## 4 Methodology for the Modernization of ATLAS-LG System Software Operational Algorithms
### Criteria for Evaluation of ATLAS-LG System Operation Results
These frequency characteristics can be determined as follows (Equation (2)):  
\[W^{0}(iw)=\frac{M_{1}w^{2}-ir_{1}w-C_{1}}{M_{1}M_{0}w^{4}-ir_{1}(M_{1}-M_{0})w ^{3}-C_{1}(M_{1}+M_{0})w^{2}}, \tag{2}\]  
where \(M_{1}\) is the weight above the spring distributed for one wheel (kg), \(M_{0}\) is the wheel weight (kg), \(f_{1}\) is the fluctuation suppression index of the weight above the springs (N-s/m), and \(C_{1}\) is the spring set elasticity distributed for one wheel (N/m). The frequency characteristics of the dynamic force transformations into the rail indentation can be described as follows (Equation (3)):  
\[W_{y}(0,iw)=\frac{1}{EI(r_{1}-r_{2})(r_{1}-r_{3})(r_{1}-r_{4})}+\frac{1}{EI(r_ {2}-r_{1})(r_{2}-r_{3})(r_{2}-r_{4})}, \tag{3}\]  
where \(r_{j}\) (\(j\) = 1, 2, 3, 4) represents the roots of the algebraic equation with complex indexes, as given in Equation (4). \[r^{4}+\frac{mV^{2}}{EI}r^{2}-\frac{fV+2 imwV}{EI}r+\frac{u-mw^{2}+ifw}{EI}=0, \tag{4}\]  
where \(E\) is the rail steel elasticity module (N/m\({}^{4}\)), \(i\) is the rail moment of inertia (m\({}^{4}\)), \(m\) is the given track weight (kg/m), \(f\) is the distributed track inhibition index (N-s/m\({}^{2}\)), \(u\) is the rail base elasticity module (Pa), and \(V\) is the travel speed (m/s). Root numbers are selected using the following method (presence of two roots with negative real parts and two roots with positive real parts is proven using the Raus criterion \(Re\), as given in the Equation (5)):  
\[Ref_{1}<0,\ Ref_{2}<0,\ Ref_{3}>0,\ Ref_{4}>0, \tag{5}\]  
where \(f_{1},f_{2},f_{3},f_{4}\) represent the fluctuation suppression index of the weight above the springs (N-s/m). If we know the wheel center movement trajectory \(\xi(Vt)\), after the inverse Laplace transformation, the dynamic force \(P_{d}\) can be calculated as follows (Equation (6)):  
\[P_{d}=W_{p}(s)\xi. \tag{6}\]  
If the wheel moves when a flat spot is present, the movement trajectory is described as follows (Equation (7)):  
\[\xi(Vt)=\left\{ \begin{array}{c}0,\ t\leq 0;\\ \frac{m^{2}_{0}t^{2}}{2},\ 0\leq t\leq\frac{l}{2w_{0}r};\\ \frac{r(\frac{l}{l-w_{0}}\ t)^{2}}{2},\ \frac{l}{2w_{0}r}\leq t\leq\frac{l}{w_{0}r};\\ 0,\ t\geq\frac{l}{w_{0}r},\end{array} \right. \tag{7}\]  
where \(w_{0}=V/r\) is the wheel angular velocity (l/s), \(l\) is the length of a \"flat spot\" type defect (for flat or compression spot, \(l\cong 2\sqrt{2r\delta}\)) (mm), \(\delta\) is the flat spot depth of a compression spot height (mm), and \(r\) is the wheel radius (m). After applying Laplace transformation to this expression, we get Equation (8). \[\xi.=\frac{W_{0}}{S^{2}}le^{-sT}-\frac{W_{0}^{2}}{S^{3}}r+\frac{W_{0}^{2}}{S^ {3}}re^{-2 sT}, \tag{8}\]where \(T=\frac{1}{2V}\) is the half-time of wheel center \"movement\" over the gap (s). In this case, the dynamic force after Laplace transformation can be described using Equation (9). \[P_{d}=W_{p}(s)\xi=\frac{W_{p}(s)}{s^{2}}w_{0}le^{-sT}-\frac{W_{p}(s)}{s^{3}}w_{0} ^{2}r+\frac{W_{p}(s)}{s^{3}}w_{0}^{2}re^{-2 sT}. \tag{9}\]  
The system transition impulse function can be described as follows (considering the layout of special points of the expression \(\frac{W_{p}(s)}{s^{2}}\) in Equation (10)):  
\[h(t)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}\frac{W_{p}(iw)}{-w^{2}}e^{iwt}dw. \tag{10}\]  
When it is determined and denoted by \(W_{P_{d}}(iw)=\frac{W_{p}(iw)}{-W^{2}}\), the function \(h(t)\) can be described as follows (Equation (11)):  
\[h(t)=\frac{1}{\pi}\int_{0}^{+\infty}\Bigl{[}ReW_{P_{d}}(iw)\cos(wt)-ImW_{P_{d }}(iw)\sin(wt)\Bigr{]}dw. \tag{11}\]  
The individual system reaction to the increase of approach acceleration can be calculated using the impulse transition function (Equation (12)).