## 4 Methodology for the Modernization of ATLAS-LG System Software Operational Algorithms
### Criteria for Evaluation of ATLAS-LG System Operation Results
\[H(t)=\int_{0}^{t}h(\tau)d(\tau). \tag{12}\]  
Increases in dynamic force over the timeline, considering the delay, can be expressed as follows (Equation (13)):  
\[P_{d}(t)=w_{0}lh(t-T)-w_{0}^{2}rH(t)+w_{0}^{2}rH(t-2T). \tag{13}\]  
In the case of step \"up\" with the \(\delta\) value, the wheel center movement trajectory can be expressed as follows (Equation (14)):  
\[\xi(Vt)=\left\{ \begin{array}{c}0,\ t\leq 0;\\ -\delta+\frac{m_{0}^{2}(t-T)^{2}}{2},\ 0\leq t\leq T;\\ -\delta,\ t\geq T.\end{array} \right. \tag{14}\]  
Performing Laplace transformation calculations identical to the calculations for the \"flat spot\" indicated above, we eventually get the following expression for calculating the dynamic force over the timeline (Equation (15)):  
\[P_{d}(t)=w_{0}\frac{l}{2}h(t)-w_{0}^{2}rH(t)+w_{0}^{2}rH(t-T). \tag{15}\]  
In this case, the length of the time interval where the wheel center movement trajectory changes is determined using Equation (16). \[l=2\sqrt{2r\delta}. \tag{16}\]  
Thus, performing consecutive calculations of \(W^{0}(iw)\), \(r_{j}\) (using the Descartes-Euler solution of quartic equations), \(W_{y}(0,\,iw)\), and \(h(t)\), when frequency range \(H(t)\) is sufficient to determine the dynamic relation of the impact, we acquire all the necessary data to determine the dynamic forces that occur in the case of a wheel flat spot. On the basis of the performed investigation, the normative recommendations for the evaluation of vertical forces transmitted from train wheels to rails measured by the ATLAS-LG measurement system were prepared. It is recommended to accept the following rolling stock culling criteria which exceed the normatives of the strike impact for the track:1. Vertical forces must be evaluated according to the size of \(P_{derivative}\), hereinafter referred to as the derivative value of the vertical force adapted to conditions of all rolling stock speeds = 80 km/h and static axle load of freight wagons = 230 kN. 2. The derivative value of the vertical force \(P_{derivative}\) is calculated if the speed of the rolling stock ranges from 30 to 80 km/h on a certain section with the installed ATLAS-LG system. 3. If the speed exceeds 80 km/h, the \"derivative\" value of the vertical force \(P_{derivative}\) is equal to the maximum value recorded by the ATLAS-LG system during rolling of the analyzed wheel. 4. The operation range of ATLAS-LG system does cover speeds less than 30 km/h. 5. The \"derivative\" value of the vertical force is calculated according to the following equations: - for freight wagons (Equation (17)): \[P_{derivative}=P_{max}+\left(115-P_{average}\right)+\left(2.72+0.0028\cdot P_{ average}\right)\cdot\left(80-v\right),\] (17) - for locomotives and passenger wagon (Equation (18)): \[P_{derivative}=P_{max}+2.5\cdot\left(80-v\right),\] (18) - special rolling stock (Equation (19)): \[P_{derivative}=P_{max}+2.7\cdot\left(80-v\right),\] (19) where \(P_{max}\) is the maximum value of the vertical force (kN), \(P_{average}\) is the average value of the vertical force (kN), and \(v\) is the speed at which the carriage passes a distance section (km/h). Two limits are set:  
\"Danger 1\"--the train speed is limited to 40 km/h and the detected wheelset must be rolled out at the nearest point of service, regardless of whether it has defects. This limit value is determined when the \"derivative\" values of the following vertical forces are exceeded: for freight wagons, 450 kN; for locomotives and passenger wagons, 350 kN; for special rolling stocks, 400 kN. \"Danger 0\"--the detected wheelset is checked at the nearest service station. After encountering defects that exceed the geometric parameters of normative indicators, the wheelset must be rolled out, and, if there are defects with parameters below the normative indicators, the wagon can continue to run at the determined speed.