The problem of planar oscillations of a pendulum with variable length suspended on the Moon’s surface is considered. It is assumed that the Earth and Moon (or, in the general case, a planet and its satellite, or an asteroid and a spacecraft) revolve around the common center of mass in unperturbed elliptical Keplerian orbits. We discuss how the change in length of a pendulum can be used to compensate its oscillations. We wrote equations of motion, indicated a rule for the change in length of a pendulum, at which it has equilibrium positions relative to the coordinate system rotating together with the Moon and Earth. We study the necessary conditions for the stability of these motions. Chaotic dynamics of the pendulum is studied numerically and analytically.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Price excludes VAT (USA)
Tax calculation will be finalised during checkout.
Pearson, J., Anchored lunar satellites for cislunar transportation and communication, J. Astronaut. Sci., 1979, vol. 27, no. 1, pp. 39–62.
Beletskii, V.V. and Levin, E.M., Mechanics of a lunar tether system, Kosm. Issled., 1982, vol. 20, no. 5, pp. 760–764.
Levin, E., Dynamic Analysis of Space Tether Missions, vol. 126 of Adv. Astronaut. Sci., 2007, San Diego, CA: AAS Publ. Office.
Beletsky, V.V. and Levin, E.M., Dynamics of Space Tether Systems, vol. 83 of Adv. Astronaut. Sci, 1993, San Diego, CA: AAS Publ. Office.
Burov, A.A. and Kosenko, I.I., Relative equilibria of an orbital station in the vicinity of triangle libration points, Dokl. Akad. Nauk, 2007, vol. 416, no. 3, pp. 335–337.
Burov, A.A., Kononov, O.I., and Guerman, A.D., Relative equilibria of a moon-tethered spacecraft, Adv. Astronaut. Sci., 2011, vol. 136, pp. 2553–2562.
Burov, A.A., Kosenko, I.I., and Guerman, A.D., Dynamics of a moon-anchored tether with variable length, Adv. Astronaut. Sci., 2012.
Schiehlen, W., Über die Lagestabilisirung künstlicher Satelliten auf elliptischen Bahnen, Doctoral (Eng.) Dissertation, Stuttgart: Technische Hochschule, 1966.
Schiehlen, W., Über den Drallsatz für Satelliten mit im Innern bewegten Massen, Z. Angew. Math. Mech., 1966, vol. 46 (special edition), pp. T132–T134.
Schiehlen, W. and Kolbe, O., Gravitationsstabilisierung von Satelliten auf elliptischen Bahnen, Ing.-Arch., 1969, vol. 38, no. 6, pp. 389–399.
Sarychev, V.A., Issues of Orientation for Satellites, Itogi Nauki Tekh., Ser.: Issled. Kosm. Prostr., Moscow: available from VINITI, 1978, no. 11.
Polyanskaya, I.P., Oscillations of a satellite with compensating devices in elliptical orbit, Kosm. Issled., 1982, vol. 20, no. 5, pp. 674–681. [Cosmic Research, p. 474].
Burov, A.A., Oscillations of a vibrating dumbbell in elliptical orbit, Dokl. Akad. Nauk, 2011, vol. 437, no. 2, p. 186.
Burov, A.A. and Kosenko, I.I., Planar oscillations of a body with variable mass distribution in elliptical orbit, Dokl. Akad. Nauk, 2011, vol. 440, no. 6, pp. 760–764.
Djebli, A., El Bakkali, L., and Pascal, M., On fast retrieval laws for tethered satellite systems, Acta Astronaut., 2002, vol. 50, no. 8, pp. 461–470.
Djebli, A., Pascal, M., and El Bakkali, L., Laws of deployment/retrieval in tether connected satellites systems, Acta Astronaut., 1999, vol. 45, no. 2, pp. 61–73.
Djebli, A., Pascal, M., and El Bakkali, L., On deployment dynamics of tethered satellite systems, Revue de Mécanique Appliquée et Théorique, 2000, vol. 1, no. 1, pp. 13–39.
Pascal, M., Djebli, A., and El Bakkali, L., A new deployment/retrieval scheme for a tethered satellite system, intermediate between the conventional scheme and the crawler scheme, J. Appl. Math. Mech., 2001, vol. 65, no. 4, pp. 689–696.
Djebli, A. and Pascal, M., A new method for the orbital modification of a tether connected satellite system, Acta Mechanica, 2004, vol. 167, nos. 1–2, pp. 113–122.
Mantri, P., Mazzoleni, A.P., and Padgett, D.A., Parametric study of tethered satellite systems, J. Spacecraft and Rockets, 2007, vol. 44, no. 2, pp. 421–424.
Wong, B. and Misra, A., Planar dynamics of variable length multi-tethered spacecraft near collinear Lagrangian points, Acta Astronaut., 2008, vol. 63, nos. 11–12, pp. 1178–1187.
Pizarro-Chong, A. and Misra, A.K., Dynamics of multi-tethered satellite formations containing a parent body, Acta Astronaut., 2008, vol. 63, nos. 11–12, pp. 1188–1202.
Guerman, A.D., Equilibria of muitibody chain in orbit plane, J. Guidance, Control and Dynamics, 2003, vol. 26, no. 6, pp. 942–948.
Guerman, A.D., Spatial equilibria of muitibody chain in a circular orbit, Acta Astronaut., 2006, vol. 58, no. 1, pp. 1–14.
Guerman, A.D., Smirnov, G.V., Paglione, P., and Seabra, A.M., Stationary configurations of tetrahedral tethered satellite formation, J. Guidance, Control and Dynamics, 2008, vol. 31, no. 2, pp. 424–428.
Burov, A.A., Guerman, A.D., and Sulikashvili, R.S., Relative equilibria of a tetrahedral structure with rigid and tethered elements, Adv. Astronaut. Sci., 2008, vol. 129, pp. 1665–1674.
Burov, A.A., German, A.D., and Sulikashvili, R.S., On orbital motion of a tetrahedral gyrostat, Prikl. Mat. Mekh., 2010, vol. 74, no. 4, pp. 594–609.
Burov, A.A., Guerman, A.D., and Sulikashvili, R.S., Dynamics of a tetrahedral satellite-gyrostat, AIP Conference Proceeding: Numerical Analysis and Applied Mathematics, American Institute of Physics, 2010, vol. 1281, pp. 465–468.
Burov, A.A., German, A.D., and Sulikashvili, R.S., Established motion of gyrostats with equal moments of inertia in a central force field, Prikl. Mat. Mekh., 2011, vol. 75, no. 5, pp. 738–744.
Original Russian Text © A.A. Burov, A.D. German, I.I. Kosenko, 2014, published in Kosmicheskie Issledovaniya, 2014, Vol. 52, No. 4, pp. 307–312.
According to the author personal communication.
Rights and permissions
About this article
Cite this article
Burov, A.A., German, A.D. & Kosenko, I.I. On plane oscillations of a pendulum with variable length suspended on the surface of a planet’s satellite. Cosmic Res 52, 289–294 (2014). https://doi.org/10.1134/S0010952514040029
- Chaotic Dynamic
- Relative Equilibrium
- Cosmic Research
- Libration Point
- Elliptical Orbit