Some approximations in the dynamic shell equations

A theoretical analysis was performed on the linear dynamic equations of thin cylindrical shells to find the error committed by the Donnell assumption and the neglect of inplane inertia. The Donnell approximation was found to be valid at high frequencies as compared to the ring frequencies, for all admissible sets of boundary conditions for finite length thin shells. The error from neglecting tangential inertia is appreciable for long circumferential and axial wave lengths, independent of shell thickness. The effect of boundary conditions was investigated from an exact solution of the linear eigenvalue problem. The inplane boundary conditions proved to be very influential in the neighborhood of the minimum frequency. An approximate technique which accounts for the inplane boundary conditions was then developed and shown to be in good agreement with the exact solution. Finally, the effect of an elastic end ring on the eigenfrequencies was studied. The out-of-plane and torsional rigidities of the ring were found to govern the overall shell stiffness. Considerable mode inter action was noticed at low circumferential wave numbers for low values of ring stiffness. The computed eigenfrequencies were found to be in good agreement with the experimental results.