# Incipient Fault Feature Extraction for Rotating Machinery Based on Improved AR-Minimum Entropy Deconvolution Combined with Variational Mode Decomposition Approach

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## Abstract

**:**

## 1. Introduction

## 2. Improved Autoregressive Minimum Entropy Deconvolution (Improved-AR-MED)

#### 2.1. Preprocessing: AR Model-Based Method

_{n}{n = 1, 2,…, N}, the p-th order AR(p) model is defined as follows [26]:

- Step 1:
- AR model order p. The optimal order of AR model can be calculated by the Akaike information criterion (AIC) [27].
- Step 2:
- Step 3:
- Residual error ${\epsilon}_{n}$. Perform step1 ahead prediction on the potentially faulty vibration data and calculate the residual prediction error.

#### 2.2. Improved AR Minimum Entropy Deconvolution

- (1)
- Assume initial filter $g{(n)}^{(0)}$ as a centered impulse, i.e., $g{(n)}^{(0)}={[0,0,\cdot \cdot \cdot ,1,\cdot \cdot \cdot 0,0]}^{T}$.
- (2)
- Solve $x(n)=g{(n)}^{(i-1)}*y(n)$.
- (3)
- Solve ${f}^{(i)}(l)=a{\displaystyle \sum _{n=1}^{N}{x}^{3}(n)y(n-l)}$.
- (4)
- Solve for new filter coefficients, i.e., ${g}^{(i)}={A}^{-1}{f}^{(i)}$.
- (5)
- Compute the error criterion, $Err={\Vert {g}^{(i)}-{g}^{(i-1)}\Vert}_{2}^{2}$, if Err ≤ tolerance, the iteration process is finished, or Err > tolerance, then $i=i+1$ and repeat the process from step 2.

**Definition.**

## 3. Variational Mode Decomposition (VMD) and Example Discussion

#### 3.1. Review of VMD

^{1}Gaussian smoothness of the demodulated signal [25]. Unlike EMD and LMD methods, the number of IMF components to be involved in the decomposition is chosen prior to the decomposition processing. Here, the IMF component u

_{k}(t) can be expressed as follows:

_{k}(t) can be evaluated with following steps:

- Step 1:
- For each IMF component u
_{k}(t), the real signal is converted to the analytic signal by means of the Hilbert transform to obtain a unilateral frequency spectrum as follows:$$[\delta (t)+\frac{j}{\pi t}]*{u}_{k}(t)$$ - Step 2:
- For each IMF component u
_{k}(t), the frequency spectrum of u_{k}(t) is translated to baseband by using modulation property to the respective estimated center frequencies. That is:$$[(\delta (t)+\frac{j}{\pi t})*{u}_{k}(t)]{e}^{-j{\omega}_{k}t}$$ - Step 3:
- The bandwidth of each IMF component u
_{k}(t) is estimated through the H^{1}Gaussian smoothness of the demodulated signal, i.e., the squared L^{2}-norm of the gradient. So the resulting is written as a constrained variational problem as follows:$$\{\begin{array}{l}\mathrm{min}\{{\displaystyle \sum _{k}{\Vert {\partial}_{t}[\delta (t)+\frac{j}{\pi t}]*{u}_{k}(t){e}^{-j{w}_{k}t}\Vert}^{2}}\}\\ S.t.{\displaystyle \sum _{k}{u}_{k}=f}\end{array}$$**Dirac**function, k is the number of IMF models, t is time script and symbol * denotes convolution, respectively.

