On the precession of the elliptic mode shape of a circular ring owing to nonlinear effects

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.