Dynamics and potential applications of a lunar space tethered system

Journal Pre-proof Dynamics and potential applications of a lunar space tethered system Liu JiaFu, Liu Bin, Wu ZhiGang, Jiang JianPing, Tian LiFeng PII:

S0094-5765(20)30032-1

DOI:

https://doi.org/10.1016/j.actaastro.2020.01.021

Reference:

AA 7845

To appear in:

Acta Astronautica

Received Date: 28 April 2019 Revised Date:

8 December 2019

Accepted Date: 14 January 2020

Please cite this article as: L. JiaFu, L. Bin, W. ZhiGang, J. JianPing, T. LiFeng, Dynamics and potential applications of a lunar space tethered system, Acta Astronautica, https://doi.org/10.1016/ j.actaastro.2020.01.021. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier Ltd on behalf of IAA.

Dynamics and potential applications of a lunar space tethered system Liu JiaFu*, Liu Bin, Wu ZhiGang*, Jiang JianPing, Tian LiFeng School of Aeronautics and Astronautics, Sun Yat-Sen University Correspondence: [email protected], [email protected]

Abstract: Nonlinear dynamics and potential applications of a lunar space tether system, with one end connected to the Moon’s surface and the other with a large floating counterweight, are investigated in the paper. Dynamical equations of the system with a massless viscoelastic tether considering the planar libration and elastic elongation within the elliptical Earth-Moon restricted three-body (EEMRTB) system are established. The equilibria and stability of the dynamical system are explored for the system at L1 side without considering small perturbations caused by eccentricity to facilitate the rough selection of the length and elastic coefficient of the tether. The potential applications including harvesting the Rayleigh Damping Dissipation power and accessible regions within the Earth-Moon system will be studied based on the dynamic analysis of the full nonlinear dynamics. The allowed parameter regions (the tether system will neither collide the Moon/Earth nor rotate about the Moon) are identified exactly using numerical analysis, moreover, the performance of the potential applications and other related references for building a tether system are explored at L1 and L2 numerically. It is suggested that one should design a space tether based energy harvester at L1 side, release an unpowered spacecraft with high mechanical energy at L2 side respectively. Some conclusions on selecting the parameters and evaluating the performance of the space tether based platform are obtained based on numerical calculations. The research will be useful for the Moon infrastructure constructions. Keywords: Space tether system, elliptical Earth-Moon restricted three-body system, energy harvesting, accessible region analysis

1

Introduction

Space tether systems have various potential applications and are under intensive investigations recently. The promising space applications include utilizing space tether system connected to the Earth/Moon for payload launch and transportation [1-2], payloads orbit transfer utilizing Motorised momentum exchange tethers (MMETs) [3], space elevators from the Earth to the altitude beyond the geostationary orbit (GEO) altitude to launch and transport payloads [4-5], asteroids deflection [6], malfunction spacecraft deorbiting and maintenance using the electrodynamic tether (EDT) [7] and space tether robots net [8] etc.. The current research will focus on the utilization of space tether system with a massless elastic super-long tether for energy harvesting and region access within the Earth-Moon system based on the work in [9]. The detailed reviews of space tether system missions, dynamics and control progress can be found in [10]. Concerning stability and equilibria of a space tether system, KRUPA et al. investigated the relative equilibria, stability and bifurcation diagram of a space tethered system with a large rigidity (even infinity) in [11]. Ref. [12] treated a space tethered system as a mechanical system and focused on the relative equilibria, stability and nonlinear dynamical behavior such as chaotic

