###### Abstract
For that reason, it would desirable that the ship operator knew the elapsed period of time in which the ship begins to roll with both a constant roll angle and angular velocity (same frequency). These issues are studied in the present paper. Regarding the periods of time in the rolling motion, relevant research works were carried out in a short-time domain [8] and using CFD [15; 18]. [PERSON] et al. [15] studied extreme roll and capsizing, which is coupled with a rapid increase in yaw angle. [PERSON] et al. [9] also concluded that the rolling angle due to restoring momentum gradually decreases over time. However, in that paper, the range of time that could be considered depreciable could not be obtained. Furthermore, the IMO's most important regulation for calculating the minimum necessary forces to be counteracted by the lashing system of non-standardized cargo is the Cargo Stowage and Securing Code [19]. This tool considers accelerations as the main parameters to be taken into account, which in turn depend on the ship's loading condition. These transverse accelerations depend on angular velocities and therefore, on the maximum rolling angles. Although the literature [20] assumes that, at the initial state of ship design, an accurate evaluation of the ship's performance in real sea conditions becomes more imperative, in our opinion, it also becomes essential for ship operators to know the overall ship's behaviour during this motion and under the influence of beam waves. In fact, as [PERSON] et al. [21] concluded in their study, during the ship's rolling motion, an understanding of the ship's natural period (i.e., the loading condition) is essential in order to avoid the effect of tank sloshing under different wavelengths, and vice versa, unlike the case of pitching motion. In our opinion, it is essential for ship operators to know, on the one hand, the average time needed to reach critical rolling angle amplitudes and, on the other hand, the moment when the frequency of rolling motion is constant. The lack of knowledge of both parameters can compromise the ship's safety, triggering cargo shifting, damage to crewmembers and passengers, and even damage to the ship, in case of severe rolling. For this reason, in the present paper, new mathematical models of the ship's behaviour in a long-time domain under the influence of a full spectrum of registered waves are proposed. A similar approach was carried out by other authors [22], although they focused only on damaged ships during a short-time domain. Furthermore, as the tank test involves high economic costs and long preparation periods, at present, the prediction of a ship's behaviour using mathematical models is more and more commonly applied [23]. Moreover, as the weather conditions faced during sea passage may vary, the shipmaster has to make multiple decisions in order to improve the safety of navigation. Therefore, in the present research, a guide on the navigational parameters to be modified (heading and ship's speed) is proposed. The new and original mathematical relations proposed in the present paper are applicable to the ship's behaviour during rolling both from a dynamic and static point of view in order to be as useful as possible to ship operators. This is in opposition to the traditional approach [24] where two approaches were used, i.e., from initial stability (static for \(\theta<10^{\circ}\)) and dynamic stability (\(\theta>10^{\circ}\)). The present research, in the first two approaches, refers to regular beam waves, at zero-speed conditions, and uncoupled from other motions. These are considered the most unfavourable conditions when the ship is stopped. In fact, considering certain initial assumptions, the procedure was followed by many relevant and similar research works which were very similar to that proposed in the present work [7; 20; 23; 25; 26]. In a sense, the present research can be considered a continuation of previous works carried out by the authors of [27; 28].