A study on the diagnosis of compound faults in rolling bearings based on ITD-SVD

2.1. ITD algorithm

ITD algorithm aims to represent a complex signal with the sum of some proper rotation components (PR) and the one of a residual trend term. Given $X_t$ is vibration signal; $L$, baseline signal operator; $L{X}_t$, baseline extraction operator, replaced by $L_t$; $H$, extraction operator of proper rotation component; $L_p^t$, residual trend component; $H_t$, rotational component, and specific process of ITD algorithm is shown in Fig. 1.

Fig. 1ITD algorithm diagram

1) Vibration signal $X_t$ is represented as:

2) Figure out $X_k$, all the extreme points of $X_t$ and related time ${\tau }_k$, in which ($k=$ 1, 2,…, $M$), $M$ represents the number of extreme points. Linear gain parameter $\alpha$ is within $0<\alpha <1$, with 0.5 as a usual option [19]:

3) Linear transformation is performed between adjacent extreme points, as shown in Eq. (3):

4) Replaces $X_t$ with $L_t$ and repeat the step 1-3 until $L_t$ becomes a monotonic function or constant. Given the number of iterations is $p$, $X_t$ can be represented as Eq. (4) after ITD decomposition:

2.2. SVD algorithm

As for the rotational component signal $H=\left(H_i\right)$, $i=$ 1, 2,…, $N$ after ITD decomposition, build a Hankel matrix $H\mathrm{\text{'}}$:

Singular value decomposition of $H\mathrm{\text{'}}$ is as follow:

In the Eq. (6), $U$ and $V$ is orthogonal matrix; $V^T$, transposed matrix of $V$; $S$, diagonal matrix composed by singular value. Singular value bears the following relation: ${\sigma }_1>{\sigma }_2>...>{\sigma }_k\gg {\sigma }_{k+1}>{\sigma }_{k+2}>{\sigma }_r$ (in which $\gg$ means being much greater than). ${\sigma }_k$ represents the $k$th singular value obtained by decomposition, $k=$1, 2,…, $r$. $r$ is the order of matrix $H\mathrm{\text{'}}$.

As the singular value of each subsequent term after the term $k+1$ is significantly smaller than previous $k\text{'}$ singular values [23], the singular values of term $k+1-r$ can be regarded as the singular value to noise component [24]. That means the key to denoising of singular value lies in the option of k, the number of reconstructed terms. As in the Hankel matrix, the vectors of one row lag behind the previous row with only one data point, a sick nature can be observed in the Hankel matrix without noise signal due to the correlation of adjacent rows: the previous $k\text{'}$ singular values are larger and present obvious mutation in some point, while the singular values after that are close to zero. On the contrary, the singular values with noise signal show no correlation between adjacent rows, namely without obvious mutation, but present a parallel line close to lateral axis. For that, difference spectrum of singular value is introduced to determine mutation point $k$. It describes the variation of sequence. As the value of difference sequence is large, the literature [25] has defined difference spectrum $d_i$ as:

Given: difference spectrum sequence $d=\left(d_1,d_2,\dots ,d_q\right)$, in which singular values of first $k$ terms are the ones corresponding to signal and from $k+1$ to the last term, the singular values correspond to noise. For that, the $k$th element $d_k$ is the maximum peak of difference spectrum sequence. Choose the first $k$ singular values to build a new diagonal matrix $S\mathrm{\text{'}}$ and rebuild original signal $H\mathrm{\text{'}}\mathrm{\text{'}}$:

2.3. ITD-SVD

In this paper, a study has been made to probe into the characteristic extraction and fault diagnosis of compound faults from rolling bearings in aero-engines. The specific ITD-SVD algorithm is shown in the scheme D in Fig. 2. The scheme A in Fig. 2 refers to rebuilding signal with rotational components obtained from ITD algorithm (directly eliminate residual trend terms); scheme B refers to selection of characteristic signals according to rotational components associated with maximum kurtosis; scheme D refers to ITD-SVD algorithm proposed in this paper.

Fig. 2Method flow diagram

ITD is combined with SVD to extract the characteristics of compound faults in rolling bearings and specific process is shown as follow:

Step 1: vibration signal is decomposed into several PR components and residual trend components based on ITD algorithm;

Step 2: each rotational component is denoised based on singular value difference spectrum algorithm;

Step 3: vibration signal is reconstructed with rotational components denoised;

Step 4: reconstructed signal is subjected to envelope analysis based on Hilbert envelope spectrum; characteristics of compound faults are extracted and fault identification is studied.

