## 2 Methods and the Data
However, the model includes neither the stagnation of sea waves nor the movement of the freight in the car due to the oscillations of the train ferry. The model also did not include the relationship between the ship's head and stern. This assumption was based on the fact that the head is about \(2.5^{\circ}\) of a given wave height. It should also be noted that the modelling of the dynamic load of the open car included the vessel's nameplate displacement, taking into account its gross tonnage. These restrictions were based on the experimental research of loading on the bearing structure of an open car transported via sea while rigidly fixed on the deck, as was described in the authors' previous studies. The research included a full-scale experiment during which the train ferry moved in the Black Sea [16]. Mathematical model (1) was verified with [PERSON]'s criterion. On the basis of the calculations of the mean squared error \(S_{d}\) = 6.94 and the dispersion of adequacy \(S_{d}\) = 35.42, the actual value of [PERSON]'s criterion, \(F_{p}\) = 5.11, was obtained, which was less than its tabular value, \(F_{t}\) = 5.41. Thus, the hypothesis on the adequacy of the model was not rejected. The approximation error was about 2%. The calculation was based on the technical characteristics of the train ferry Geroi Shipki. Mathematical model (1) includes the characteristics of the disturbing force typical of the Black Sea: a sea wave height of 8 m, a length of 120 m, and a wind pressure of 1.47 kPa, according to the reference data on the hydrometeorological characteristics of the sea [17]. A 12-757 open car manufactured by Kryukovsky Railway Car Building Works was taken as the prototype (Figure 5). The main technical characteristics of the car are shown in Table 1.  
The calculation assumed that the honeycomb panel was made of an elastic, viscous material. The viscous resistance coefficient was in the range of 150-165 kN\(\cdot\)s/m, and the stiffness was in the range of 500-520 \(\rm{kN/m}\). These parameters were determined via mathematical modelling using the sequential selection method. This mathematical model was solved using MathCad software, version 2015 [17; 18; 19; 20]. The equations were reduced to the form:  
\[F(t,y)=\left[ \begin{array}{c}y_{3}\\ y_{4}\\ \frac{p^{\prime}\cdot\frac{b}{2}+A_{y}\cdot\frac{b}{2}\cdot F(t)-1}{\left|\frac {1}{D_{2}^{2}}\left(3-2\cdot\frac{b}{2}\right)\right|},y_{3}\\ \frac{p_{k}\cdot\frac{b}{2}+F_{y}-\frac{b}{2}\cdot y_{0}-\frac{b}{2}\cdot y_{4} }{l_{0}}\end{array} \right], \tag{3}\]  
\[Z={{\it{k}}{{\it{fixed}}}\left({{\it{Y}}0,{\it{tn}},{\it{tk}},{\it{n}}^{\prime},F}\right)}\]  
where \(y_{1}=q_{1};y_{2}=q_{2};y_{3}=\dot{y}_{1};y_{4}=\dot{y}_{2}\). The function _rfixed_ is built into the software. This function was used to solve differential equations using the Runge-Kutta method with a constant pitch [21]. The arguments of the function are as follows:  
\(Y0\) is the vector of the initial conditions of K elements (\(k\) is the number of equations in the system);  
\({\it{tn}}\) and \({\it{tk}}\) are the left and right boundaries of the interval where the solution is sought, respectively;  
\({\rm{n}}^{\prime}\) is the number of points within the interval (\({\it{tn}}\), \({\it{tk}}\)) at which the solution is sought;  
\(F\) is the vector consisting of K elements, which contains the first derivative of the required functions. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Unit** & **Value** \\ \hline Carrying capacity & t & 69.0 \\ Tare weight & t & 25.0 \\ Design speed & km/h & 120 \\ Dimensions & – & 1–VM (0–T) \\ Base & mm & 8670 \\ Length along the coupler axes & mm & 13,920 \\ Number of axes & – & 4 \\ Two-axle bogie & – & 18-100 \\ Inside body length & mm & 11,228 \\ Inside body width & mm & 2964 \\ Inside body height & mm & 2315 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The main technical characteristics of the car. Figure 5: Open car model 12-757.