EA038209B1 - Damping device for stabilizing an electrodynamic rope system in the near-earth space - Google Patents

Abstract

The invention relates to space technology and can be used to stabilize a space rope system in the near-Earth space. For the electrodynamic rope system to function in the near-Earth space in the mode of orientation along the local vertical, collectors are placed to collect charges of opposite signs at the ends of the rope. When the charges interact with the Earth magnetic field, a moment of Lorentzian forces arises, which has an orienting effect on the rope. To achieve an asymptotically stable orientation of the rope along the local vertical, a charge value on one of the collectors is changed in accordance with the conditions defined by a current orientation of the rope, and parameters of the rope are selected in accordance with the conditions that also take into account the influence of disturbing moments, which makes it possible to solve the problem of stabilizing the rope in a position along the local vertical.

Description

The invention relates to the field of space technology and can be used to stabilize the space tether system in near-earth space in order to increase the efficiency of its functioning in the process of cleaning up space debris.

The theoretical developments carried out to date and tests carried out in open space allow us to assert that electrodynamic tether systems (EDTS), including conductive cables interacting with the Earth's magnetic field, can be used as sources of the Ampere traction force in near-Earth space [1]. In particular, EDTS can be used as a promising source of traction force, which does not require fuel consumption, for solving the urgent problem of descent from orbit of spent elements of space systems [2-11].

From the analysis of the directions of the current flowing along the cable and the magnetic induction of the Earth's magnetic field, it follows that the most efficient is the cable operating in the mode of a conductor with a current oriented in the near-earth space along the local vertical. This orientation of the tether is stable in the central Newtonian gravitational field [1]. At the same time, it was found that under the action of the moment of Ampere forces, the vertical orientation of the cable is destroyed [1]. The problem of EDTS instability is known and is considered by specialists as critically important [8,12]. A number of works [2-15] are devoted to solving this problem, in which it is concluded that control should be used to combat dynamic instability. Among the possible approaches to its solution, various options for controlling the strength of the current flowing along the cable are proposed. For example, in [12,13] it is proposed to use devices for periodically switching off the current flowing through the cable.

However, in most cases, the EDTS should function under conditions that imply a continuous flow of current along the cable in one direction, for example, to create the above-mentioned traction force in order to remove space debris [2-11] or for the operation of the EDTS in the power generator mode [16] ... Therefore, a common disadvantage of the above-mentioned methods for controlling an electrodynamic cable is the periodic shutdown of the current flowing through the cable, or switching the direction of the current, which reduces the efficiency of the EDTS and limits the possibilities of their use.

Known patents [17, 18], in which it is proposed to abandon the periodic shutdown of the current flowing through the cable, or switching the direction of the current, and generate control signals based on measurements of the parameters of the current state of the cable and subsequent calculations. Measurements of the voltage and current in the cable and, possibly, the tension of the cable, are used in [17, 18] to form such variations in the amperage that coincide with the values of the current intensity induced by the geomagnetic field with undesirable components of the cable velocity. In addition, in accordance with the algorithm [17, 18], it is required to integrate the system of differential equations of motion of the EDTS to generate a control signal for the formation of a control current pulse. The disadvantage of the approach [17, 18] is the complexity of the control system caused by the need to perform measurements to assess the state of the cable during movement, which, in turn, reduces its reliability.

The closest to the claimed device is the patent [19], selected as a prototype, which does not require a control system for the current transmitted through the cable, and also does not require devices for measuring the cable tension, and offers to separate the charges at the ends of the cable and use the resulting moment Lorentz forces as an additional stabilizing moment. The device proposed in [19] creates a restoring torque used to stabilize the EDTS. The existence, uniqueness and stability of the taut equilibrium position along the local vertical are proved. However, to ensure the asymptotic stability of the vertical position of the cable, along with the restoring one, a dissipative moment is also required. Therefore, the disadvantage of the approach [19] is that it does not disclose a method for realizing the dissipative moment, and numerical calculations are performed for a very small model dissipative moment, which is realized in a passive way.