4.1. Scheme A: (shown in Fig. 2, marked by blue)

According to scheme A, vibration signal is broken up by ITD algorithm, with residual trend components eliminated and then signal is reconstructed based on the obtained PR components. Extract characteristics of compound faults based on reconstructed signal and make fault diagnosis, and the result is shown in Fig. 5. Fig. 5(a) is the time domain waveform of original signal and Fig. 5(b) is Hilbert envelope spectrum; Fig. 5(c) is the time-domain waveform of rotational components after ITD decomposition; Fig. 5(d) and (e) is the time-domain waveform of reconstructed rotational components and Hilbert envelope spectrum.

It can be found from the analysis of Fig. 5(d)-(e) that no matter the direct envelope analysis of vibration signal or ITD-based algorithm, it is impossible to effectively extract characteristic frequency of rolling bearings according to Hilbert envelope spectrum of reconstructed rotational components (without residual trend components), which means it cannot effectively determine the type of faults.

4.2. Scheme B (shown in Fig. 2, marked by green)

Vibration data of Section 4.1 was reused to facilitate comparative analysis. According to scheme B, kurtosis of each rotational component (obtained by ITD decomposition) was calculated and the one with maximum kurtosis was regarded as optimal rotational component, based on which characteristics of compound faults were extracted. The kurtosis of each rotational component is shown in Table 3.

Fig. 5Compound faults feature extraction of rolling bearing – scheme A

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It can be found from Table 3 that PR2 is chosen as optimal rotational component according to the principle of maximum kurtosis. To reduce the influence of noise in signal, we took the self-correlation function of PR2 and extracted the characteristics of compound faults of rolling bearings with the Hilbert envelope spectrum of self-correlation function. The result is shown in Fig. 6. Fig 6(a) and (b) is the self-correlation function of PR2 rotational component and Hilbert envelope spectrum.

Fig. 6Compound faults feature extraction of rolling bearing – scheme B

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As shown in Fig. 6(b), ITD algorithm is combined with kurtosis and a distinct 820 Hz frequency component can be observed in the Hilbert envelope spectrum of optimal rotational component associated with maximum kurtosis. This frequency component is consistent with the 15-time frequency (820/15 = 54.7 Hz) of characteristic frequency (54.2 Hz) of rolling elements in rolling bearings. It can be observed from scheme B that faults occurred in rolling elements, but this scheme cannot effectively extract the characteristic frequency of inner and outer rings, which means the combination of ITD algorithm and kurtosis unable to make effective judgment of compound faults of rolling bearings.

4.3. Scheme C (shown in Fig. 2, marked by yellow)

Vibration data of Section 4.1 was used (compound faults occurred in outer and inner ring and rolling elements) to facilitate comparative analysis. The PCA method is used to realize data fusion (the two channels vibration data in the horizontal and vertical direction is selected). To reduced noise, the autocorrelation function is selected. The results obtained by Hilbert envelope spectrum analysis are shown in the Fig. 7. Among them, Fig. 7(a) is the time domain signal after PCA, Fig. 7(b) is the autocorrelation function of Fig.7(a), and Fig. 7(c) is the Hilbert envelope spectrum of Fig. 7(b).

By analyzing Fig. 7(c), it can be concluded that: in Hilbert envelope spectrum can only find 3 times frequency (166 Hz) and 15 times frequency (820 Hz) of rolling element fault characteristic frequency (54.2 Hz) by using PCA method, while the inner ring and outer ring feature frequency cannot be find. In other words, scheme C (PCA) cannot accurately identify compound fault diagnosis types of rolling bearings.

5.1. Characteristic extraction of compound faults in rolling bearings

To make a comparative analysis with the scheme A/B/C in Section 4.1-4.3 in terms of the effect of characteristic extraction, the characteristic extraction basic on ITD-SVD algorithm took the same vibration data as Section 4.1. The result is shown in Fig. 8, in which Fig. 8(a1)-(a4) is the denoised rotational component by SVD (obtained by ITD algorithm, as shown in Fig. 5(c)); Fig. 8(a5) is the reconstructed time-domain after denoising (based on Fig. 8(a1) to (a4)); Fig. 8(a6) is the Hilbert envelope spectrum of reconstructed signal.

After analyzing Fig. 8(a6) and comparing it with Fig. 5(e), 6(b) and 7(c) we can find that:

Based on the ITD-SVD (scheme D) algorithm, the frequency components of 138 Hz and 284 Hz, 820 Hz were observed to have larger amplitude in Hilbert envelope spectrum. It can be found from the analysis that these frequencies are relative to the characteristic frequency (138.1 Hz) of inner ring faults, the difference of 4-time frequency of characteristic frequency (79.9 Hz) of outer ring and inversion (31 Hz) ((284+31)/4 = 78.8 Hz), and the 15-time frequency (820/15 = 54.7 Hz) of characteristic frequency of faults in rolling elements. These frequency characteristics are consistent with the type of compound faults of rolling bearings. That means the proposed ITD-SVD algorithm can effectively extract the characteristics of compound faults of rolling bearings and precisely determine the type of faults.

