# Determination of the Loading of a Flat Rack Container during Operating Modes
## 5 Determination of the Dynamic Loading of a Container in the Vertical Plane  
The dynamic loading of a container was determined with mathematic modelling. The design diagram is presented in Figure 4. The research took into account that the flat car was loaded with four containers. The link between the containers and the frame of a flat car was modeled as elastic.  
The differential equations of motion of a flat car loaded with containers look like:  
\[M_{1}\cdot\ddot{q}_{1}+C_{1,1}\cdot q_{1}+C_{1,2}\cdot q_{2}+C_{1,3}\cdot q_{3} =-F_{FR}\cdot\Big{(}sign\Big{(}\dot{\phi}_{1}\Big{)}+sign\Big{(}\dot{\phi}_{2} \Big{)}\Big{)}-\sum_{i=1}^{4}F_{z}, \tag{1}\]  
\[M_{2}\cdot\ddot{q}_{2}+C_{2,1}\cdot q_{1}+C_{2,2}\cdot q_{2}+B_{2,2}\cdot\dot{q }_{2}=F_{FR}\cdot sign\Big{(}\dot{\phi}_{1}\Big{)}+k(\eta_{1}+\eta_{2})+\beta \big{(}\dot{\eta}_{1}+\dot{\eta}_{2}\big{)}, \tag{2}\]  
Figure 4: Design diagram of the bearing structure of a flat car loaded with containers.  
Figure 3: Location of elastic elements in the cross bearers of a container.  
\[M_{3}\cdot\ddot{q}_{3}+C_{3,1}\cdot q_{1}+C_{3,3}\cdot q_{3}+B_{3,3}\cdot\dot{q}_{ 3}=F_{FR}\cdot sign\Big{(}\dot{\delta}_{2}\Big{)}+k(\eta_{3}+\eta_{4})+\beta\big{(} \dot{\eta}_{3}+\dot{\eta}_{4}\big{)}, \tag{3}\]  
\[M_{4}\cdot\ddot{q}_{4}=F_{z}-M_{4}\cdot g,\,F_{z}=-k_{k}(q_{1}-q_{4}), \tag{4}\]  
where \(M1\)--mass of the bearing structure of a flat car; \(M_{2}\), \(M_{3}\)--mass of the first and second bogies facing the engine, accordingly; \(M_{4}\)--container mass; \(C_{ij}\)--elasticity characteristics of the elements of the oscillation system determined by the rigidity coefficients of the springs \(k_{T}\); \(k_{b}\)--rigidity of the elastic elements in the side beams of the frame; \(k_{k}\)--rigidity of the elastic linkage which models the interaction between a container and a flat car; _Bij_--dissipation function; \(a\)--half-base of a bogie; \(k\)--track rigidity; \(\beta\)--damping coefficient; \(F_{FR}\)--absolute friction force in a spring group; \(\delta_{i}\)--deformation of elastic elements of spring suspension; and \(\eta(t)\)--track irregularity.  
The study assumed that a flat car loaded with containers moved on the elastic viscous track [28; 29; 30].  
Differential Equations (1)-(4) were solved in the MathCad software [31; 32; 33] and was determined as  
\[F(t,y)=\left[\begin{array}{c}y_{2}\\ y_{4}\\ y_{6}\\ y_{8}\\ -f_{FR}\big{(}sign\big{(}\dot{i}_{1}\big{)}+s\beta t\big{(}\dot{i}_{2}\big{)} \big{)}-\frac{\sum\limits_{i=1}^{4}F_{z}-C_{1,1}\cdot y_{1}-C_{1,2}\cdot y_{3} -C_{1,3}\cdot y_{5}}{M_{1}}\\ \frac{F_{FR}\cdot sign\big{(}\dot{i}_{1}\big{)}+k(\eta_{1}+\eta_{2})+\beta \big{(}\dot{\eta}_{1}+\dot{\eta}_{2}\big{)}-C_{2,1}\cdot y_{1}-C_{2,2}\cdot y_{ 3}-B_{2,2}\cdot y_{4}}{M_{2}}\\ \frac{F_{FR}\cdot sign\big{(}\dot{i}_{2}\big{)}+k(\eta_{3}+\eta_{4})+\beta \big{(}\dot{\eta}_{3}+\dot{\eta}_{4}\big{)}-C_{3,1}\cdot y_{1}-C_{3,3}\cdot y_ {5}-B_{3,3}\cdot y_{6}}{M_{4}}\\ \frac{F_{z}-M_{4}\cdot g}{M_{4}}\\ Z=rkfixed(Y0,\,tn,\,tk,\,n,\,F),\end{array}\right], \tag{5}\]  
where \(Y0\)--vector with the initial conditions, \(tn\) and \(tk\)--values determining the initial and final integration variable, \(n^{\prime}\)--fixed number of stages, \(Q\)--symbolic vector [34; 35].  
Here  
\[y_{1}=q_{1},y_{3}=q_{3},y_{5}=q_{5},y_{7}=q_{7},\]  
\[y_{2}=\dot{y}_{1},y_{4}=y_{3},y_{6}=\dot{y}_{5},y_{8}=\dot{y}_{7}.\]  
The initial displacements and speeds were taken equal to zero [36; 37; 38]. The results of the calculation are shown in Figure 5.  
Figure 5: Acceleration acting on the container in the vertical plane.