motion. Moreover, the deployment and retrieval of a sub-satellite with control were also addressed. Concerning a long space tethered system with its center of mass at geostationary altitude, the radial equilibria and stability of a tapered Earth orbiting tether were explored and the stabilization by adding a sufficient heavy counterweight at geostationary altitude was also performed [13]. It was also pointed out that the stabilization was impossible for a tether with an over-small axial elasticity since no equilibria are available because the elastic force will fail to balance the centrifugal force. Xu et al. utilized the theory of space manifold dynamics to investigate the equilibrium configurations of tethered three-body system [14]. The periodic Lyapunov orbits were identified using an iteration procedure and the corresponding invariant manifolds were also presented. The rotation of a space tether system connected to a surface of one of the primaries in double system was studied in [15]. The possibilities to control the tether system’s rotation by changing the length of the rigid tether were explored within the elliptic restricted three-body system. The results were useful for the development of a planet elevator or payload transportation system. The relative equilibria and stability of a pendulum with a tip mass and rigid tether connected to asteroids were studied in [16]. The effect of connecting positions and tether’s length on the equilibria was addressed for a spheroidal asteroid. Some nonlinear dynamical analysis of space tethered systems was also investigated. Baoyin et al. investigated the mechanical energy transfer between the orbit and libration motions of a spinning tether system and an averaging method was used to analyze the coupled dynamics analytically [17]. The nonlinear interactions between the pitch motion of a sub-satellite and the system’s in-plane libration, and the pitch motion of a main-satellite and the system’s libration were focused on analytically and verified numerically respectively in [18] and [19]. Some complicated nonlinear dynamical motions such as coexistent quasiperiodic and quasiperiodic motions were also presented. Yu et al. focused on the Earth orbiting space tethered system with the perturbations by the J2 perturbation and eccentricity etc. [20]. The equilibria and stability were studied and the nonlinear phenomenon such as bifurcations, quasi-periodic oscillations, and chaotic motions were detected numerically. The vibro-impact dynamics was explored for an in-plane space tether system and some periodic motions and the corresponding stability were studied using the composite Poincare map [21]. Besides the numerical studies, chaotic pitch libration of a space tether system was predicted analytically using the Melnikov method and the validation was supported numerically by analyzing the Poincare maps etc. [22]. It was also found that the chaotic libration could be suppressed by adding damping. Furtherly, an extended time-delay autosynchronization (ETDAS) was used to stabilize chaotic librations to periodic ones and the verification was validated experimentally [23]. Space tethered system based energy harvesting and transportation are of great importance to humankind for deep space explorations. NASA studied an Electrodynamic Tethers (EDTs) based propulsion and power generation system at Jupiter [24] and this dual utilization proposal seems attractive. The design and analysis of a long space tower with one end connecting the Earth’s surface and the other extending beyond the geostationary height were performed in [1]. It was proposed to exploit the Earth’s rotational energy to launch payloads without consuming propulsions. Besides proposing to build an Earth elevator, Pearson proposed to build a tapered space tethered elevator system connecting the Moon’s surface at the near and far sides within the Earth-Moon restricted three-body system for space transportation and communication purposes [25]. Ref. [2] presented some details for developing a lunar space elevator system at L1

side for transporting lunar materials to high Earth orbits, at L2 side to afford a communication and scientific observation platform between the Moon and Earth. A top-level system analysis to evaluate the potential payoffs of lunar space elevators was also performed. McInnes and McInnes et al. investigated to use orbital towers with a massless tether and large tip mass to carry out payloads ascending from the Earth’s surface to the height beyond the GEO altitude [26, 27]. The dynamics considering the friction between the payloads and massless tether was focused on. In this paper, a space tethered system is built connecting the Moon’s surface and a large tip mass floating within the Earth-Moon three-body system. As proposed in [2, 9], the system will be operated at the near and far sides, i.e., L1 and L2 sides, respectively. The space tidal effects from the Earth and Moon were considered as external periodic forces. The potential applications such as harvesting the mechanical energy from the Earth-Moon system by adding a damper within the proposed tether system and accessing more regions within the Earth-Moon system, are proposed, both by exploiting the space tidal effects [9]. The paper is organized as follows. Sec. 2 gives dynamic modeling of a space tether system connected to the Moon’s surface with a floating tip mass. The elongation and libration dynamical equations of the space tether system within the EEMRTB system will be established. The corresponding non-dimensional dynamics will be obtained to facilitate the analysis. Sec. 3 will take the space tether system moving at L1 side as an example and the equilibria and stability will be explored by setting the eccentricity zero. This will provide a reference for selecting the parameters such as the tether’s length and rigidity. Sec. 4 will analyze the potential applications of the space tether system for energy harvesting and accessible region analysis within the Earth-Moon system. Some detailed analysis concerning these two applications will be addressed and some suggestions are presented based on numerical calculations.

2 Dynamic modeling of a lunar space tethered system The space tethered system consists of a massless viscoelastic tether, one end connecting the Moon’s surface and the other connecting a large tip mass P. It is assumed that an operation of space tethered system is failed when the tether rotates around the Moon or collide the Earth/Moon. E and M denote the Earth and Moon, and c is their common centroid as detailed in Fig. 1. In the paper, the libration angle
and the length of elastic tether  are taken as the generalized coordinates.

XI

E

θ c

YI

Re

M xr

Rm

rP L1

α l

P

yr reP r

Fig. 1 The space tether system in the EEMRTB system

The fixed and rotating reference frames are established as shown in Fig. 1, denoted as          .  points to the perigee,   points to the direction of the angular velocity of the Earth-Moon system about c.  points from c to M.  coincides with   .  is the true anomaly of the Earth-Moon system rotating about c.  ,    are the positions of M, E and P relative to c.  = 
, 
, 0!" and  denote the positions of P relative to M and E respectively.  = #1 + & '( , 0,0!" is the position of M relative to E with p indicating the semi-latus rectum, expressed as # = 1 − & * and a and e are the orbit semi-major axis and eccentricity respectively, while the superscript and subscript “T” and “r” representing transpose of a matrix or vector and the components in the rotating frame. One can arrive at the expression for the velocity of P as follows.

