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16-04-2024 | Original Paper

Dynamics of a heavy pendulum of variable length with a movable suspension point

Authors: Alexander A. Burov, Vasily I. Nikonov

Published in: Acta Mechanica

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Abstract

Dynamics of a pendulum of variable length with a movable suspension point performing a predetermined motion is considered. The laws of changing the length of the pendulum and the motion of the suspension point are indicated, at which the equation of motion can be reduced to the equation of motion of a mathematical pendulum of constant length. The result obtained is extended to the case of a spherical pendulum.
Literature
1.
go back to reference Beletskii, V. V.: The motion of a satellite about its center of mass. Nauka, Moscow (1965). (in Russian); (English translation: Beletskii, V.V.: Motion of an artificial satellite about its center of mass. Jerusalem: Israel program for scientific translations. 261 p. (1966)) Beletskii, V. V.: The motion of a satellite about its center of mass. Nauka, Moscow (1965). (in Russian); (English translation: Beletskii, V.V.: Motion of an artificial satellite about its center of mass. Jerusalem: Israel program for scientific translations. 261 p. (1966))
2.
go back to reference Schiehlen, W.: Über die Lagestabilisirung künstlicher Satelliten auf elliptischen Bahnen. Diss. Dokt.-Ing. (Doctoral’s Dissertation in Engineering) (Technische Hochschule, Stuttgart), (1966) Schiehlen, W.: Über die Lagestabilisirung künstlicher Satelliten auf elliptischen Bahnen. Diss. Dokt.-Ing. (Doctoral’s Dissertation in Engineering) (Technische Hochschule, Stuttgart), (1966)
3.
go back to reference Schiehlen, W.: Über den Drallsatz fur Satelliten mit im Innern bewegten Massen. Z. Angew. Math. Mech. 46(S1), T132 (1966) Schiehlen, W.: Über den Drallsatz fur Satelliten mit im Innern bewegten Massen. Z. Angew. Math. Mech. 46(S1), T132 (1966)
4.
go back to reference Schiehlen, W., Kolbe, O.: Gravitationsstabilisierung von satelliten auf elliptischen Bahnen. Ingenieur-Archiv 38(6), 389–399 (1969) Schiehlen, W., Kolbe, O.: Gravitationsstabilisierung von satelliten auf elliptischen Bahnen. Ingenieur-Archiv 38(6), 389–399 (1969)
5.
go back to reference Polyanskaya, I.P.: Satellite oscillations with compensating devices in an elliptical orbit. Kosm. Issled. 20(5), 674–681 (1982) Polyanskaya, I.P.: Satellite oscillations with compensating devices in an elliptical orbit. Kosm. Issled. 20(5), 674–681 (1982)
6.
go back to reference Sarychev, V.A.: Questions of Orientation of Artificial Satellites, Itogi Nauki Tekh., Ser. Issled. Kosm. Prostranstva 11, VINITI, Moscow, [in Russian] (1978) Sarychev, V.A.: Questions of Orientation of Artificial Satellites, Itogi Nauki Tekh., Ser. Issled. Kosm. Prostranstva 11, VINITI, Moscow, [in Russian] (1978)
7.
go back to reference Burov, A.A.: Oscillations of a vibrating dumbbell on an elliptic orbit. Doklady Phys. 56(3), 182–185 (2011) Burov, A.A.: Oscillations of a vibrating dumbbell on an elliptic orbit. Doklady Phys. 56(3), 182–185 (2011)
8.
go back to reference Burov, A.A., Kosenko, I.I.: Planar vibrations of a solid with variable mass distribution in an elliptic orbit. Doklady Phys. 56, 548–552 (2011) Burov, A.A., Kosenko, I.I.: Planar vibrations of a solid with variable mass distribution in an elliptic orbit. Doklady Phys. 56, 548–552 (2011)
9.
go back to reference Burov, A., Kosenko, I.: On planar oscillations of a body with a variable mass distribution in an elliptic orbit. Proc. Inst. Mech. Eng. Part C: J. Mech. Eng. Sci. 225(10), 2288–2295 (2011) Burov, A., Kosenko, I.: On planar oscillations of a body with a variable mass distribution in an elliptic orbit. Proc. Inst. Mech. Eng. Part C: J. Mech. Eng. Sci. 225(10), 2288–2295 (2011)
