Static deformation of space elevator tether due to climber

Acta Astronautica 111 (2015) 317–322

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Static deformation of space elevator tether due to climber$ Stephen Cohen a,n, Arun K. Misra b,1 a b

Vanier College, 821 Sainte-Croix, Montréal, Québec, Canada H4L 3X9 McGill University, Canada

a r t i c l e i n f o

abstract

Article history: Received 24 December 2014 Accepted 16 February 2015 Available online 25 February 2015

As higher strength to density ratio materials become available, the construction of a space elevator on Earth becomes more plausible. Though many fundamental aspects of the mechanical behaviour of a space elevator have been previously analysed, several details have not been rigorously explored. This paper examines the deformation of the tether from its nominal state when it is loaded with a climber at any altitude. Using an assumed modes numerical approach, the equilibrium conditions governing the static deformation of the elevator tether are derived, taking into account the presence of a climber. These discretised equations are solved numerically to determine the static deformation of the tether. A spectrum of statically deformed tether profiles is presented. Strain, stress and tension profiles are also computed and discussed. In general, when a climber is present below GEO, the extension (and stress) of the portion of tether below it is reduced and that above it is largely unaffected. When a climber is present above GEO, the extension (and stress) of the portion of tether below it is increased (that above it remains largely unaffected). Finally, the absolute displacement of the apex anchor (counterweight) is plotted against climber locations. & 2015 IAA. Published by Elsevier Ltd. All rights reserved.

Keywords: Space Elevator Tether deformation Stress Tension

1. Introduction A space elevator consists of a long tether that stretches from the surface of the Earth somewhere on the equator to a counterweight located beyond the geosynchronous altitude. The tether is in tension due to the counteracting gravitational and centripetal loads. Climbers may then be used to scale the tether transporting payloads to space. The first mechanical analysis of the space elevator was documented in 1975 [1]. In this paper, the profile of the cross-sectional area of the tether that would allow it to have constant stress was derived, though it did not account for the

☆

This paper was presented during the 65th IAC in Toronto. Corresponding author. E-mail addresses: [email protected] (S. Cohen), [email protected] (A.K. Misra). 1 Tel.: þ1 514 398 6288. n

http://dx.doi.org/10.1016/j.actaastro.2015.02.017 0094-5765/& 2015 IAA. Published by Elsevier Ltd. All rights reserved.

nominal strain that manifests in the deployed tether. Since this time, several analyses have been conducted on the space elevator, including that of Edwards [2], which investigates several important design features associated with it. In 2013, a more detailed assessment of the futuristic transportation system was compiled [3]. Thus far, there exist more dynamic analyses of the space elevator than static ones. Such dynamic analyses include a modal analysis of the tether without a climber [4] and the effects of a moving climber on the tether (ignoring tether elasticity) [5]. However, the effect of a stationary climber that is located at any given location on the tether has yet to be documented. This study uses a numerical approach to investigate the longitudinal deformation that the tether experiences when a climber is stationed anywhere along its length. The ensuing stress profile is found and used to impose conditions on a climber, such as its allowable mass.

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derived

2. Static equilbrium conditions

  σ 0 Am =g0 eF ðL0 Þ mc ¼   2  2  ½RE þL0 ð1 þ ε0 Þ=RG  RG =½RE þL0 ð1 þ ε0 Þ RE =RG

2.1. Static model The space elevator tether is modelled as a continuum with the tether elements located within the domain given by 0osoL0, where L0 is the length of the tether when it is unloaded, as shown in Fig. 1. The longitudinal extension of each tether element is given by u(s). The climber and counterweight locations (de and L0, respectively) are defined without including any extension. As such, the actual altitude of the climber, for example, is given by de þ usjde . The Earth's radius and angular velocity are denoted as RE and Ω, respectively. Not shown in the space elevator model is the crosssectional area of the tether, which varies along its length according to the following function AðsÞ ¼ Am eF ðsÞ

ð1Þ

where Am is the maximum cross-sectional area of the tether (at the geosynchronous altitude) and (

