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On a toroidal pendulum

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Abstract

We consider the problem of motion of a heavy particle on the surface of a torus with horizontal axis of rotation.

On nondevelopable surfaces other than surfaces of revolution with vertical axis, the solution is known only for the surface of an elliptic paraboloid [1].

To solve the problem on the surface of a torus with horizontal axis of rotation, we use the method of reduction of equations of motion proposed in [2]. We construct the asymptotics of the general and periodic solutions and show that one can use this asymptotics when studying the motion of a heavy particle on an elliptic torus.

We obtain the stability conditions in the first approximation for the particle motion along the outer equator and the lower meridian of the torus.

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References

  1. S. A. Chaplygin, Complete Collection of Works, Vols. 1–13 (Izd-vo AN SSSR, Leningrad, 1933–1935) [in Russian].

    Google Scholar 

  2. A. P. Blinov, “On theMotion of a Mass Point on a Surface,” Izv. Akad. Nauk.Mekh. Tverd. Tela, No. 1, 23–28 (2007) [Mech. Solids (Engl. Transl.) 42 (1), 19–23 (2007)].

  3. F. G. Tricomi, Differential Equations (Hafner, New York, 1961; Izd-vo Inostr. Lit.,Moscow, 1962).

    MATH  Google Scholar 

  4. G. N. Duboshin, Celestial Mechanics. Fundamental Problems and Methods (Fizmatgiz, Moscow, 1963; Translation Div., Wright-Patterson Air-Force Base, Fairborn, Ohio, 1969).

    Google Scholar 

  5. B. P. Demidovich, Lectures on Mathematical Theory of Stability (Nauka, Moscow, 1967) [in Russian].

    Google Scholar 

  6. J. K. Hale, Oscillations in Nonlinear Systems (McGraw Hill, New York, 1963; Mir,Moscow, 1966).

    MATH  Google Scholar 

  7. N. E. Zhukovskii, “Finiteness Conditions for Integrals of the Equation d 2 y/dx 2 + py = 0,” in Complete Papers, Vol. 1 (Gostekhizdat, Moscow-Leningrad, 1948) pp. 246–253 [in Russian].

    Google Scholar 

  8. I. G. Malkin, Several Problems of Theory of Nonlinear Oscillations (Gostekhizdat, Moscow, 1956) [in Russian].

    Google Scholar 

  9. V. Ph. Zhuravlev and D. M. Klimov, Applied Methods in Vibration Theory (Nauka, Moscow, 1988) [in Russian].

    Google Scholar 

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Authors and Affiliations

  1. Timiryazev Moscow Agriculture Academy, Russian State Agricultural University, Timiryazevskaya 49, Moscow, 127550, Russia

    A. P. Blinov

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  1. A. P. Blinov
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Additional information

Original Russian Text © A.P. Blinov, 2010, published in Izvestiya Akademii Nauk. Mekhanika Tverdogo Tela, 2010, No. 1, pp. 28–33.

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Blinov, A.P. On a toroidal pendulum. Mech. Solids 45, 22–26 (2010). https://doi.org/10.3103/S0025654410010048

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  • DOI: https://doi.org/10.3103/S0025654410010048

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