The objective of this invention is to improve the device for stabilizing the vertical position of the cable, increasing its efficiency (robustness) and speed by creating an active control torque of a dissipative nature. The solution of this problem makes it possible to expand the area of applicability of EDTS operating in the orientation mode along the local vertical.

The technical result of the claimed device is the stabilization of the EDTS in the position when the cable is stretched along the local vertical. The specified technical result is implemented by the claimed device, the diagram of which is illustrated in FIG. 1-2. FIG. 1 shows the orbital coordinate system Cξηζ, which is the basic system for solving the problem of stabilizing the EDTS. Point C is the center of mass of the EDTS. In the nominal mode, the z axis, directed along the taut cable, is collinear with the Сф axis directed along the radius vector of the point С relative to the attracting center. The axes Cξ and Cn are directed, respectively, along the positive transversal to the orbit and along the normal to the orbital plane. FIG. 2 shows a diagram of the claimed device; in fig. 3 shows the angles defining the orientation of the z-axis of the tether in the orbital coordinate system; in fig. 4 shows the boundary of the region of existence of oblique equilibrium positions of the EDTS; in fig. Figures 5-15 show the results of numerical modeling of the EDTS stabilization process.

The claimed device is illustrated in FIG. 2, on which 1 is a cable; 2 - a positive charge collector connected to the end body 3 of the cable by means of an electrically insulating attachment 4; 5 - electronic emitter (at least one), electrically connected to the collector 2 and located inside the electrically insulating mount 4; 6 - end body at the opposite end of the cable, connected to the negative charge collector 7 by means of an electrically insulating attachment 8; 9 - electronic emitter (at least one), electrically connected to the end body 6 and located inside the electrically insulating mount 8; 10 - electrically insulating mount, inside which there is an electronic emitter (at least one) 11, electrically connected to the negative charge collector 7; 12 - control unit, the operation of the emitter 11, connected to the negative charge collector 7 by means of an electrically insulating attachment 10; 13 - measuring device included in the control unit 12.

The operation of the claimed device is illustrated in FIG. 2. Collectors - devices for collecting electric charges are connected to the ends of the conductive insulated cable 1 (Fig. 2). Collector 2, located at the upper end of the cable (the one that is farther from the Earth), by means of electrically insulating fasteners 4 is connected to the end body 3 of the conductive cable 1. Collector 2 receives a positive charge supported by one or more electronic emitters (for example, field electronic emitters, cold electron emitters based on nanoporous carbon, or Hall ion sources) 5, transmitting a negative charge to the end body 3. At the opposite end of the cable (the one that is closer to the Earth), the end body 6 is connected to the collector 7 by means of an electrically insulating attachment 8. The collector 7 receives negative charge, supported by one or more electronic emitters 9, transmitting a negative charge from the end body 6. In accordance with the control algorithm, the collector 7 gives off a part of the negative charge to the near-earth space using the electronic emitters 11 based on the control signals generated by the control unit 1 2, including a measuring device 13 for measuring the angle of deviation of the cable from the vertical and the time derivative of this angle, as well as a calculating and control device.

The technical result achieved by the claimed invention consists in improving the device for stabilizing the vertical position of the cable, increasing its efficiency (robustness) and speed.

This technical result is achieved by the fact that the placement of a positively charged collector 2 (Fig. 2) at the upper end of the cable 1 (Fig. 2) and a negatively charged collector 7 (Fig. 2) at the lower end of the cable leads to the excitation of a moment of Lorentz forces acting on collectors, and exerting an orienting effect on the cable, and a change in the magnitude of the charge at the lower end of the cable in accordance with the current value θ of the angle of deviation of the cable from the vertical and the derivative & of this angle in time leads to the excitation of a moment having a dissipative character. When the conditions for the EDTS parameters are met, the indicated moments ensure the existence and asymptotic stability of the vertical equilibrium position of the cable and solve the problem of stabilizing the cable in this position.

