ α x 1 L 1 z = ± e (3) These satisfy the physical expectation of symmetry, given the opposing velocities in a pair of tues. We can solve for α, leaving γ as free variales. One form of the solution leads to travelling waves at the olts velocity, ut the real exponential terms require further examination. Write ( m ( m Tu V γ ) M = (4) 1 ) u This leads to the following equations for α: M mγ = ± (5) M m γ ( ) Mα pα ε m γ α = M (6) This quadratic equation in α has the solution: Mp ± M p 4Mε ( M m γ ) α = (7) ( M m γ ) The quantity γ is a free variale, which may e complex. Therefore, we can make no assumptions aout the sign of the real part of α. Since real positive values of α are possile, these will tend to dominate, leading to exponential growth in the deformation of the space cale when winds distur it. For this reason, the supports descried in section Support Structure are proposed. Support Structure In equations (19) (), the dominant factor tends z to e m V. The greater the curvature of a tue, x the more centrifugal force is created as olts travel round the curve at velocity V. Previous calculations (reference 5) have shown that a likely maximum wind force on a tue is aout 5N. The introduction of an aluminium pipe increases this to aout 1N. The arrangement of tethers supports needs to cope with this force plus the centrifugal effect, ut that effect needs to e limited y adding stiffness to the tues at the elevations at which winds occur, typically up to 1 km. 15 km is taken as a safer working assumption. The added stiffness is provided y attaching an aluminium pipe to each tue comprising the space cale. The proposed mass of this stiffening pipe is 4 kg/metre, as shown in section Calculations for Pipes. Fine tethers that are closely spaced are more effective than coarse tethers widely spaced. The wider the spacing, the greater is the reliance placed on the stiffening pipe. pair of tethers every metre is proposed, at an angle of 45 to the tues, with one on the right one on the left. These are sloping tethers, they rise from the main cale to the support. The comination of tethers the support hold the weight of the stiffening pipes also provide lateral staility. The proposed material for the tethers is Kevlar. The average weight of a pair of sloping tethers is 8 N (see section Calculations for Tethers ). ecause of the angle of the support, the average spacing of these is aout 1. metres, giving an average load per metre of 4 N or nearly 4 kg. To minimize sagging of the sloping tethers, vertical tethers are placed at 1 km intervals along each sloping tether. Overall, the tethers form a network, with the vertical tethers earing the weight of the sloping tethers. The load per metre of the vertical tethers works out at 4 N (see section Calculations for Tethers ). The net effect of the arrangements of tethers is to keep the movement under strong winds elow.5 metres (see section Expansion Displacement ). 5

6 There is a support at each end of the space cale, they are themselves space cales in miniature. Each support is stailized y eing tethered to the ground. In addition to earing the load of the stiffening pipes tethers, it must ear the forces in these groundattached tethers. These forces are in alance, so the effect of the ground-attached tethers is to doule the load on the support. Furthermore, the support requires its own stiffening pipes at 4 kg/metre. The net effect is to place a load of 14 kg/metre on the support. Using the methods of the section Space Cale Version, this leads to a olt velocity in the support of 3.1 km/sec. The amit radius for the support is 5 metres. The support rises to 15 km also extends over a horizontal range of 15 km. The next section gives the details y which these numers are calculated. Calculations for Tethers Each pair of sloping tethers ears the weight of a metre length of pipe, aout 4 kg, amounting to nearly N on each tether. In the mid position, they also ear equal opposite tensions of N, giving a total tension of, or nearly 3N. Under cross winds, they will move laterally, so that either of them may ear nearly the whole load, up to 4 N horizontally the same vertically. This increases the total tension from aout 3 N to aout 6 N. The weight w of a tether of height h required to support tension T satisfies 4 g( T wh) w = ρ (8) S Here, S is the strength (345 MPa), ρ is the density ( kg/m 3 ), g is the acceleration due to gravity, 4 is a safety factor. Hence w= N/metre for a sloping tether. Hence the weight of an average 1 km tether is 14 N or 8 N per pair, as stated in section Support Structure. vertical tether of a typical 1 km height supports 1, sloping tethers, giving a maximum tension of N, requiring a weight of 3 N/metre or N altogether. The vertical tethers are spaced along the support at intervals of approximately 7 metres, so the average load per metre on