Nonlinear Oscillations of a Spring Pendulum at the 1 : 1 : 2 Resonance: Theory, Experiment, and Physical Analogies

Abstract

Nonlinear spatial oscillations of a material point on a weightless elastic suspension are considered. The frequency of vertical oscillations is assumed to be equal to the doubled swinging frequency (the 1 : 1 : 2 resonance). In this case, vertical oscillations are unstable, which leads to the transfer of the energy of vertical oscillations to the swinging energy of the pendulum. Vertical oscillations of the material point cease, and, after a certain period of time, the pendulum starts swinging in a vertical plane. This swinging is also unstable, which leads to the back transfer of energy to the vertical oscillation mode, and again vertical oscillations occur. However, after the second transfer of the energy of vertical oscillations to the pendulum swinging energy, the apparent plane of swinging is rotated through a certain angle. These phenomena are described analytically: the period of energy transfer, the time variations of the amplitudes of both modes, and the change of the angle of the apparent plane of oscillations are determined. The analytic dependence of the semiaxes of the ellipse and the angle of precession on time agrees with high degree of accuracy with numerical calculations and is confirmed experimentally. In addition, the problem of forced oscillations of a spring pendulum in the presence of friction is considered, for which an asymptotic solution is constructed by the averaging method. An analogy is established between the nonlinear problems for free and forced oscillations of a pendulum and for deformation oscillations of a gas bubble. The transfer of the energy of radial oscillations to a resonance deformation mode leads to an anomalous increase in its amplitude and, as a consequence, to the break-up of a bubble.