½RE þsð1 þ ϵ0 Þ2 3 RG   FðsÞ ¼ hRG ð1 þ ϵ0 Þ 2 RE þ sð1 þ ϵ0 Þ 2R2G R2E

) ð2Þ

Here, RG is the geosynchronous radius and ε0 is the nominal strain that manifests in the deployed space elevator tether, and is the ratio of the nominal stress, σ0, and the modulus of elasticity of the tether material, E. Finally, h, the characteristic height of the tether, is given by h ¼ σ 0 =γ g 0

ð3Þ

where γ is the volumetric density of the tether material and g0 is the surface gravity of the Earth. Eq. (1) was derived in a previous study [4], and is the area profile that ensures that the deployed, nominally stretched tether has constant stress throughout it. In the paper, the appropriate mass of the counterweight was also

ð4Þ

2.2. Static equations The conditions describing equilibrium for the model shown in Fig. 1 may be found by ensuring a force balance on all tether elements. Such an analysis must be done in two parts: once for all tether elements below the climber, and again for all tether elements above the climber. This is because there is a discontinuity in the tension profile of the tether at the climber's location. In this paper, an alternative approach will be used. Dynamic equations of motion for a system very similar to that described by Fig. 1 have already been derived by Cohen and Misra [4] using the Lagrangian approach. This previously analysed model allowed for lateral deformations as well, though these will be neglected here. The model did not, however, include a climber. The longitudinal extension function can be expressed as a sum of the nominal deformation (that experienced when no climber is present) and the additional deformation that the climber's presence causes, v(s) uðsÞ ¼ A 0 s þ vðsÞ

ð5Þ

This additional deformation can then be expressed as a linear combination of N admissible basis functions, ψi(s), each with a specific weight, ai, as follows XN vðsÞ ¼ a ψ ðsÞ ð6Þ i¼1 i i The static equilibrium conditions for a space elevator without a climber are presented in Eq. (7) with all of its parameters non-dimensionalized (to improve numerical computations). They are obtained by zeroing the dynamic

Fig. 1. Model of deformed tether with counterweight and climber.

S. Cohen, A.K. Misra / Acta Astronautica 111 (2015) 317–322

terms of the governing equations found in the previous study [4]. There are N equations, as k goes from 1 to N. Z 1 Z 1 2 2 dψ ∂V dψ Ω eFðξÞ k dξ þ ϵ0 Ω eFðξÞ k dξ dξ ∂ξ dξ 0 0 Z M p

1

 1 eFðξÞ ψ k ξð1 þ ε0 Þ þ þV dξ

Λ

0

 1 M c ψ k1 1 þ ε0 þ þ V 1

Λ

3 Z þM p

β Λ

0

3 þM c

β Λ



eFðξÞ ψ k

1



ξð1 þ ε0 Þ þ Λ1 þ V ψ k1

1 1 þ ε0 þ Λ þV1

2 dξ

2 ¼ 0

ð7Þ

Eq. (7) contains six terms. The first two terms arise due to the tension in the tether. The following two terms are associated with the centripetal effects, while the final two are due to gravitational ones. Table 1 describes all of the non-dimensional parameters that appear in Eq. (7). In order to update the equilibrium conditions to include the presence of a climber, two additional terms must be added. They represent the centripetal and gravitational effects that act on the climber, and they are analogous to the two terms associated with the counterweight (they simply have a different mass and position along the tether). All terms associated with the gravitational force can then be expanded binomially, discarding third and higher order terms. If this is done, and the terms representing the nominal case are cancelled out, the static equilibrium conditions become     K ik Ak ¼ f k ð8Þ where

β f k ¼ Me Λ

ψ ke

2 De ð1 þ ε0 Þ þ Λ1 ð10Þ

Λ

and K ik ¼ Ω

Z

1 0

2

8 R1 >

3 > < Mp 0

β Λ

> > :

eF ðξÞ ψ i ψ k

dξ þM c

9

ψ i1 ψ k1 > > 3 = ð1 þ ε0 þ Λ1 Þ

½ξð1 þ ε0 Þ þ Λ ψ ie ψ ke þ Me ½De ð1 þ ε0 Þ þ Λ1 3 1 3

> > ;

