A new method for rolling bearing compound fault diagnosis based on WT-PCA method

2.1. Wavelet transform theory

Signals can be resolved into low-frequency portions $A_n$ (approximate) and high-frequency portions $D_n$ (detailed) by WT, one of common signal processing methods. $n$ represents the number of layers of WT and the number is 4 in this paper:

Given finite energy function is $\psi \left(t\right)$, $a$ elastic factor, $b$ shift factor, and continuous wavelet basis function ${\psi }_{a,b}\left(t\right)$ can be represented as:

Discrete wavelet is to discretize parameters $a$ and $b$, namely, $a=2j$, $b=k*2j$, $(j,k\in Z)$. Discretized wavelet function is:

2.2. PCA theory

PCA is a statistic dimensionality reduction analysis. It works by retaining the principal components of larger variances while neglecting other components, so as to drastically cut down the dimensionality of signal at the cost of less information.

Supposing that $X=\left[x_1,x_2,\dots ,x_n\right]$ contains $n$ samples and each sample includes $m$ features, covariance matrix of $X$ is:

In which, $E\left(X\right)=\frac{1}{n}\sum _{i=1}^n{x}_i$ is the mean value of sample space; assuming characteristic value of $C_x$ is ${\lambda }_1$, ${\lambda }_2$,…, ${\lambda }_m$, eigenvector is ${\mu }_1\ge {\mu }_2\ge \cdots \ge {\mu }_m$; making $U=[{\mu }_1,{\mu }_2,\cdots ,{\mu }_m]$, sample space is reconstructed as:

Sample space $X$ is transformed into characteristic sample space $Y$ by Eq. (5); the $y_{ii}$ in $Y$ represents the $j$-th main component in sample $x_i$. Accumulated contribution rate is defined as:

When $\varphi \left(L\right)\ge \text{0.95}$, the $U_L=[{\mu }_1,{\mu }_2,\cdots ,{\mu }_L]$ composed by previous $L$ eigenvectors is treated as low-dimension projection space, namely the completion of dimensionality reduction.

2.3. Characteristic frequency of rolling bearings

As the fault type of bearings is one-to-one relative to characteristic frequency, a fault type is usually determined by characteristic frequency. Calculation formula of characteristic frequency of rolling bearings is shown as Eq. (7):

where $r$otation frequency – $f_r$, inner ring fault frequency – $f_i$, outer ring fault frequency – $f_o$, rolling element fault frequency – $f_b$, retainer fault frequency – $f_c$ and variable $f_n$ is rotate speed; $Z$ the number of rolling elements; $\alpha$ the angle of stress direction and perpendicular line of inner and outer ring; $d$ and $D$ the diameter and pitch diameter of rolling bearings.

2.4. Diagnosis scheme

In this paper, a signal analysis method combining WT and PCA is proposed to identify the combined faults of rolling bearings. The flow consists of 5 steps shown in Fig. 1:

(1) Acceleration and vibration signal of rolling bearings is decomposed into 4 layers by sym4 wavelet;

(2) Figure out the kurtosis of high-frequency signal D1 – D4;

(3) Choose the signals relative to the largest and second largest kurtosis and form a 2D target matrix;

(4) Reduce the dimensionality of 2D matrix with PCA to make a new 1D matrix;

(5) Reduce the noise of 1D matrix with self-correlation function and extract characteristic frequency of combined faults through Hilbert demodulation spectrum.

For comparative analysis, the signal components of the largest and second largest kurtosis are chosen to form a 2D target matrix (red block diagram, scheme B) and compare with the signal component of the largest kurtosis (green block diagram, scheme B) to highlight the advantages of proposed WT-PCA method.

Fig. 1Flow chart of diagnostic methods

3.1. Compound fault diagnosis of outer ring and inner ring-scheme A

Firstly, randomly chose the acceleration signal as an example for analysis, which was picked up by vertical sensors during combined fault of outer and inner ring, and rotate speed of rolling bearing was 1538 r/min. According to Eq. (7), we can have the characteristic frequencies of rolling bearings shown in Table 1.

