# Feature Extraction Method for Hydraulic Pump Fault Signal Based on Improved Empirical Wavelet Transform

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Feature Energy Ratio

_{1}+ E

_{2}+ …+ E

_{n})/E

_{1}, E

_{2}, …, E

_{n}are respectively the energy, and they are respectively presented in fault feature frequency and its harmonics, and E is the total energy of the signal.

#### 2.2. Improved Empirical Wavelet Transform

_{1}, x

_{2}, …, x

_{n}) be a signal, and X(ω) is its Fourier amplitude spectrum in the frequency domain, and then the power spectrum of the signal is denoted as P, and it is defined as

_{coefficient}can be obtained. Thus, the bad influence of interference components on adaptive segment acquirement is much reduced.

_{coefficient}. It is supposed that the Fourier support [0, π] is segmented into N contiguous segments, which means that there are N + 1 boundaries. However, 0 and π are always used in definition and it is need to be found N − 1 extra boundaries. To find the boundaries, local maxima value of power spectrum are selected, and the values are sorted in decreasing order (0 and π are excluded). It is assumed that the algorithm found M maxima, and two cases can appear:

_{n}= [ω

_{n−1}, ω

_{n}], and it can be found that ${U}_{\mathit{n}=\mathbf{1}}^{\mathit{N}}$ Λ

_{n}= [0, π]. Centered around each ω

_{n}, a transition phase (the gray hatched areas on Figure 2) T

_{n}of width 2τ

_{n}is defined.

_{n}. Then ∀n > 0, it can be defined that the empirical scaling function and the empirical wavelets by expressions of Equations (4) and (5) respectively.

_{n}[ω

_{n+}

_{1}− ω

_{n}/ω

_{n+}

_{1}+ ω

_{n}], β(x) = 35x

^{4}− 8x

^{5}+ 70x

^{6}− 20x

^{7}.

_{1}, P

_{2}, …, P

_{L}, and thus L decomposition results can be got. Each result is comprised of some component modes, and the mode corresponding to the biggest FER value FER

_{coefficient, max}is compared with that corresponding to the second biggest FER value FER

_{coefficient, max}, and the comparison result is denoted in Equation (11).

_{coefficient}= (FER

_{coefficient, max}− FER

_{coefficient, secondmax})/FER

_{coefficient, secondmax}

_{coefficient}is denoted as A

_{max}, and thus A

_{max}corresponds to the best decomposition result, and the corresponding threshold value is the best decomposition value. Therefore, the mode to FER

_{coefficient, max}contains the richest fault feature information.

#### 2.3. The Flowchart of IEWT

## 3. Results and Discussion

#### 3.1. Simualtion Study

#### 3.1.1. The Simulated Signal

_{1}(t) + x

_{2}(t)

_{1}(t) and x

_{2}(t). x

_{1}(t) is used to simulate an impact signal caused by a fault, and it is an impact signal with periodic exponential attenuation, and its periodicity is 16 Hz, and attenuation function is e

^{−100t}sin(510πt) in one periodicity. x

_{2}(t) is a cosine signal, and its periodicity is 20 Hz, and x

_{2}(t) is used to simulate an interference signal of low frequency harmonic. The sampling frequency is 10,240 Hz and sampling time is 1 s.

#### 3.1.2. The Simulated Result Analysis Based on EWT

_{3}in the time domain. Compared with the wave of x

_{1}(t) in Figure 4b, it is shown that the wave of F

_{3}is distorted. Thus, the periodic impact feature information of F

_{3}is much different from that of x

_{1}(t).

_{2}in the time domain, and it is seriously interfered with. Compared with the wave of x(t) in Figure 4a, the feature information of F

_{2}is unexpectedly high similar with that of original signal x(t), but it is known that F

_{2}is just a mode obtained by EWT, and thus the decomposition is wrong.

_{1}in the time domain, and its amplitudes are very small. By comparing it with the waves of the three signals of x(t), no signal is the same as the F

_{1}, and its wave is also seriously distorted.

_{3}is a little similar with the impact signal x

_{1}(t) among the three modes.

_{3}in the frequency domain. The fault feature information at fault feature frequency 16 Hz and its harmonics 32 Hz, 48 Hz, 64 Hz, 80 Hz and 96 Hz are all obvious.

_{3}is a little similar with the impact signal x

_{1}(t) in the time domain, there are only two signals in the simulated signal x(t), and there are three modes of F

_{1}, F

_{2}, and F

_{3}in the decomposition result, and thus x(t) is over-decomposed by EWT.