The operability of the claimed device is provided by sources of electricity converted from light with the help of solar cells, which are part of the EDTS (not shown in Fig. 2 due to their popularity and widespread use). Sources of electricity ensure the operation of the emitters to maintain the charges on the collectors and prevent the discharge of the collectors in the process of moving through the plasma of the near-earth space.

The essence of the claimed invention is as follows. For an EDTS, the center of mass of which (point C) moves at a speed relative to the Earth's magnetic field (EMF), characterized by magnetic induction, the accumulation of electrostatic charges q1 and q ₂ on the EDTS collectors leads to the appearance of a moment of Lorentzian forces determined by the formula [21]

M _L = PxA \ v _c xB), (1) exerting an orienting effect on the EDTS under certain conditions and used as a restoring moment along with the gravitational moment [20].

In this case, the bodies attached to the ends of the cable are considered pointlike, and their rotational motion can be neglected due to the small ratio of the characteristic size of the body to the length of the cable. Elastic deformations of the cable are also not considered. The tether is considered to be stretched along the z-axis with the unit vector, and its librational motion relative to the center of mass does not affect its orbital motion. When P = v _r = Ρ (ω _η - ω _Ρ ) ξ _η , under the assumptions ^{1 1 ζ Δ} '° ^{υ tu} where zi is the distance from the center of mass of the system to the center of mass of the loads attached to the cable at the lower end, z ₂ - distance from center of mass

- 2 038209 systems to the center of mass of the cargo attached to the cable at the upper end, ω ₀ is the orbital angular velocity of the center of mass of the EDTS, ω _E is the angular velocity of the Earth's daily rotation, R is the radius of the orbit of the center of mass of the EDTS. Under the conditions of modeling the EMF by a direct magnetic dipole ^ ~ 'where g ⁰ 1 = - 29556.8 nT is the Gaussian coefficient, RE is the average radius of the Earth.

Analysis of the mathematical model of the EDTS allows one to obtain the conditions under which the mode of stabilized movement of the EDTS is achieved. It has been established (for more details - in the appendix) that the existence, uniqueness and asymptotic stability of the equilibrium position of a cable of length I in a taut state along the local vertical is ensured by choosing such parameters of the cable and the aforementioned masses attached to it, for which the distances from the center of mass of the system to the centers of charged collectors are the same and the inequality

I! Oo ~ ^ω Ε) (^ 20 - ^ 10) _> γ / 2)

4lR ² (Dg (m ₂ + m _Q / 6) 'where I is the length of the cable, m ₀ is the mass of the cable, m ₂ is the mass of weights attached to the cable at the upper end, q ₂₀ is the constant part of the charge on the upper EDTS collector, q ₁₀ is the constant part of the charge on the lower collector of the EDTS The total charge on the lower collector 7 is the sum of the constant part q ₁₀ and the variable (controllable) part # 1 "determined by the formula ^- ^^ 2" where # 2 is a small, generally speaking, variable part of the charge on the upper collector 2, and the coefficient k _q changes according to the law

2d x. [d> 0 for &> 0 k _q = --- 3-sign & -l, j.

^ ² and the parameter d takes the values A ⁼ θ ⁿ P ^and

The claimed invention has been tested by computer simulation on the basis of the applicant's mathematical and mechanical faculty - St. Petersburg State University. Examples of approbation are given below.

Example 1.

First, a numerical analysis of inequality (2) was carried out for a set of EDTS with the following values of the orbital radius and electrostatic parameters: R = 7-106 m, q ₁₀ = -5-10 ' ² C, q20 = 5-10' ^{2 C.} The result is shown in FIG. 4, where the values of m2 + m _0/6 (kg) are plotted along the horizontal axis, and the values of I (m) are plotted along the vertical axis. If the lengths and masses are such that the representing point lies below the curve on the graph, then inequality (2) is satisfied. For example, let I = 100 m, the linear mass of the rope γ = 2-10-3 kg / m, m1 = m ₂ = 30 kg. Then inequality (2) is satisfied. An example confirms the validity of inequality (2) for an extensive set in the space of parameters of EDTS.