10.
go back to reference Burov, A.A., Guerman, A.D., Kosenko, I.I.: Tether orientation control for lunar elevator. Celestial Mech. Dyn. Astron. 120, 337–347 (2014) Burov, A.A., Guerman, A.D., Kosenko, I.I.: Tether orientation control for lunar elevator. Celestial Mech. Dyn. Astron. 120, 337–347 (2014)
11.
go back to reference Stephenson, A.: On a new type of dynamical stability. Mem. Proc. Manchester Literary and Phil. Soc. 52(2)(8), 1–10 (1908) Stephenson, A.: On a new type of dynamical stability. Mem. Proc. Manchester Literary and Phil. Soc. 52(2)(8), 1–10 (1908)
12.
go back to reference Stephenson, A.: On an induced stability. Phil. Mag. 15, 233–236 (1908) Stephenson, A.: On an induced stability. Phil. Mag. 15, 233–236 (1908)
13.
go back to reference Kapitsa, P.L.: The dynamic stability of a pendulum with an oscillating suspension point. J. Exp. Theor. Phys. 21(5), 588–597 (1951) Kapitsa, P.L.: The dynamic stability of a pendulum with an oscillating suspension point. J. Exp. Theor. Phys. 21(5), 588–597 (1951)
14.
go back to reference Kapitsa, P.L.: A pendulum with a vibrating suspension. Usp. Fiz. Nauk 44(1), 7–20 (1951) Kapitsa, P.L.: A pendulum with a vibrating suspension. Usp. Fiz. Nauk 44(1), 7–20 (1951)
15.
go back to reference Markeyev, A.P.: The equations of the approximate theory of the motion of a rigid body with a vibrating suspension point. J. Appl. Math. Mech. 75(2), 132–139 (2011) Markeyev, A.P.: The equations of the approximate theory of the motion of a rigid body with a vibrating suspension point. J. Appl. Math. Mech. 75(2), 132–139 (2011)
16.
go back to reference Yakubu, G., Olejnik, P., Awrejcewicz, J.: On the modeling and simulation of variable - length pendulum systems: a review. Archiv. Comput. Method. Eng. 29, 2397–2415 (2022) Yakubu, G., Olejnik, P., Awrejcewicz, J.: On the modeling and simulation of variable - length pendulum systems: a review. Archiv. Comput. Method. Eng. 29, 2397–2415 (2022)
17.
go back to reference Neishtadt, A.I., Sheng, K.: Bifurcations of phase portraits of pendulum with vibrating suspension point. Commun. Nonlinear Sci. Numer. Simul. 47, 71–80 (2017) Neishtadt, A.I., Sheng, K.: Bifurcations of phase portraits of pendulum with vibrating suspension point. Commun. Nonlinear Sci. Numer. Simul. 47, 71–80 (2017)
18.
go back to reference Polekhin, I. Yu.: The spherical Kapitza-Whitney pendulum. Regul. Chaotic Dyn. 27(1), 65–72 (2022) Polekhin, I. Yu.: The spherical Kapitza-Whitney pendulum. Regul. Chaotic Dyn. 27(1), 65–72 (2022)
19.
go back to reference Wright, J.A., Bartuccelli, M., Gentile, G.: Comparisons between the pendulum with varying length and the pendulum with oscillating support. J. Math. Anal. Appl. 449, 1684–1707 (2017) Wright, J.A., Bartuccelli, M., Gentile, G.: Comparisons between the pendulum with varying length and the pendulum with oscillating support. J. Math. Anal. Appl. 449, 1684–1707 (2017)
20.
go back to reference Krasilnikov, P., Gurina, T., Svetlova, V.: Bifurcation study of a chaotic model variable-length pendulum on a vibrating base. Int. J. Non-Linear Mech. 105, 88–98 (2018) Krasilnikov, P., Gurina, T., Svetlova, V.: Bifurcation study of a chaotic model variable-length pendulum on a vibrating base. Int. J. Non-Linear Mech. 105, 88–98 (2018)
21.
go back to reference Markeev, A.: Stability of an equilibrium position of a pendulum with step parameters. Int. J. Non-Linear Mech. 73, 12–17 (2015) Markeev, A.: Stability of an equilibrium position of a pendulum with step parameters. Int. J. Non-Linear Mech. 73, 12–17 (2015)
