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Mathematics and Mechanics

Russian Academy of Sciences
 Founded
in January 1936
(Translated from 1958)
Issued 6 times a year
ISSN 0021-8928
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IssuesArchive of Issues2014-1pp.30-38

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A.A. Zevin, "The theory of parametric oscillations," J. Appl. Math. Mech. 78 (1), 30-38 (2014)
Year 2014 Volume 78 Issue 1 Pages 30-38
Title The theory of parametric oscillations
Author(s) A.A. Zevin (Dnepropetrovsk, Ukraine, zevin@westa-inter.com)
Abstract The correctness of the existing definitions of the parametric oscillations of linear and non-linear systems is discussed. The possibility of an erroneous choice of the mathematical parametric model instead of the autooscillatory model, connected with the existence in such systems of the same periodic solutions, is pointed out. Some non-local properties of parametric oscillations in Hamiltonian systems are established. It is shown, in particular, that stability regions are convex with respect to the frequency of the parametric excitation (i.e., all the points between the boundaries of neighbouring instability regions correspond to stable solutions). At the critical frequencies of parametric resonance the well-known Rayleigh and Zhuravlev theorems on the behaviour of the frequencies of natural oscillations when the stiffness and inertia changes are generalized. Some additional assertions on the limits of the first instability region for the Hill vector equations are established.
Received 20 November 2012
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