The kinetic and potential energies of an incompressible fluid having a free surface in a cylindrical tank subjected to translation, rotation, and simple bending are derived in the forms
where the qi (t) are generalized co-ordinates. The results are illustrated by a calculation of the frequency equation for coupled bending and free-surface motions. It is found that if the mass of the empty tank is small compared with that of the liquid and the depth of the liquid is equal to the tank diameter the presence of the free surface increases the bending frequency (relative to that calculated with a plane cap that coincides with the liquid surface and is normal to the axis of the cylinder) by about 5 per cent for cantilever bending oscillations and 27 per cent for free-free bending oscillations.
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