# Optimized Detection Algorithm for Vertical Irregularities in Vertical Curve Segments
## 3 Investigation of the Relationship Between Vertical Curves and Track Irregularities
### Impact of Train Operating Speed on Detection Accuracy  
This section examines the impact mechanism of train operating speed on detection accuracy. Considering that the operating speed of most urban rail transit systems does not exceed 160 km/h, the test conditions were configured accordingly, as detailed in Table 1.  
After obtaining virtual dynamic responses under varying speed conditions, virtual irregularities were computed. Given the extensive track length, a 500 m segment in both directions from the centre of the vertical curve was selected to include the complete vertical curve and allow the precise analysis of errors between virtual and excitation irregularities. This segment was analyzed in both the time and frequency domains. In Figure 4a, the mileage ranges 1600-1900 m and 2300-2600 m represent straight gradient sections, while 1900-2300 m corresponds to the vertical curve segment.  
Figure 4a demonstrates that the virtual irregularities align well with the excitation irregularities across straight gradient sections and the central part of the vertical curve,  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Operating** & **Train Speed** & **Radius \(R\) (m)** & **Length \(L\) (m)** & **Gradient Difference** & **Single-Side Gradient \(i\)** \\
**Conditions** & \(V\) (km\(\cdot\)h\({}^{-1}\)) & **Radius \(R\) (m)** & **Length \(L\) (m)** & **Difference** & **(\%)** \\  & & & & & \\  & & & & & \\
1 & 80 & & & & \\
2 & 120 & 10,000 & 400 & 40 & \(\pm\)20 \\
3 & 160 & & & & \\ \hline \hline \end{tabular}
\end{table}
Table 1: Experimental conditions for speed variation.  
Figure 4: Time–frequency domain comparison of virtual and excitation irregularity metrics at different detection speeds (**a**) Comparison of error standard deviation (**b**) Comparison of power spectral density.  
with the error standard deviation remaining within 0.1 mm. However, notable spikes occur near the transition points--tangent-to-curve and curve-to-tangent points--of the vertical curve. The spike amplitude begins to increase approximately 100 m before the transition point, reaching a peak at the transition point itself, and then gradually decreasing within 100 m beyond it. Additionally, the spike amplitude is more pronounced at the curve-to-tangent transition point. With increasing speed, the error standard deviation shows minimal variation, indicating that train speed has a limited effect on the amplitude accuracy of virtual irregularity detection at vertical curves, consistent with previous findings in the literature [18].  
Figure 4b illustrates that within the wavelength range of 1-100 m, the PSD curves of the virtual and excitation irregularities exhibit strong alignment, with similar trends and amplitude. Speed variations have no significant effect on the overall shape of the PSD curve in this range. However, for wavelengths exceeding 100 m, the PSD values of the long-wavelength components in the virtual irregularities increase noticeably, with a slight rise in PSD values observed as the speed decreases.