# Dynamic Load Modelling within Combined Transport Trains during Transportation on a Railway Ferry
## 3 Mathematical Modelling of the Dynamic Load of Containers
That is, when \(y_{1}=\theta_{1},\ y_{2}=\dot{\theta}_{1},\ y_{3}=\theta_{2},y_{4}=\dot{\theta}_{2}, \ y_{5}=\theta_{3},y_{6}=\dot{\theta}_{3}\), it follows that  
\[Q(t,\ y)=\left| \begin{array}{c}y_{2}\\ y_{4}\\ y_{6}\\ p_{w}^{\prime}\frac{y_{2}+\Lambda_{\theta}\cdot\frac{\mathbb{B}}{2}\cdot \mathbb{F}(t)-\left(\Lambda_{\theta}\cdot\frac{\mathbb{B}}{2}\right)y_{2}}{l_ {0}^{0}}\\ \frac{p_{wag}^{\prime}\frac{y_{2}}{2}+\Lambda_{\text{aug}}^{\text{c}}+M_{ \text{aug}}^{\text{h}}}{l_{\text{aug}}^{0}}\\ \frac{p_{c}^{\prime}\cdot\frac{\mathbb{R}}{2}-\beta l_{c}\cdot\dot{q}_{3}+M_{c }^{\text{aug}}}{l_{0}^{0}}\end{array} \right., \tag{6}\]  
where \(Y_{0}\) is a vector containing the initial conditions, \(tn\) and \(tk\) are values that define the initial and final integration variable, n' is fixed number of steps, and \(Q\) is character vector. The initial displacements and velocities were assumed to be zero. The coefficient of viscous resistance should be in this case in the range 0.5-1.2 kN\(\cdot\)s/m. On the basis of the calculations, we found that the greatest acceleration values occurred when the course angles of the wave in relation to the body of the railway ferry were \(\chi=60^{\circ}\) and \(\chi=120^{\circ}\). At the same time, the maximum acceleration of the container was about 1.5 m/s\({}^{2}\) (see Figure 4) of the flat wagon 0.4 m/s\({}^{2}\) (see Figure 5). The numerical values of accelerations were given without taking into account the component of gravitational acceleration. Figure 4: Accelerations acting on the load-bearing structure of the flat wagon when transported on a railway ferry. The total amount of acceleration can be determined:  
\[\ddot{\theta}_{t}=\ddot{\theta}_{prt}+g\cdot\sin\theta, \tag{7}\]  
where \(\ddot{\theta}_{prt}\) is acceleration, which applies to the regular position of the flat wagon with containers on the deck; g is gravitational acceleration; and \(\theta\) is roll angle of the railway ferry. Considering the hydrometeorological characteristics of the Black Sea area and the surface projection of a railway ferry of the \"Heroes of Shipka\" type, we obtained a value of \(\theta\) = 12.2\({}^{\circ}\) for the roll angle of a railway ferry. The roll value was calculated for the case of static wind action on the surface projection of a railway ferry [33]. Then, the total amount of acceleration that acts on the container was 3.57 m/s\({}^{2}\) (0.36 g), and on the load-bearing structure of the flat wagon was 2.47 m/s\({}^{2}\) (0.25 g). Therefore, the introduction of a viscous connection between the load-bearing structure of the flat wagon and the containers made it possible to reduce the dynamic load by 38% and 23%, respectively, compared to the typical interaction scheme. Dynamic load is the main factor that affects the stability of a flat wagon with containers when transported on a rail ferry. Consequently, due to the decrease in the dynamic load that acts on the container placed on the flat wagon, its stability is also improved during oscillations of the railway ferry.