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A Journal of Russian Academy of Sciences
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IssuesArchive of Issues2008-5pp.698-708

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A. G. Petrov and A. V. Fomichev, "Nonlinear 3D vibrations of a heavy material point on a spring at resonance," Mech. Solids. 43 (5), 698-708 (2008)
Year 2008 Volume 43 Number 5 Pages 698-708
DOI 10.3103/S0025654408050026
Title Nonlinear 3D vibrations of a heavy material point on a spring at resonance
Author(s) A. G. Petrov (Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences, pr-t Vernadskogo 101, str. 1, Moscow, 119526, Russia, petrov@ipmnet.ru)
A. V. Fomichev (Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences, pr-t Vernadskogo 101, str. 1, Moscow, 119526, Russia)
Abstract We consider the problem of nonlinear 3D vibrations of a heavy material point suspended on a weightless spring at a 1:1:2 frequency resonance. To construct an asymptotic solution, we use the Hamiltonian normal form method. Just as in the plane problem, this asymptotic solution describes the periodic process in which the vertical vibration energy passes into the horizontal vibration energy. For an arbitrarily small nonzero angular momentum with respect to the vertical axis, an effect typical of 3D systems manifests itself. The projection of the trajectory of the point onto the horizontal plane (xy) is an ellipse of constant area with axes varying in time. For certain initial conditions, the ellipse almost degenerates into straight-line segments. The direction of the straight line does not vary on the time interval where the vibration energy is in the horizontal mode and then varies almost by a jump on the interval where the vibration energy is transferred into the vertical mode. The analytic results are in good agreement with numerical solutions of equations of motion of the system.
Received 22 January 2007
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