## 4 The Determination of the Dynamic Loading on the Removable Module Structure  
The loading on the removable module at the operational modes can be reduced through mounting viscoelastic bonds in the fittings [22]. The authors have substantiated this solution with the fact that when the dynamic loading on the removable module (container) exceeds the friction forces between the fittings and the fitting stops, the dynamic loading on the module can considerably exceed the normative values. The dynamic loading of the removable module placed on a flat wagon during a shunting impact was determined with mathematical modelling. The design model is given in Figure 8. The parameter \(P_{M}\) is the forces emerging in the contact areas between the fittings and the fitting stops. The dynamic loading on the removable module was studied in the plane coordinates. It was assumed in the mathematical model that the removable module was loaded with conditional freight at the maximum allowable loading capacity. The natural displacements of the freight relative to the removable module were neglected. The 13-401M flat wagon was taken as the prototype. Figure 8: Design diagram of a flat wagon loaded with two removable modules. Figure 7: Cross-section of the rectangular tube: \(H\)—height; \(B\)—width; \(S\)—wall thickness. The mathematical model for determination of the dynamic loading of the removable module had the form:  
\[\left\{ \begin{matrix}M_{FW}\dddot{q}_{1}=P_{l}-\sum\limits_{i=1}^{n}\Big{(}F_{ FR}\cdot sign\big{(}\dot{q}_{1}-\dot{q}_{2}\big{)}+C_{f}\cdot(q_{1}-q_{2})+\beta_{f} \cdot(\dot{q}_{1}-\dot{q}_{2})\Big{)},\\ M_{M}\dddot{q}_{2}=\Big{(}F_{FR}\cdot sign\big{(}\dot{q}_{1}-\dot{q}_{2}\big{)}+ C_{f}\cdot(q_{1}-q_{2})+\beta_{f}\cdot(\dot{q}_{1}-\dot{q}_{2})\Big{)}.\end{matrix} \right. \tag{3}\]  
where \(M_{FW}\)--the gross mass of the flat wagon;  
\(P_{l}\)--the value of the longitudinal force on the coupler (3.5 MN);  
\(n\)--the number of removable modules placed on the flat wagon;  
\(F_{FR}\)--the friction force between the fitting stops and the module fittings;  
\(M_{M}\)--the mass of the loaded removable module;  
\(C_{f}\)--the rigidity of spring elements in the fittings of the removable module;  
\(\beta_{f}\)--the coefficient of the viscous resistance in the removable module fittings;  
\(q_{1}\), \(q_{2}\)--the coordinates corresponding to the displacements of the flat wagon and the removable module relative to the longitudinal axis, respectively. Mathematical model Equation (3) was solved in MathCad with the Runge-Kutta method [23; 24; 25] at the initial conditions equalling zero [26; 27; 28]. The integration of differential equations was carried out with a fixed step. In this case, the built-in function rkfixed (\(\gamma_{0}\), \(tn\), \(tk\), \(n\), \(F\)) was used, where \(\gamma_{0}\) is a vector containing the initial conditions, \(tn\) and \(tk\) are the initial and final integration variables, respectively, \(n\) is the number of steps; \(F\) is a character vector. Mathematical model Equation (3) was solved in the following way:  
\[F(t,y)=\left[ \begin{matrix}y_{2}\\ y_{4}\\ \frac{P_{l}-\sum_{i=1}^{n}(F_{FR}\cdot sign(y_{2}-y_{4})+C_{f}\cdot(y_{1}-y_{3 })+\beta_{f}\cdot(y_{2}-y_{4}))}{\frac{M_{FW}}{F_{FR}\cdot sign(y_{2}-y_{4})+ C_{f}\cdot(y_{1}-y_{3})+\beta_{f}\cdot(y_{2}-y_{4})}}\\ \frac{M_{FW}}{M_{M}}\end{matrix} \right], \tag{4}\]  
Whereas \(y_{1}=q_{1}\), \(y_{2}=\dot{q_{1}}\),the minimum element size was 16 mm. Steel 09C2 Cu was taken as the structural material. The main technical characteristics of the steel are presented in Table 1.  
The model was secured in the areas of support on the bogies. The calculation included the vertical loading \(P_{v}\) on the removable modules at the full loading capacity (Figure 10). The longitudinal load \(P_{l}\) equalling 3.5 MN was applied to the rear coupler support of the flat car. The fittings of the removable module had the bonds with the rigidity 20 kN/m and the viscous resistance coefficient 30 kN\(\cdot\)s/m.