-( ./ − 
/ 
− / 
+  /
+ = ,-( ./ + 
/ 
+ /
+  /
0 1

0 

where + is the velocity of P, -( = 1 /1 + 1 with 1 and 1 denote the masses of E and M respectively. The overhead dot represents the derivative with respect to time. One can arrive at the kinetic, gravitational and elastic energies, and the Rayleigh dissipative function of the space tether system as 1 3 = 1+ ∙ + 2 6 = 67 + 6 - 1 - 1 67 = − − || | + |

1 : − ; *  > ; 6 = 92 0  ≤ ; 1 *   /  > ; Ψ = 92 ? 0  ≤ ;

2

where 3 , 67 , 6 and Ψ are the kinetic, gravitational and elastic energies, and the Rayleigh

dissipative function respectively. - and - are the gravitational constants of the M and E, : and ; are the elastic coefficient and constant original length of the tether. ? is the damping coefficient. Ψ is the power harvested during the elongation of the viscoelastic tether. It should be noted that 6 and Ψ will be zero when  ≤ ;. It is evident the tether will be tensionless and tensional when  ≤ ; and  > ; respectively. In the subsequent section, the numerical simulations will be performed considering the expressions for 6 and Ψ as clarified in Eq. (2) and this section. One can substitute Eq. (2) into the following Lagrange equation to establish the dynamics for the tether system.  BC BC A F− − G? = H 3

@ BD/ E BDE " where C = 3 − 6 is the Lagrangian of the space tethered system, DE = ,
!" , D/ E = J /,
/ K

and G? = J?  /, 0K are the generalized displacement, velocity vectors and the damping force "

vector respectively. The dimensional full nonlinear dynamics for the elongation and oscillation, or  and
degree of freedoms (DOFs), are as follows. L +

: - ?  / +  − ; + * − / * − 
/ * − 2/
/ + 1 1 

. *

-  + .


+

*

M + 2.
*

+ -( .L 
+ 2-( ./ /
+ -( .L 
= 0

− -( ./ *


4

2 //

2 /
/ -( ./
/
-( ./
/ 
-( ./ *
-( .L 
2-( ./ /
-( .L 
− − + + + − +         - .
+ L − M =0 . * + 2.
+  * * 5

st nd The dynamical equations are non-autonomous as the 1 and 2 derivatives of the true anomaly  with respect to time exist explicitly in Eqs. (4-5). The complicated coupling between the libration and elongation motions is evident. It is noteworthy that the coupled libration and elongation space tethered system is gyroscopic as the dynamics is established in the rotating / / in the frame and the Coriolis accelerations are introduced naturally such as 2/
/ and 2 
L +

preceding dynamics. One can take the reciprocal of the mean motion of the Earth and Moon about “c”, the instantaneous distance between the two primaries . and the total mass of the two primaries M=me+mm as time, distance and mass units respectively to make the dimensional dynamical equations in Eqs. (4-5) non-dimensional. The dimensionless full nonlinear coupled dynamics can be obtained as follows. P&  S  * P 2QR AP + 1 + &F R * TP − P; U P -( 
+ + − − P
S * − 2P
S − * * V 1 + &

1 + &

1 + &

1 + &

 +

P * 1

+ + &

1 +

-( TP + 
U

& T1 + P *

+

M 2P
U*

=0

P
SS + 2
S % 1 P S ) -(
S 
) )

-(
& 
-( &
% -( 
) 1 % &

1 % &

S

-( 


1 % & T1 % 2P
%

M P * U*

(6)

0

(7) The preceding two equations are just the corresponding non-dimensional dynamics. The prime indicates the derivative with respect to the true anomaly .

Here the explanation for the transition from the dimensional dynamics, i.e., Eqs. (4-5), to the corresponding dimensionless ones, i.e., Eqs. (6-7) will be presented as a supplement. One should extract the factors -W /. * or equivalently -W 1 % & 2 !/#* for all the terms in Eqs. (4-5) by referring to the following expressions/definitions. .

# -W -W &1 % & * , ./  X & , .L  # #* 1 % &

2- & 1 % & 3 /  X 3 1 % & 2 , L  ) # #3

?  , Y  Z-W /#M , R  YE /Y[ P  , YE  Z:/1, Q  21YE [ .

Each term in the dimensionless dynamics Eqs. (6-7) can be obtained by performing the procedure stated above. In the preceding expressions and Eqs. (6-7), P  /. is the dimensionless length of the tether and P; 

original length of the tether with d 

0

1)&2

\] (^_`ab

W(' c



0 #



 #

 d % d& is the dimensionless

. One can utilize ;   to represent

the natural length of the tether with  representing the distance between the Moon and the natural L1/L2 point in the Earth-Moon system for the tether system operating at the near/far side respectively. One can therefore assert that the dimensionless quantity is utilized to measure the natural length of the tether, instead of the dimensional quality ;. Utilizing to measure the length of a super-long tether is much more convenient. It can be seen that the P; is varying when . is used as the length unit. Some simple denotations are introduced to simplify the expressions such as YE  Z:/1, Q  ? /21YE , Y[  Z-W /#M and R  YE /Y[ . The prime represents the derivative with respect to , e is the eccentricity of the moon orbiting the Earth.