Example 2.

A series of numerical experiments has been carried out to simulate the process of stabilization of the EDTS in the mode of vertical positioning of the cable. To illustrate the numerical simulation of the stabilization process, an example of an EDTS design was chosen, containing a cable 200 m long, having a linear mass γ = 2-10 ^-3 kg / m and a conducting current I = 2A. The mass at the lower end of the cable is m1 = 55 kg, the mass at the upper end of the cable is m ₂ = 50 kg. The center of the collector charge at the lower end of the cable has the coordinate zi = -95 m, the center of the collector charge at the upper end of the cable has the coordinate z ₂ = 105 m. The center of mass of the EDTS moves in a circular equatorial orbit of radius R = 7-106 m.

FIG. 5-8 illustrate the movement of EDTS with zero charges on the collectors (the stabilization system does not work). At the initial moment of time, the cable was deflected from the local vertical by an angle l rad in the plane (η, ζ) and released without an initial angular velocity relative to the orbital coordinate system. The vertical position of the cable turns out to be unstable, which is clearly demonstrated by the behavior of the guiding cosine γ ₃ (Fig. 5), which deviates over time from the target value γ ₃ = 1 under the action of the Ampere moment, as well as the behavior of the modulus orthogonal to the cable of its relative angular velocity component, referred to to the orbital angular velocity (Fig. 6). Hereinafter, in all figures, a dimensionless angle is plotted along the horizontal axis - the argument of latitude u = ω ₀ t. Note that during the entire time of motion the magnitude of the perturbing Ampere moment remains small (Fig. 7) - 2 orders of magnitude less than the value of the gravitational moment (Fig. 8), which tends to stabilize the EDTS along the local vertical. The model dissipative moment I ^{_ _} Vvzya \ s), taken exactly in the form that was previously proposed in [19] (J is the tensor of inertia of the EDTS) with a dimensionless small parameter h = 0.001, changing with time, qualitatively repeats the curve on the graph fig. 6 and takes values in the range from 0.0005 to 0.0025 N, that is, during the entire movement it remains an order of magnitude less than the disturbing Ampere moment and is not able to eliminate the instability of the vertical position of the cable.

FIG. 9-10 illustrate the motion of EDTS with nonzero charges on the collectors (q1 = - 0.05 C, q ₂ = 0.05 C) in accordance with the approach proposed in [19]. All other parameters of the EDTS and the initial conditions of its motion are retained unchanged, as in the case of Fig. 5-8, corresponding to zero charges on the collectors. Numerical integration results (coinciding with the corresponding

- 3 038209 results from [19]), show that the direction cosine γ ₃ slowly tends to the target value γ ₃ = 1 (Fig. 9), and the modulus of its relative angular velocity component orthogonal to the tether, referred to the orbital angular velocity, tends to zero (Fig. 10).

FIG. 11-15 illustrate the movement of an EDTS with an active cable damping system in accordance with the claimed invention. As in the previous cases, at the initial moment of time the cable was deflected from the local vertical by an angle l rad in the plane (η, ζ) and released without the initial angular velocity relative to the orbital coordinate system. The results of numerical integration show that the direction cosine γ3 tends to the target value γ3 = 1 (Fig. 11) an order of magnitude faster than in the absence of active damping (Fig. 9), and the modulus of the tether component of its relative angular velocity referred to the orbital angular velocity tends to zero (Fig. 12) an order of magnitude faster than in the absence of active damping (Fig. 10).