#### 3.1.3. The Simulated Result Analysis Based on IEWT

_{coefficient}, and threshold value THVA is set as coefficient × mean(P

_{coefficient}), where coefficient is an integer and mean (P

_{coefficient}) is mean spectrum value of P

_{coefficient}.

_{1}and F

_{2}in each decomposition result, which means that there are two contiguous segments and three boundaries. Thus, there is no over-decomposition and mode mixing, and it is no need to compute comparison between the FER

_{coefficient, max}and FER

_{coefficient, secondmax}in Step 4.

_{2}(the highest-order mode) is the biggest in each result, which means that each F

_{2}contains the richest fault feature information. FER value of each F

_{2}in all decomposition results is revealed in Figure 8.

_{2}is shown in Figure 9a, and it displays the periodic impact feature information in the time domain. It is compared with the wave of x

_{1}(t) in Figure 4b, and there is nearly no difference between them. Therefore, it is concluded that periodic impact feature information of F

_{2}is high similar with that of x

_{1}(t) in the time domain.

_{1}, and it is contaminated by little interferences. By comparing the wave of F

_{1}with that of x

_{2}(t) in Figure 4c, it can be seen that feature information of F

_{1}is high similar with that of cosine signal x

_{2}(t) in the time domain.

_{2}in the frequency domain. The amount of the fault feature information at fault feature frequency 16 Hz and its harmonics 32 Hz, 48 Hz, 64 Hz, 80 Hz and 96 Hz is very large, thus the fault feature information is extracted effectively, and there are nearly no interference components in other frequencies.

_{2}got by IEWT and F

_{3}got by EWT are modes which contain the most of fault feature information, thus the two are compared with each other in the time and frequency domains. The similarity between time waves of F

_{2}and that of x

_{1}(t) is higher than that between time waves of F

_{3}and that of x

_{1}(t) in the time domain; the fault feature information amount of F

_{2}is larger than that of F

_{3}in the frequency domain.

#### 3.2. Application to Fault Signals of Hydraulic Pump

#### 3.2.1. Experimental Scheme

_{z}at frequency of 50 kHz. The two kinds of fault feature frequency are 171.5 Hz [35]. The experimental system is displayed in Figure 11.

#### 3.2.2. The Application to the Loose Slipper Fault Signal Based on EWT

_{13}corresponds to maximum FER value, thus the mode contains the largest amount of fault feature information, and F

_{13}is displayed in Figure 13.

_{13}is displayed, and the periodic impact feature information is a little obvious, but the periodicity and amplitude information is very irregular. The fault feature information of F

_{13}in Figure 13a is low, similar with that of the original loose slipper fault signal in Figure 12.

_{13}is displayed in Figure 13b. In the frequency domain, the fault feature information at fault feature frequency 171.5 Hz and some of its harmonics is obvious, and there are some noises in the other frequencies.

#### 3.2.3. The Application to the Loose Slipper Fault Signal Based on IEWT

_{coefficient}, the threshold value THVA is set as coefficient × mean (P

_{coefficient}), coefficient is an integer, and mean (P

_{coefficient}) is mean value of P

_{coefficient}.

_{coefficient, max}of the highest-order mode is compared with FER

_{coefficient, secondmax}of a certain mode in each result. If the above two FERs are very close, there is a real possibility that there is over-decomposition and mode mixing in this result. Comparison result of FER

_{coefficient, max}and FER

_{coefficient, secondmax}in each result is demonstrated in Figure 16.

_{max}= 46.89% in Figure 16, and it can be also seen that F

_{11}is the highest-order mode in Figure 14, which means that there are 11 contiguous segments and 12 boundaries in the result. FER

_{8, max}corresponding to F

_{11}is 0.5644, and FER

_{4, max}corresponding to F

_{4}is 0.3842. FER

_{8, max}is 46.90% bigger than FER

_{4, max}, and thus there is a real possibility that there is no over-decomposition and mode mixing in the case of coefficient = 8. The best decomposition result of IEWT can be obtained, as displayed in Figure 17 and Figure 18.

