## 5 Application in Order Tracking for Fault Diagnosis of Gearbox  
As the rotational speed can be well estimated based on the proposed GAD method, it is further applied in order tracking for a gearbox with a local fault under non-stationary conditions. A gearbox test setup with a spalling fault under non-stationary conditions \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Items**} & \multicolumn{4}{c}{**Preset Rotation Speed**} \\ \cline{2-5}  & **310 r/min** & **410 r/min** & **520 r/min** & **620 r/min** \\ \hline Encoder Z & 307.90–311.22 & 412.35–412.78 & 519.97–520.72 & 621.95–622.55 \\ Estimated average speed & 309.47–313.15 & 408.43–416.12 & 516.14–524.58 & 617.36–627.47 \\
**Error** & \(\mathbf{-0.74}\)****–\(\mathbf{0.45}\)**\%** & \(\mathbf{-0.83}\)**–\(\mathbf{1.02\%}\)** & \(\mathbf{-0.92}\)**–\(\mathbf{0.82\%}\)** & \(\mathbf{-0.81}\)**–\(\mathbf{0.79\%}\)** \\ Encoder A & 306.23–313.84 & 406.74–417.63 & 512.80–527.14 & 613.22–630.44 \\ Estimated instantaneous speed & 303.20–318.64 & 401.39–426.50 & 500.25–537.17 & 605.45–636.76 \\
**Error** & \(\mathbf{-3.61}\)**–\(\mathbf{3.63\%}\)** & \(\mathbf{-4.85}\)**–\(\mathbf{3.72\%}\)** & \(\mathbf{-3.55}\)**–\(\mathbf{3.59\%}\)** & \(\mathbf{-3.80}\)**–\(\mathbf{3.15\%}\)** \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The errors of the estimated rotational speed based on the GAD method under stable working conditions. Figure 14: The estimation errors based on the proposed method: (**a**) the estimated average speed and (**b**) the estimated instantaneous speed. is constructed, including a two-stage gearbox, a driven motor, a magnetic powder brake and several couplings and control units, as shown in Figure 15. The teeth numbers of the two-stage gearbox are 23/39 and 25/53, respectively. The photograph of the wireless in situ acceleration test system is shown in Figure 16: a sensor board containing four MEMS accelerometers in the tangential direction is mounted at one end of the gear, and a ESP8266 chip is used for wireless transmission of the measured signals. Furthermore, an ICP accelerometer is mounted at the bearing housing; both the vibration signals from MEMS and the ICP sensor are acquired synchronously. The fault gear with the MEMS accelerometer is mounted at the second stage of the gearbox. The detailed design parameters of the gear pair are listed in Table 2.  
The seeded spalling fault is shown in Figure 17, where \(w_{s}/d_{s}/l_{s}\) represent the width/depth/ length, respectively. In this study, the width and depth are set as 3 mm and 1 mm. The length of the spalling fault is set as 15 mm (small fault), 30 mm (medium fault) and 60 mm (large fault), the seeded spalling fault is highlighted in Figure 17 by a red dashed frame. A total of six datasets are acquired, corresponding to two distinct load conditions of 20 Nm and 40 Nm, with a speed ranging from 500 r/min to 900 r/min for each type of spalling fault condition. These datasets are summarized in Table 3. The sampling frequency is \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Value** & **Parameter** & **Value** \\ \hline Module (mm) & 3 & Pressure angle (\({}^{\circ}\)) & 20 \\ Tip clearance (mm) & 0.25 & Addendum (mm) & 1 \\ Tooth width (mm) & 60 & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The parameters of the gears. Figure 16: A photograph of the wireless in situ acceleration test system. Figure 15: (a) A schematic illustration of the test system and (b) photograph of the test system. 5120 Hz and the time length is about 6 s. The signal acquired from in situ accelerometer #1 is used for analysis. In this study, the #1 dataset is analyzed as an example to show the effectiveness of the proposed method. The time domain waveform of the sped-up data is shown in Figure 18. An obvious increasing trend in the amplitude is observed as the rotation speed grows. In the zoomed-in plot, obvious sine waveforms are found.