## 4 Novel Proposed Method Application for Naturally-Developed Fault Diagnosis  
For the novel method application, experimental signals are collected from a lab fixed-axis gearbox test rig with naturally-developed bearing and gear faults. The dataset for testing high-speed (HS) shaft gear and bearings is from an end of life test case with an overall duration of 548 h. Detail end of life test case information can be found in [31]. The intermediate speed (IS) shaft and its gears and bearings and the low speed (LS) shaft and its gear and bearings are from an ended implanted test case. Wear-in duration for the IS and LS shaft-related components lasted 10 h with different load and speed combinations. Figure 7 illustrates a schematic of the experimental gearbox test rig. It includes a two-stage fixed-axis gearbox, a 4-kW three-phase asynchronous motor for driving the gearbox and a magnetic powder brake for loading. The motor speed can be adjusted for different values. The NI data acquisition system consists of four IPEE accelerometers (Dytran 3056B4--Dytran Instruments Inc., Fraser, MN, USA), PXI-1031 mainframe, PXI-4472B data acquisition cards and LabVIEW software (National Instrument Inc., Austin, TX, USA). In addition, a tachometer and torque sensor is installed in the input shaft for acquiring the speed and load information. The internal structure of the gearbox is depicted in Figure 7. The gearbox has three shafts supported by rolling element bearings. The LS shaft gear has 81 teeth and meshes with the IS shaft gear with 18 teeth. The IS shaft gear with 64 teeth meshes with the HS shaft with 35 teeth. Therefore, the overall gear ratio of the gearbox equals 8.22:1. In this case, study datasets with a 20-Hz input shaft rate, a 20-kHz sampling frequency, a 12-s sampling duration, a 199-Nm and 405-Nm loads are used. After the test, the gearbox was dismounted in order to check the bearings' condition. It was found that all bearings have faults at different levels. However, only the HS shaft and LS shaft bearings have obvious faults. Both HS shaft bearings had the outer race fault shown in Figure 8a,b, and the right bearing in addition had the inner race fault and ball fault shown in Figure 9a,b, respectively. Faults of the IS shaft bearing were not obvious faults. The left LS shaft bearing had the inner race fault and ball fault shown in Figure 10a,b, respectively. The HS and IS shaft are supported by SKF 6205 bearings and the LS shaft by SKF 6208 bearings. The fundamental faults frequencies of SKF 6205 under a 1-Hz rotating speed are BPFO (3.585 Hz), BPFI (5.415 Hz), BSF (2.357 Hz) and FTF (0.398 Hz). For SKF 6208, they are BPFO (3.578 Hz), BPFI (5.423 Hz), BSF (2.337 Hz) and FTF (0.398 Hz). Therefore, the actual fault frequencies can be acquired through these fundamental frequencies multiplied by the rotating frequency. The bearing types and the characteristic frequencies for a 20-Hz input shaft rate are concerned and summarized in Table 1.  
Figure 7: A schematic of the experimental gearbox test rig. It is very difficult to diagnose the bearing fault of the LS shaft because of the low speed and requirement of special techniques. In this case, only HS bearing faults under loads of 199 Nm and 405 Nm are considered. Except the bearing faults, the HS shaft gear with 35 teeth had a slight tooth wear fault. The IS shaft gear with 64 teeth had a chipped tooth fault, and the IS shaft gear with 18 teeth had a severe tooth wear fault shown in Figure 11. For a gear transmission with fixed axes, the gear \begin{table}
\begin{tabular}{c c c c c} \hline
**Bearing Type** & **BPFO** & **BPFI** & **BSF** & **FTF** \\ \hline
6205 & 71.70 & 108.30 & 47.17 & 7.96 \\
6208 & 8.70 & 13.19 & 5.68 & 0.96 \\ \hline \end{tabular}
\end{table} 
Table 1: Characteristic frequencies of bearings (Hz). BPFO: ball pass frequency outer race; BPFI: ball pass frequency inner race; BSF: ball spin frequency; FTF: fundamental train frequency. Figure 8: High speed (HS) shaft bearing outer race faults: (**a**) left bearing; (**b**) right bearing.