_{11}is more obvious than that of F

_{1}–F

_{10}, and that of F

_{11}is high similar with that of the original loose slipper fault signal in Figure 12. The spectrum of F

_{11}is shown in Figure 18a, the amplitudes at fault feature frequency 171.5 Hz and its harmonics are all obvious; in Figure 18a–k, the amplitudes at the above frequencies are not all extracted in the spectrum of F

_{1}–F

_{10}, and there are many interference components in other frequencies. Thus, F

_{11}contains the largest amount of fault feature information.

_{13}got by EWT in Figure 13a, the periodic impact feature information of F

_{11}got by IEWT is very obvious and regular in Figure 17a, and the amplitudes of F

_{11}are also much higher than those of F

_{13}. By comparing with F

_{13}got by EWT in Figure 13b, it can be seen that the amplitudes of F

_{11}got by IEWT at the fault feature frequency and its harmonics are all extracted, and the amount of fault feature information is much larger than that of F

_{13}.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

IEWT | improved empirical wavelet transform |

EWT | empirical wavelet transform |

FER | feature energy ratio |

WT | wavelet transform |

STFT | short-time Fourier transform |

EEMD | ensemble empirical mode decomposition |

SVR | support vector regression |

CAFs | cyclic autocorrelation functions |

WPT | wavelet packet transform |

ELM | extreme learning machine |

LMD | local mean decomposition |

LTSA | local tangent space alignment |

FFT | fast Fourier transform |

IMF | intrinsic mode function |

EMD | empirical mode decomposition |

THVA | threshold value |

P_{coefficient} | a new spectrum distribution got after threshold processing |

FER_{coefficient, max} | the biggest FER value in case of the coefficient in a decomposition result |

FER_{coefficient, secondmax} | the second biggest FER value in case of the coefficient in a decomposition result |

coefficient | an integer and equals to 1, 2, …, L |

A_{coefficient} | comparison result between FER_{coefficient, max} and FER_{coefficient, secondmax} |

A_{max} | maximum value of A_{coefficient} |

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**Figure 5.**The results of the simulated signal based on empirical wavelet transform (EWT). (

**a**) F

_{3}; (

**b**) F

_{2}; (

**c**) F

_{1}.

**Figure 8.**The feature energy ratio (FER) value distribution diagram of all F

_{2}based on the simulated signal.

**Figure 9.**The best decomposition results of the simulated signal based on IEWT in the time domain. (

**a**) F

_{2}; (

**b**) F

_{1}.

**Figure 11.**The experiment system of the swash plate axial plunger pump. (

**a**) Schematic diagram; (

**b**) swash plate axial plunger pump; (

**c**) data acquisition equipment.

**Figure 15.**The FER value distribution diagram of all highest-order modes based on the loose slipper fault signal.

**Figure 17.**The best decomposition result of loose slipper fault signal based on IEWT in the time domain. (

**a**) F

_{11}; (

**b**) F

_{10}; (

**c**) F

_{9}; (

**d**) F

_{8}; (

**e**) F

_{7}; (

**f**) F

_{6}; (

**g**) F

_{5}; (

**h**) F

_{4}; (

**i**) F

_{3}; (

**j**) F

_{2}; (

**k**) F

_{1}.

**Figure 18.**The best decomposition results of the loose slipper fault signal based on IEWT in the frequency domain. (

**a**) F

_{11}; (

**b**) F

_{10}; (

**c**) F

_{9}; (

**d**) F

_{8}; (

**e**) F

_{7}; (

**f**) F

_{6}; (

**g**) F

_{5}; (

**h**) F

_{4}; (

**i**) F

_{3}; (

**j**) F

_{2}; (

**k**) F

_{1}.

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Zheng, Z.; Wang, Z.; Zhu, Y.; Tang, S.; Wang, B.
Feature Extraction Method for Hydraulic Pump Fault Signal Based on Improved Empirical Wavelet Transform. *Processes* **2019**, *7*, 824.
https://doi.org/10.3390/pr7110824

**AMA Style**

Zheng Z, Wang Z, Zhu Y, Tang S, Wang B.
Feature Extraction Method for Hydraulic Pump Fault Signal Based on Improved Empirical Wavelet Transform. *Processes*. 2019; 7(11):824.
https://doi.org/10.3390/pr7110824

**Chicago/Turabian Style**

Zheng, Zhi, Zhijun Wang, Yong Zhu, Shengnan Tang, and Baozhong Wang.
2019. "Feature Extraction Method for Hydraulic Pump Fault Signal Based on Improved Empirical Wavelet Transform" *Processes* 7, no. 11: 824.
https://doi.org/10.3390/pr7110824