Stability of the triple inverted physical pendulum described in the article of academician V.N. Chelomey (1983)

The article presents a solution to the stability problem of the inverted vertical position of the triple physical pendulum described in the famous scientific publication of Academician V.N. Chelomei 1983. The pendulum is stabilized in the inverted position by vertical monogarmonic vibrations of the suspension axis. The authors analyze some mathematical models of pendulum systems and identify the most suitable form of the model for this research. With the help of the Floquet theory, the authors solve the stability problem of the pendulum inverted position by calculating monodromy matrices and multipliers. The authors are the first to determine experimentally the stability boundary of the multilink (triple) pendulum in a wide range of excitation parameters. The authors obtained experimental results using an installation built purposely for the experimental determination of the stability region of the research subject. The divergence between the calculation and experimental results is no more than 5% for the vast majority of the boundary points.

References

[1] Сhelomey V.N. Paradoxes in mechanics, vibrations caused. Doklady Akademii Nauk SSSR [Proc. of the USSR Akademy of Sciences], 1983. vol. 270, no. 1. pp. 62-67 (in Russ.).

[2] Stephenson A. On a New Type of Dynamical Stability. Memoirs and Proceedings of the Manchester Literary and Philosophical Society, 1908, vol. 52, no. 8, part II, pp. 1-10.

[3] Bogolyubov N.N. Perturbation Theory in Nonlinear Mechanics. Sb. Inst. Stroitelnoy mehaniki AN USSR [Collection of Papers of the Institute of Structural Mechanics Academy of Science of Ukraine], 1950, vol. 14, no. 2. pp. 9-34 (in Russ.).

[4] Kapitsa P.L. Dynamic Stability of a Pendulum with a Vibrating Suspension Point. Zh. Eksp. Teor. Fiz. [J. Exp. Theor. Phys.], 1951, vol. 21, no. 5, pp. 588-597 (in Russ.).

[5] Kapitsa P.L. The Pendulum with a Vibrating Suspension. Usp. Fiz. Nauk [Sov. Phys.-Usp.], 1951, vol. 44, no. 1. pp. 7-20 (in Russ.).

[6] Arkhipova I.M., Luongo A., Seyranian A.P. Vibrational Stabilization of the Upright Statically Unstable Position of a Double Pendulum. J. of Sound and Vibration, 2012, vol. 331. pp. 457-469.

[7] Sorokin V.S. Analysis of Motion of Inverted Pendulum with Vibrating Suspension Axis at Low-Frequency Excitation as an Illustration of a New Approach for Solving Equation without Explicit Small Parameter. International J. ofNon-Linear Mechanic, 2014, vol. 63. July, pp. 1-9.

[8] Markeev A.P. On the Stability of Nonlinear Oscillations Coupled Pendulums. Izv. Akad. Nauk, Mekh. Tverd. Tela [Mech. Solids], 2013, no. 4, pp. 20-30 (in Russ.).

[9] Beletskiy V.V., Levin E.M. Dinamika kosmicheskikh trosovykh system [The Dynamics of Space Tether Systems]. Moscow, Nauka Publ., 1990. 336 p.

[10] Alpatov A.P., Beletskiy V.V., Dranovskiy V.I., Zakrzhevskiy A.E., Pirozhenko A.V., Troger G., Horochilov V.S. Dinamika kosmicheskikh sistem s trosovymi i sharnirnymi soedineniyami [Dynamics Space Systems and Rope Swivels]. Moscow-Izhevsk, NIC Regul. i khaot. dinamika. Inst. Komp’yut. issl. Publ., 2007. 559 p.

[11] Strizhak T.G. Metod usredneniya v zadachakh mehaniki [The Averaging Method in Mechanics Problems]. Kiev-Donetsk., Vysh. Shkola, 1982. 250 p.

[12] Acheson D.J., Mullin T. Upside-Down Pendulums. Nature, 1993, vol. 366, pp. 215-216.

[13] Mathieu E. Memoiresur le mouvement vibratoirediine membrane de formeelliptique. Jour, de Math. Pureset Appliques (Jour. De Liouville), 1868, vol. 13, pp. 137-203.

[14] McLachlan N.V. Theory and Application of Mathieu Functions. Oxford. Clarendon, 1947. 401 p. (Russ. ed.: Mak-Lahlan N.V. Teoriya i prilogenie funktsiy Mat’e. Moscow, Inostr. Lit. Publ., 1953. 475 p.).

[15] Hamilton W.R. Second Essay on a General Method in Dynamics. Philos. Trans. Roy. Soc. London, 1835, part I, pp. 95-144.

[16] Andronov A.A., Leontovich M.A. On the Vibrations of a System with Periodically Varying Parameters. Jurn. Russk. Fiziko-Him. Obtshestva, Chast Fizich. [J. of Russian Physical and Chemical Society. Part Physical.], 1927, vol. 59, no. 5-6, pp. 429-443 (in Russ.).

[17] Bardin B.S., Markeev A.P. On the Stability of Equilibrium of the Pendulum with Vertical Oscillations of the Point of Suspension. Prikl. Mat. Mekh. (PMM) [J. Appl. Math. Mech.], 1995. vol. 59, no. 6, pp. 923-929 (in Russ.).

[18] Chelomey S.V. On Two Problems of Dynamic Stability of Oscillatory Systems Set Academicians P.L. Kapitsa and V.N. Chelomey. Izv. Akad. Nayk SSSR. Mekh. Tverd. Tela [Mech. Solids], 1999, no. 6, pp. 159-166 (in Russ.).

[19] Otterbein S. Stabilisierung des n-Pendels und der IndischeSeiltrick. Archive for Rational Mechanics and Analysis, 1982, vol. 78, pp. 381-393.

[20] Floquet G. Sur les equations differentielles lineaires а coefficients periodiques. Ann. Sci. Ecole Norm. Sup., 1883, vol. 12, pp. 47-89.

[21] Chetaev N.G. Ustoichivost’ dvizheniya [Stability Movement]. Moscow, Nauka Publ., 1990. 176 p.