ð11Þ

In Eqs. (8–11), both i and k go from 1 to N. Parameters Me and De denote the non-dimensional mass and position, respectively, of the climber. Additionally, ψ ie denotes shape function i evaluated at the climber's position (ξ ¼De), while ψ i1 denotes shape function i evaluated at the counterweight's position (ξ ¼1). Eq. (8) consists of a stiffness matrix (Eq. (11)) multiplied by a column vector composed of the weights for the basis functions (Eq. (9)) equal to a forcing vector (Eq. (10)). The forcing vector would be zero if the mass of the climber was ! ! zero, which leads to Ak ¼ 0 , meaning that there is no additional deformation (nominal and unloaded state). One other case where the forcing vector is zero is when the   climber is situated at De ¼ β 1 =Λð1 þ ε0 Þ, which corresponds to the position that nominally extends to the geosynchronous altitude. One final zero deformation case occurs, somewhat surprisingly, at De ¼0. While the net load on a climber located at the base is not zero, all admissible basis functions, ψi, must equal zero at this location. It is also noteworthy that two terms associated with the climber appear in the stiffness coefficients. Thus, strictly speaking, the additional deformation has a non-linear relation with the climber mass. However, the terms associated with the climber are far smaller than the others appearing in the stiffness coefficients, and so all climberinduced tether deformation is approximately proportional to the mass of the climber. 3. Numerical results

ð9Þ

 1 M e ψ ke De ð1 þ ε0 Þ þ

2

 M c ψ i1 ψ k1 M e ψ ie ψ ke

3.1. Tether deformation

Ak ¼ ½A1 ; A2 ; A3 …T

3

319

dψ dψ k eF ðξÞ i dξ  M p dξ dξ

Z

1 0

eF ðξÞ ψ i ψ k dξ

Table 1 Non-dimensional parameters. Symbol

Meaning

Equation

ξ V Ai Λ Mp Mc β Ω

Position on ribbon Climber-induced extension Weights for basis functions Tether length Measure of tether mass Counterweight mass Geo to Earth radius ratio Characteristic frequency ratio

ξ ¼ s=L0 V ¼ υ=L0 Ai ¼ ai =L0 Λ ¼ L0 =RE M p ¼ Am γL0 =mtot M c ¼ mc =mtot β ¼ RG =RE pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ω ¼ EAm =ðmtot L0 Þ=Ω

In order for Eq. (8) to be evaluated, admissible basis functions must be stipulated. A previous study [4] found that the following basis functions are convenient for the representation of the longitudinal extension  

  1 ψ i ξ ¼ sin i  πξ ; i ¼ 1; 2; …; N ð12Þ 2 while these basis functions are admissible (they enforce zero displacement at the base yet allow for non-zero displacement at the tip), they are suitable for this study for yet another reason: they will allow good approximation for the step in tension that inevitably arises due to the presence of a climber. This study assumes the same numerical parameters that were assumed in the most recent comprehensive space elevator analysis [3]. The tether material has a density of 1300 kg/m3 and an elastic modulus of 1000 GPa. The unloaded tether length is set at 100,000 km. The nominal stress in the deployed tether is set to 35.19 GPa, resulting in a taper ratio (ratio of cross-sectional area at GEO vs the base) of exactly six and a characteristic height of 2762 km. The nominal strain throughout the vacant tether is 0.03519 as a result, and the counterweight's actual altitude when there is

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no climber present is 103,519 km due to the nominal linear extension. Finally, the maximum cross-sectional area is set to 62.8 mm2. As a result of these specifications the mass of the tether becomes 6355 metric tons while that of the counterweight must be 1878 metric tons. The mass of the climber has been set to 50 metric tons. It will be shown that this is the approximate upper limit of what this space elevator model can support at any altitude while maintaining equilibrium. Finally, the number of basis functions (and N by N size of the stiffness matrix) is set to 50. In general, including more basis functions leads to more precise results, particularly for approximating the step in strain, stress and tension across the tether at the climber's location. However, including too many basis functions (say, 100) leads to ill-conditioning of the numerical computations, and spurious results. Fig. 2 shows several longitudinal extension profiles, which result from solving Eq. (8) and inputting the re-dimensionalized weights into Eq. (6) for various climber locations. The shown extension profiles do not include the nominal extension (which is more than an order of magnitude greater) inherent to deployment. As indicated by the large bullets, climber locations have been set at 10,000 km, 20,000 km, all the way until 100,000 km. These locations refer to the unloaded (undeployed) tether locations. Consider the climber at the tip, which adds an additional longitudinal extension of about 65 km. The actual altitude of the tip therefore changes from 103,519 km to 103,584 km (there is a nominal, steady state extension of 3519 km at the tip). A fairly simple trend is observed when examining Fig. 2. The presence of a climber causes a change in tension for the tether elements below it, but not above it. Thus, the portion of tether above where the climber is situated displaces as a unit. When the climber is located below GEO, the tension in the tether below it is reduced, and it experiences a decrease in extension. When the climber is located above GEO, the tension in the tether below it is increased, and it experiences an increase in extension. Not shown in Fig. 2 are the particular profiles resulting from placing a climber at the

