# Dynamic Load Modelling within Combined Transport Trains during Transportation on a Railway Ferry
## 3 Mathematical Modelling of the Dynamic Load of Containers  
To determine the dynamic loads that act on the supporting structure of the flat wagon and container, we carried out mathematical modelling. The calculation scheme is shown in Figure 3, wherein the angular movements of the railway ferry relative to the longitudinal axis, which passes through its center of gravity, are taken into account. This type of oscillation in seafaring practice is called rolling. In the \"dynamics of wagons\", such vibrations are called lateral pitching. When transporting wagons by sea, the lateral rolling of a railway ferry has the greatest impact on their stability with respect to decks. Figure 3: Diagram of a flat wagon with a container. Figure 2: Flat wagon of model 13-401: (**a**) typical design of a flat wagon; (**b**) improved flat wagon design. At the same time, we considered that fixing the wagon relative to the deck of the railway ferry should be carried out according to a standard scheme. In other words, to de-load the spring suspension of the flat wagon, we used four mechanical stop-jacks, which were installed under the pivot beams of the frame. Eight chain binders were also used, which were attached to the supporting structure of the wagon at one end, and to the deck eye at the other. We also installed break shoes under the end wagons in the couplings. The end wagons in the couplings also interact with dead-end stops. A mathematical model developed by the authors that describes the movement of the supporting structure of a flat wagon with containers during oscillations of a railway ferry is given below:  
\[\left. \begin{array}{l}I_{S}^{0}\ddot{q}_{1}+\left(\Lambda_{0}\cdot\frac{B}{2 }\right)\dot{q}_{1}=p_{w}^{\prime}\cdot\frac{b}{2}+\Lambda_{0}\cdot\frac{B}{2 }\cdot\dot{F}(t),\\ I_{uw}^{0}\cdot\ddot{q}_{2}=p_{uang}^{\prime}\cdot\frac{b_{n}}{2}+M_{uw}^{c}+M_ {tang}^{uag},\\ I_{c}^{0}\cdot\ddot{q}_{3}=p_{c}^{\prime}\cdot\frac{b_{c}}{2}-\beta\cdot h_{c }\cdot\dot{q}_{3}+M_{c}^{uag},\end{array} \right\} \tag{1}\]  
where \(q_{1}\approx\theta_{n}\)--generalized coordinate that corresponds to the angular movement around the longitudinal axis of a railway ferry; \(q_{2}\approx\theta_{w}\)--generalized coordinate, which corresponds to the angular movement around the longitudinal axis of the flat wagon; \(q_{3}\approx\theta_{c}\)--generalized coordinate that corresponds to the angular movement around the longitudinal axis of the container; \(I_{S}^{0}\)--moment of inertia; \(B\)--width; \(h\)--depth; \(\Lambda_{0}\)--coefficient of resistance to vibrations; \(p_{w}^{\prime}\)--wind load on the surface projection; \(F(t)\)--law of action of the effort that causes the movement of a railway ferry with wagons placed on its decks; \(I_{w}^{0}\)--moment of inertia of the flat wagon; \(h_{w}\)--height of the side surface of the flat wagon; \(p_{uang}^{\prime}\)--wind load on the side surface of the flat wagon; \(M_{ung}^{c}\)--moment of forces that occurs between the flat wagon and the deck of a railway ferry during angular movements relative to the longitudinal axis; \(M_{ung}^{h}\)--moment of forces that occurs between the flat wagon and containers during angular movements relative to the longitudinal axis; \(I_{c}^{0}\)--moment of inertia of a container; \(h_{c}\)--height of the side surface of the container; \(p_{c}^{\prime}\)--wind load on the side surface of the container; \(M_{c}^{uag}\)--moment of forces that occurs between the container and the flat wagon during angular movements relative to the longitudinal axis; \(\beta\)--coefficient of viscous resistance between the load-bearing structure of the flat wagon and the container. It is important to note that this mathematical model has no analogues, since the proposed technical solutions were developed by the authors of the article. Previously, the issues of transporting combined trains by sea, as well as the improvement of the supporting structures of flat wagons for safe transportation by sea, were not considered.