22.
go back to reference Burov, A.A., Nikonov, V.I.: On the motion of the pendulum in an alternating sawtooth force field. Int. J. Bifurc. Chaos 30(9), 2050135 (2020) Burov, A.A., Nikonov, V.I.: On the motion of the pendulum in an alternating sawtooth force field. Int. J. Bifurc. Chaos 30(9), 2050135 (2020)
23.
go back to reference Binet, M.: Mémoire sur la variation des constantes arbitraires dans les formules générales de la dynamique, et dans un systéme déquations analogues plus étendues. J. de l’Ecole Polytechnique 17, 1–94 (1841) Binet, M.: Mémoire sur la variation des constantes arbitraires dans les formules générales de la dynamique, et dans un systéme déquations analogues plus étendues. J. de l’Ecole Polytechnique 17, 1–94 (1841)
24.
go back to reference Levi-Civita, T.: Sulle trasformazioni delle equazioni dinamiche, Annali di Matematica, 1896, Vol. 24, pp. 255 – 300 (English translation: Levi-Civita T., On the Transformations of the Dynamical Equations, Regular and Chaotic Dynamics,, Vol. 14, Nos. 4-5, pp. 580-614.) (2009) Levi-Civita, T.: Sulle trasformazioni delle equazioni dinamiche, Annali di Matematica, 1896, Vol. 24, pp. 255 – 300 (English translation: Levi-Civita T., On the Transformations of the Dynamical Equations, Regular and Chaotic Dynamics,, Vol. 14, Nos. 4-5, pp. 580-614.) (2009)
25.
go back to reference Levi-Civita, T.: Sur la résolution qualitative du problème restreinte de trois corps. Acta Mathematica 30, 305–327 (1906) Levi-Civita, T.: Sur la résolution qualitative du problème restreinte de trois corps. Acta Mathematica 30, 305–327 (1906)
26.
go back to reference Nechville, V.: Sur une nouvelle forme des équations différentielles du probléme restreint elliptique. CRAS182, 310–311 (1926) Nechville, V.: Sur une nouvelle forme des équations différentielles du probléme restreint elliptique. CRAS182, 310–311 (1926)
27.
go back to reference Markeeva, A.A., Levin, M.A.: Limit equations of motion for mechanical systems with vibrating elements. Moscow Univ. Mech. Bulletin 70(5), 112–121 (2015) Markeeva, A.A., Levin, M.A.: Limit equations of motion for mechanical systems with vibrating elements. Moscow Univ. Mech. Bulletin 70(5), 112–121 (2015)
28.
go back to reference Burov, A.A., Guerman, A.D., Nikonov, V.I.: Asymptotic invariant surfaces for non-autonomous pendulum-type systems. Regul. Chaotic Dyn. 25(1), 121–130 (2020) Burov, A.A., Guerman, A.D., Nikonov, V.I.: Asymptotic invariant surfaces for non-autonomous pendulum-type systems. Regul. Chaotic Dyn. 25(1), 121–130 (2020)
29.
go back to reference Demidovich, B.P.: Problems in Mathematical Analysis. Lomonosov Moscow State University, Moscow (1997). ([in Russian]) Demidovich, B.P.: Problems in Mathematical Analysis. Lomonosov Moscow State University, Moscow (1997). ([in Russian])
30.
go back to reference Morozov, A.D., Dragunov, T.N., Boykova, S. A. & Malysheva, O.V.: Invariant Sets For Windows: Resonance Structures, Attractors, Fractals And Patterns. World scientific series on Nonlinear Science. Series A. Vol.37. World Scientific, 272 p. (1999) Morozov, A.D., Dragunov, T.N., Boykova, S. A. & Malysheva, O.V.: Invariant Sets For Windows: Resonance Structures, Attractors, Fractals And Patterns. World scientific series on Nonlinear Science. Series A. Vol.37. World Scientific, 272 p. (1999)
Metadata
Title
Dynamics of a heavy pendulum of variable length with a movable suspension point
Authors
Alexander A. Burov
Vasily I. Nikonov
Publication date
16-04-2024
Publisher
Springer Vienna
Published in
Acta Mechanica
Print ISSN: 0001-5970
Electronic ISSN: 1619-6937
DOI
https://doi.org/10.1007/s00707-024-03926-x

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