A Method to Simplify Solution of Stability Problem for Parametrically Stabilized Statically Unstable Pendulum Systems

The authors propose a method that simplifies solution of a linear problem of stability of statically unstable N-link inverted pendulums with parametrically stabilized harmonic oscillations of the pivot axis. The difference of the method from the conventional approaches (S. Otterbein, D. Acheson, S.V. Chelomei) is in a wider range of application, ease and efficiency of solving the problem of stability, and a higher accuracy in determining boundary lines of stability area of the pendulum systems. The method is based on the well-known and extensively researched Mathieu equation that describes dynamics and stability of a simple mathematical pendulum in the bottom and top positions relative to the vertical equilibrium point. The solution of the stability problem for Mathieu equation in its canonical form (by N.V. McLachlan) can be easily reduced to the analysis of Ince-Strutt diagram. In this paper Ince-Strutt diagram is applied to inverted pendulum systems with an arbitrary number of degrees of freedom.

References

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

[2] Acheson D.J. A pendulum theorem. Proceedings of the Royal Society A, 1993, vol. 443, pp. 239–245.

[3] Chelomei S.V. O dvukh zadachakh dinamicheskoi ustoichivosti kolebatel’nykh sistem, postavlennykh akademikami P.L. Kapitsei i V.N. Chelomeem [On two problems of dynamic stability of oscillatory systems set academicians P.L. Kapitsa and V.N. Chelomey]. Izvestiia Rossiiskoi Akademii nauk. Mekhanika tverdogo tela [A Journal of Russian Academy of Sciences. Mechanics of Solids]. 1999, no. 6, pp. 159–166.

[4] Demidovich B.P. Lektsii po matematicheskoi teorii ustoichivosti [Lectures on mathematical theory of stability]. St. Petersburg, Lan’ publ., 2008. 480 p.

[5] Mak-Lakhlan N.V. Teoriia i prilozheniia funktsii Mat’e [Theory and application functions Mathieu]. Moscow, Inostrannaia literatura publ., 1953. 475 p.

[6] Uitteker E. T., Vatson Dzh. N. Kurs sovremennogo analiza [Course of Modern Analysis]. Moscow, Editorial URSS publ., 2015. 864 p.

[7] Spravochnik po spetsial’nym funktsiiam s formulami, grafikami i matematicheskimi tablitsami [Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables]. Ed. Abramovits M., Stigan I. Moscow, Nauka publ., 1979. 832 p.

[8] Beitmen G., Erdeii A. Vysshie transtsendentnye funktsii [Higher transcendental functions]. Vol. 3. Moscow, Nauka publ., 1967. 300 p.

[9] Abramov A.A., Kurochkin S.V. Calculation of solutions to the Mathieu equation and of related quantities. Computational Mathematics and Mathematical Physics, 2007, vol. 47, no. 3, pp. 397–406.

[10] Kashevarov A.V. Mathieu functions and coulomb spheroidal functions in the electrostatic probe theory. Computational Mathematics and Mathematical Physics, 2011, vol. 51, no. 12, pp. 2137–2145.

[11] Arkhipova I.M., Luongo A., Seyranian A.P. Vibrational stabilization of the upright statically unstable position of a double pendulum. Journal of Sound and Vibration, 2012, vol. 331, iss. 2, pp. 457–469.

[12] Tsimring Sh.E. Spetsial’nye funktsii i opredelennye integraly [Special functions and definite integrals]. Moscow, Radio i sviaz’ publ., 1988. 272 p.

[13] Tablitsy dlia vychisleniia funktsii Mat’e: sobstvennye znacheniia, koeffitsienty i mnozhiteli sviazi [Tables for the calculation of Mathieu functions: eigenvalues, ratios and multipliers communication]. Moscow, VTs AN SSSR publ., 1967. 279 p.