Vibration signal was decomposed by sym4 wavelet. After calculation, we could have the kurtosis D1-D4: 6.42, 6.478, 7.054 and 4.131. Details are shown in Fig. 3(a). Fig. 3(b) the low-frequency and high-frequency time domain of Fig. 4(b) after wavelet transform.

Fig. 3a) D1-D4 kurtosis values- outer ring and inner ring fault, b) results of wavelet decomposition of four layers, c) D1-D4 kurtosis values-Outer ring and rolling ball compound fault

a)

b)

c)

It can be found from the kurtosis of each component shown in Fig. 4 that the kurtosis corresponds to D3, D2, D1 and D4 component in descending order; characteristics were extracted by scheme A, namely, to obtain self-correlation function of D3 component and then extract the characteristic frequency of rolling bearings with Hilbert spectrum envelope based on the self-correlation function. As shown in Fig. 4, 4(a) is the time domain of original vibration signal, 4(b) Hilbert spectrum envelope of 4(a); 4(c) the Hilbert spectrum envelope of self-correlation function of component D3.

Fig. 4Results of compound fault diagnosis of outer ring and inner ring-scheme A

a) Time-domain acceleration signal

b) Hilbert envelope spectrum of Fig. 4(a)

c) Hilbert envelope spectrum

It can be concluded from the analysis of Fig. 4(b) and (d) that:

(1) In Fig. 4(b), Hilbert spectrum envelope of rolling bearing contains too much noise signal to extract characteristic frequencies as effective fault information is covered by strong noise background.

(2) In Fig. 5(d), according to scheme A in Fig. 1 based on the principle of maximum kurtosis, the characteristic frequency of inner ring fault can be found (multiple characteristic frequency of inner ring or the sum of frequency doubling and rotating frequency: 8$f_i$ (896 Hz), 10$f_i+f_r$ (1154 Hz), 15$f_i$ (1685 Hz) and 18$f_i$ (2027 Hz)), but it is unable to extract the characteristic frequency of axonometric outer ring or its multiple frequency component.

3.2. Compound fault diagnosis of outer ring and inner ring-scheme B

The paper combines wavelet transform and PCA algorithm (scheme B, as shown in Fig. 1). Firstly, make wavelet decomposition of vibration signal (4 layers); secondly, choose the signal components of the largest and second largest kurtosis after wavelet transform (namely, D3 and D2), and use them to form a 2D target matrix; thirdly, reduce the dimensionality of 2D matrix with PCA; fourthly, strive for self-correlation function of 1D matrix and extract characteristics of combined faults with Hilbert spectrum envelope. The result is shown in Fig. 5, in which Fig. 5(a) shows the time domain of dimensionality reduction of 2D target matrix; 5(b) time domain waveform of denoised self-correlation function; 5(c) Hilbert demodulation spectrum.

The following conclusions can be obtained by analyzing Fig. 5:

(1) After analyzing Fig. 5(a), the time domain diagram after dimensionality reduction, it can be observed that the signal comprises of distinct periodic impulse signal, which indicates that the layering and dimensionality reduction method proposed is effective for combined fault diagnosis.

(2) By analyzing Fig. 5(c), we can find characteristic frequency of inner ring and its frequency doubling, 8$f_i$ (896 Hz), 10$f_i+f_r$ (1154 Hz) and 10$f_i$ (1128 Hz). Meanwhile, characteristic frequency of outer ring and related frequencies can also be extracted, 13$f_o$ (854 Hz), 13$f_o+f_r$ (880 Hz), 18$f_o$ (1170 Hz) and 18$f_o+f_r$ (1196 Hz).

By scheme B, the WT-PCA algorithm proposed by this paper can precisely extract the characteristic frequency of combined fault of inner and outer rings or its frequency doubling, and the performance is obviously superior to scheme A.

Fig. 5Results of compound fault diagnosis of outer ring and inner ring- scheme B

a) Time domain signals D3 and D2(PCA)

b) Time domain of the auto-correlation function of Fig. 5(a)

c) Hilbert envelope spectrum of Fig. 5(b)