3. The Equilibria and stability It is important to determine the rough allowed parameter regions for a successful operation of the system. Therefore, the equilibria and stability analysis will be performed before carrying out the dynamic analysis. One should note that the analysis in this subsection merely present a rough estimation for locating the parameters including R and . The exact parameter region should be determined by numerical calculations. The equilibria and stability of a space tether system attached to the Moon’s surface have been studied in [9, 29]. In [9], the artificial equilibria at L2 side are plotted as a function of the

dimensionless natural length and elasticity of the tether numerically, but the corresponding stability was not focused on. In [29], merely the axial elastic elongation was studied and the equilibria/stability were investigated numerically for a space tether system at L1/L2 side. In this paper, the artificial equilibria and the corresponding stability at L1 side for a tether system with two degrees of freedom (axial elastic elongation and in-plane libration) will be focused on. The equilibria can be determined without considering the excitation and derivative related terms as follows. )P ) -( 
%

% P*

1

-( P % 


+ P*

-( 
−

+

M 2P
*

-( 


+ R * P − R * d = 0 8

M

1 + 2P
+ P * *

= 0 9

In the paper, the equilibria at L1 side are taken as an example and therefore
 = g with the subscript ‘&’ representing the equilibria. By substituting
 = g into the preceding two equations, one can arrive at the following equation to determine the equilibria at the natural L1 side. -(

 -( − P + * − + R * P − R * = 0 10a

P 1 − P * #


 = g 10b

It is easy to write the equilibria vector j as P , 0, g, 0 for a tether system with two degrees of freedom in this paper different from the one in [29]. It can be seen that Eq. 10(a) is identical to the one of Eq. (12) of Ref. [29]. Therefore, it is expected that the P in Eq. 10 (a) will be the same as the one in [29]. The stability of equilibria can be determined by checking 1 JkTlm UK with n = 1 − 4 and k representing the real part of complex number lm . lm is determined by |op − lq| = 0 with q representing a 4 × 4 identity matrix and op is the Jacobi of the dynamics at j with e=0. We further write Eq. 10(a) as follows. "

for P ∈ xW , 0.9z. ?

# -( * v P = R *  AR P − P + -( + P * − 1 − P * F 11

  t # -( uRP = X AP − + − -( F 12

 t #P −   P* 1 − P * s

By analyzing Eq. (11), it can be seen that P is ↓ when R ∈ 0, 2.69384 , ↓→↑→↓, ↑→↓ and ↑ when R ∈ 2.69384, 3.3624353 , R ∈ 3.3624353, 44.45709728705823

and R ∈ 44.45709728705823, ∞ respectively. By analyzing Eq. (12), one can see RP is ↑ and ↑→↓→↑ when ∈ 0, 0.90293 and

∈ 0.90293, 0.999 respectively when < 1. RP is ↓→↑ and ↓ when ∈ 1, 5.611

and ∈ 5.611, 5.9399 respectively. The denotations ↓ and ↑ mean P or RP is monotone decreasing and monotone increasing with P . We will plot P and RP for some representing R and

in Figs. 2 and 3.

First, P is plotted adopting R  0.1, 1, 2.69384, 3, 3.3624353, 10.470441 and 44.45709728705823 respectively.

Fig. 2. Equilibria and stability analysis results based on Eq. (11)

One can see that equilibria beyond certain distance from moon are forbidden when R < 10.470441 by observing Fig. 2(a). Positive values in Fig. 2(b) means that equilibria are physically realizable using a tether. Space tether can merely afford tensions. Although the region of allowed equilibria increase with R as seen from Fig. 2(a), it is more difficult to keep space tether taut when increasing R as seen from Fig. 2(b). Equilibria positions and lengths nearly coincide and extremely small oscillations nearby are permitted if we want to keep space tether always taut. We merely consider the condition with taut tether and consider other cases as failed ones. It can be estimated that extremely large elastic potential is required when P approaching 0.9 as the increment and decrement of gravitational potential of the Earth and centrifugal potential. Equilibria within certain range will be stable when R > 2.69384 by observing Fig. 2(d). It is concluded that stable range corresponds to  /P > 0 in Fig. 2(c, c1). Stable region will increase with R when R > 2.69384. Furthermore, equilibria are stable within / , 0.9

when R > 44.45709728705823. One can never select overlarge/small R by considering the abovementioned analysis. Second, RP is plotted adopting = 0.01, 0.5, 0.90293, 0.999, 1.5, 3.5 and 5.93 respectively.

