Susceptibility of the roll equation to the bifurcation phenomenon depending on the damping coefficient value and form of the roll damping formula

This paper deals with the susceptibility of the roll equation to the bifurcation phenomenon depending on the damping coefficient value and form of the roll damping formula. Generally, the bifurcation phenomenon depends mainly on the shape of the righting arm curve (GZ curve), but roll damping also has a significant impact. The commonly used formulas for roll damping are presented, as well as values of the linear equivalent roll damping coefficient, calculated according to the simple Ikeda method. Values of the linear equivalent roll damping coefficient were calculated for a wide spectrum of roll amplitudes and roll frequencies for two ships. The loading conditions for these ships were selected to show different GZ curve characteristics. One ship has a softening spring characteristic and the second has a hardening spring characteristic. For these two ships, a number of calculations of roll spectra are presented where the bifurcation phenomenon occurs. Calculations were made for different damping coefficient values and forms of the roll damping formula.