Rolling element bearings are widely used in various machinery to support rotation shaft and they are prone to failures. Once a bearing fails, it accelerates failures of other adjacent components and results in unexpected machine breakdown. To prevent machine breakdown and reduce unnecessary economic loss, bearing fault must be detected as early as possible. Besides spectral kurtosis, empirical mode decomposition, cyclostationarity, etc., wavelet transform has proven to be an effective method for identification of different bearing faults because it aims to highlight the inner product between an artificial wavelet function and a signal to be analyzed. In the application of wavelet transform, optimization of wavelet parameters attracts much attention because proper selection of wavelet parameters can maximize performance of wavelet transform and extract impulses caused by bearing faults in the case of interruption from other strong low-frequency vibration components and heavy noises. Compared with other optimization methods, such as genetic algorithm, particle swarm optimization, etc., an analytic and fast Bayesian inference on optimal wavelet parameters for an optimal wavelet filtering for bearing fault diagnosis is proposed in this paper. Prior to Bayesian inference, a state space model of wavelet parameters should be constructed to reflect the relationship between wavelet parameters and measurements. Here, measurements are monotonically increasing kurtosis values, which are able to quantify bearing fault signals. The first kurtosis value and initial wavelet parameters are provided by the fast kurtogram, which is a fast algorithm that can be used to locate one of resonant frequency bands for further demodulation with envelope analysis. For other measurements, they are generated by artificial extrapolations of the first kurtosis value. To iteratively infer posterior probability density functions of wavelet parameters and track the artificial measurements, an unscented transform based Bayesian method is introduced. As the iteration number increases, posterior probability density functions of wavelet parameters converge. Then, the optimal wavelet parameters can be found to conduct an optimal wavelet filtering so as to isolate bearing fault signals from other strong low-frequency vibration components. At last, squared envelope analysis and Fourier transform are utilized to demodulate bearing fault signals enhanced by the proposed method and to identify bearing fault characteristic frequencies, respectively. One real case study is used to illustrate how the proposed method works and to demonstrate that the proposed method can be effectively and efficiently used to extract bearing fault signatures. Additionally, a comparison with the fast kurtogram is conducted to show the proposed method is better than the fast kurtogram for bearing fault diagnosis.
Assimilation of multiple linearly dependent data vectors