It is stated previously that P > P; holds when generating equilibria using taut space tethers. It is evident that P > P; holds when < 1 in Fig. 3(b.1). In Fig. 3 (a.1), circle, diamond, and pentagram are used as restriction boundaries for n=1.5, 3.5 and 5.93 to ensure Fig. 3. Equilibria and stability analysis results based on Eq. (12)

P > P; . It can be seen that all equilibria are available when < 1 in Fig. 3(b.1), but the stable equilibria start to appear merely when > 0.9023 as shown in Fig. 3(b.2, b.3). The stable region becomes larger with the increment of as observed in Fig. 3(b.2, b.3). For the case > 1, the allowed equilibria region decreases with n as shown in Fig. 3(a.1). Moreover, it seems that the allowed equilibria region for < 1 is larger than the one for

> 1 by comparing Fig. 3(b.1) with Fig. 3(a.1). However, one can see that the range and location of the stable equilibria region for < 1 and > 1 are quite different as observed in Fig. 3(a.2, b.2). It is also concluded that the structures of the autonomous dynamics in this paper and [29] are similar since the equilibria and stability are similar.

4

Energy harvesting and accessible regions analysis

For energy harvesting, it seems that using the steady-state magnitude of the power output as a measure to evaluate the performance of a space tether based energy harvester is not so reasonable and it seems that utilizing the energy harvested per orbit of the Earth-Moon system revolving their common mass center (i.e., one lunar month) is more reasonable after the motion enters into a steady state. One can calculate the average power harvested for a harvester entering into steady state to measure its performance. The corresponding expressions are detailed as *E„^b] 1 1 *E„^b] Ψ , ΨW = ƒ Ψ ,  = 1,2,3 ⋯ 13

‚= ƒ 2 g b]

b]

where ‚ is the energy harvested in one month, and ΨW indicates the average power, both for a space tether with steady states. Therefore, the space tether system has already entered into steady state at ;. One can see that the international units of ‚ and ΨW are Joule and Watt respectively. In the paper, we use :‚ ∙ ℎ and ‚ @@. In the subsequent numerical analysis part, we will present some examples and explanations for selecting ‚ and ΨW to evaluate the performance of the harvester instead of Ψ. For accessible regions analysis, the Jacobi’s integral is a direct and useful measure and one can get the expression for the dimensional Jacobian ‡? as follows. |+ |* / * - 1 - 1 ‡? = − | |* − − 14

|| | + | 2 2 All variables can be found in the previous sections. The first term corresponds to the dimensional kinetic energy, and the remaining two terms are the potential energies, both expressed in  . It is convenient to use a dimensionless Jacobi’s integral ‡ with & = 0 to measure the accessible regions, as follows. * * P P x 
− P
S 
z + xP
S 
+  
z 1 ‡= − -( + P
* + P
* ! − 2 2 P -( − Z1 + 2P
+ P *

15

All variables can be found in previous sections. The first and remaining terms correspond to

the kinetic and potential energies respectively in the preceding equation. The Jacobi’s integral provides an energy measure for assessing feasible regions available for an unpowered space vehicle with a certain initial condition. 4.1 The results at L1 side The dynamic responses of the space tether system will be determined by three important parameters, i.e., R, and Q. In the paper, we will present ˆR, with different Q. ˆ is the physical variables that we are interested in. First, R ∈ 2.85, 5.5! with Q  1 is selected. The corresponding allowed region of will be determined. Moreover, the steady-state magnitudes of ΨR, , ‡R, , R, , tensions and strains within the tether will be presented to check the performance, give some design references. The steady-state magnitude of power output within the allowed parameter region is detailed in Fig. 4 (a). Specially, the parameter ranges of and R with power output larger than 100 kW are given together with the corresponding power output as presented in Fig. 4 (b, c).

Fig. 4 The steady-state magnitude of power output of the tether based harvester at L1 side

The allowed lower and upper boundaries of can be found in Fig. 4 (a) when R ∈ 2.85, 5.5! and Q  1. It is evident that the area of the allowed parameter region increases with R. The tip mass will collide the Moon/Earth when is selected a little smaller/larger than the values at the lower/upper boundary. Besides, the tip mass will collide the Moon/Earth when R is selected a little larger/smaller than the values at the upper/lower boundary. These conclusions are intuitive. It can be seen that the steady-state magnitude of power output will be maximized when locates at the upper boundary for each R. The parameter region of and R is detailed in Fig. 4 (b-c) for the power output larger than 100 kW. It is evident that the area of the parameter region is very small and the robustness of the harvester is weak as the power output will decrease dramatically with little change of R or . The conclusion that one should build a long and stiff space tether holds but the risk of collision with the Earth increases and the cost of building a long tether is also expensive. To build a high-performance, reliable and robust harvester demands one select parameters elaborately, keep the parameters invariable exactly. As discussed above Eq. (13), we use ‚ , R and ΨW  , R to evaluate the performance

of the harvester. The arguments and R will be selected near the parameter region as shown in Fig. 4(b, c) because the harvester with the parameters within this region may make the power output large. Therefore, one can present ‚ , R and ΨW  , R as follows.

