Since minimum correlated generalized Lp/Lq deconvolution (MCG-Lp/Lq-D) cannot simultaneously have a good sensing ability to repetitive fault impulses and a strong robustness to noise when processing low SNR signals, a new deconvolution method, minimum improved correlated generalized Lp/Lq deconvolution (MICG-Lp/Lq-D), is proposed in this article. The proposed method is designed based on a novel norm, the improved correlated generalized Lp/Lq (ICG-Lp/Lq) norm. The ICG-Lp/Lq norm not only has the low-order statistics form that is similar to the correlated generalized Lp/Lq (CG-Lp/Lq) norm used in MCG-Lp/Lq-D, but also has the high shift-order signal correlation construction similar to the correlated kurtosis (CK) norm used in maximum correlated kurtosis deconvolution (MCKD). This guarantees that the proposed norm is robust enough to strong noise and can improve its sensing ability for repetitive impulses simultaneously. This feature of the ICG-Lp/Lq norm ensures that MICG-Lp/Lq-D can suppress outliers and extract weak impacts when processing low SNR signals. Simulated and experimental case studies verify the performance of the proposed method. The results indicate that the proposed method is robust enough to strong noise and very sensitive to repetitive impulses simultaneously, which indeed overcomes the aforementioned defects of MCG-Lp/Lq-D. Therefore, MICG-Lp/Lq-D can be considered a promising tool for detecting repetitive impulses.

Repetitive impact detection (RID) is a core topic in bearing fault detection. The deconvolution-based method is an effective method in RID. Many deconvolution algorithms have been developed for this purpose. Simplified sparse filtering (SSF) is a typical deconvolution-based method, which has been successfully applied in RID due to its remarkable performance and fast computing speed. SSF generates an inverse filter by minimizing the generalized Lp/Lq (G-Lp/Lq) norm. The performance of SSF depends on the properties of G-Lp/Lq. However, G-Lp/Lq cannot characterize the periodicity of a specific period. Therefore, SSF cannot effectively detect multiple fault components at the same time and achieve compound fault detection. Additionally, when the signal-to-noise ratio (SNR) of the fault signal is extremely low, SSF might not recover the repetitive impacts from the measured signals. In order to improve these two shortcomings of SSF, we extend the G-Lp/Lq norm from time domain to signal correlation domain, and propose a correlated generalized Lp/Lq norm (CG-Lp/Lq). The proposed CG-Lp/Lq norm uses the enhanced periodicity which is obtained by calculating the correlation of repetitive impacts to realize the characterization of repetitive impacts with a specific period. Based on the proposed CG-Lp/Lq, a new deconvolution algorithm, the minimum correlated generalized Lp/Lq deconvolution (MCG-Lp/Lq-D) algorithm, is designed. The superiority of the proposed MCG-Lp/Lq-D is validated by both simulated and experimental data. The results demonstrate that MCG-Lp/Lq-D can recover the repetitive impacts, submerged in heavy noise, from the vibration mixture, and it can achieve compound fault detection.

Sparse representations based on convolutional sparse dictionary learning (CSDL) provide an excellent framework for extracting fault impulse response caused by bearing faults. In order to achieve fast dictionary learning, most CSDL-based fault diagnosis techniques recommend truncating the original data. However, the choice of truncation length is very difficult. An improper truncation length will lead to the problems of fault pattern rupture and uneven sparsity distribution. By contrast, if the data are not truncated, these problems will not occur. However, this will result in significant memory and computation consumption of CSDL. In order to overcome these problems, a novel CSDL method, a fast CSDL (FCSDL) algorithm, is proposed by combining the local orthogonal matching pursuit (OMP) algorithm with the conjugate gradient least-squares (CGLS) algorithm. The new method can achieve fast dictionary learning without truncating data and occupies very little memory. On this basis, an adaptive bearing fault diagnosis method based on envelope spectrum Kurtosis optimization is further proposed. When the sparsity is unknown, the new method can accurately search for the optimal sparsity and quickly recover the fault impact submerged in noise. The performance of the proposed fault diagnosis method is verified by using simulated signals, open bearing data, and wheelset bearing experimental data. It is compared with the union of a convolutional dictionary learning algorithm (UC-DLA) to highlight the advantages of the proposed method. The test code of reproducible research can be downloaded at https://github.com/aresmiki/FastCSDL .

Sparse representation has been widely used in bearing fault impact detection, which can find the impact that best matches the fault waveform from the pre-defined dictionary and recover the fault impulse waveform. However, the current dictionary of sparse representation and the efficiency of sparse representation algorithm need to be improved. In order to accurately detect the fault impulse in the original signal, a convolutional sparse coding using pathfinder algorithm-optimized orthogonal matching pursuit with asymmetric Gaussian chirplet model (CSC-OAGCM) is proposed in this paper. A new time-frequency atom prototype, AGCM, is used to match the fault impulse waveform. The specific application steps of the proposed algorithm are as follows: Firstly, a convolution dictionary is constructed with atoms generated by AGCM. Subsequently, based on the convolution dictionary, a pathfinder algorithm-optimized orthogonal matching pursuit algorithm is used to solve the sparse representation and optimize the atomic parameters to achieve the best approximation of the original signal. In other words, the proposed method detects the convolutional sparse patterns in the signal. A simulation signal, two sets of mixed signals of experimental data collected from the experimental platform and an axle box vibration signal collected from the actual operating train are used to verify the effectiveness of proposed method. Additionally, the spectral kurtosis and empirical wavelet transform are also used to process these signals, and their processing results are compared with those obtained by the proposed method to demonstrate the superiority of the proposed method.

The health status of bearing is directly related to the safe operation of rotating machinery. Bearing fault detection technology is of great significance to reduce or eliminate safety accidents. Singular value decomposition (SVD), as an effective low-rank approximation tool of data matrix, is widely used in bearing fault detection. The construction of trajectory matrix and the selection of singular value are two important factors that affect the performance of SVD-based fault diagnosis methods. In this paper, a new matrix low-rank approximation tool similar to SVD, shifted rank-1 reconstruction (SR1R), is introduced for fault diagnosis. SR1R uses periodic segment matrix (PSM) as trajectory matrix, and its signal reconstruction does not need to select singular value. In addition, a high oscillation region (HOR) detection method based on variance evaluation is proposed and applied to signal reconstruction of SR1R. The fault impulse detection method combining these two methods is called HOR-SR1R. The proposed method cannot only detect the fault impulses, but also suppress the noise between adjacent fault impulses. By contrast, the traditional SVD cannot eliminate the noise between adjacent fault impulses. The effectiveness of proposed method is validated by simulated signals, wheelset bearing data and open data set. Compared with two SVD-based fault diagnosis methods, Hankel matrix-based SVD and PSM-based SVD, the superiority of the proposed method is highlighted.