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According to A. W. Leissa in Vibration of shells, a shell is “a three-dimensional body, which is bounded by two closely spaced curved surfaces, the distance between the surfaces being small in comparison with the other dimensions” [1].

Adding on to this from the Encyclopedia Brittanica [ ], the shape of the curved surfaces allows for the transmission of applied loads to supports in more than two directions. These loads are carried by the development of compressive, tensile and shear stresses that act in the plane of the surface. Where efficiency means that for the same cross sectional area subjected to the same loading conditions, a beam undergoes the least deflection, it is found theoretically and experimentally that the cylindrical shell is the most efficient structure for bending in any direction. Given this, there is a huge potential for the uses of cylindrical shells and the need for further understanding of the structure in response to various loading conditions has led to an extensive study into the structure itself.

More specifically in the context of present day mechanical and civil engineering, the phenomenon of vibrating thin cylindrical shells is a particular area that has received much attention. This is due to the prevalent use of shafts in modern rotating machinery and the need for higher operating speeds of such machinery.

Vibration is the periodic back and forth motion of an object under dynamic excitation in mechanics. It usually becomes a problem when it is excessive or when the natural frequency of the vibrating structure coincides with that of the exciting source, resulting in resonance.

Shell theories are usually used to study the vibration characteristics of rotating cylindrical shells, all of which are affected by factors such as anisotropy, initial stresses, variable thickness, surrounding media (e.g., water, air), large (nonlinear) deflections, shear deformation, rotary inertia, and non-homogeneity (including laminated composites) just to name a few. This presents a huge scope for research studies.
The first published work on a rotating cylindrical shell was by Bryan [ ], where the rotating ring was considered and the phenomena of traveling modes was discovered. Later works include that of DiTaranto and Lessen [ ], which investigated the effects of Coriolis forces on an infinitely long and isotropic cylindrical shell, and that of Srinivasan and Lauterbach [ ], which looked into the effects of both Coriolis forces and travelling modes in rotating isotropic cylindrical shells. Till this point, these papers mainly dealt with the analysis of the natural frequency of vibration.

However, one vibration characteristic that is also crucial and relevant in the study of cylindrical shells is the critical speed of the rotating shell. Zinberg and Symonds [ ] were the firsts to obtain, through experiments, critical speed results for rotating shells. The results obtained also proved the advantages of using shells made of orthotropic materials over aluminium alloy shells. The results of Zinberg and Symonds were further build on by dos Reis et al. [ ] where a finite element approach was used to obtain the critical speeds of the shell. Following that, a paper by Kim and Bert [ ] presented a simplified theory for analysing the first critical speed of a composite cylindrical shell. Results obtained using different shell theories were compared.

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