Fig. 5 ΨW  , R and ‚ , R with Q  1 at L1 side

The conclusion that one should select both and R as large as possible under the condition of successful operations of the system to generate as much electricity as possible is arrived at by analyzing the results in Fig. 5 (a, b). It is also seen that the magnitude of ΨW in Fig. 5 (a) is one order smaller than the one in Fig. 4 (a). Actually, adopting Ψ in Fig. 4 as a measure to evaluate the performance of the harvester is not reasonable and ΨW is a suitable measure to evaluate the harvester’s performance and this will be discussed subsequently. The largest power output and energy harvested per month are about 50 kW and 40 000 kW*h when R  5.5 and  2.555 as indicated in Fig. 5. The steady-state magnitude of the tether’s length  presented in Fig. 6 (a) is of importance as it will give one a restriction without disturbing the spacecrafts in high Earth orbits or even colliding the Earth. The steady-state magnitude of the Jacobi’s integral ‡ in Fig. 6 (b) provides a measure of region accessible for an unpowered satellite.

Fig. 6 (a). The steady-state magnitude of the tether’s length (b). The Jacobi’s integral J

It can be seen that the largest steady-state magnitudes of tether’s length is about 2.68 r 10‰ 1 when R  5.5 and  2.555. This length seems fine as the tip mass will neither disturb space vehicles in high Earth orbits nor collide the Earth. But both large R and

will lead to risk. It is evident that ‡ is not large (the largest is merely about -1.59) and the unpowered tip mass can access not very large regions when it is released. The steady-state magnitudes of the tensions and elastic strains within the tether are detailed in Fig. 7 (a, b).

Fig. 7 The magnitudes of the elastic tensions and strains

It seems that the magnitude of the tensions within the tether is large when R and are selected to construct the most efficient harvester. The tension approaches about 1.29 r 10V Š and one should build a tether with enough material strength considering this value. Fortunately, by observing the results in Fig. 7(a, b), it is evident the largest steady-state magnitude of the

strain happens when R  3.05,  1.168, but one can build a most efficient harvester when R  5.5,  2.555. Second, Q  2 and meanwhile R ∈ 2.85, 5.5! is selected. One can compare the results in this case with the preceding one with Q  1 to check how the dynamic responses are affected by Q. Merely the steady-state magnitudes of ‚ and ΨW are given in Fig. 8.

Fig. 8 The steady-state magnitudes of ‚ and ΨW with Q  2

In Fig. 8, we merely present the results within the parameter region with very large ‚ and ΨW , i.e., near the upper boundary. It can be seen that the allowed parameter region varies with Q by comparing the result in Fig. 8 with the one in Fig. 5. Specifically, larger or smaller R falls in the allowed parameter region when larger Q is adopted. For example, R  5.5,  2.64 with Q  2 and R  5.5,  2.555 with Q  1 are at the allowed region boundaries respectively. The largest steady-state of ‚ and ΨW within yellow regions are smaller than the corresponding ones in Fig. 5 and this indicates that increasing Q from 1 to 2 is not helpful for harvesting more energy. Overlarge damping will be beneficial for successful operation of the tether system with longer length or smaller rigidity. 4.2 Comparisons with the results at L2 side In this subsection, the dynamic responses at L2 side will be investigated and comparisons including the energy harvesting and accessible region will be made with the results at L1 side discussed in the preceding subsection. Intuitively, the tether system will librate dramatically at L2 side as the Earth will attract the tip mass. This will be beneficial to accessing large regions within the Earth-Moon system. To verify this, some numerical calculations will be performed for the space tether system at L2 side. First, dynamic responses for Q  4 will be presented. Specifically, the steady-state magnitudes of ‡, Ψ, tensions and strains will be given in Fig. 9 (a-d). The results in Fig. 9 (b, c and d) are extensions of Figs. (4) and (6) in Ref. [9]. In [9], some asymptotic analysis was performed and the effectiveness and accuracy of the results obtained using the multiscale analysis were verified numerically. In this work, the ranges for n and ϖ are selected as n [1.044,

4] and ϖ [1.5, 5] compared the ones of n [1.1, 2.5] and ϖ [1.1, 2.5] in Ref. [9]. The correctness of the results in this paper can be supported by comparing the results in Fig. 9 (b-d) with the corresponding ones in Figs. 6 of [9] as follows.