Fig. 2. Extension profiles resulting from climbers situated at 10,000 km intervals.

base or at GEO, both of which result in a flat line (zero deviation). One final observation is that there is an abrupt change in curvature of the longitudinal deviation at the climber. This is expected, as it indicates a step in strain at this location. 3.2. Corresponding strain, stress, tension The strain of a given tether element may be found by differentiating the total extension function with respect to space as follows     d ε ξ dV εξ ¼ 0 þ dξ dξ ¼ ε0 þ π

  

1 1 cos i πξ Ai i  2 2 i¼1 N X

ð13Þ

The stress of a given tether element is simply its strain multiplied by the elastic modulus. The tension as a function of dimensional space is the product of stress and cross-sectional area, both of which vary along the tether TðsÞ ¼ AðsÞσ ðsÞ

ð14Þ

In Fig. 3, the strain, stress, and tension profiles are plotted for a climber located at 10,000 km, at GEO, and at the counterweight. For the strain and stress profiles, the climber located at GEO results in a flat line at the nominal value. The tension profile for the GEO case is also unchanged from its nominal one, which follows the shape of the cross-sectional

Fig. 3. Strain, stress, and tension profiles for three particular climber locations: 10,000 km (solid), GEO (dash), and tip (dot).

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area function (Eq. (1)). A climber at the 10,000 km location yields the bottom (solid) curve in each of the three plots, while a climber at the tip yields the top (dotted) one. Additional profiles were not presented as they would overlap greatly and be hard to interpret. For the two non-GEO cases, a clear step is observed at the location of the climber for each of the strain, stress, and tension plots (these steps are not easily observable as they are small compared to the absolute tensions). Though sharp, the steps are not instantaneous as they should be; this is due to the discretisation technique employed in this study. The selected basis functions are continuous, as are their derivatives. As a result, the strain function, shown in Eq. (13), can never have a discontinuity. It is observed that while a climber stationed at the tip causes a larger magnitude of deviation than one located at 10,000 km, that of 10,000 km causes a larger change in strain, stress, and tension. However, it must be noted that from an absolute stress point of view, a climber at the tip poses a larger problem, as it increases the nominal stress at the base by a non-negligible amount (about 7%). Depending on material allowable strength, this particular scenario could, in theory, dictate the maximum permissible climber mass. In practice, however, the effect on material stress of a higher mass at the tip can be easily negated by manually lowering the counterweight's position by a small amount. The two particular steps in tension that occur at the climber locations (10,000 km and 100,000 km, respectively) are quite small with respect to the tension variation that exists in the nominal space elevator. They are thus not easily observed in Fig. 3. Instead of plotting the absolute tension, Fig. 4 plots the change in tether tension across a climber at a given location per unit mass of climber. The step in tension per unit mass is given by the difference in the gravitational and centripetal accelerations acting on the climber at any location

321

denoted by de. Fig. 4 is generated using the domain: 0 o de oL0 . The only location on the tether without a step in tension is GEO. The tension step is very large near the base, where the gravitational pull is strong. At the base, the step in tension per unit mass is about 10 N/kg. This location of maximum step in tension coincides with the location of minimum nominal tension (due to minimum area of cross-section). This location effectively determines the maximum allowable climber mass. For the space elevator modelled in this paper, having a maximum cross-sectional area of 62.8 mm2, the corresponding tension at the base is about 400,000 N. If the step in tension is any larger than this amount at this location, the space elevator cannot achieve equilibrium. For this reason, the maximum climber mass in this case is 40,000 kg. The 50,000 kg climber used for numerical computations in this paper is thus slightly higher than what would be practically feasible. The theoretical tension steps of about þ50 kN (climber at 10,000 km) and  25 kN (climber at tip), though difficult to see, are present in their respective tension plots in Fig. 3.

