Spring analogy of non-linear oscillations of a bubble in a liquid at resonance

Two non-linear oscillatory systems are considered. The first is a point mass on a spring with vertical vibration of the suspension point with a frequency that coincides with the frequency of free vertical oscillations and is two times greater than the frequency of free horizontal oscillations. The friction force in the spring is taken into account. For an initial deviation of the point mass from the vertical, after a long enough time the energy of the vertical oscillations is almost completely transferred into the energy of horizontal oscillations. Using an averaging method, an asymptotic solution is constructed, describing the transient process setting up a periodic solution. Comparison of the analytical solution with the numerical one demonstrates its high accuracy. The second system is an axisymmetrical bubble in a liquid under the variable pressure. An analogy between this system and the previous one is established. Vibration of the suspension point of a spring pendulum corresponds to variable liquid pressure, and the vertical and horizontal oscillation modes of the swinging spring correspond to the radial and deformational oscillation modes of the bubble, and the ratio of the frequencies of these modes is also taken to be equal to two. The friction force in the spring corresponds to energy dissipation under radial oscillations of the bubble. In our calculations of energy dissipation, we take into account the liquid viscosity, thermal dissipation, and acoustic radiation due to liquid compressibility. During transfer of the energy of the radial oscillations, the amplitude of the resonant deformational mode of the bubble oscillations grows anomalously, which makes it possible for the bubble to break up with small energy dissipation under the action of a time-varying external pressure field.