## 3 Fault Diagnosis Method
### Improved AdaBoost Algorithm
To be exact, they are the vibration data at rotating speed of 2 Hz, 4 Hz, 6 Hz, 8 Hz of the bogie rolling bearing in the running department of the rail train at normal, inner fault, outer fault, and rolling fault, respectively, and the sampling frequency is 51,200 Hz, the value is vibration acceleration, the unit is m/s\({}^{2}\). The number of samples for various sample set is 80. To increase the reliability of the experiment, the rolling bearing standard data set of the Western Reserve University is increased. The standard data set is Fan End (FE) accelerometer data and the sampling frequency is 12 kHz. The damage of the bearing is a single point of damage by electric discharge machining (EDM), and the diameter of the damage is 0.1778 mm. The data set has a rotating speed of 1796r/min, which is equivalent to 30 Hz, and the motor load is 0 hp. The following Tables 2 and 3 show the specific detailed information of experimental data. Figure 5: Typical failure of train rolling bearings. (**a**) experiment platform; (**b**) Inner fault; (**c**) Outer fault; (**d**) Rolling fault. The normal rolling bearing time-domain diagram and fault rolling bearing timing diagram are shown in Figure 6a,b. The amplitude of the vibration acceleration of the normal bearing, which has periodicity, fluctuates around the zero points. The upper and lower fluctuation range of the amplitude is small. Also, the amplitude has Root Mean Square (RMS), Kurtosis, and Energy characteristics. The RMS is the root mean square value of the vibration amplitude. The Kurtosis is the fourth-order central moment of normalized original data, which can effectively reflect the distribution characteristics of the vibration signal. When the amplitude vibration law changes due to a malfunction or other reasons, the value of this statistic is higher than the normal value. The Energy is the sum of the squares of the amplitudes of the data signals. After a component fails, it often causes an increase in the amplitude of the vibration, increasing the amount of energy obtained. Therefore, we can see from Figure 6 that the RMS, Kurtosis, and Energy of the amplitude of the normal bearing are smaller than that of the faulty bearing. \begin{table}
\begin{tabular}{c c c c} \hline
**Rolling Velocity** & **Category** & **Sampling Frequency** & **Number of Data Samples** \\ \hline \multirow{4}{*}{2 Hz} & Normal & 51,200 Hz & 80 \\  & Inner fault & 51,200 Hz & 80 \\  & Outer fault & 51,200 Hz & 80 \\  & Rolling fault & 51,200 Hz & 80 \\ \hline \multirow{4}{*}{4 Hz} & Normal & 51,200 Hz & 80 \\  & Inner fault & 51,200 Hz & 80 \\  & Outer fault & 51,200 Hz & 80 \\  & Rolling fault & 51,200 Hz & 80 \\ \hline \multirow{4}{*}{6 Hz} & Normal & 51,200 Hz & 80 \\  & Inner fault & 51,200 Hz & 80 \\  & Outer fault & 51,200 Hz & 80 \\  & Rolling fault & 51,200 Hz & 80 \\ \hline \multirow{4}{*}{8 Hz} & Normal & 51,200 Hz & 80 \\  & Inner fault & 51,200 Hz & 80 \\ \cline{1-1}  & Outer fault & 51,200 Hz & 80 \\ \cline{1-1}  & Rolling fault & 51,200 Hz & 80 \\ \hline \end{tabular}
\end{table} 
Table 2: The sample-set information on rail transit experimental data. Figure 6: Time-domain map of rail transit experimental data samples (4 Hz). (**a**) Time-domain diagram of Normal bearings; (**b**) Time-domain diagram of Outer fault. The normal bearing time-domain diagram and fault bearing timing diagram of the sample of the standard data set are shown in Figure 7.

Each type of sample, \(x_{i},i=1,2,\ldots,80\) (\(80\) is the number of samples), is converted to obtain the local frequency \(v_{ij}\) of all samples. \(i=1,2,\ldots,80,j=0,1,\ldots,m\), \(i\) represents the \(i\)-th sample, \(v_{ij}\) represents the \(j\)-th local frequency of the \(i\)-th sample, and m is the number of intervals of the local spectrum. Then, find the maximum value Max (\(v_{ij}\)) and the minimum value Min (\(v_{ij}\)) of the local frequency in the sample, and use Max (\(v_{ij}\)) and Min (\(v_{ij}\)) as the span of all local frequency.