Fig. 9 The steady-state magnitudes of ‡, Ψ, tensions and strains; the results in Ref. [9] for ψ, tension and strain in (b, c and d) denoted as red solid circles

The forbidden parameter region is surrounded by triangle lines as seen in Fig. 9. The remaining region corresponds to the allowed region. The distributions of the forbidden and allowed regions are different from the ones at L1 side. The tether system will rotate about the Moon when adopting the parameters within the forbidden regions and this failure operation mode is different from the one at L1 side. Roughly, it can be seen that Ψ is much smaller in Fig. 9 (b) than the ones in Fig. 4, which indicates that one should construct an energy harvester at L1 side rather than at L2 side. To validate this furtherly, the corresponding ‚ and ΨW will be given for different Q later in Fig. 10. However, compared to the steady-state magnitude of ‡ in Fig. 6, the Jacobi’s integral reaches about -1.45 as shown in Fig. 9 and this means that all positions within the Earth-Moon system are accessible with an unpowered tip mass. The steady-state magnitudes of the tensions and strains are also presented to give material strength and fatigue design references when constructing a space tether system. The correctness of the results in this paper can be supported by observing and analyzing the results in Fig. 9 (b-d). It can be seen that the results in Ref. [9] coincide with the ones in this paper. To further investigate the power output and energy harvesting performance in detail, the steady-state magnitudes of ‚ and ΨW are plotted for Q  1, 4,  7 as follows to check the magnitudes.

Fig. 10 The steady-sate magnitudes of ‚ and ΨW for Q  1, 4,  7 at L2 side

It can be seen that the steady-state magnitudes of ‚ and ΨW are much smaller than the ones at L1 side by comparing the results in Fig. 10 with the ones in Figs. 5 and 8. Therefore, it is suggested that one should build a space tether based energy harvester at L1 side rather than L2 side. The steady-state magnitudes of Jacobi’s integral for Q  7, 4 and 1 are presented in Fig. 11 (a-c) respectively. The parameter regions with the steady-state magnitudes of Jacobi’s integral larger than -1.5675 are extracted and shown in Fig. 11 (d).

Fig. 11 The forbidden parameter regions surrounded by triangle lines and the allowed regions for Q  7, 4 and 1 in (a), (b) and (c) respectively; the parameter regions for 2‡ ‹ 3.315 summarized in (d)

The forbidden and allowed parameter regions are different for each Q as seen in Fig. 11. It seems that the upper boundaries move to larger and smaller R with the increment of Q as shown in Fig. 11. It is evident that the values of ‡ are large near the upper boundaries. One can set ‡  )1.5675 and extract to show the steady-state magnitudes of ‡R, ‹ )1.5675 and its corresponding parameter regions in Fig. 11 (d) [28]. An unpowered space vehicle with ‡ ‹ )1.5675 can access everywhere within the Earth-Moon system. Similar to the energy harvesting mission at L1 side, one should select the parameters extremely near the allowed parameter boundary to get large ‡ and the robustness of the tether system should be strong to withstand operation failure (the tether will rotate about the Moon). It is known that one utilizes ‚ and ΨW to evaluate the performance of an energy harvester. One merely needs to focus on the steady-state motions with proper parameters. ‚

l(m)

dα/dt(rad/s)

J

J

dl/dt(m/s)

α(rad)

and ΨW are accumulation or integration with respect to time/true anomaly as presented in Eq. (13). There is no need to focus on specific time history of the motion. But for releasing a payload with large ‡, one should determine the release time or release window besides focusing on the steady-state motions with proper parameters. Chaotic or other irregular motions are not preferred for releasing payloads with large ‡ because it is difficult to decide the launch time for a non-periodic or even chaotic motion. For energy harvesting, one merely focuses on generating large ‚ and ΨW under the condition of reliable operation of the tether system. Any kind of steady-state motions are preferred as long as ‚ and ΨW are large, even chaotic motions are acceptable. Following the discussion in the preceding paragraph, we select R = 2.3, = 1.748 and Q = 7 to perform a specific time domain simulation and to show the launch window available for an unpowered satellite with ability for accessing everywhere within the Earth-Moon system. The time histories of the libration, elongation and the Jacobi’s integral are detailed in Fig. 12.

Fig. 12 The time histories of the periodic steady-state
, 
/@, , /@ and ‡

It is obvious that the tether system’s steady-state motion is periodic and the steady-state periodic motion with several periods is presented in Fig. 12. The largest ‡ reaches about -1.442 as denoted by stars. Moreover, the bold red lines indicate the launch window of an unpowered satellite accessible for everywhere within the Earth-Moon system. The width of the launch window is about 0.8532 within each 2π, or about 3.7098 days per lunar month. The period of the libration is 2π, about 27.32166 days. One can see that the period of the libration is rather long compared to the ones of the fundamental period about 150 hrs-160 hrs of an Earth tether elevator [30]. The difference is mainly from the distinct dynamically environment of an Moon and Earth elevator. One important phenomenon is one may puzzle that one will wait for a long time for the launch window as indicated in the last figure. It is suggested that one can select the initial conditions at the minimal value of ‡ as shown in the middle below figure. It may be easy to put the satellite at this low mechanical energy level. The waiting time will be dramatically reduced and the steady-state motion will appear very soon. It is emphasized again that the better the performance of the space missions (energy harvesting and accessible regions analysis) achieved, the riskier the tether system will be failed to operate. Therefore, it is suggested that one should design a tether system with enough robustness. Here we will also present some results concerning some periodic and triple- periodic motions etc. to see how the motion properties are influenced by the parameters of the tether

system. First,  1.2, Q  0.024 and Y  2.1 are selected for a tether system at L2 side. The phase planes and its Poincare sections for elongation and libration motions are presented as follows.

