# Acoustic Roughness Measurement of Railhead Surface Using an Optimal Sensor Batch Algorithm
## 2 Measurement of Rail Acoustic Roughness and Method
### Chord Offset Synchronizing  
As depicted in Figure 2, the chord offset synchronizing method assumes that the measurement system, whose chord length is L and sensor position is d, moves on the rail in sinusoidal form \((y(x)=Ysin(kx)=Ysin(2\pi x/\lambda))\) and has a random wavelength. Equations (1) and (2) show the coordinates of two wheels in contact with the rail.  
\[y_{r}=Ysin(kx_{r}) \tag{1}\]  
\[y_{f}=Ysin(kx_{f}) \tag{2}\]  
Here, Y is the amplitude of the sinusoidal rail, while k is \(2\pi/\lambda\). Based on the Equations (1) and (2), and the structure of Figure 2, the sensor position on the rail (P in Figure 2) and the sensor position on the platform (Q in Figure 2) can be calcuated as shown below.  
\[P=y((1-\alpha)x_{r}+\alpha x_{f}) \tag{3}\]  
\[Q=(1-\alpha)y(x_{r})+\alpha y(x_{f}) \tag{4}\]  
where \(\alpha=d/L=d^{\prime}/(x_{f}-x_{r})\), \(d^{\prime}\) is the projecton of \(d\) to the \(x\)-axis, P and Q are \(y\)-axis coordinate of the sensor position on rail and platform respectively. Since the actual value \(z(x)\) measured by the sensor is the actual distance between the sensor position on the platform and the vertical rail waveform, it can be expressed as follows:  
\[z(x) =P-Q\] \[=y((1-\alpha)x_{r}+\alpha x_{f})-\left\{(1-\alpha)y(x_{r})+\alpha y (x_{f})\right\} \tag{5}\] \[=Ysin(k(1-\alpha)x_{r}+\alpha kx_{r})-(1-\alpha)Ysin(kx_{r})- \alpha Ysin(kx_{f})\]  
where \(y(x)=Ysin(kx)\) from the Equations (1) and (2).  
Therefore, amplitude \((Z=|z(x)|)\) and phase \((\Phi=phase(z(x)))\) can be obtained from the sensor measurement and the transfer function of the measured value for the rail surface roughness can be defined as follows:  
\[H(\lambda)=Z(\lambda)/Y(\lambda) \tag{6}\]  
The transfer function in Equation (6) can calculate the relative amplitude and phase angle of the measured rail surface value and can predict amplification of the measured sensor value and the change in phase angle according to sensor position. When measuring rail surface roughness using a sensor, the measured value can differ from actual value due to amplification or attenuation according to wavelength (\(\lambda\)), equipment length (L), and sensor position (d) even when analyzing the measured value by reflecting platform inclination. Moreover, it is difficult to compensate for the built-in measurement  
Figure 2: Single-sensor-based chord offset synchronizing model (\(d\) is the distance between \(x_{r}\) and the sensor, and \(d^{\prime}\) is the projection of \(d\)) (Adapted and modified from [17]).  
error of the conventional trolley-type mobile measurement system. Therefore, the recently proposed chord offset synchronization method uses multiple sensors with consideration given to actual platform length (L) and measured wavelength range, mutually compensating for amplification or attenuation to reduce measured value errors [17]. This has the advantage of a compensated value that differs according to relative distance between the sensors, more accurate measurement than the existing analysis method throughout the various wavelength ranges on the rail surface when using optimal sensor layout, and analysis of rail cracks or irregularities. However, to increase measurement accuracy, the method requires at least three sensors in various positions to be synchronized to the measurement from the same position.  
Figure 3 shows the multi-sensor-based chord offset synchronizing model that calculates the measurement value and phase difference using two or more sensors. Calculating batch positions for three sensors on the platform with wheelbase (L = 716 mm) and wheel diameter (D = 100 mm) using this model results in \(d_{1}\) = 222 mm, \(d_{2}\) = 327 mm, and \(d_{3}\) = 432 mm. Here, di is the distance between the rear wheel and the \(i^{th}\) sensor [17].