## 4 Research Methods
### Symmetrized Dot Pattern (SDP)  
The SDP, as an algorithm, can map the time domain waveform to the symmetric space of points. The 2D scatter diagram is drawn in polar coordinates. The SDP can graphically describe the signal amplitude and frequency variation of time domain signals. The SDP calculates time domain signal data to find the positions of data in the polar coordinate space. The mirror symmetry image is drawn to display the variation between different vibration signals.  
Even when there were subtle signal changes, there would be significant differences in the symmetric pattern of output. Figure 9 shows the image of normal gearbox vibration time-domain signals. The appearance has snowflake-like sextuple symmetry (also known as snowflake diagrams). The differences among different vibration signals can be visualized, and the SDP can process noise signals effectively. Figure 9 is the snowflake diagram of a normal gearbox.  
The time-domain vibration signals of the gearbox were captured by the capturing system and transformed into a 2D feature image by SDP. Figure 10 shows the schematic diagram of SDP. The time-domain vibration signals of the gearbox were transformed in  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters Name** & **Parameters** \\ \hline low-pass cutoff frequency & 15,175 Hz \\ High-pass cutoff frequency & 63 Hz \\ center frequency & 7619 Hz \\ bandwidth & 15,112 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Bandpass filter parameters.  
Figure 8: Rusty gearbox vibration signal spectrum analysis: (**a**) unfiltered spectrum; (**b**) filtered spectrum.  
Figure 9: Vibration signals and snowflake diagram of a normal gearbox: (**a**) vibration signals; (**b**) snowflake diagram.  
the polar coordinate space of point \(P(\psi(i),\partial(i),\rho(i))\), where \(\psi(i)\) is the polar coordinate radius of the pattern; \(\rho(i)\) is the initial angle line of clockwise rotation; and \(\partial(i)\) is the initial counterclockwise rotation angle line, as expressed by Equations (2)-(4).  
The gearbox time-domain vibration signal is \(VIB_{x}=\{x_{1},x_{2},x_{3},\ldots,x_{i}\}\), where \(x_{i}\) is No. \(i\) sampling point in vibration time-domain signal \(VIB_{x}\), and \(x_{i+\Delta T}\) is No. \(i+\Delta T\) sampling point of vibration time-domain signal after interval time \(\Delta T\). The \(x_{i}\) is substituted in Equation (2) to obtain the radius of \(x_{t}\) from the image center in polar coordinates. The \(x_{i+\Delta T}\) is substituted in Equations (3) and (4). \(\partial(i)\) is the initial angle line of the clockwise rotation of \(x_{i+\Delta T}\) in polar coordinates, and \(\rho(i)\) is the initial counterclockwise rotation angle line of \(x_{i+\Delta T}\) in polar coordinates.  
\[\psi(i)=\frac{x_{i}-x_{min}}{x_{max}-x_{min}} \tag{2}\]  
\[\partial(i)=\beta-\frac{x_{i+\Delta T}-x_{min}}{x_{max}-x_{min}}\,J \tag{3}\]  
\[\rho(i)=\beta+\frac{x_{i+\Delta T}-x_{min}}{x_{max}-x_{min}}\,J \tag{4}\]  
In the vibration time-domain signals of the wind turbine gearbox, if the vibration value at time \(t\) is \(x_{t}\), and the vibration value at time \(t+\Delta T\) is \(x_{t+\Delta T}\). \(x_{t}\) and \(x_{t+\Delta T}\) were substituted in Equations (2)-(4), respectively, and transformed into the points in the polar coordinate space. As a result, the snowflake feature map based on vibration signals can be generated by changing the rotation angle.  
Wherein \(x_{min}\) and \(x_{max}\) are the minimum and maximum values of vibration signals, respectively. \(\Delta T\) is the interval time, and the range is 1 to 10. \(\beta\) is the initial rotation angle. \(J\) is the magnification of the rotation angle, generally smaller than the initial rotation angle \(\beta\). According to the test results, when \(\beta=60^{\circ},\ \Delta T=3,\ J=30\), the formed image was the optimal wind turbine gearbox vibration feature extraction map.