- (1)
- Initialization, $\{\stackrel{\wedge}{{u}_{k}^{1}}\}$, $\{\stackrel{\wedge}{{\omega}_{k}^{1}}\}$, $\{\stackrel{\wedge}{{\lambda}^{1}}\}$, n;
- (2)
- Execute a loop, n = n + 1;
- (3)
- For all $\omega \ge 0$, update $\stackrel{\wedge}{{u}_{k}}$ is as follows:$$\stackrel{\wedge}{{u}_{k}^{n+1}}(\omega )\leftarrow \frac{\stackrel{\wedge}{f}(w)-{\displaystyle \sum _{i<k}\stackrel{\wedge}{{u}_{i}^{n+1}}(\omega )}-{\displaystyle \sum _{i>k}\stackrel{\wedge}{{u}_{i}^{n}}(\omega )}+\frac{\stackrel{\wedge}{{\lambda}^{n}}(\omega )}{2}}{1+2\alpha {(\omega -{\omega}_{k}^{n})}^{2}},k\in \{1,K\}$$
- (4)
- Update $\stackrel{\wedge}{{\omega}_{k}}$ is as follows:$${\omega}_{k}^{n+1}\leftarrow \frac{{\displaystyle {\int}_{0}^{\infty}\omega {\left|\stackrel{\wedge}{{u}_{k}^{n+1}}(\omega )\right|}^{2}d\omega}}{{\displaystyle {\int}_{0}^{\infty}{\left|\stackrel{\wedge}{{u}_{k}^{n+1}}(\omega )\right|}^{2}d\omega}},k\in \{1,K\}$$
- (5)
- Update $\lambda $ is as follows:$$\stackrel{\wedge}{{\lambda}^{n+1}}(\omega )\leftarrow \stackrel{\wedge}{{\lambda}^{n}}(\omega )+\tau (\stackrel{\wedge}{f}(\omega )-{\displaystyle \sum _{k}\stackrel{\wedge}{{u}_{k}^{n+1}}(\omega )})$$
- (6)
- Iterate steps (2) to (5) until the residual satisfies a given stopping criterion. That is:$$\sum _{k}{\Vert \stackrel{\wedge}{{u}_{k}^{n+1}}-\stackrel{\wedge}{{u}_{k}^{n}}\Vert}_{2}^{2}/{\Vert \stackrel{\wedge}{{u}_{k}^{n}}\Vert}_{2}^{2}}<\epsilon $$

#### 3.2. Example Discussion

_{1}(t), x

_{2}(t) and x

_{3}(t) are shown in Figure 2. Examples of IMFs and Product functions (PF) modes obtained respectively by the EEMD, LMD and VMD are illustrated in Figure 3a,c,e. The Hilbert spectrum plots of the mono-components derived from the three decomposition methods are shown in Figure 3b,d,f, respectively. It can be found from Figure 3a–d that the IMFs and PFs derived from EEMD and LMD method are all anamorphic with serve scale-mixing problem. While in the decomposition of VMD method (see Figure 3e,f), it can be clearly found that three IMFs components, i.e., IMF1(t), IMF2(t) and IMF3(t) are corresponding to the x

_{1}(t), x

_{2}(t) and x

_{3}(t) of the original AM-FM signal x(t), respectively. Moreover, the three horizontal lines (instantaneous frequency are 4 Hz, 14 HZ and 30 Hz) in Figure 3f has a narrower bandwidth than that of the EEMD and LMD methods. Therefore, the comparison results illustrate that the IMF components can be accurately extracted from the mixed signal by using the VMD method, which is much closer approximation to the real components of x(t).

## 4. Simulation Analysis for Weak Fault Extraction of Rotating Machinery

_{0}= 1 denotes the amplitude of modulating signal, a = 0.1 is a damping coefficient, sampling point N = 8192, t = 1/2000:1/2000:0.4096, sampling frequency Fs = 20,000 Hz, the frequencies of the above signal are as follows: f

_{1}= 20, f

_{2}= 35, and f

_{3}= 2000 represents the natural frequency of excited structure. Moreover, a stationary white noise x

_{3}(t) was added to obtain the simulated signals with SNR = −6dB.

- (1)
- AR-MED is optimizing the norm Kurtosis which prefers a solution of a single impulse or deterministic impulse. For the bearing fault signals, a series of periodic impulses features maybe not suit for this approach.
- (2)
- There are still a heavy noise remained in the output signal that generated by AR-MED filter, which means the impulsive convolved in the suppressed resonance frequency bands cannot be detected by AR-MED.

_{IMF1}= 2.2802, K

_{IMF2}= 6.7916, K

_{IMF3}= 3.2967, K

_{IMF4}= 3.0748, K

_{IMF5}= 3.0472 and K

_{IMF6}= 3.1231. Hence, the IMF2 component was elected to extract the fault characteristic frequency.