Fig. 13 The phase planes (black lines) and corresponding Poincare sections (blue dotts) of a tether system with  1.2, Q  0.024 and Y  2.1 at L2 side

It can be seen that a triple-periodic motion exists for the elongation and libration motion. We can get one-periodic and quasi-periodic motions by selecting Q  0.03 and Q  0.019 respectively, meanwhile keeping the remaining parameters unchanged. The corresponding results are given in Fig. 14.

Fig. 14 The phase planes (black lines) and corresponding Poincare sections (blue dotts) of a tether system with  1.2 and Y  2.1 at L2 side for Q  0.03 in (a, b) and Q  0.019 in (c, d)

It is obvious that a one-periodic motion exists when Q is 0.03 by observing the results in Fig. 14 (a and b). The motion will become quasi-periodic when Q is 0.019 by observing the results in Fig. 14 (c and d). By analyzing the results in Figs. 13 and 14. It is concluded that a one-periodic motion will become a triple-periodic one when Q is reduced from 0.03 to 0.024, a triple-periodic one will become a quasi-periodic one when Q is reduced from 0.024 to 0.019. This means the stability

of a motion will be influenced by the damping of the system. A larger damping will be beneficial for stabilizing a motion (making a motion more regular).

5

Conclusions

The paper treated nonlinear dynamics and potential applications of a space tether system connected to the Moon’s surface with a large tip mass and massless viscoelastic tether within the EEMRTB system. Dynamical equations considering the planar libration and elongation of the tether system excited by the eccentricity of the Earth-Moon system were established and numerical calculations were performed thereafter. Two potential applications including energy harvesting and accessible regions analysis were addressed and the following conclusions were obtained. 1. The equilibria and stability analysis of the system at L1 side provided a necessary but not sufficient condition for selecting R and , because the success or failure for operating the system is also highly influenced by the parameters such as Q and & related terms. 2. It is suggested that one should build a space tether based energy harvester at L1 side although the motion region and allowed parameter regions are restricted to make the system between the Earth-Moon without colliding any primary nor rotating about the Moon. A more appropriate and reasonable measure for evaluating the performance of the harvester, i.e., the energy harvested per lunar month or the average power output, is proposed, instead of the power of Rayleigh damping dissipation. 3. The energy harvesting results within the allowed parameter regions were presented and it was indicated that one should design the most efficient harvester with both R and as large as possible. The system’s parameters should be kept accurately during the operations because the robustness of the system is weak, indicating the performance even the success or failure will be changed dramatically even the parameters change just a bit. 4. It is concluded that the steady-state magnitude of the Jacobi’s integral ‡ of the tip mass can be large enough to access anywhere within the Earth-Moon system at L2 side. The allowed parameter regions with ‡ larger than the critical value -1.5675 are adjacent to the boundaries separating the allowed and forbidden parameter regions (the system will rotate about the Moon when parameters are in the forbidden regions) and this indicates that more optimal performance, more risk. Moreover, a robust tether should be constructed. 5. Energy harvesting mission deals with the integration of steady-state power output within certain time duration while the region access analysis concerns about not only steady-state magnitude of ‡, but also the specific release time and release time window, and the waiting time duration for the launch window. It is suggested that one should build a tether system with proper parameters and initial conditions to make the steady-state motion regular rather than chaotic, meanwhile reducing the waiting time for payload release. 6. There exist one-, triple- and quasi-periodic motions for a tether system and the motion properties influenced by the damping are studied and it is found that a larger damping is beneficial for stabilizing the motion (making the motion more regular).

6

Acknowledgement This work was support by the National Natural Science Foundation of China (Project

number: 11302134, 91748203, 11872381). The authors highly appreciate the above financial supporting. The authors would like to thank the reviewers and the editor for their comments and constructive suggestions that helped to improve the paper significantly.

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1. Periodic gravity tides are exploited for energy harvest and mechanical energy raising.

2. Energy harvest at L1 and L2 sides are studied and comparisons are made.

3. A new and reasonable measurement to evaluate harvester s performance is proposed.

3. Mechanical energy raising (accessible region analysis’ at both sides is investigated.

4. Detailed analysis for releasing an unpowered payload to access more regions is presented.

5. Suggestions for constructing a reliable/efficient tether system for space missions are given.

To whom it may concern,

All the authors declare here that we have no conflicts of interest to this work.

Thanks

Jiafu Liu, Bin Liu, Zhigang Wu, Jianping Jiang, Lifeng Tian Dec. 9, 2019