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IssuesArchive of Issues2011-4pp.508-518

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O.V. Kholostova, "On Stability of Relative Equilibria of a Double Pendulum with Vibrating Suspension Point," Mech. Solids. 46 (4), 508-518 (2011)
Year 2011 Volume 46 Number 4 Pages 508-518
DOI 10.3103/S0025654411040029
Title On Stability of Relative Equilibria of a Double Pendulum with Vibrating Suspension Point
Author(s) O.V. Kholostova (Moscow Aviation Institute (State University of Aerospace Technologies), Volokolamskoe sh.4, GSP-3, A-80, Moscow, 125993 Russia, kholostova_o@mail.ru)
Abstract We consider the motions of a double pendulum consisting of two hinged identical rods. The pendulum suspension point is assumed to perform harmonic vibrations of arbitrary frequency and arbitrary amplitude in the vertical direction. We carry out a complete nonlinear analysis of the stability of the four pendulum relative equilibria on the vertical.

The problem on the stability of the relative equilibria of the mathematical pendulum in the case where the suspension point performs vertical harmonic vibrations of arbitrary frequency and arbitrary amplitude was considered in a linear setting [1-3] and a nonlinear setting [4, 5]. In the case of small-amplitude rapid vertical vibrations of the suspension point, linear and (mathematically not fully rigorous) nonlinear stability analysis of the relative equilibria was carried out for an ordinary pendulum [6-9] and a double pendulum [10, 11]. In [12], for the same case of rapid vibrations, stability conditions in the linear approximation were obtained for the four relative equilibria of a system consisting of two physical pendulums. In the special case of a system consisting of two identical rods, the problem was solved in the nonlinear setting.
Keywords double pendulum, relative equilibrium, Mathieu equation, stability resonance
References
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2.  N. W. Mac-Lachlan, Theory and Applications of Mathieu Functions (Clarendon Press, Oxford, 1947; Izd-vo Inostr. Lit., Moscow, 1953).
3.  J. J. Stoker, Nonlinear Vibrations in Mechanical and Electrical Systems (Wiley, New York-London, 1950; Izd-vo Inostr. Lit., Moscow, 1953).
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9.  P. L. Kapitsa, "Dynamical Stability of a Pendulum with Vibrating Suspension Point," Zh. Éksp. Teor. Fiz. 21 (5), 588-597 (1951) [Soviet Phys. JETP (Engl. Transl.)].
10.  A. Stephenson, "On Induced Stability," Phil. Mag. Ser. 7 17, 765-766 (1909).
11.  T. G. Strizhak, Methods for Studying 'Pendulum'-Type Dynamical Systems (Nauka, Alma-Ata, 1981) [in Russian].
12.  O. V. Kholostova, "On the Motions of a Double Pendulum with Vibrating Suspension Point," Izv. Akad. Nauk. Mekh. Tverd. Tela, No. 2, 25-40 (2009) [Mech. Solids (Engl. Transl.) 44 (2), 184-197 (2009)].
13.  A. P. Markeyev, "A Constructive Algorithm for the Normalization of a Periodic Hamiltonian," Prikl. Mat. Mekh. 69 (3), 355-371 (2005) [J. Appl. Math. Mech. (Engl. Transl.) 69 (3), 323-337 (2005)].
14.  V. A. Yakubovich and V. M. Starzhinskii, Parametric Resonance in Linear Systems (Nauka, Moscow, 1987) [in Russian].
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17.  V. I. Arnold, V. V. Kozlov, and A. I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics (Springer-Verlag, New York, 1988; Editorial URSS, Moscow, 2002).
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20.  A. P. Markeev, "Stability of Planar Rotations of a Satellite in a Circular Orbit," Izv. Akad. Nauk. Mekh. Tverd. Tela, No. 4, 63-85 (2006) [Mech. Solids. (Engl. Transl.) 41 (4), 46-63 (2006)].
Received 23 April 2009
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