# Dynamic Load Modelling within Combined Transport Trains during Transportation on a Railway Ferry
## 3 Mathematical Modelling of the Dynamic Load of Containers
The moment of inertia of a vessel was determined according to the classical formula [33]  
\[I_{s}^{0}=\frac{D}{12g}\big{(}B^{2}+4z_{g}^{2}\big{)}, \tag{2}\]  
where \(D\) is ship displacement, and \(z_{g}\) is coordinate of the center of gravity. The coordinate of the center of gravity of a vessel with cargo on board is determined by [33]  
\[z_{g}=\frac{1}{D+m_{1}+m_{2}+\cdots+m_{n}}\sum D\cdot z_{c}+m_{1}z_{1}+m_{2}z_ {2}+\cdots+m_{n}z_{n}, \tag{3}\]  
where \(m_{1},\ m_{2},\ \ldots,\ m_{n}\) are cargo masses, and \(z_{1},\ z_{2},\ \ldots,\ z_{n}\) are distances of the center of gravity of the cargo from the main plane (keel). The distance of the center of gravity of the vessel above-water surface from the current waterline is given by  
\[z_{c}=\frac{S_{1}z_{c}+m_{1}z^{\prime}_{1}+m_{2}z^{\prime}_{2}+\cdots+m_{n}z^ {\prime}_{n}}{S_{1}+S_{2}+\cdots+S_{n}}, \tag{4}\]  
where \(S_{1},\ S_{2},\ \ldots,\ S_{n}\) are areas of figures into which the side projection of the surface of the ship is divided, and \(z^{\prime}_{1},\ z^{\prime}_{2},\ \ldots,\ z^{\prime}_{n}\) are distances of the center of gravity of areas \(S_{1},\ S_{2},\ \ldots,\ S_{n}\) from the current waterline. Determination of the coefficient of resistance to vibrations of a railway ferry was carried out according to the method given in [33; 34]. When determining the accelerations that act on the flat wagon with containers, the course angles of the wave in relation to the body of the railway ferry are taken into account  
\[\chi=k\lambda\cdot L\cdot\cos(\alpha), \tag{5}\]  
where \(k\lambda\) is a coefficient that depends on the shape of the vessel contours, \(L\) is length of the vessel, and \(\alpha\) is the angle of the wave relative to the body. During the model building process, the impact action of sea waves was not taken into account. The motion of the wave was described as a trochoidal law. When determining the moments of forces arising between the flat wagon and the deck, as well as between the flat wagon and the container, we took into account the horizontal components of the gross weight of the flat wagon and the container, respectively. Since container trains are placed on the decks of a railway ferry and repeat its trajectory during oscillations, we took into account the technical characteristics of the vessel when creating and solving the mathematical model (1). Accelerations that act directly on the railway ferry and on container trains placed on decks depend on these characteristics. Calculations were made for the railway ferry \"Heroes of Shipka\", running in the Black Sea waters. The main technical characteristics of the railway ferry are given in Table 1[35]. The parameters of the disturbing effect (sea waves) were determined on the basis of the reference literature (Table 2) [36]. The solution of the system of differential Equation (1) was performed using the Runge-Kutta method in the MathCad environment [37; 38; 39; 40; 41; 42; 43]. For this purpose, the transition was made from systems of second-order differential equations to systems of first-order differential equations, followed by the use of standard algorithms for solving systems using the rkfixed function of Mathcad. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter Name** & **Value** \\ \hline Length, m: & \\ Maximum & 184.25 \\ Between perpendiculars & 170 \\ Width & 26 \\ Height, m: & \\ To the upper deck & 15.2 \\ To the main deck & 9 \\ Draft, m & 6.5 \\ Deadweight, t & 12,889 \\ Displacement, t & 23,744 \\ Speed, knots & 18.6 \\ Wagon capacity, pes. & 108 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Main technical characteristics of the ship ferry “Heroes of Shipka”. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter Name** & **Numerical Value** \\ \hline Sea wave height, m & 8 \\ Sea wave period, s & 9 \\ Heading angle of a wave, degree & 0–180 \\ Wind pressure on the surface projection of the vessel, kPa & 1.47 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters of sea waves used in mathematical modelling.