## 4 Method Validations Using Simulations and Experiments
### Simulation Signal Analysis  
Health and planet fault simulation signals were generated using the dynamic model described in Section 2. Planetary gearboxes under health condition and planet gear fault condition were simulated respectively. During the simulation, initial phase differences between planet gears and sensor were 0.51, 2.08, 3.65, and 5.22 radians, rotating frequency of the carrier is 10 Hz, and simulation duration is 20 s to generate sufficient simulation for analysis. Other parameters that used during simulation are same as those in Table 1. Shown in Figure 10 are vibration signal and tachometer signal (tachometer is located at the carrier). Because of heavy noises, it is hard to observe modulation effect caused by planetary movements directly in Figure 10. The \(SNR\) (signal noise ratio) of the simulation signal is \(-6\) dB. According to \(f_{c}\) and \(f_{s}\), there are 2560 data points in each carrier cycle. To make sure that data points collected during a cycle is an integer multiple of ring gear tooth number, a resampling rate of 2592 is used during angular resampling, which helps minimize the interpolation error since 2592 is the nearest multiple of \(N_{r}\) to 2560. Thus, DPT (data per tooth) equals to 27. Correspondingly, extraction index in each cycle are 210, 858, 1506, and 2154 according to  
\[N_{n}^{*}=\text{round}(\frac{\psi_{n}}{\Delta\theta}), \tag{19}\]  
where \(\Delta\theta=2\pi/L\), which equals to 0.0024 radians. \(L\) is the resampling rate. During data extraction, a window width of 2 gear tooth is used since \(N_{p}=34\) while P\({}_{\text{HTR}}=17\). Such a window ensures that extracted data covers all planet gear teeth while avoiding data segments overlapping. Thus, \(2\times\text{DPT}=54\) data points are extracted at every extraction index. Figure 11 shows how data are extracted and labeled with planet gear numbers. Hunting tooth ratio of planet P\({}_{\text{THR}}=17\), which means at every 17 carrier cycles, extracted data will correspond to same gear tooth. In this case, 200 cycles data are generated, correspondingly, 200 data pieces are extracted; thus, 11 Sets can be obtained by dividing every 17 data pieces into a group, and last 13 data pieces are ignored since signal of a full planet gear cycle cannot be obtained from them. Figure 10: Simulation signal waveform (vibration and tachometer). The next step is data mapping. In this step, planet gear meshing regularity is used to guarantee that data segments are mapped to the corresponding location in AHM. AHM is a matrix that has DPT \(\times N_{p}\) rows and \(P_{\text{HTR}}\) columns, and in each column, data piece extracted is mapped to corresponding position with help of tooth index. Taking planet gear P1, e.g., the planet gear tooth that meshes at the sensor in first carrier cycle is 8, which is shown in Table 2. Accordingly, the first data segments in each group should be mapped to tooth 8 in AHM. In second carrier cycle, the planet gear tooth that meshes at the sensor is 2; thus, the second data segments should be mapped to tooth 2 in AHM. Shown in Figure 12 is an AHM of P1. Data segments are mapped to their corresponding gear tooth number. Summing of the AHM will be planet gear dynamic responses in a planet gear cycle, which is shown as black waveform at the right side of the AHM in Figure 12.  
By lining up rebuilt dynamic responses, combination signal can be obtained. Dynamic responses of planet gear P1 can be obtained after applying TSMA algorithm. Lastly, MED filter is used for enhancement detection of fault-induced periodic impulses within the CVS signal. Shown in Figure 13 is raw simulation signal and its envelope order spectrum. Envelope spectrum is a widely used impulse period detection method, and it has long been proven effective in fault diagnosis studies. For better comparison, frequency axis of spectrum is normalized using planet  
Figure 11: Data extraction in cycle 1–3. Figure 12: An example of Assembly Holding Matrix (AHM). gear fault character frequency.