## 5. Experimental and Engineering Application Verification

#### 5.1. Case 1—Bearing Fault Experimental Setup

#### 5.2. Case 1—Results and Discussion of Bearing Single Fault

_{0}, 3f

_{0}), which shows that the BPFO peak is highlighted while the noise interference can be ignored. By comparing Figure 11d and Figure 12d, as the failure progresses, the amplitudes of the BPFO peaks increase, thus making the feature monitoring easier and more accurate.

#### 5.3. Case 2—Gearbox Fault Experimental Setup

_{1}= 55 and Z

_{2}= 75, respectively, i.e., gear number ratio is Z

_{1}:Z

_{2}= 11:15, the pressure angle is 20° and face width of teeth is 20 mm. The overall instrument configuration and the real equipment layout figure of the gearbox test rig are shown in Figure 16a,b, respectively. The pitting fault frequency f

_{pitting}of the bull gear is f

_{pitting}= 17.96 Hz and the wear fault frequency of pinion gear is f

_{wear}= 33.41 Hz.

#### 5.4. Case 2—Results and Discussion of Gearbox Multiple-Fault

_{fitting}= 17.96 Hz) and its harmonic components (49.06 Hz ≈ 3 × f

_{fitting}, the pinion gear wear fault frequency (31.25 Hz ≈ f

_{wear}) are clearly observed in the envelope spectrum. The result indicates that there are local defects on the gear pair that match with the engineering application. Therefore, it can be concluded that the proposed hybrid method is able to accurately identify the multiple weak faults caused by a gear-pair, which is particularly appropriate for incipient multiple fault feature extraction.

## 6. Conclusions

- (1)
- Improved AR-minimum entropy deconvolution provides a solution for the filter that is an optimal solution, which can filter out the high-frequency and background noise and enhance the peak value of the multiple transient impulses. With the improved AR-Minimum entropy deconvolution method, the location of multiple transient impulses can be deconvolved effectively, which is well-suited for the fault nature of rotating machinery.
- (2)
- As a new adaptive decomposition technique, the filtered signal is further decomposed by the VMD method, and the illusive components and model mixing problems are eliminated based upon the VMD method. Moreover, the VMD is more robust to analyze noisy signal on numerical simulation and actual data than the EEMD and LMD approach.
- (3)
- The proposed method has then been applied to experimental and engineering application signals obtained by accelerometers. Both the single-fault of bearing outer race and multiple-fault of gearboxes have proven the validity of the proposed method compared to other conventional methods. Meanwhile, the number of IMF modes requires predetermining before VMD decomposition, which leads to a lot of inconvenience in the engineering application, so an adaptive calculation algorithm should be established in this regard in future works.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 3.**Decomposition results and time-frequency distribution of the multivariate signal with noise using EEMD, LMD and VMD methods. (

