Wavelet Transform Multifractal Analysis

Multiple research papers have demonstrated the use of WT-based multifractal analysis to diagnose faults in rotary machinery. In the case of ball bearings, when multiple faults with different diameters and depths are present on both inner and outer races, the time/ frequency spectrum could distinguish a very few of them (Ref. 9). The paper focuses on comparing a variety of induced faults in ball bearings using the WT modulus maxima method. It shows that different fault conditions would carry the fault information in the form of a distribution of singularities. The multifractal spectrum and scaling exponents can be clearly distinguished. One interesting application of multifractal analysis is in material testing. To identify progression of the failure in a structural composite material, the multifractal spectrum and fractal leaders could be deployed to observe nonstationary changes in the vibration signal (Ref. 10). In this experiment, vibratory signals generated during a composite bending test were recorded. The test progressed in three stages until the failure of the specimen. Each stage generated fractals that were plotted on a singularity spectrum to differentiate from each other. The holder coefficients increase with increased bending and thus would help to indicate the initial or final stage of the failure. The gear fault analyses were performed on spur gear pairs to demonstrate the use of WT (Refs. 11,12). For these analyses, multiple gear pairs showing different stages of the bending fatigue failure progression were mounted on test rigs, and vibration samples were analyzed. For instance, Ref. 11 shows how the distribution of the scaling exponents changes from a healthy gear with no cracks to the fatigue crack magnitude of 15 percent, 50 percent, and 75 percent of the gear tooth root, and subsequently relates to tooth loss. Such progression of the failure would not be easy to distinguish by observing FFT and the power spectral density graphs. These studies deploy fault detection algorithms to perform gear diagnostics. Along with gear fault diagnosis, the WT method has also been used to simulate gear misalignment due to assembly error and variation in clearance (Ref. 13). These faults, when recorded in separate tests and plotted on the multifractal spectra, show differences in the singularity distributions. Singularity subsets, called cumulants, were fed to machine learning methods, such as neural network (NN), K-NN, and support vector machine (SVM), to classify the gearbox state. In the case of Cycloidal gearbox fault diagnosis, the benefits of the WT are demonstrated by comparing the vibration data gathered from the faulty and non-faulty Cyclo reducers (Ref. 14). The paper argues that the conventional FFT falls short in distinguishing these two reducers. To create a fault in the gearbox, two rotary components, in this case ring gear housing pins, were removed (refer to Figure 1). Plots of multifractal spectra and two-cumulant dispersions show clear separations of faulty and non-faulty data. The removal of the components may represent an assembly error. However, it does not specifically depict flaws such as component fracture, wear, or manufacturing error (clearance due to inadequate tolerance). Another paper on cycloidal fault diagnosis uses actual worn-out parts, including the disc and eccentric bearing, but only considers FFT in vibration analysis (Ref. 15). The current paper intends to extend the aforementioned analyses and perform WT multifractal analysis on a simulated worn-out cycloidal disc. The objective of this paper is to diagnose a cycloidal gearbox by utilizing discrete wavelet transform analysis and compare the findings with FFT.

A signal X(t) in 1d can be decomposed with WT by using a known function or mother wavelet, }0(t), into scales that correspond to frequencies. When plotted against the time shift on the X-axis, a time-frequency graph can be achieved (refer to Figure 2B). In this work, discrete wavelet transform (DWT) is utilized as it is computationally more efficient than continuous wavelet transform (CWT).

The coefficients are then further used to derive Wavelet Leaders L(j, k) in the multifractal analysis, as they are the largest coefficients in a certain time neighborhood (local supremum) and possess significant qualities to construct a multifractal formalism (Ref. 16).

Wavelet leaders multifractal formalism (WLMF) in plotting the multifractal spectrum and log cumulants.

Here, when the cumulant c2=0, the signal is monofractal and has the same scaling in entire data. when c2≠0, the signal is multifractal.

The local regularities are described by Hölder exponent h, and the multifractal spectrum D(h) represents the distribution of singularities in the signal.