## 5 Conclusions  
In this study, mathematical models were obtained based on several theoretical assumptions of a ship's synchronous rolling. These models refer to the worst conditions considered, i.e., the ship in static condition and under the influence of regular pure beam and trochoidal waves. After analysing and carrying out a set of simulations for ships with different intact stability, expressed as the ship's natural roll period (Td), determined by the specific loading condition, and wave periods (Tw), relevant results were obtained. It was determined that for each half-Td (\(1/2\cdot\) Td), the amplitude of the rolling angle increases by a constant value that depends on the maximum slope of the wave. Therefore, as a function of Td, the margin available to reach a given maximum allowed angle varies according to the ship's loading condition. Furthermore, it is important to notice that as the ship approaches synchronous rolling, the relative maximum rolling angle reached changes from occurring in the middle of the wave slope to the crest or trough of the wave. Then, if the ship operators have good knowledge and previous training about this circumstance, with good observation of the clinometer on board and the wave state, undesirable situations can be avoided. For that, the officer on watch should recognize as soon as possible the conditions conducive to synchronism phenomena in order to alter the ship's heading to change the encounter period and eliminate this danger. Therefore, the effort by the officer on watch to bring the ship's heading into the waves should be made well in advance. Another theoretical solution that requires more time would be to alter the position of the vertical centre of gravity of the ship (KG) in order to change the metacentric height (GM), as well as ballasting, de-ballasting or transferring ballast. By doing so, the ship's natural rolling period (Td) also changes, making it a non-synchronous value. However, the proposed solution to be used as a countermeasure is only feasible from a theoretical point of view, because according to the present results, synchronous roll occurs so rapidly that it is not technically possible to alter the vertical position of the centre of gravity sufficiently in advance. Finally, and although it may not be as useful and simple to be used by ship operators, the generic rule of the periodicity with which the maximum rolling angles are reached was obtained, as a  
Figure 23: A ship’s rolling motion just as the synchronism phenomenon occurs. function of two known parameters, the ship's loading condition, represented by the ship's natural roll period (Td), and the sea state (Tw). The proposed control system does not ensure an exact rolling angle because the effect of time does not allow for obtaining an exact rolling angle. However, setting 0.25 rad allows for not exceeding the Td = 16 s maximum in this study. Similarly, the results shown in the red curves have shifted and overlapped with the initial map at different Td values according to the value of rolling angle. Therefore, it is possible to solve the ship's rolling motion (ship's stability) inversely and accurately by means of 10-layer neural networks. In order to rectify the limitations of this study, which was carried out under theoretical conditions, future studies could be guided to perform simulations in towing tanks and validate the results. Regarding putting the results and the proposed control system into practice by ship operators, virtual simulators for training could be used in such a manner that with the visual observation of the clinometer and the ship's position along the wave profile, they can predict when the ship is approaching dangerous situations. Conceptualization, J.M.P.-C. and J.A.O. ; methodology, J.M.P.-C. and J.A.O. ; validation, J.M.P.-C. and J.A.O. ; formal analysis, J.M.P.-C. and J.A.O. ; investigation, J.M.P.-C. and J.A.O. ; data curation, J.M.P.-C. and J.A.O. ; writing--original draft preparation, J.M.P.-C. and J.A.O. ; writing--review and editing, J.M.P.-C. and J.A.O.