Importance of geometric non-linearity and follower pressure load in the dynamic analysis of a gossamer structure

Vibration analysis of a gossamer or inflated structure poses special problems, usually not encountered in a conventional metallic or composite structure. In an inflated structure, internal pressure is a major source of strength and rigidity. In the past, most of the studies conducted on the vibration analysis of gossamer structures used inaccurate or approximate theories in modelling the internal pressure. The inexactness in these theories arises due to (1) exclusion of the follower pressure loads, and (2) approximations in the geometric non-linearity. Taking cues from the earlier work done in this area and using line-of-curvature co-ordinates, we re-derive the governing equations for vibration analysis of a shell under pressure, and point out the shortcomings of the previous approximate theories. The same governing equations were derived earlier by Budiansky using tensors. Thereafter, a free-vibration analysis of an inflated torus with free boundary condition is performed using the accurate and the approximate shell theories. It can be seen that the natural frequencies and the mode shapes obtained from the approximate theories are significantly different from those obtained from the accurate shell theory. Since the boundary condition of the torus is free, the vibration analysis should yield six zero frequencies corresponding to the six rigid-body modes. It is shown here that while the accurate theory does give six zero frequencies, the approximate theories do not.