Rolling of Ships

ROLLING OF SHIPS A ship at sea is subjected to the action of waves which set up rolling, pitching and other oscillatory motions, of a more or less complex character. Of these, in general, rolling is the most important, and it is sufficient in considering rolling to neglect the influence of other oscillations.

Unresisted Rolling.

It is convenient first to consider the ship as rolling in still water and to neglect the influence of resist ance ; when it follows from a simple investigation that for small angles of inclination the period of roll—the period being defined as the time of a single complete oscillation, i.e., out to out and back again—is given by the expression where K is the mass radius of gyration of the ship about the axis of rotation and in is the metacentric height.

The theory now generally accepted for the unresisted rolling of ships amongst waves is due to Froude, and is based on the proposition that the forces acting on a ship among waves tend to place her normal to the wave sub-surface passing through the ship's centre of buoyancy. To simplify the investigation Froude also assumed that the wave profile was a curve of sines, instead of the trochoid which more nearly represents actual waves, and that the ship was rolling broadside on in a regular series of similar waves of given dimensions and given period. It was also assumed that the ship's rolling in still water was isochronous, and that the period of rolling was given by the theoretical expression already referred to, viz.: For the reasons underlying the assumptions made and for the complete investigation of the equations of motion, etc., reference can be made to Froude's papers in Trans. I.N.A. (1861 and 1862). It may be stated broadly that the ship oscillates as in still water, but has superposed on these oscillations a series of oscillations governed by the wave-slope and the relation existing between the period of the ship and that of the wave. An interesting case arises when the ship's period is equal to the period of the wave, when at the passage of each successive wave crest and hollow the inclination of the ship is increased, so that but for the effect of resistances and the departure from synchronism at large angles of roll, she would inevitably capsize.

Resisted Rolling.

It remains to consider the practical case of actual rolling of ships in which resistance has important effects. In still water resistance causes a degradation of the amplitude of roll until the ship finally comes to rest. A similar degradation is caused when the ship rolls amongst waves. After the theoretical consideration of unresisted rolling already described, attention was given by Froude and others, first to resisted rolling in still water, and the method adopted was to roll an actual ship in still water to as large an amplitude as could be obtained and then observe the diminishing amplitude at each successive roll until she eventually came to rest.

A curve plotted with number of rolls as abscissae and amplitude of roll as ordinates gave a curve of declining angles. From this curve a second curve was made, called the curve of extinction, with angles of roll as abscissae and angle lost per swing as ordinates. Particulars of such curves obtained by experiment on H.M.S. "Revenge" can be seen in Sir W. White's I.N.A. paper of 1895. With these curves as data Froude proceeded to investi gate the relation between the degradation of amplitude and the resistances causing such degradation. Making the assumption that the resistance to rolling varies as the angular velocity, it is readily shown that the period is slightly increased and the ampli tude progressively diminished by the resistance. In actual cases although part of the resistance varies as the angular velocity, part also varies as the square of the angular velocity. This leads to an equation of motion not in general susceptible of analytical solu tion, but the solution can be obtained by a process of graphic integration. If it be assumed that the motion be simple harmonic then the equation of the curve of extinction can be approximately expressed by the empirical formula : where 0 = extreme angle in degrees reached at any particular oscillation, n the number of oscillations and a and b are coeffi cients which can be determined in a particular case.