# Fault Feature Extraction Using L-Kurtosis and Minimum Entropy-Based Signal Demodulation
## 4 Results and Discussion
While the proposed scheme effectively captures the overall gear meshing behavior, its ability to accurately detect and characterize the specific pitting fault remains limited in this experimental scenario. The absence of a clear fault signature suggests the need for further refinement of the scheme or exploration of alternative techniques for effective fault identification in such complex systems. The proposed method is then employed to analyze the measured signal. From Figure 5c, the maximum L-kurtosis and minimum entropy of the frequency domain are 4.32 and 1.11, respectively, for the second segment among all the segments. As a result, this band is used as the passband of the filter with a frequency ranging from 1585 Hz to 3282 Hz, as shown in Figure 5d. Applying the Hilbert transform to the filtered signal yields the frequency spectrum of the envelope signal plotted in Figure 5e, where the fault frequency (FF) of 1.25 Hz is clearly observable. Figure 5: Processing of the signal of worm gearbox. (**a**) Raw signal. (**b**) Envelope spectrum of raw signal. (**c**) Segmentation of the signal. (**d**) Reconstructed signal using band pass filter. (**e**) Envelope spectrum of the reconstructed signal. The robustness of the proposed scheme is further demonstrated through its application to a real-world scenario involving a three-stage helical gearbox. This gearbox features reduction ratios of 7, 3.2, and 3.1, resulting in a complex transmission system with multiple gear meshing frequencies. The schematic diagram of this gearbox is presented in Figure 6, whereas the detailing of the helical gearbox is tabulated in Table 2.  
To introduce a controlled fault condition, a wear tooth defect is deliberately seeded on the gear of the intermediate shaft, which possesses 41 teeth (as shown in Figure 6). Vibration data are acquired at a motor speed of 2100 rpm (35 Hz), providing a representative operating condition for analysis. The three stages of the gearbox lead to distinct gear meshing frequencies (GMFs) corresponding to the rotational frequencies of each shaft. These frequencies are calculated as 210 Hz (input shaft), 65 Hz (intermediate shaft), and 20 Hz (output shaft), based on the motor speed of 35 Hz and the gearbox reduction ratios. The acquired vibration signal from the helical gearbox is displayed in Figure 7a, while its envelope spectrum is presented in Figure 7b. Initial analysis of the envelope spectrum reveals the presence of the characteristic frequency of the motor shaft (35 Hz, the rotational frequency of the input shaft) and its harmonics. However, the expected gear meshing frequencies (210 Hz, 65 Hz, and 20 Hz) and the fault frequency related to the wear tooth defect are conspicuously absent from the spectrum. This initial observation highlights \begin{table}
\begin{tabular}{l l l} \hline \hline
**S. No.** & **Components** & **Specifications** \\ \hline
1. & Motor & DCmotor; power: 1 hp.; maximum speed: 2880 rpm. \\
2. & Helical gearbox & Three stages; helical gears; reduction ratio 7:3.2:3.1. \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Detail of helical gear test rig. Figure 6: A schematic of helical gearbox and a photograph of seeded wear tooth defect. the challenge of detecting subtle fault signatures amidst the complex vibration signals generated by the gearbox. To overcome this challenge, a segmentation strategy is implemented in the acquired signal. Segment one of the signal exhibits the highest L-kurtosis value and the lowest entropy, indicating potential significance in terms of fault information (as illustrated in Figure 7c). This segment, spanning a frequency range of 0 Hz to 5542 Hz, is selected as a passband filter for filtering the raw vibration signal (shown in Figure 7d). The filtered signal is then subjected to the Hilbert transform, and the envelope spectrum of the resulting signal reveals key frequencies (Figure 7e).