The automatic selection of an optimal wavelet filter and its enhancement by the new sparsogram for bearing fault detection

Highlights

• A new optimal wavelet filter based on genetic algorithm is designed.

• Optimal parameters of complex Morlet wavelet can be automatically determined.

• Convergence of optimal Morlet filter has been enhanced by a new sparsogram.

• Sparsity measurement value is maximized by genetic algorithm.

• A non-linear function is introduced to depress noise that is embedded in the inspected signals.

Abstract

Rolling element bearings are the most important components used in machinery. Bearing faults, once they have developed, quickly become severe and can result in fatal breakdowns. Envelope spectrum analysis is one effective approach to detect early bearing faults through the identification of bearing fault characteristic frequencies (BFCFs). To achieve this, it is necessary to find a band-pass filter to retain a resonant frequency band for the enhancement of weak bearing fault signatures. In Part 1 paper, the wavelet packet filters with fixed center frequencies and bandwidths used in a sparsogram may not cover a whole bearing resonant frequency band. Besides, a bearing resonant frequency band may be split into two adjacent imperfect orthogonal frequency bands, which reduce the bearing fault features. Considering the above two reasons, a sparsity measurement based optimal wavelet filter is required to be designed for providing more flexible center frequency and bandwidth for covering a bearing resonant frequency band. Part 2 paper presents an automatic selection process for finding the optimal complex Morlet wavelet filter with the help of genetic algorithm that maximizes the sparsity measurement value. Then, the modulus of the wavelet coefficients obtained by the optimal wavelet filter is used to extract the envelope. Finally, a non-linear function is introduced to enhance the visual inspection ability of BFCFs. The convergence of the optimal filter is fastened by the center frequencies and bandwidths of the optimal wavelet packet nodes established by the new sparsogram. Previous case studies including a simulated bearing fault signal and real bearing fault signals were used to show that the effectiveness of the optimal wavelet filtering method in detecting bearing faults. Finally, the results obtained from comparison studies are presented to verify that the proposed method is superior to the other three popular methods.

Introduction

Rolling element bearings are widely used in many machines to support rotating parts and components. Any faults in a bearing must be immediately detected to avoid fatal breakdowns, loss of production and even human casualties [1]. Rolling element bearing faults are mainly caused by localized defects on the outer race, the inner race and the roller elements. Once a fault is seeded on the surface of a bearing, a series of impacts are generated over short time intervals. These impulses excite the resonance frequencies of the bearing housing where the sensor is mounted. The modulating frequency is the bearing fault characteristic frequency, which causes the amplitude modulation phenomenon. Therefore, potential periodic bursts of exponentially decaying sinusoidal vibration occur after one of the components of a bearing proves faulty [2], [3].

The basic method for detecting the bearing fault characteristic frequency is the Fourier transform with an envelope analysis. However, the Fourier transform is difficult to directly detect non-stationary transients with weak energy. Therefore, a wavelet transform is considered an effective method for localizing non-stationary transients because the nature of wavelet transform is to calculate the similarity between a wavelet mother function and the transients to be analyzed [4], [5], [6], [7]. However, when a continuous wavelet transform (CWT) is employed to analyze non-stationary transients, too many wavelet scales used in CWT can result in considerable redundant information [8]. Besides, the computation of CWT is very time-consuming because the operations with unnecessary scales are involved. On the other hand, a wavelet transform can be also regarded as a series of filtering operations with different wavelet scales. References [9], [10], [11], [12], [13], [14], [15], [16] show that a single optimal filtering operation is adequate to extract bearing fault-related signatures. Lin and Qu [9] proposed the wavelet entropy for controlling the shape of the Morlet wavelet for bearing fault diagnosis. Nikolaou and Antoniadis [10] introduced three criteria for the selection of the parameters of a complex shifted Morlet wavelet for bearing fault diagnosis. Qiu et al. [11] considered the Shannon entropy and a periodicity detection method for the selection of the shape and the scale of Morlet wavelet for bearing fault diagnosis. He et al. [12] used differential evolution to optimize a complex Morlet wavelet for extracting envelope signals. Then, a maximum likelihood estimation-based soft-threshold method was applied to further enhance the signal to noise ratio. Su et al. [13] took the Shannon entropy of the filtered signal as the objective function and optimized the parameters of the complex Morlet wavelet using genetic algorithms to detect bearing fault characteristic frequencies. Bozchalooi and Liang [14] selected the shape factor of the complex Morlet wavelet by minimizing the ratio of the geometric mean to the arithmetic mean and determined the scale using a novel resonance estimation algorithm. Sheen [15] proposed a systematical method to design the parameters of Morlet wavelet to filter out one of the resonance modes of a bearing vibration according to the resonance frequencies of the known bearing vibration modes. Ericsson et al. [16] compared some different vibration analysis with Morlet based wavelet techniques for automatic bearing defect detection and concluded that the wavelet analysis was very suitable to bearing health condition monitoring. From the above results, it is found that the sparsity measurements, such as the kurtosis, the smoothness index and Shannon entropy, are some criteria for choosing the parameters of the Morlet wavelet.

Because an ultrasonic echo signal is similar to a vibration transient signal, the Morlet wavelet is also used to extract weak ultrasonic echo signatures in ultrasonic non-destructive testing. The major difference is that only the bandwidth of Morlet wavelet is required to be optimized for enhancing the ultrasonic time of flight resolution [17], [18], because the echoes have a center frequency similar to the emitted impulse. Considering the sparsity measurement used in non-destructive testing, Liang et al. [19] revised the blind deconvolution algorithm based on the proposed nonlinear function to improve the time resolution of the ultrasonic signal. They [20] also used a sparse solution to enhance the estimated accuracy of the time of flight of ultrasonic echoes. Chen et al. [21] employed the ensemble empirical mode decomposition method to analyze magnetic flux leakage signals and chose the most useful intrinsic mode function based on sparsity value. Therefore, the sparsity measurement used in non-destructive testing may be a potential method for the optimization of complex Morlet wavelet used in vibration analysis. Part 2 paper features an optimal method that maximizes the sparsity measurement used in non-destructive testing to choose optimal parameters for the complex Morlet wavelet through the help of a genetic algorithm that enhances the accuracy of bearing fault diagnoses. Moreover, a wavelet filtering operation with an optimal center frequency and bandwidth is required only once to extract bearing fault related signatures. By combining the optimal wavelet filtering with an envelope spectrum analysis (with or without non-linear transform), an intelligent process has been realized that automatically selects the parameters of a Morlet wavelet filter for detecting bearing fault characteristic frequencies. It should be noted that the previous research on the parameter selections of the optimal Morlet wavelet filter did not illustrate how to properly set the initial center frequencies and bandwidths for speeding up the convergence of the optimal Morlet wavelet filter [9], [10], [11], [12], [13], [14], [17], [18]. The new sparsogram reported in Part 1 paper is capable of providing the proper initial center frequencies and bandwidths for the use of genetic algorithm for the optimization of the complex Morlet wavelet filter.

The organization of Part 2 paper is given as follows. Section 2 introduces the new method by which the complex Morlet wavelet parameters are optimized through a genetic algorithm and the maximum sparsity measurement value. Section 3 presents a simulated bearing fault signal and real bearing fault signals collected from an experimental motor to investigate and validate the new method. Some comparisons with other popular methods are done in Section 4. Finally, Section 5 concludes the paper.