# Method for Theoretical Assessment of Safety against Derailment of New Freight Wagons
## 2 Method for Assessment of Theoretical Safety against Freight Wagon Derailment
From the condition for the considered boundary condition, it follows that the distance from pole \(M\), according to Equation (11), is  
\[x=x_{max}=l+\frac{R.\sigma_{b}}{2.l} \tag{11}\]  
which is typical for the \"maximum overshoot\" position. It also follows that the total reaction of the second wheel axle is zero, e.g.,  
\[Y_{2}=0 \tag{12}\]  
which is typical for the \"free settling\" position. This allows Equation (10) to be solved and to obtain specific values for \(Y_{1}\) and \(v_{1}\), which are typical for the limited state. When solving Equation (10), it is possible for \(Y_{1}\) or for \(v_{1}\) to obtain negative values. This indicates that the boundary condition is not valid for the specified track and bogie parameters. In this case, it is necessary to go to Step 3 of the current methodology. Step 2. When the bogie is in the \"maximum overshoot\" state, it will move with a speed in the interval from \(0\) to \(v_{1}\), and the pole distance will be \(x=x_{max}\). Therefore, the system of Equation (10) can be solved concerning \(Y_{1}\) and \(Y_{2}\) by setting discrete movement speed values in the specified interval. Step 3. It is assumed that the bogie is in a limited state between free settling and maximum displacement. From this, the next conditions are shown in the following Equation (13):  
\[x=x_{min}=l\quad\text{ and }\quad Y_{2}=0 \tag{13}\]  
Movement speed \(v_{2}\) and force \(Y_{1}\), in this case, can be found by solving Equation (10) under the conditions of Equation (13). Step 4. If the bogie is freely fixed (\(Y_{2}=0\)), then, in Equation (10), there are three unknowns: \(Y_{1}\), \(v\), and \(x\). In this case, the condition in Equation (14) is relevant:  
\[v_{1}\ <v<v_{2}\quad\quad\text{and}\quad\quad x_{min}<\ x\ <x_{max} \tag{14}\]  
Figure 6: Graphical representation of dependencies \(Y_{1}=Y_{1}(v)\) and \(Y_{2}=Y_{2}(v)\)[13]. Therefore, it is possible to obtain the remaining two unknowns by setting discrete values of \(v\) or \(x\) in Equation (10). The calculation process is greatly simplified when setting values of the parameter \(x\). Step 5. If the design speed of the wagon, \(v_{k}\), is higher than \(v_{2}\), it is necessary to build the third zone of the horizontal dynamic calculations, i.e., the zone of maximum displacement. In this case, the condition in Equation (15) is valid:  
\[x=x_{min}=l \tag{15}\]  
Therefore, by setting movement speed values in the interval between \(v_{2}\) and \(v_{k}\), the full reaction forces, \(Y_{1}\) and \(Y_{2}\), can be determined. The methodology proposed above allows us to determine the full reactions, \(Y_{1}\) and \(Y_{2}\), of the first- and second-wheel axles of each bogie at different speeds, curve radii, specific track parameters, different wheel loads, different bogie wheel axle distances, and other parameters. Methodology for Theoretical Determination of the Horizontal Load Force between the Inner (Non-Attacking) Wheel \(Y_{ji}\) of the Investigated Wheel Axle and the Inner Rail  
The inner wheel of the examined wheel axle does not contact its flange with the corresponding rail. Therefore, the horizontal force acting between them arises from the frictional forces, which are determined via Equation (16):  
\[Y_{ji}=\mu.Q_{ji}=\mu.\left(2.Q_{nom}-Q_{ja,min}\right) \tag{16}\]  
where \(\mu\) is the coefficient of friction between the wheel and the rail, assumed to be equal to 0.4 for clean and dry rails, \(Q_{ji}\) is the vertical load force of the inner wheel (index \(i\)) on axle \(j\), \(Q_{nom}\) is the nominal vertical load force of the wagon wheels, and \(Q_{ja,min}\) is the minimum vertical force acting on the outer (attacking) wheel of axle \(j\). It is determined in accordance with the methodology given in Section 2.3. of this paper.