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DESIGNING AND DETAILING OF BUILDING SYSTEMS. MECHANICS IN CIVIL ENGINEERING

Bifurcation points of a rotating and oscillatingmechanical system depending on one parameter

Вестник МГСУ 12/2013
  • Lenev Vlamidir Stepanovich - Moscow State University of Civil Engineering (MGSU) Candidate of Physical and Mathematical Sciences, Associate Professor, Department of Higher Mathematics, Moscow State University of Civil Engineering (MGSU), 26 Yaroslavskoe shosse, Moscow, 129337, Russian Federation; Этот e-mail адрес защищен от спам-ботов, для его просмотра у Вас должен быть включен Javascript .

Страницы 27-33

This work is dedicated to finding bifurcation points for a special mechanical system comprised of a pendulum hanging on a spherical joint. Its motion and positions depend on one parameter: the angular velocity around the axis. When the parameter grows uninterruptedly, the system passes through a discrete sequence of bifurcation points: when the parameter has a small value, the mass of pendulum meanwhile has special configuration, it makes the simple rotation near its balance position. The increase of the parameter up to the first point of bifurcation deviates the pendulum from balance: now it is a conic pendulum. The consequent increase of the parameter the friction in the hinge causes the precession and then the vertical gyroscope motion. Their parameters are calculated and there are mechanical interpretations for them. For example, at the first point of bifurcation we have congruence of rotary motion and oscillation of the pendulum. At the second point there is a change in the configuration of the conic pendulum. Physical interpretations are given at the third, the fourth, the fifth and the sixth points of bifurcation where the masses reach the first, the second, the third and the forth cosmic velocities respectively. Also the article demonstrates a new method of accelerating masses to cosmic velocities.The article is of theoretical character and the ideas of G.M.T. Tompson, V.I. Arnold, A.A. Andronov, V.S. Lenev are used in it.

DOI: 10.22227/1997-0935.2013.12.27-33

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