Each condition has 0 Nm and 15 Nm loads. Both torque and speed curves depict the steadiness of the power (proportional to the product of torque and speed) flowing through the system by generally showing opposite trends to each other. The variation in those curves over the period of three seconds confirms the randomness of the signal, hence categorizing it as a nonstationary signal. The FFT would assume the same signal as stationary, i.e., the signal not varying with time. This assumption ignores the local irregularity or fractals, and hence potentially any evidence of the fault existence. When the signal of one second period was FFT processed and plotted against the RMS amplitude in m/s2 (refer to Figures 6 and 8), the aforementioned cycloidal mesh frequencies can be observed.
A typical way to diagnose any faulty machine using vibration is to compare its pre- and post-incident data, which means comparing the healthy (Figure 6) and the damaged reducer (Figure 8) graphs. However, apart from observing the higher amplitudes across the spectrum, it was incomprehensible to establish any relation with the fault by utilizing known frequencies (highlighted in graphs). Even at 16× order (disc-pin mesh frequency), where the fault was induced, no noticeable relative data change was observed. The same time period (one second) vibration data was then analyzed with WT, by plotting the Scalogram shown in Figure 9. As discussed in the “Introduction” section, the Scalogram or “Frequency vs Time” graph generally reveals the dominant frequencies at a time instant inside the non-stationary signal. Figure 9 shows a burst in localized activities, confirming the existence of mono- or multifractality in the signal generated by the damaged gearbox. Equations 4–8 were used to perform Multifractal analysis to extract wavelet coefficients that built the Scalogram, to derive the multifractal spectrum. The multifractal spectrum is plotted in Figure 10 for all three axes of vibrations: vertical, horizontal, and axial. The axis diagram, referred to in Figure 1, has the reducer shaft axis aligned with the vibration axial direction, and the remaining axes are in the radial direction of the cycloidal disc.
Each graph in Figure 10 consists of four curves showing different operating conditions of the gearbox: a healthy reducer with no load, a healthy reducer with 15 Nm load, a damaged reducer with no load, and a damaged reducer with 15 Nm load. Based on their positioning on the graph, the curves can visibly be sorted into healthy and damaged groups (green color and red color, respectively) in Figure 10(A) and Figure 10(B). The damaged gearbox’s curves are shifted towards the right side of the multifractal spectrum, making them separated from the healthy ones. It can be argued that the main cause of this distinction was the induced damage on the cycloidal disc. The worn-out surface on the disc generated singularities affecting the wavelet coefficients, subsequently creating a deviation in the multifractal spectrum in radial directions. The axial axis graph, Figure 10(C), does not show similar shifting, as the worn-out lobe of the disc would not have energy change in this direction. The loading conditions, 0 Nm and 15 Nm, do not noticeably influence the shifting in all three graphs.
The paper compares results of the two tests conducted on the same gearbox. The only difference between the two test setups was the condition of Cycloidal disc, depicting ‘healthy’ in one and ‘faulty’ in another. The multifractal spectrum clearly distinguished the vibration data with respect to their conditions, which Fast Fourier Transform graphs could not. This paper argues that the wavelet transform based multifractal analysis approach makes the Cycloidal gearbox diagnosis less ambiguous compared to FFT. This is mainly because the collected vibration signals were time dependent and the singularities in the signal cannot be captured using FFT. The multifractal spectra also showed that the load on the gearbox output shaft does not significantly influence the curve shifting or the H?lder coefficient. For the future work, it is worth exploring the ‘worn-out’ effect of other Cycloidal gearbox components, such ring gear pins, output rollers, on the multifractal spectrum and establish the changes to be utilized in fault diagnostics.