**a**) IMF components of x(t) decomposed by using EEMD; (

**b**) Hilbert spectrum distribution of the EEMD results; (

**c**) PF components of x(t) decomposed by using LMD; (

**d**) Hilbert spectrum distribution of the LMD results; (

**e**) IMF components of x(t) decomposed by using VMD; (

**f**) Hilbert spectrum distribution of the VMD results.

**Figure 4.**Simulated signal and its envelope spectrum. (

**a**) Periodic impulse signal; (

**b**) Raw fault signal with heavy noise; (

**c**) Envelope spectrum of the raw fault signal.

**Figure 5.**Filtered signal and its envelope spectrum with AR-MED and improved AR-MED method. (

**a**) Filtered signal with AR-MED method; (

**b**) Envelope spectrum of AR-MED filtered signal; (

**c**) Filtered signal with improved AR-MED method; (

**d**) Envelope spectrum of improved AR-MED filtered signal.

**Figure 6.**Decompose results using the VMD and spectrum analysis. (

**a**) IMF components of improved AR-MED de-noised signal decomposed by VMD; (

**b**) The amplitude spectrum of each IMF component; (

**c**) Envelope spectrum of IMF2.

**Figure 7.**Bearing test rig; (

**a**) Ball bearing fault test rig; (

**b**) The schematic diagram of accelerated life test.

**Figure 8.**Time feature (

**a**) equivalent vibration intensity and (

**b**) kurtosis of bearing 1 for the whole life cycle.

**Figure 9.**Results using the proposed method of method the 3rd day’s data (2980 min); (

**a**) The original bearing vibration signal; (

**b**) The improved AR-MED filtered signal; (

**c**) The IMF components of improved AR-MED filtered signal generated by VMD method; (

**d**) The envelope spectrum of IMF2.

**Figure 10.**Results using the proposed results of the 4th day’s data (4420 min); (

**a**) The original bearing vibration signal; (

**b**) The improved AR-MED filtered signal; (

**c**) The IMF components of improved AR-MED filtered signal generated by VMD method; (

**d**) The envelope spectrum of IMF2.

**Figure 11.**Results using the proposed method method on the 5th day’s data (5860 min); (

**a**) The original bearing vibration signal; (

**b**) The improved AR-MED filtered signal; (

**c**) The IMF components of improved AR-MED filtered signal generated by VMD method; (

**d**) The envelope spectrum of IMF2.

**Figure 12.**Results using the proposed method on the 6th day’s data (7300 min); (

**a**) The original bearing vibration signal; (

**b**) The improved AR-MED filtered signal; (

**c**) The IMF components of improved AR-MED filtered signal generated by VMD method; (

**d**) The envelope spectrum of IMF2.

**Figure 14.**Results using Wavelet method. (

**a**) Sub-band components of original signal decomposed by using Wavelet; (

**b**) Kurtogram of a7 sub-band component; (

**c**) Fourier transform magnitude of the squared envelope.

**Figure 15.**Results using LMD method. (

**a**) PF components of original signal decomposed by using LMD; (

**b**) Kurtogram of PF2 component; (

**c**) Fourier transform magnitude of the squared envelope.

**Figure 16.**Gearbox test rig; (

**a**) The engineering drawing of gearbox test rig; (

**b**) The equipment layout figure of real gearbox test rig.

**Figure 17.**Raw vibration signal and its envelope spectrum. (

**a**) Raw vibration signal; (

**b**) Envelope spectrum of the raw fault signals.

**Figure 18.**AR-MED filtered signal and comparison of spectrum of modes extracted by VMD. (

**a**) Filtered signal with improved AR-MED method. (

**b**) The spectrum of modes extracted by VMD, K = 19, $\alpha $ = 2000. (

**c**) The spectrum of modes extracted by VMD, K = 20, $\alpha $ = 2000.

**Figure 19.**Results obtained by the proposed method. (

**a**) 19 IMF components of improved AR-MED filtered signal decomposed by using VMD; (

**b**) Envelope spectrum of the IMF4 component.

Mode K | Central Frequency/Hz | ||||||
---|---|---|---|---|---|---|---|

2 | 54.93 | 2000 | |||||

3 | 54.93 | 2000 | 7188 | ||||

4 | 54.93 | 2000 | 4763 | 7395 | |||

5 | 54.93 | 2000 | 3811 | 6147 | 8682 | ||

6 | 54.93 | 2000 | 3201 | 4893 | 6992 | 8748 | |

7 | 54.93 | 1699 | 2100 | 3579 | 5281 | 7188 | 8926 |

Bearing Type | Rexnord ZA-2115 |
---|---|

Rotating speed of shaft (rpm) | 2000 |

Pitch diameter (mm) | 71.501 |

Roller diameter (mm) | 8.4074 |

Roller number | 16 |

Contact angle (deg) | 15.17 |

Ball pass frequency outer (BPFO) (Hz) | 236.4 |

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## Share and Cite

**MDPI and ACS Style**

Li, Q.; Ji, X.; Liang, S.Y.
Incipient Fault Feature Extraction for Rotating Machinery Based on Improved AR-Minimum Entropy Deconvolution Combined with Variational Mode Decomposition Approach. *Entropy* **2017**, *19*, 317.
https://doi.org/10.3390/e19070317

**AMA Style**

Li Q, Ji X, Liang SY.
Incipient Fault Feature Extraction for Rotating Machinery Based on Improved AR-Minimum Entropy Deconvolution Combined with Variational Mode Decomposition Approach. *Entropy*. 2017; 19(7):317.
https://doi.org/10.3390/e19070317

**Chicago/Turabian Style**

Li, Qing, Xia Ji, and Steven Y. Liang.
2017. "Incipient Fault Feature Extraction for Rotating Machinery Based on Improved AR-Minimum Entropy Deconvolution Combined with Variational Mode Decomposition Approach" *Entropy* 19, no. 7: 317.
https://doi.org/10.3390/e19070317