Modeling added mass effects on the vibrations of air-backed, pre-deformed membranes

The study of vibrations of air-backed structures in water is necessary across a range of engineering applications, from naval to civil engineering. Common to these applications is the presence of geometric curvatures in the structure, which are responsible for rich, three-dimensional fluid-structure interactions. Toward a first theoretical understanding of these phenomena, we focus on the vibrations of hydrostatically pre-deformed, air-backed membranes. Under the assumption that the mode shapes of the membrane are only marginally affected by the fluid, we put forward a staggered solution procedure that is amenable to mathematical treatment. First, we determine the mode shapes of the pre-deformed membrane in vacuum using von Kármán theory. Then, we employ these mode shapes as an input to the potential flow problem, which is solved through perturbation theory. The proposed approach allows for an analytical study of hydrodynamic forces on circular membranes as a function of their curvature, shedding light on the added mass effect associated with the inertia of the fluid. Our results indicate that the membrane curvature enhances the added mass effect, thereby reducing the fundamental frequency of the membrane. The increase in the added mass is ascribed to the increase in the volume of fluid displaced by the membrane and to the increase in the velocity of the induced fluid flow, which causes larger dynamic loading close to the center of the membrane. Our methodology offers critical insight into the physics of fluid-structure interactions in air-backed curved structures.