Abstract
A mathematical analysis and evaluation is made of the axisymmetric vibration modes of a cylindrical-hemispherical tank partially filled with a liquid. The procedure used is similar to that employed previously by Miles. It consists of formulating generalized functions that describe compatible deformations of the shell and liquid. Kinetic and potential energy expressions are then formed and the equations governing modal motion are derived by application of Lagrange's equations. A number of calculations are made for study of the modal behavior of the tank. The results are presented in the form of modal frequency vs liquid level, with cylinder-hemisphere stiffness ratio as a parameter. Representative mode shapes showing displacements of the tank wall and free surface of the liquid are also presented. Finally, a check on the accuracy of the calculations is obtained by study of the pressure matching of the liquid-shell interfaces. The results show that the bulging behavior of the tank can be visualized as a coupling of the motions of two simpler tank configurations: 1) a tank with rigid cylindrical walls and a flexible hemispherical bottom and 2) a tank with flexible cylindrical walls and a rigid bottom.
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