5.2. Different installation positions of sensors

To analyze the sensibility of ITD-SVD algorithm to installation direction and position of sensors, we have chosen the vibration acceleration signal collected by the sensors of horizontal direction in section 4.1 (sensors in vertical direction). Rotation speed $f_n$ remains 1869 r/min and frequency $f_r$ is 31 Hz. Characteristic frequency of outer ring, inner ring, rolling elements and holder is 79.9 Hz, 138.1 Hz, 54.2 Hz and 11.4 Hz respectively. The result is shown in Fig. 9, in which Fig. 9(a) is the time domain waveform of vibration acceleration signal; Fig. 9(b) is the Hilbert envelope of original signal; Fig. 9(c) is the rotational components of acceleration signal; Fig. 9(d1)-(d4) is the time domain waveform of denoised rotational component by ITD-SVD; Fig. 9(d5) is the reconstructed time-domain after denoising (based on Fig. 9(d1)-(d4)); Fig. 9(d6) is the Hilbert envelope spectrum of reconstructed signal; fig. 9(d7) is the partial zoom view of Hilbert envelope spectrum of reconstructed signal.

Fig. 8Compound faults feature extraction of rolling bearing – scheme D-vertical direction

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It can be found from the analysis of Fig. 9(d7) that there is distinct characteristic frequency of inner ring fault 138 Hz, 3-time frequency 232 Hz (232/3 = 77.3 Hz) of characteristic frequency (79.9 Hz) of outer ring fault and 15-time frequency (805/15 = 53.6 Hz) of characteristic frequency (54.2 Hz) of faults in rolling elements. As the characteristic frequency of faults in rolling elements is modulated by rotational frequency of holder, there is the difference between the 15-time frequency of fault characteristic frequency of rolling elements and rotational frequency of holder, 794 Hz (805-11.4 = 793.6 Hz).

It can be concluded from the comparison of Fig. 7 and Fig. 8 that with ITD-SVD algorithm, we can effectively extract the characteristic frequency of compound faults of rolling bearings no matter sensors are installed in horizontal or vertical direction, and thereby identify the fault type. That means ITD-SVD algorithm is insensitive to installation direction of sensor.

Fig. 9Compound faults feature extraction of rolling bearing – scheme D-Horizontal direction

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5.3. Other compound faults

To further validate the effectiveness of ITD-SVD algorithm, we will study the characteristic extraction of other compound faults in rolling bearings based on ITD-SVD next. Compound faults of outer and inner rings (sensors were fixed vertically), outer ring and rolling elements (sensors were fixed horizontally) were randomly chosen for the study based on ITD-SVD algorithm and the result is shown in Fig. 10. Fig. 10(a1)-(b1) shows the time domain waveform of vibration signal; Fig. 10(a2)-(b2), the Hilbert envelope spectrum of original vibration signal; Fig. 10(a3)-(b3), time domain waveform of reconstructed signal with ITD-SVD; Fig. 10(a4)-(b4), Hilbert envelope spectrum of reconstructed signal; Fig. 10(a5)-(b5), partial zoom of Hilbert envelope spectrum of reconstructed signal; Fig. 10(a1)-(b5), compound faults of outer and rolling elements.

Figure out the characteristic frequencies of different compound faults according to Eqs. (10-14). The result is shown in Table 4.

From the analysis of Fig. 10(a5), we can find some distinct frequency components, like 129 Hz, 589 Hz and 24 Hz in the Hilbert envelope spectrum based on ITD-SVD algorithm. These frequency components are relative to 2-time and 9-time frequency (589/9 = 65.4 Hz) of outer-ring characteristic frequency 65 Hz, the sum of 2-time frequency of inner-ring characteristic frequency 112 Hz and inversion 25 ((241-25)/2 = 108 Hz), which is consistent with the state of compound faults of inner and outer rings.

Fig. 10Compound faults feature extraction of rolling bearing – scheme D: outer ring and inner ring compound fault (a1-a5); outer ring and rolling element compound fault (b1-b5)

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From the analysis of Fig. 10(b5), we can find some distinct frequency components, like 53 Hz and 240 Hz in the Hilbert envelope spectrum based on ITD-SVD algorithm. It can be found from the analysis that these frequency components are relative to the 3-time frequency (240/3 = 80 Hz) of fault characteristic frequency (52.6 Hz) of rolling elements and that (77.5 Hz) of outer-ring. The result is consistent with the state of compound faults of rolling elements and outer ring.

That is to say the proposed ITD-SVD algorithm can precisely and effectively extract characteristic frequencies of different compound faults in rolling bearings and thereby determine the type of faults.