# Nonlinear vibrations and stability of shells with fluid-structure interaction

**At LMSSC, Cnam, Paris, July 5th 2004**

Marco Amabili

Professor, Department of Industrial Engineering, University of Parma, Italy

Presentation

Shells coupled to flowing fluids are widely used in engineering. Vibrations are a major problem for thin shells due to excitations of many different kinds, including flow excitation. In some applications, the vibration response of shells calculated by linear theory is inaccurate. When the shell displacement becomes comparable to the shell thickness, a geometrically nonlinear theory should be used.

A number of phenomena are discussed for large-amplitude vibrations of circular cylindrical shells and doubly-curved panels forced by harmonic radial excitation, including internal resonances, i.e. relationships of type 1:1, 1:2 and 1:3 between natural frequencies of different modes of the shell, which affect the nonlinear response of the system. Computational aspects related to solution methods, different shell theories, boundary conditions, convergence of the solution and reduced-order models are addressed.

When shell structures are coupled to internal, external or annular flowing fluids, mild and catastrophic instability phenomena can happen by increasing the flow velocity. Flutter instability (oscillation) is generally reached for supersonic flow, as shown by NASA experiments and calculations. For subsonic flow, catastrophic buckling can be reached for flow velocities much smaller than predicted by linear theory. In fact, linear calculations give a pitchfork bifurcation, indicating divergence of the shell. However, strong subcritical static bifurcations come out from this point and are associated to the divergence of the system. The bifurcated branches become stable after a folding, which occurs for a flow velocity considerably smaller than the critical flow velocity calculated by linear theory. Therefore, by giving enough perturbation to the shell, it is possible to buckle it for a flow velocity much smaller than the one predicted by linear theory. This behaviour was modelled recently for the first time and is very important for calculation of aerodynamic stability of shell structures conveying or immersed in subsonic flow.