Stability of Floating of Vessels with Cross Sections in the Form of Elliptical and Hyperbolic Segments

Two problems on the stability of the trivial equilibrium position of floating vessels with cross sections in the form of elliptic and hyperbolic segments are considered. Examples on the stability of floating bodies are reviewed, and the key principles of its investigation by analytical statics methods are outlined. For each of the presented problems, by means of quite serious mathematical constructions, an exact expression for the potential energy is obtained within the accepted configuration, and its quadratic approximation near the equilibrium state under study is calculated. On its basis, the stability conditions of the equilibrium state in terms of three dimensionless parameters are determined and the limiting cases are also analyzed. The intermediate expressions and final results obtained as a result of discussion of each of the problems are compared, and their common and distinctive features are identified. The found solutions are illustrated as families of boundaries of stability regions on the plane of two dimensionless parameters at different values of the third parameter. These results are of fundamental theoretical importance and can prove useful for practical applications.