Smoothness index-guided Bayesian inference for determining joint posterior probability distributions of anti-symmetric real Laplace wavelet parameters for identification of different bearing faults

• Anti-symmetric real Laplace wavelet is analyzed and chosen as a wavelet function.
• A smoothness index based state space model is built.
• Minimization of the smoothness index leads to the optimal wavelet parameters.
• Posterior probability distributions of wavelet parameters are graphically displayed.
• A Bayesian inference method is proposed for bearing fault diagnosis.

Rolling element bearings are one of the most common components used in machines and they are used to support rotating shafts. Unexpected bearing failures may cause machine breakdown which results in economic loss. Detection of bearing faults is crucial to prevent bearing failures. Vibration based signal processing methods have been proven to be effective in identifying different bearing faults. Among different signal processing methods, wavelet analysis is widely investigated because it is able to highlight the similarity between wavelet functions with different wavelet parameters and impulses caused by bearing faults. In wavelet analysis, two topics are of great concern. The first is how to choose a suitable wavelet mother function for bearing fault diagnosis. In recent studies, an anti-symmetric real Laplace wavelet or impulse response wavelet has been experimentally proven to have a high similarity with impulses caused by bearing faults. Therefore, anti-symmetric real Laplace wavelet is chosen as the wavelet mother function in this paper. The second is how to determine the optimal wavelet parameters so as to enhance the ability of wavelet analysis. Based on the anti-symmetric real Laplace wavelet, smoothness index based Bayesian inference is proposed in this paper to establish joint posterior probability density functions of wavelet parameters, which reflect graphical relationships between wavelet parameters. The smoothness index is chosen because it is not only able to quantify bearing fault signals, but also has upper and lower bounds, compared with other metrics, such as wavelet entropy, Shannon entropy, kurtosis, sparsity measurement, etc. For Bayesian inference, a general particle filter is adopted to iteratively calculate and update joint posterior probability density functions of wavelet parameters. Once the joint posterior probability density functions of wavelet parameters are available, the optimal wavelet parameters are determined and an optimal wavelet filtering is conducted to extract bearing fault signatures. Real case studies are investigated to illustrate how the proposed method works. The results show that the proposed method can determine joint posterior probability density functions of wavelet parameters and is effective in identifying different bearing faults.