## 2 Multisensor Array Layout Scheme Based on the Spatial Distribution Characteristics of Rail Strain  
### Establishment of Simulation Model  
#### 2.1.1 Wheel-Rail Force Simulation Model  
Multibody dynamics (MBD) is the science that studies the laws of motion for a multibody system, which is generally composed of several flexible or rigid objects connected to each other. MBD simulation has been widely used in the design, homologation test, and research of railway transportation [26]. The MBD software, Simpack, was used to calculate the wheel-rail forces under different wheel flat conditions. As is shown in Figure 1, the vehicle model had 38 degrees of freedom and consisted of a vehicle body, two bogies, and four wheelsets. The modeling parameters are shown in Table 1.  
In actual conditions, the rails are not perfect, which also affects the wheel-rail forces. Therefore, we used a sample of vertical track irregularity measured on a subway in Nanjing, China as the initial excitation of the wheel-rail contact. As shown in Figure 2, the total length of the sample is 700 m and the interval is 0.25 m.  
Figure 1: Vehicle-track system dynamic model. Figure 2: Track irregularity sample. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Type** & **Parameter** & **Value** & **Unit** \\ \hline \multirow{3}{*}{Mass} & Vehicle body & 33,593 & kg \\  & Bogie & 3730 & kg \\  & Wheel & 1730 & kg \\ \hline \multirow{3}{*}{Moment of inertia} & Vehicle body (around x/y/z axis) & 50,000/1,101,000/1,099,000 & kg\(\cdot\)m\({}^{2}\) \\  & Bogie (around x/y/z axis) & 1194/876/2099 & kg\(\cdot\)m\({}^{2}\) \\  & Wheel (around x/y/z axis) & 1042/137/1042 & kg\(\cdot\)m\({}^{2}\) \\ \hline \multirow{3}{*}{Stiffness} & Primary suspension & \multirow{3}{*}{\(17.62\times 10^{6}/9.62\times 10^{6}/5.35\times 10^{6}\)} & N/m \\  & (longitudinal/lateral/vertical) & & \\  & Secondary suspension & & \\  & (longitudinal/lateral/vertical) & & \\ \hline \multirow{3}{*}{Damping} & Secondary suspension (lateral) & 5000 & N-s/m \\  & Secondary suspension (vertical) & 5000 & N-s/m \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the vehicle model. The wheel flats are simulated by defining the deviation of wheel radius. As shown in Figure 3, the equation of the undamaged wheel (plotted in blue) is expressed as follows:  
\[z^{2}+y^{2}={R_{0}}^{2} \tag{1}\]  
where \(z\) and \(y\) are the longitudinal and vertical position of a certain point relative to the center of the wheel and \(R_{0}\) is the nominal radius of a perfect wheel. According to the generation mechanism, a wheel with flats is approximately equivalent to cutting a plane from a certain position on the wheel tread [27]. Therefore, the equation of the flat (plotted in red) can be expressed as follows:  
\[y=R_{0}-h \tag{2}\]  
where \(h\) is the depth of the wheel flat. Convert the above equations into polar coordinates, thus the deviation of wheel radius of a damaged wheel can be expressed as follows:  
\[\Delta R(\varphi)=\left\{ \begin{array}{c}0,\ \varphi\geq\frac{\alpha_{R}}{2} \ \text{or}\ \varphi\leq-\frac{\alpha_{R}}{2}\\ R_{0}-\left|\frac{R_{0}-h}{\cos(\varphi)}\right|,-\frac{\alpha_{R}}{2}<\varphi< \frac{\alpha_{R}}{2}\end{array} \right. \tag{3}\]  
where \(\varphi\) is the angle between the radial line and the vertical line through the center of the wheel, \(\alpha_{R}\) is central angle corresponding to the wheel flat region and equals \(2\times\cos^{-1}[(R_{0}-h)/R_{0}]\). By defining the radius deviation of wheels according to Equation (3), the geometric information of the wheel flats is input into the Simpack model. In this section, we have set nine damage conditions of different flat depths from 0 to 0.5 mm. Only one of the eight wheels has wheel flat and the vehicle runs at a speed of 20 m/s with a sampling frequency of 4 kHz. The calculated wheel-rail forces of different damage conditions are shown in Figure 4. The occurrence of wheel flats induces intermittent peaks to the load history and the amplitude increases as the depth grows.