## 4 Technology Validation Via Simulation for Fault Diagnosis of Gear Fitting  
The mathematical model, used to simulate gear vibrations under undamaged and damaged conditions is a lumped mass model, which considers spur gears as rigid disks, coupled along the line of action through a time varying mesh stiffness, as shown in Figure 10[31]. Number of teeth is equal to 31 for a pinion and 39 for a gear. Vibration data for each case are sampled using a sampling frequency of 80 kHz and provided recordings of 25 s duration. The speed of the gear shaft is equal to 307.7 rpm (5.1 Hz). Two sets of simulations were performed: the first set contained no tooth damage, and the second set contained 15% relative pitting size, which is a ratio between the damaged and total tooth areas; pitting is distributed equally among all 39 teeth. The pitting level is simulated by reducing the tooth meshing stiffness. Authors in [31] have developed an accurate rotation speed estimation method from the gear mesh harmonics appearing in the vibration signal. To extract the instantaneous speed, a narrow-band filter is centered at the mesh frequency. The instantaneous speed profiles are used for angular resampling procedure to transform vibration signals from time into the angular domain. The angular resampled vibration signals were time-synchronous averaged to isolate vibrations, related to a single gear, from another mechanical component. The main parameter of the TSA is the number of rotational segments to be averaged, which is defined via minimization of the TSA variance. For this purpose, a TSA convergence analysis is performed, which revealed a reliable TSA estimate, based on 20 segments. From the entire vibration signal a total of 127 shaft rotation segments were obtained. Since the TSA is calculated from 20 consecutive segments and s performed without overlapping, a total of 6 TSA signals were obtained for further analysis. The spectral kurtosis (SK) is estimated for 6 consequent segments of the gear residual signal obtained after removing mesh harmonics from the TSA signal. Figure 11 shows SK segments for (a) fault-free and (b) damaged cases, which is estimated using a Hamming window of size equal to half mesh period. In the damaged case, the SK shows values above 3 within the frequency band of 0 Hz to 6 kHz compared to undamaged case, which are related to the resonant frequencies of the structure and should therefore be considered for filtering. It is also evident, that in the fault-free case, the SK acquired values above 0.5 within 0 Hz to 9 kHz, which indicates weak non-stationary behavior even in the case of no damage. The SK threshold for filtering is set to 1. Figure 12 shows the diagnostic features for all 39 teeth and 6 segments. The feature values for the damaged case (b) are significantly higher compared to the features from undamaged case (a) and have a mean feature value equal to \(6.2\times 10^{-3}\) compared to the undamaged case mean feature value of \(10^{-6}\). Several orders of magnitude higher feature values confirm good separation between the undamaged and damaged cases. Figure 10: Dynamic model of spur gear pair [31]. Figure 11: Spectral kurtosis for consequent gear vibration segments. combination with a threshold 3. In this case, the FC values mostly remain below 0.5. The reason for a poor performance with half mesh period and 3 SK threshold is related to a SK threshold: if a threshold being set too high, potentially above SK amplitude values for a damaged case, then the estimated frequency bands based on the SK values remaining above the threshold will be narrow and reduce to zero when the threshold and SK peak values are the same. Figure 14 shows results obtained by kNN classification of the diagnostic features, extracted using the SK window of half mesh period and filtering threshold of \(1\). For supervised training, a set of diagnostic features, extracted from the fault-free case is used for k-means clustering.