The vibrations of bubbles and balloons

Bubbles and balloons are two examples of structures that feature a pressure difference across the skin, a thin, tensioned membrane, and a doubly curved interface surface. While mathematical models have been formulated for bubble vibrations, no such model exists for balloon vibrations. This paper reviews a model of bubble vibrations, and compares its predicted natural frequencies and modeshapes to those of a rubber balloon. It is shown that the bubble model consistently underpredicts the balloon's natural frequencies, and it is concluded that the nonlinear elasticity present in the balloon skin accounts for this result.

Vibration of pressurised, elastic, spherical shells

Balloons are one example of pressurised, elastic, spherical shells. Whilst analytical solutions exist for the vibration of pressurised spheres, these models only incorporate constant tension in the membrane. For elastic shells, changes in curvature will result in restoring forces that are proportional to the elasticity in the membrane; hence the assumption of constant tension is not valid. This paper describes an analytical solution for the natural frequencies of an elastic spherical shell subject to internal pressure. When the membrane tension is set to zero, the results are shown to converge to the analytical solution for a spherical shell, and when the skin elasticity is neglected, the results converge to the constant-tension solution. This analytical solution is used to predict the natural frequencies of a small balloon, based on a value for the elastic modulus that is determined using biaxial tensile testing. These predictions are compared to experimental measurements of balloon vibrations using impact hammer testing, and good agreement is seen.