3.3. Counterweight displacement

In Eq. 15, G is the universal gravitational constant, while Me is the mass of the Earth. The location of the climber is

Finally, Eq. (8) is solved for a climber stationed at 1000 km intervals. The ensuing displacement of the counterweight (the tip) is then plotted against these climber locations in Fig. 5. As expected, the maximum extension occurs when a climber is located at the tip. A 50 t climber stationed there causes about 65 km of additional extension. Less trivial is the climber location associated with the maximum negative deviation. When a climber is located at about 3000 km altitude, the outer tip experiences a drop of about 48 km. This critical point is a consequence of two things. First, as seen in Fig. 4, there is a large net load on climbers in the lower portion of the tether. However, for the drop in tension below the climber to cause a large deflection, there must be a significant amount of tether below it to experience this change in strain. Additionally, it may be concluded that the tether position wherever the climber is situated deflects by the same

Fig. 4. Change in tether tension across climber per unit mass of climber.

Fig. 5. Counterweight deflection vs climber location.

ΔT ¼

GM e 2

ðR þde Þ

 ΩðR þde Þ

ð15Þ

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amount as the tip, because the entire portion of tether above the climber shifts as a unit. Thus, a plot of climber position extension vs climber location has the same shape as that of Fig. 5 (note that the bold points on Fig. 2 follow the same curve as Fig. 5). 4. Future work This study used a single, continuous function, described by Eq. (6), to approximate the additional tether extension brought on by the presence of a climber. While this simplifies the analysis, it is not efficient from a numerical perspective, as a large number of basis functions are required to capture the ensuing step in tension. It would be more numerically efficient to separate the deformation into two separate functions on either side of the climber, and account for the step in tension implicitly. To build upon this study, the static deformation of the tether when multiple climbers are stationed along its length concurrently should be evaluated, as this is anticipated for space elevator operations. 5. Conclusions This study evaluated the static deformation of a 100,000 km space elevator tether when a climber is stationed somewhere along its length. Depending on the climber location (above or below GEO), the portion of tether below it experiences an increase or decrease in strain, while that above it is largely unaffected. The magnitude for longitudinal extension caused by a static climber is of the order of tens of kilometres. The only two locations causing no effect are at the base and at GEO. With the exception of these two critical locations, the climber causes a step in tension, stress, and strain to

manifest across the tether at its location. Fundamentally, the maximum allowable climber mass is that which erases the tension at the base. For the space elevator parameters used in this study, a maximum climber mass of 40,000 kg is determined. When a 50 t climber is stationed at the tip of the tether, the stress at the base increases from its nominal amount (35.19 GPa) to about 37.65 GPa. This effect must be accounted for when assessing the tether's supporting capability. Finally, the largest negative and positive tip displacements for a 50 t climber occur when it is situated at an altitude of about 3000 km (causing about  48 km of deflection) and at the tip (causing about þ 65 km of deflection). Though these amounts are relatively small compared to the nominal extension experienced by the tether, they represent variations that occur during the operation of the space elevator, while the nominal extension is permanent, and occurs during initial deployment of the tether.

Acknowledgements The authors would like to thank Dr. Blaise Gassend for his contributions to this paper. References [1] J. Pearson, The orbital tower: a spacecraft launcher using the Earth's rotational energy, Acta Astronaut. 2 (1975) 785–799. [2] B.C. Edwards, E.A. Westling, The Space Elevator: a Revolutionary Earth-to-Space Transportation System, BC Edwards, 2003. [3] P.A. Swan et al., Space elevators: an assessment of the technological feasibility and the way forward, in: Proceedings of the IAA, 2013. [4] S.S. Cohen, A.K. Misra, Elastic oscillations of the space elevator ribbon, J. Guid. Control Dyn. 30 (2007) 1711–1717. [5] S.S. Cohen, A.K. Misra, The effect of climber transit on the space elevator dynamics, Acta Astronaut. 64 (2009) 538–553.