Note that during the entire time of motion the value of the control dissipative moment remains small (Fig. 13) - an order of magnitude less than the value of the restoring Lorentzian moment (Fig. 14), which tends to stabilize the EDTS along the local vertical. This ratio of the values of the restoring and damping moments is consistent with the need to maintain a negative charge at the lower end of the cable. Numerical calculation of the value of q1 in the mode of controlled movement of the EDTS confirms that the charge q1 remains negative during the entire movement (Fig. 15) and, over time, as the vertical position of the cable stabilizes, tends to a stationary value of -3-10 ^{-2 C.}

We also note that in the considered examples of numerical modeling, the worst values of the EDTS parameters are selected, for which condition (2) is not satisfied. Moreover, even the relation z1 = -z ₂ does not hold, which allows us to consider the Ampere moment as a constantly acting perturbation that naturally arises in this problem. Comparison of the graphs shown in FIG. 9 and FIG. 11 as well as FIG. 10 and FIG. 12, indicates that the inclusion of the inventive damping device, which ensures the creation of an active control torque of a dissipative nature, in the EDTS stabilization system makes it possible to increase the efficiency (robustness) and speed of the system in comparison with the prototype and, thereby, improve the device for stabilizing the vertical position of the cable. The solution of this problem, in turn, makes it possible to expand the area of applicability of EDTS operating in the orientation mode along the local vertical.

As the above examples have shown, computer simulation of EDTS stabilization processes in the vertical cable mode, analytical studies and numerical experiments, the claimed device can actively generate a control torque of a dissipative nature, which increases the efficiency (robustness) and speed of the stabilization system in comparison with the prototype, and also allows the EDTS system to operate in the absence of control of the current flowing through the cable, which increases the efficiency of the EDTS itself. At the same time, the claimed device, as shown by example 2, makes it possible to ensure the asymptotic stability of the vertical position of the cable due to the effectively realized possibility of separating charges at the ends of the cable and active control of the charge on the lower collector by using the resulting moment of the Lorentz forces as a control stabilizing moment. All this makes it possible to more efficiently provide the nominal mode of motion of the EDTS for removing space debris compared to analogs.

- 4 038209

List of used literature:

1. Beletsky V.V., Levin E.M. Dynamics of space tether systems. M., Science,

1990,336 s.

2. Forward R.L. Electrodynamic drag terminator tether, Appendix K of high strength-toweight tapered Hoytether for LEO to GEO payload transport, Final Report on NASA SBIR Phase I Contract NAS8-40690,10 July 1996.

3. Forward R.L., Hoyt R.P., Uphoff C. Application of the Terminator Tether ™ electrodynamic drag technology to the deorbit of constellation spacecraft, Paper AIAA 983491, 34th Joint Propulsion Conference and Exhibition, Cleveland, OH, July 13-15.1998.

4. Forward R.L., Hoyt R.P. Terminator Tether ™: a spacecraft deorbit device, Journal of Spacecraft and Rockets 37 (2000) 187-196.

5. Cosmo M.L., Lorenzini E.C. (Eds.) Tethers in Space Handbook, 3rd ed., Smithsonian Astrophysical Observatory, Cambridge, MA, USA, 1997.

6. Forward R.L. et.al. Electrodynamic tether and method of use, U.S. Pat. No. 6,116,544, Sep. 12, 2000.

7. Vannaroni G., Dobrowolny M., De Venuto F. Deorbiting with electrodynamic tethers: comparison between different tether configurations, Space Debris 1 (2001) 159-172.

8. less L., Bruno C. et al. Satellite de-orbiting by means of electrodynamic tethers part I: general concepts and requirements, Acta Astronautica 50 (2002) 399-406.

9. less L., Bruno C. et al. Satellite deorbiting by means of electrodynamic tethers Part II: system configuration and performance, Acta Astronautica 50 (2002) 407-416.

10. Ishige Y., Kawamoto S., Kibe S. Study on electrodynamic tether system for space debris removal, Acta Astronautica 55 (2004) 917-929.

11. Yamaigiwa Y., Hiragi E., Kishimoto T. Dynamic behavior of electrodynamic tether deorbit system on elliptical orbit and its control by Lorentz force, Aerospace Science and Technology 9 (2005) 366-373.

12. Zhong R., Zhu Z.H. Libration dynamics and stability of electrodynamic tethers in satellite deorbit, Celestial Mechanics and Dynamical Astronomy 116 (2013) 279-298.

