1. Introduction
Bearings are widely used in rotating machinery, and bearing failures are the most frequent problem. Failures may be catastrophic or may cause major downtime, which will result in production loss and even personal injury or death [
1,
2], hence, it is significant to analyze and diagnose bearing faults. In bearing fault diagnosis, the fault signal is always non-linear and non-stationary, and the fault signal collected is always submerged in background noise, and the energy distribution of the fault signals is unknown, so it’s hard to make a diagnosis due to the weak energy distribution of fault signals. Therefore, it is of vital significance to carry out effective fault diagnosis on the bearings.
For bearing fault diagnosis, the mainstream methods are to analyze the vibration signals collected from the time domain, frequency domain, time-frequency domain, phase-space dissimilarity measurement and other methods [
3,
4,
5,
6,
7]. Previous studies indicated that there are mainly two bearing diagnosis methods: one is discriminant model-based, and the other one is fault characteristic frequency-based. The general process of model-based involves collecting the vibration signals in the fault state, and setting up an effective discriminant model for fault diagnosis by means of signal filtering [
8] and feature extraction [
9]. This method will usually combine some statistics and machine learning methods, such as PCA, CDA, neural networks (NN), support vector machine (SVM) etc. and other intelligent methods [
10]. With constant development of deep learning, some methods such as CNN and DBN, etc. are employed to build discriminant models of bearing faults, and good effects are achieved [
11,
12]. However, in engineering applications, such methods also show their deficiencies: whether the fault can be effectively diagnosed or not depends on the accuracy of the discriminant model, and the accuracy of discriminant model in turn largely depends on the effective extraction of features, and the effective extraction of features is based on good signal filtering. During the whole procedure, if any link of the procedure is not ideal it will seriously affect the fault diagnosis result.
Another fault diagnosis method is a fault characteristic frequency-based method, namely the signal demodulation method. The general process of characteristic frequency-based methods is: filter the acquired signal first, then decompose the signal to extract fault signals or strengthen the fault signals. After that, a demodulation method such as envelope demodulation or morphological demodulation, is used to demodulate the signal to identify the fault characteristic frequencies for fault diagnosis. The signal decomposition is the basis of this method, and the empirical mode decomposition (EMD) is one of the most classical decomposition methods [
13,
14]. As further studies on decomposition method were carried out, researchers have proposed a series of decomposition methods, such as variational model decomposition [
15], intrinsic time-scale decomposition [
16], singular value decomposition, singular spectrum decomposition [
17,
18], multifractal detrended fluctuation analysis [
19] etc. Among these decomposition methods, the singular value decomposition (SVD) method is an effective fault diagnosis method. It is a non-linear filtering method, which is able to eliminate random noise components from a signal and obtain a relatively pure fault signal. Additionally, SVD has superior stability and invariability, and the singular value decomposed by it can reflect the intrinsic properties of signals and improve the Signal to Noise Ratio (SNR). It is suitable for fault diagnosis against strong background noise signals. At present, this method has good effectiveness in bearing fault diagnosis [
13,
17,
18]. However, there are two problems not well solved yet: one is how to decide the embedded dimension (length of time window) of the trajectory matrix, and the other is how to choose the best singular value. The embedded dimension is used to construct the trajectory matrix, based on which the SVD, feature recombination and signal restoration are performed; the quality of the embedded dimension affects the final analysis results to a large extent, and currently the selection mainly relies on experience. In SVD fault diagnosis, the improper selection of singular values will significantly influence the final result. The singular value representing background and noise signals is set to zero to achieve the effect of background noise elimination. However, in the example of bearing faults at low-speed rotation, the SNR is low, and the fault signal energy is weak, so the energy distribution of fault signals and background noise are unknown; therefore, it is important to effectively select the singular value. Previously, the singular value selection depended on experiments or trial and error, which always generated relatively large errors. Some studies have elaborated on this problem [
20,
21,
22]. Some researchers tried to seek the singular value by constructing a proper singular spectrum and identifying the turning point. For example, Zhao et al. [
23] proposed selecting a singular value using difference spectra. The performance of this method, however, will be reduced against a strong background of noisy signals, as the method mainly focuses on the maximum peak position of the constructed singular spectrum, which may result in the loss of important information about other peaks. Other researchers proposed selecting an effective singular value based on the asymptotic relationship between singular values and vectors of the signal matrix and the observed matrix [
20]. The filtered signal matrix is reconstructed by minimizing the asymptotic loss. Its performance is superior to the conventional reduction of singular values accomplished by thresholding methods [
24]. However, some assumptions must be met to use this method, such as the orthogonally invariance of the signal noise. The assumption is difficult to satisfy in engineering applications.
In view of the aforementioned issues, the discussed fault methods are not ideal against the background of strong noise. Due to the above problems, this study proposes an effective method to select singular values and applied it in fault diagnosis under low-speed rotation. First of all, according to the reconstruction theory of chaos phase space, the embedded dimension of the trajectory matrix can be reconstructed with the FNN method. The phase space reconstructed with this method can be used to characterize the dynamic features of the motive power system. After the trajectory matrix is determined, the trajectory matrix is subject to SVD, and different SVs acquired from SVD are combined to change the decomposed signals back to one-dimensional signals, which are compared with signals in a normal state; the designed evaluation function of statistical information is used to compare paired signals as well as the similarities between restored signal and signal in normal state. The SV combination with maximum similarities is considered to represent the background signal and noise signal, and the remaining SVs are considered to represent the fault signal. The SV representing the fault signal changes the decomposed signal back to a one-dimensional signal, and effective analysis can be performed on the faults using spectrum envelope modulation. This method has been verified with simulation experiments and engineering experiment, and compared with the published SVD-based method and Fast Kurtogram method to verify the effectiveness of the method.
The remainder of the study is organized as follows:
Section 2 describes the mechanism of SVD filtering;
Section 3 describes the importance of SV selection for the diagnosis and bearing envelope analysis;
Section 4 describes the methods proposed in the study;
Section 5 proves the effectiveness of the methods proposed with simulation experiments and engineering experiments, as well as comparison with the published SVD-based method and Fast Kurtogram;
Section 6 presents the conclusions.