# Acoustic Roughness Measurement of Railhead Surface Using an Optimal Sensor Batch Algorithm
## 2 Measurement of Rail Acoustic Roughness and Method
### Chord Offset Synchronizing with Two Sensors  
Although using multiple sensors can increase measured value accuracy and minimize the impact of external disturbances on the platform wheel, as shown in the spectrum analysis of rail surface roughness (see Figures 10 and 12 at later field test), at least three sensors are needed on the platform. Therefore, we sought the same accuracy of three sensors even when using two sensors on a platform in the case that the wavelength range to be measured has a limitation.  
First, it was assumed the wheelbase between the front and rear wheels of the platform to be 716 mm and specified the wavelength related to rolling noise of the rail surface to be 5-500 mm. The following equation is obtained when calculating the transfer function of the sensor measurement to the rail surface roughness value \(H(\lambda)=Z(\lambda)/Y(\lambda)\) for each sensor position change and calculating the sum of squared residuals (SSR) using the residual, \(r=H(\lambda)-\widehat{H(\lambda)}\), when the amplification ratio is 1.  
\[SSR=\sum_{i=1}^{n}r_{i}^{2}=\sum_{i=1}^{n}\left(H_{i}(\lambda)-\widehat{H_{i}( \lambda)}\right)^{2}=\sum_{i=1}^{n}\left(H_{i}(\lambda)-1\right)^{2} \tag{7}\]  
where \(H_{i}(\lambda)\) is the observed response value at the \(i^{th}\) measured point and \(\widehat{H_{i}(\lambda)}\) is the fitted response value at the \(i^{th}\) measurement. The fitted response becomes one when input and output is coincident. To minimize amplification ratio of the measured value, we can calculate the sensor position that minimizes SSR obtained above. Figure 4 shows the results of SSR calculation by changing the sensor position for two sensors (\(S_{1}\) and \(S_{2}\)) on the above platform. The results show that the positions of the  
Figure 3: Multi-sensor-based chord offset synchronizing model (\(\alpha_{i}=d_{i}/L\), and \(d_{i}\) is the distance from \(x_{r}\) to the \(i^{th}\) sensor (\(i\) = 1, 2, 3)) (Adapted and modified from [17]).  
two sensors with minimum SSR at the critical wavelength range (5-500 mm) where the rolling noise occurs are \(d_{1}\) = 255 mm and \(d_{2}\) = 380 mm from the rear wheel.  
Figure 5 shows a comparison of the transfer function values of the chord offset synchronization method using three sensors and that of the two sensors with the minimum SSR. As shown in the graph, the total SSR with the two sensors has much less value at the 300 mm or higher wavelength band which is the main interest, although the SSR values of the transfer function at lower wavelength are irregular.  
However, applying the optimized two sensor positions to the measurement platform with the drive wheel (wheel diameter of 100 mm) can affect the measured value significantly in the frequency distortion (\(\lambda=2\pi R=314\) mm) associated with drive wheel eccentricity. Therefore, we determined the sensors whose residual calculation in \(\lambda\) = 314 mm converged to 0 first and then calculated the final positions to minimize the SSR in the wavelength range of the total rolling noise. The calculated positions of the two sensors are \(d_{1}\) = 230 mm and \(d_{2}\) = 380 mm, and the SSR value, in that case, is 26.16. Figure 6 shows a comparison of these results with the transfer functions values of the existing sensor positions. Since the amplitude of the transfer function using recalculated two sensors approaches to 1  
Figure 4: Results of SSR as the location of two sensors changes(Minimum SSR at \(d_{1}\) = 255 mm and \(d_{2}\) = 380 mm).  
Figure 5: Minimized residual results with two sensors using the chord offset synchronization.  
as shown in the red line of Figure 6, the overamplified spectrum respond at a wavelength of 314 mm can be reduced in comparison with that of two sensors with a minimum SSR. These analytical results will be verified through field experiment.