13. Corsi J., less L. Stability and control of electrodynamic tethers for de-orbiting applications, Acta Astronautica 48 (2001) 491-501.

14. Pelaez J., Ruiz M., Lopez-Rebollal 0., Lorenzini E.C., Cosmo M. A two bar model for the dynamics and stability of electrodynamic tethers, Journal of Guidance, Control and Dynamics 25 (2002) 1125-1135.

15. Larsen M.B., Blanke M. Passivity-based control of a rigid electrodynamic tether, Journal of Guidance, Control, and Dynamics 34 (2011) 118-127.

16. Roberts et al. Tether power generator for Earth orbiting satellites, U.S. Pat. No. 4,923,151

Bl, Mar. 1, 1998.

17. Levin E.M., Carroll J.A. Apparatus for observing and stabilizing electrodynamic tethers, U.S. Pat. No. 6,755,377 Bl, Jun. 29, 2004.

18. Levin E.M., Carroll J.A. Method for observing and stabilizing electrodynamic tethers, U.S. Pat. No. 6,758,443 Bl, Jul. 6, 2004.

19. Tikhonov A.A. Patent RU - No. 2666610 for the invention "Device for stabilizing an electrodynamic cable system for removing space debris", Priority 08/22/2017, Date of state. registration in the State. the register of inventions of the Russian Federation 09/11/2018.

- (prototype)

20. Beletsky V.V. Satellite motion relative to the center of mass in a gravitational field. M., ed. Moscow University, 1975.308 p.

21. Petrov K.G., Tikhonov A.A. The moment of Lorentz forces acting on a charged satellite in the Earth's magnetic field. 4.2: Calculation of the moment and the assessment of its components // Vesta. St. Petersburg, un-that. Ser. 1. 1999. Issue. 3 (No. 15). S. 81-91.

Claims (2)

CLAIM 1. A damping device for stabilizing an electrodynamic cable system in near-earth space, containing a conducting electrically insulated cable, one end of the cable is equipped with a collector of positively charged particles connected to the end body of the cable through an electrical insulator, inside which there is at least one electron emitter that generates an electron beam from the collector positively charged particles, the second end of the insulated cable is equipped with a collector of negatively charged particles connected to the end body of the cable through an electrical insulator, inside which there is at least one electron emitter, which generates an electron beam from the end body of the cable, characterized in that the collector of negatively charged particles is equipped with an electrical insulator cylindrical shape, inside which there is at least one - 5 038209 electronic emitter, and on the outside of the electrical insulator there is a control unit for the operation of the emitter, while the control unit contains a measuring device for measuring the angle of deviation of the cable from the vertical, and the negatively charged collector has a controlled charge with a value of q1, where: H \ = Chu ~ k ^ _r <0, 2d ik = - Th sign # for &> 0 k _q = -1 for &<0 θ is the angle of deviation of the cable from the vertical, is the derivative of the angle θ with respect to time, q ₁₀ is a constant part of the charge on a negatively charged collector, ^ 2 is a small part of the charge q ₂ on a positively charged collector, d is a positive constant, the choice of q ₁₀ , ^ 2 and d does not violate the inequality q1 < 0, I is the length of the cable. 2. A damping device for stabilizing an electrodynamic tether system in near-earth space according to claim 1, characterized in that the distances from the center of mass of the system to the centers of charged collectors are the same, and the length of the cable and the charges of the collectors correspond to the inequality: I £ 1 I (^ 0 ~ ^ω Ε) (^ 20 ~ 010) _> | ΜΚ ² ωΙ (τη ₂ + m _Q / 6) 'where m ₀ is the mass of the cable, m2 is the mass of the positively charged collector with the electrical insulator and the end body of the cable connected to it, q20 is the constant part of the charge on the positively charged collector, RE is the mean radius of the Earth, g ⁰ ₁ = -29556.8 nT is the Gaussian coefficient, ω ₀ is the orbital angular velocity of the center of mass, ω _Ε is the angular velocity of the Earth's daily rotation, R is the radius of the orbit of the center of mass.