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V.Ph. Zhuravlev, "On the Precession of the Elliptic Mode Shape of a Circular Ring Owing to Nonlinear Effects," Mech. Solids. 50 (1), 1-5 (2015)
Year 2015 Volume 50 Number 1 Pages 1-5
DOI 10.3103/S002565441501001X
Title On the Precession of the Elliptic Mode Shape of a Circular Ring Owing to Nonlinear Effects
Author(s) V.Ph. Zhuravlev (A. Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences, pr. Vernadskogo 101, str. 1, Moscow, 119526 Russia, zhurav@ipmnet.ru)
Abstract The Foucault pendulum, which maintains the plane of its vibrations in inertial space, loses this property as soon as the trajectory ceases to be flat. If the pendulum end circumscribes an elliptic trajectory instead of a straight line segment, then this ellipse precesses in the same direction as the material point circumscribes the ellipse itself. In this case, the angular velocity of the ellipse precession is proportional to its area and can be explained by the nonlinearity of the equations of vibrations of a mathematical pendulum [1].

A similar phenomenon takes place in an elastic inextensible ring, which is a representative of the "generalized Foucault pendulum" family [1]. If a standing wave is excited in an immovable ring, then this wave is immovable with respect to the ring only in the case of zero quadrature, but if the quadrature is nonzero, then the standing wave precesses with respect to the ring with a velocity proportional to the quadrature value.

As in the case of the classical pendulum, this phenomenon can be explained by the nonlinearity of the ring regarded as an oscillatory system.

In the present paper, we obtain an explicit formula for calculating the angular velocity of such a precession.
Keywords generalized Foucault pendulum, elastic inextensible ring, mode shape precession
References
1.  V. Ph. Zhuravlev, "The Controlled Foucault Pendulum as a Model of a Class of Free Gyros," Izv. Akad. Nauk. Mekh. Tverd. Tela, No. 6, 27-35 (1997) [Mech. Solids (Engl. Transl.) 42 (6), 21-28 (1997)].
2.  V. Ph. Zhuravlev and D. M. Klimov, Hemispherical Resonator Gyro (Nauka, Moscow, 1985) [in Russian].
3.  C. C. Glynn, "On the Resonant Nonlinear Traveling Waves in a Thin Rotating Ring," Int. J. Non-Lin. Mech. 17 (5/6), 327-349 (1982).
4.  V. Ph. Zhuravlev and D. M. Klimov, Applied Methods in the Vibration Theory (Nauka, Moscow, 1988) [in Russian].
Received 27 March 2013
Link to Fulltext http://link.springer.com/article/10.3103/S002565441501001X
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