the support is 4 N, as stated in section Support Structure. Expansion Displacement s the tension increases, a tether exps according to the formula Tl e = (9) σe Here, e is the extension in metres, l is the length without tension, σ=w/ρ is the area of cross section, E is the modulus of elasticity (179 Gpa). For a tension increasing from 3 N to 6 N in a cross wind, the extension of a sloping tether comes to 1.7 metres. sloping tether forms a catenary, increased tension changes the geometry, leading to further movement that must e added to the extension. If χ χ are the horizontal displacements at the top ottom of the catenary, H is the horizontal component of the tension, then the length l height h satisfy 8 H wχ w l = χ sinh sinh (3) w H H H wχ w h = χ cosh cosh (31) w H H Using stard equalities for hyperolic functions including w w cosh ( χ χ ) sinh ( χ χ ) = 1 (3) H H we otain w w sinh ( χ χ ) = l h (33) H H The presence of the vertical tethers enales us to consider each length of 1km separately. Since l h at 45 slope, the value of the right-h side when H= is approximately.. If the tension doules, this value reduces to.1. If χ' χ' are the equivalent displacements when H doules, we otain 1 χ χ sinh. = 1.5 (34) 1 χ χ sinh.1 This gives a horizontal displacement of.5 metre over a typical 1 km height. dded to the expansion effect of equation (9), the total movement under the increased tension is under.5 metres, as stated in section Support Structure. Calculations for Pipes The final consideration for staility is the force on a tue caused y its curvature as olts travel through it at velocity V. s in equation (19), this is z Fc = mv (35) This force is added to the external force F w due to wind to give F=F c F w. Now the deflection of a tue of length K under a load of F Newtons per metre is given y 9 z Fx FKx EI = (36) Here, I is the moment of area. For a tue of inner outer radii r 1 r, I is given y 4 4 ( r 1 r ) I = π (37) 8 Comining (35) (36), we can otain an overall force per metre for a length K of tue etween tethers y integrating F c to give 6

7 ( Fx FKx) m V L FK mv F c = dx KEI = (38) 1EI Previous calculations (see section Support Structure ) showed the maximum force in a jetstream wind to e aout 1 N. The tether calculations were ased on a maximum lateral force of 4 N, a ratio of 4:1, so we want to set F = F c Fw = 4Fw (39) Comining equations (39) (38) gives 4 4 8mV K r1 r = (1.1) 9πE Taking E=7 1 1 Pa for aluminium, K=1 metre, r =4 cm, V m as efore (V= m/sec, m =1 kg), we otain an outer radius r 1 of 8.5 cm. Since the mass per metre is ρπ(r 1 - r ), ρ= kg/m 3, we otain a mass per metre of 4 kg, as stated in the section Support Structure. 4 Lofstrom, K., The Launch Loop, I Paper , July Knapman, J., High-ltitude Electromagnetic Launcher Feasiility, I Journal of Propulsion Power, Vol., 6, pp Post, R. F. ender, D.., mient- Temperature Passive Magnetic earings for Flywheel Energy Storage Systems, Seventh International Symposium on Magnetic earings, Zurich, Switzerl, ug., 7 Post, R. F. Gurol, S., The Inductrack pproach to Electromagnetic Launching Maglev Trains, 6 th International Symposium on Magnetic Levitation Technology, Turin, Italy, Oct., 1 8 Synge, J. L. Griffith,.., Principles of Mechanics, McGraw-Hill ook Company, Inc., New York, 1959, section olton, W., Mechanical Science ( nd Edition), lackwell Science Ltd, Oxford, 1998, section 4.. CONCLUSION This approach to the prolem of staility in the space cale is rather heavyweight. It only deals with lateral staility, which is the most significant prolem. Overall, the space cale is vertically stale ecause of the influence of gravity, ut vertical oscillations still need to e accounted for. In principle, with a constant cross wind, it is possile to align the space cale so that its curvature exactly counteracts that cross wind; this means deflecting the cale towards the wind so that the wind pushes it ack into line. However, such stale conditions are rare, it has not proved possile to devise a dynamic scheme to take advantage of this idea. nother area of concern is the possiility of perturations aove the level of the proposed tethers supports. s there is little air movement at these altitudes the air is very thin, there is less of a prolem. However, over the long term, some disturances are likely at very high altitudes, a suitale means must e devised to deal with them. This report presents work in progress. Further investigation may lead to other solutions. REFERENCES 1 Edwards,. C. Westling, E.., The Space Elevator: Revolutionary Earth-to-Space Transportation System, Spageo, San Francisco, Jan. 14, 3 Forward, R., Indistinguishale from Magic, aen Pulishing Enterprises, Riverdale, NY, 1995, pp Hyde, R., Earthreak: Review of Earth-to- Space Transportation, in Mark, H. Wood, L., Energy in Physics, War Peace, Kluwer cademic Pulishers, 1988, pp