A semi-analytical method for free vibration of semi-closed shells of revolution with variable curvature

This study is concerned with the free vibration analysis of semi-closed shells of revolution with variable curvature, with a focus on the analytical solution to the natural frequency of such shells. A semi-closed shell of revolution with variable curvature is firstly divided into some narrow small shell segments, then the first small shell segment with a closed shell apex is approximated with a semi-closed conical shell, while the remaining small shell segments are approximated with a series of progressively larger truncated conical shells. Since each small conical shell is part of the large semi-closed shell of revolution with variable curvature and has to vibrate in sync with the large shell, the analytical solution to the axisymmetric free vibration of the large shell can be transformed into the analytical solutions to the synchronous free vibration of these small conical shells. The general solutions for the natural frequencies of all small conical shells are analytically derived under the condition that conical shells have zero meridional curvature. The undetermined constants in the general solutions are determined by letting all natural frequencies of these small conical shells be equal. Finally, the analytical solution for the axisymmetric free vibration of semi-closed shells of revolution with variable curvature is presented and is proved to be basically reliable by conducting a confirmatory experiment based on polymer 3D-printing technique.