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IssuesArchive of Issues2016-6pp.623-631

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A.P. Markeev, "Stability in a Case of Motion of a Paraboloid over a Plane," Mech. Solids. 51 (6), 623-631 (2016)
Year 2016 Volume 51 Number 6 Pages 623-631
DOI 10.3103/S0025654416060017
Title Stability in a Case of Motion of a Paraboloid over a Plane
Author(s) A.P. Markeev (Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences, pr. Vernadskogo 101, str. 1, Moscow, 119526 Russia, markeev@ipmnet.ru)
Abstract We solve a nonlinear orbital stability problem for a periodic motion of a homogeneous paraboloid of revolution over an immovable horizontal plane in a homogeneous gravity field. The plane is assumed to be absolutely smooth, and the body-plane collisions are assumed to be absolutely elastic. In the unperturbed motion, the symmetry axis of the body is vertical, and the body itself is in translational motion with periodic collisions with the plane.

The Poincaré section surface method is used to reduce the problem to studying the stability of a fixed point of an area-preserving mapping of the plane into itself. The stability and instability conditions are obtained for all admissible values of the problem parameters.
Keywords collision, mapping, stability
References
1.  I. G. Malkin, Theory of Stability of Motion (Nauka, Moscow, 1966) [in Russian].
2.  P. Appel, Theoretical Mechanics, Vol. 2 (Fizmatgiz, Moscow, 1960) [in Russian].
3.  A. P. Markeev, "Stability of Motion of a Rigid Body Colliding with a Horizontal Plane," Izv. Ross. Akad. Nauk. Mekh. Tverd. Tela, No. 5, 32-40 (1997) [Mech. Solids (Engl. Transl.) 32 (5), 27-34 (1997)].
4.  V. Ph. Zhuravlev and D. M. Klimov, Applied Methods in Theory of Vibrations (Nauka, Moscow, 1988) [in Russian].
5.  V. Ph. Zhuravlev and N. A. Fufaev, Mechanics of Systems with Unilateral Constraints (Nauka, Moscow, 1993) [in Russian].
6.  A. P. Ivanov, Dynamics of Systems with Mechanical Collisions (MEzhd. Progr. Obrazov., Moscow, 1997) [in Russian].
7.  H. Poincaré, Selected Works, Vol. 2: New Methods of Celestial Mechanics (Nauka, Moscow, 1972) [in Russian].
8.  A. P. Markeev, "Area-Preserving Mappings and Their Applications to the Dynamics of Systems with Collisions," Izv. Ross. Akad. Nauk. Mekh. Tverd. Tela, No. 2, 37-54 (1996) [Mech. Solids (Engl. Transl.) 31 (2), 32-47 (1996)].
9.  A. P. Markeev, "On Stability of Fixed Points of Area-Preserving Mappings," Nelin. Din. 11 (3), 503-545 (2015).
10.  A. P. Markeev, Libration Points in Celestial Mechanics and Space Dynamics (Nauka, Moscow, 1978) [in Russian].
11.  J. Moser, Lectures on Hamiltonian Systems (Amer. Math. Soc., Providence, 1968; Mir, Moscow, 1973).
12.  C. L. Siegel and J. K. Moser, Lectures on Celestial Mechanics (Springer, New York, 1971; NITs "Regular and Chaotic Dynamics", Izhevsk, 2001).
13.  V. I. Arnold, V. V. Kozlov, and A. I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics (Editorial URSS, Moscow, 2002) [in Russian].
14.  A. P. Markeev, "A Method for Analytically Representing Area-Preserving Mappings," Prikl. Mat. Mekh. 78 (5), 611-624 (2014) [J. Appl. Math. Mech. (Engl. Transl.) 78 (5), 435-444 (2014)].
Received 27 October 2015
Link to Fulltext http://link.springer.com/article/10.3103/S0025654416060017
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