Stability of floating vessels with cross sections in the form of elliptic and hyperbolic segments

The paper considers two problems on the stability of the trivial equilibrium position of floating vessels with cross sections in the form of elliptic and hyperbolic segments. A review of examples on the stability of floating bodies is given and the key principles of its study by methods of analytical statics are outlined. For each of the presented problems, by means of rather serious mathematical constructions, an exact expression for the potential energy is obtained within the framework of the accepted configuration, and its quadratic approximation is calculated near the equilibrium state under study. On its basis, stability conditions of the equilibrium position are established in terms of three dimensionless parameters, and limit cases are also analyzed. A comparison of intermediate expressions and final results obtained in the process of discussing each of the problems is carried out, and their common features and distinctive features are revealed. The found solutions are illustrated as families of boundaries of stability regions on the plane of two dimensionless parameters for different values of the third parameter. These results are of fundamental theoretical importance and may be useful for practical applications.