## 2 Materials and Methods
### Ship's Rolling Motion in Calm Waters in a Resistant Environment
Solving for operational convenience:  
\[2\lambda\ =\ \frac{\mathrm{A_{R}}}{\mathrm{I_{g}}}\ =\ \frac{\mathrm{A_{R}} \cdot\mathrm{g}}{\mathrm{D}\cdot\mathrm{k}^{2}}, \tag{5}\]  
and replacing Equation (5) in Equation (3), we find:  
\[\frac{\mathrm{d}^{2}\theta}{\mathrm{dt}^{2}}\ +\ 2\lambda\ \frac{\mathrm{d} \theta}{\mathrm{dt}}\ +\ \frac{\mathrm{D}\cdot\mathrm{GM}}{\frac{\mathrm{g}}{\mathrm{g}}\cdot\mathrm{k}^{2 }}\cdot\theta\ =\ 0, \tag{6}\]\[\frac{\mathrm{d}^{2}\theta}{\mathrm{d}t^{2}}\;+\;2\lambda\;\frac{\mathrm{d}\theta}{ \mathrm{d}t}\;+\;\frac{\mathrm{g}\cdot\mathrm{G}\mathrm{M}}{\mathrm{k}^{2}}\; \cdot\;\theta\;=\;0. \tag{7}\]  
Furthermore, knowing that:  
\[\omega^{2}\;=\;\frac{\mathrm{g}\cdot\mathrm{G}\mathrm{M}}{\mathrm{k}^{2}}, \tag{8}\]  
and introducing Equation (8) in Equation (7), the following is expressed:  
\[\frac{\mathrm{d}^{2}\theta}{\mathrm{d}t^{2}}\;+\;2\lambda\;\frac{\mathrm{d} \theta}{\mathrm{d}t}\;+\;\omega^{2}\;\cdot\;\theta\;=\;0, \tag{9}\]  
where Equation (9) is the differential equation of the rolling motion in calm waters with resistance. According to other studies of damped oscillatory motions, the necessary auxiliary equation for the analysis and development of this equation is:  
\[\mathrm{m}^{2}\;+\;2\lambda\cdot\mathrm{m}\;+\;\omega^{2}\;=\;0, \tag{10}\]  
The solutions to which are:  
\[\mathrm{Ig}\cdot\frac{\mathrm{d}^{2}\theta}{\mathrm{dt}^{2}}\ +\ \mathrm{A_{R}}\ \frac{\mathrm{d}\theta}{\mathrm{dt}}\ +\ \mathrm{D}\cdot\mathrm{GM}\cdot\theta\ =\ 0, \tag{2}\]0  
\[\mathrm{Ig}\cdot\frac{\mathrm{d}^{2}\theta}{\mathrm{dt}^{2}}\ +\ \mathrm{A_{R}}\ \frac{\mathrm{d}\theta}{\mathrm{dt}}\ +\ \mathrm{D}\cdot\mathrm{GM}\cdot\theta\ =\ 0, \tag{2}\]1  
\[\mathrm{Ig}\cdot\frac{\mathrm{d}^{2}\theta}{\mathrm{dt}^{2}}\ +\ \mathrm{A_{R}}\ \frac{\mathrm{d}\theta}{\mathrm{dt}}\ +\ \mathrm{D}\cdot\mathrm{GM}\cdot\theta\ =\ 0, \tag{2}\]2  
The difference (\(\lambda^{2}-\omega^{2}\)) gives rise to three different cases with different solutions. It is, therefore, necessary to analyse the three possibilities and, afterwards, develop the case corresponding to the rolling motion. In the first case, when (\(\lambda^{2}-\omega^{2}\)) > 0, the damping factor (\(\mathrm{A}_{\mathrm{R}}\)) is high, and an overdamped system is produced. In this case, the solution for Equation (9) represents a smooth and non-oscillating rolling motion. In the second case, when (\(\lambda^{2}-\omega^{2}\)) = 0, its equation represents a non-oscillating motion. However, any decrease in the damping force (\(\lambda\)) converts to (\(\lambda^{2}-\omega^{2}\)) in a value negative, which would be the third case. These conditions are called critically damped. Therefore, when (\(\lambda^{2}-\omega^{2}\)) < 0, the roots of Equations (12) and (13) are negatives and, in consequence, they give rise to complex numbers:  
\[\mathrm{Ig}\cdot\frac{\mathrm{d}^{2}\theta}{\mathrm{dt}^{2}}\ +\ \mathrm{A_{R}}\ \frac{\mathrm{d}\theta}{\mathrm{dt}}\ +\ \mathrm{D}\cdot\mathrm{GM}\cdot\theta\ =\ 0, \tag{2}\]3  
\[\mathrm{Ig}\cdot\frac{\mathrm{d}^{2}\theta}{\mathrm{dt}^{2}}\ +\ \mathrm{A_{R}}\ \frac{\mathrm{d}\theta}{\mathrm{dt}}\ +\ \mathrm{D}\cdot\mathrm{GM}\cdot\theta\ =\ 0, \tag{2}\]4  
Consequently, Equation (9) can be expressed as follows:  
\[\mathrm{Ig}\cdot\frac{\mathrm{d}^{2}\theta}{\mathrm{dt}^{2}}\ +\ \mathrm{A_{R}}\ \frac{\mathrm{d}\theta}{\mathrm{dt}}\ +\ \mathrm{D}\cdot\mathrm{GM}\cdot\theta\ =\ 0, \tag{2}\]5  
where Equation (16) describes the ship's own oscillatory motion, called underdamped. Therefore, this is the most interesting case for the present research.