## 3 Modeling and Signal Analysis Method
### Joint Time-Frequency Method based on EEMD and WVD  
In order to investigate the simulated axle-box acceleration signals in the time-frequency domain, an efficient algorithm based on EEMD and WVD is proposed: by decomposing a noise-assisted multi-component signal into a number of intrinsic mode functions (IMFs), and then analyzing by using the Wigner-Ville transformation, one cannot only suppress cross-term interference but also exploit the advantages of the WVD. Figure 3 shows a flowchart describing the methodology to identify wheel polygons. First, to remove additive noise or unwanted signals from the raw vibration signals, adaptive noise cancellation [22] is \begin{table}
\begin{tabular}{c c c} \hline \hline
**Component** & **Parameter** & **Value** \\ \hline \multirow{4}{*}{Carbody} & Mass (kg) & 4.0\(\times\) 10\({}^{4}\) \\  & Moments of pitch inertia (kg\(\cdot\)m\({}^{2}\)) & 2.0\(\times\) 10\({}^{6}\) \\  & Moments of roll inertia (kg\(\cdot\)m\({}^{2}\)) & 1.0\(\times\) 10\({}^{5}\) \\  & Moments of yaw inertia (kg\(\cdot\)m\({}^{2}\)) & 2.0\(\times\) 10\({}^{6}\) \\ \hline \multirow{4}{*}{Bogie} & Mass (kg) & 2.0\(\times\) 10\({}^{3}\) \\  & Moments of pitch inertia (kg\(\cdot\)m\({}^{2}\)) & 2.5\(\times\) 10\({}^{3}\) \\  & Moments of roll inertia (kg\(\cdot\)m\({}^{2}\)) & 1.5\(\times\) 10\({}^{3}\) \\  & Moments of yaw inertia (kg\(\cdot\)m\({}^{2}\)) & 3.5\(\times\) 10\({}^{3}\) \\ \hline \multirow{4}{*}{Wheelset} & Mass (kg) & 1.5\(\times\) 10\({}^{3}\) \\  & Moments of pitch inertia (kg\(\cdot\)m\({}^{2}\)) & 120 \\  & Moments of roll inertia (kg\(\cdot\)m\({}^{2}\)) & 800 \\  & Moments of yaw inertia (kg\(\cdot\)m\({}^{2}\)) & 800 \\  & nominal rolling radius (m) & 0.46 \\ \hline \multirow{4}{*}{Axlebox} & Mass (kg) & 50.0 \\  & Moments of pitch inertia (kg\(\cdot\)m\({}^{2}\)) & 5.0 \\  & Moments of roll inertia (kg\(\cdot\)m\({}^{2}\)) & 1.0 \\  & Moments of yaw inertia (kg\(\cdot\)m\({}^{2}\)) & 5.0 \\ \hline \multirow{4}{*}{CHN60 rail} & Elastic modulus (N\(\cdot\)m\({}^{-2}\)) & 2.1\(\times\) 10\({}^{11}\) \\  & Poisson ratio & 0.3 \\  & Density(kg\(\cdot\)m\({}^{-3}\)) & 7800 \\ \hline \multirow{4}{*}{Rail fastenings} & Elastic stiffness(MN\(\cdot\)m\({}^{-1}\)) & 50 \\  & Damping coefficient (kN\(\cdot\)s\(\cdot\)m\({}^{-1}\)) & 60 \\  & Longitudinal spacing(m) & 0.65 \\ \hline \multirow{4}{*}{Concrete slab layer} & Elastic modulus(N\(\cdot\)m\({}^{-2}\)) & 3.9\(\times\)10\({}^{10}\) \\  & Poisson ratio & 0.2 \\ \cline{1-1}  & Length \(\times\) width \(\times\) thickness (m) & 6.45 \(\times\) 2.55 \(\times\) 0.20 \\ \cline{1-1}  & Density(kg\(\cdot\)m\({}^{-3}\)) & 2500 \\ \hline \multirow{4}{*}{CA mortar layer} & Elastic modulus(N\(\cdot\)m\({}^{-2}\)) & 7.0\(\times\)10\({}^{9}\) \\ \cline{1-1}  & Poisson ratio & 0.167 \\ \cline{1-1}  & Thickness(m) & 0.03 \\ \cline{1-1}  & Density(kg\(\cdot\)m\({}^{-3}\)) & 2590 \\ \hline \multirow{4}{*}{Foundation} & Elastic modulus(N\(\cdot\)m\({}^{-2}\)) & 5.0\(\times\)10\({}^{9}\) \\ \cline{1-1}  & Poisson ratio & 0.2 \\ \cline{1-1}  & Width \(\times\) thickness(m) & 3.25 \(\times\) 0.3 \\ \cline{1-1}  & Density(kg\(\cdot\)m\({}^{-3}\)) & 2500 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of vehicle and track systems. used to pre-process the raw signal. Vibration signals caused by rail irregularities are then filtered and noise is reduced by using an adaptive filter. Figure 3: Flow chart showing algorithm for joint time-frequency analysis based on Ensemble Empirical Mode Decomposition (EEMD) and Wigner-Ville distribution (WVD). Next, add Gaussian white noise to the original vibration signal \(x(t)\) and decompose the conditioned signal by EEMD into a series of IMFs and a residual:  
\[x(t)=\sum_{i=1}^{n}c_{i}(t)+r_{n}(t) \tag{12}\]  
where \(c_{i}(t)\) is an IMF, \(i\) is the number of modes, and \(r_{n}(t)\) is the corresponding residual.