*2.3. The Flowchart of IEWT*

Firstly, the power spectrum of the loose slipper fault signal is calculated. Secondly, the threshold processing is applied to eliminate the power spectrum of the interference components, and segments can be acquired. Thirdly, the signal can be decomposed by IEWT based on the segments. Lastly, the best decomposition result can be selected by *FER*.

The flowchart of IEWT is shown in Figure 3.

**Figure 3.** The flowchart of improved empirical wavelet transform (IEWT).

#### **3. Results and Discussion**

#### *3.1. Simualtion Study*

#### 3.1.1. The Simulated Signal

In order to verify the superiority and effectiveness of IEWT, a simulated signal is defined as

$$\mathbf{x}(t) = \mathbf{x}\_1(t) + \mathbf{x}\_2(t) \tag{12}$$

The signal consists of two component modes of *x*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−100*<sup>t</sup>* sin(510π*t*) 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.

The signals *x*(*t*), *x*1(*t*) and *x*2(*t*) are displayed in Figure 4.

**Figure 4.** The time domain wave of the simulated signal. (**a**) *x*(*t*); (**b**) *x*1(*t*); (**c**) *x*2(*t*).

3.1.2. The Simulated Result Analysis Based on EWT

For demonstrating the superiority and effectiveness of IEWT, the simulated signal is also decomposed by EWT. After the decomposition analysis, three modes can be obtained, which means that there are three contiguous segments and four boundaries. Thus, there is over-decomposition, and the decomposition result is shown in Figures 5 and 6.

**Figure 5.** The results of the simulated signal based on empirical wavelet transform (EWT). (**a**) *F*3; (**b**) *F*2; (**c**) *F*1.

**Figure 6.** Power spectrum density of *F*3.

Figure 5a shows the wave of *F*<sup>3</sup> in the time domain. Compared with the wave of *x*1(*t*) in Figure 4b, it is shown that the wave of *F*<sup>3</sup> is distorted. Thus, the periodic impact feature information of *F*<sup>3</sup> is much different from that of *x*1(*t*).

Figure 5b displays the wave of *F*<sup>2</sup> 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*<sup>2</sup> is unexpectedly high similar with that of original signal *x*(*t*), but it is known that *F*<sup>2</sup> is just a mode obtained by EWT, and thus the decomposition is wrong.

Figure 5c depicts the wave of *F*<sup>1</sup> 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.

Thus, it is shown that only *F*<sup>3</sup> is a little similar with the impact signal *x*1(*t*) among the three modes. Figure 6 shows the spectrum of *F*<sup>3</sup> 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.

It can be concluded from the above analysis that although the *F*<sup>3</sup> 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*<sup>3</sup> in the decomposition result, and thus *x*(*t*) is over-decomposed by EWT.

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

The power spectrum of the simulated signal is denoted as *Pcoe*ffi*cient*, and threshold value *THVA* is set as *coe*ffi*cient* × mean(*Pcoe*ffi*cient*), where *coe*ffi*cient* is an integer and mean (*Pcoe*ffi*cient*) is mean spectrum value of *Pcoe*ffi*cient*.

The simulated signal is decomposed by IEWT in [1 122] of the *coe*ffi*cient* range, and mode numbers of 122 decomposition results based on different *coe*ffi*cients* are displayed in Figure 7.

**Figure 7.** The distribution diagram of mode numbers based on the simulated signal.

It can be seen from Figure 7 that the signal can be effectively and correctly decomposed into only two modes of *F*<sup>1</sup> and *F*<sup>2</sup> 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 *FERcoe*ffi*cient, max* and *FERcoe*ffi*cient, secondmax* in Step 4.

In order to get the best decomposition result, *FER* of each mode is computed based on each of the 122 results. According to the above *FER*, it can be seen that the *FER* value of *F*<sup>2</sup> (the highest-order mode) is the biggest in each result, which means that each *F*<sup>2</sup> contains the richest fault feature information. FER value of each *F*<sup>2</sup> in all decomposition results is revealed in Figure 8.

**Figure 8.** The feature energy ratio (*FER*) value distribution diagram of all *F*<sup>2</sup> based on the simulated signal.

It can be known from Figure 8 that *FER* values in [1 122] of the *coe*ffi*cient* range are all above 0.9, and their trend tends to be stable in the range, and all of the biggest values are 0.936 in [61 122]. Thus, the best decomposition result is in [61 122], and all of the results are the same in the range, so only the best result based on *coe*ffi*cient* = 61 is shown in Figures 9 and 10.

**Figure 9.** The best decomposition results of the simulated signal based on IEWT in the time domain. (**a**) *F*2; (**b**) *F*1.

**Figure 10.** Power spectrum density of *F*2.

The wave of *F*<sup>2</sup> 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*<sup>2</sup> is high similar with that of *x*1(*t*) in the time domain.

Figure 9b displays the feature information of cosine curve in *F*1, and it is contaminated by little interferences. By comparing the wave of *F*<sup>1</sup> with that of *x*2(*t*) in Figure 4c, it can be seen that feature information of *F*<sup>1</sup> is high similar with that of cosine signal *x*2(*t*) in the time domain.

Figure 10 shows the spectrum of *F*<sup>2</sup> 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.

It can be known that *F*<sup>2</sup> got by IEWT and *F*<sup>3</sup> 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*<sup>2</sup> and that of *x*1(*t*) is higher than that between time waves of *F*<sup>3</sup>

and that of *x*1(*t*) in the time domain; the fault feature information amount of *F*<sup>2</sup> is larger than that of *F*<sup>3</sup> in the frequency domain.

From the above analysis, it is concluded that the segment can be set according to Fourier power density spectrum in the effective way, and the right mode number can be got, and then one mode which contains the richest feature information can be selected based on *FER*, and thus the simulated signal can be best decomposed by IEWT. Moreover, there is no over-decomposition and mode mixing, so IEWT performs much better than EWT.

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

#### 3.2.1. Experimental Scheme

An experiment was performed to swash plate axial plunger pump whose type was 10MCY14-1B. Its rotational speed was set as 1470 r/min, and outlet pressure of the pump was set as 15 MPa. The signals of loose slipper fault are sampled by accelerometer *az* 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.

**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.

The length of the loose slipper fault signal is 0.2 s, and the signal is shown in Figure 12.

**Figure 12.** The loose slipper fault signal.

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

In order to validate the superiority and effectiveness of IEWT, the loose slipper fault signal is firstly decomposed by EWT, and 58 modes can be obtained, so there is serious over-decomposition. The *F*<sup>13</sup> corresponds to maximum *FER* value, thus the mode contains the largest amount of fault feature information, and *F*<sup>13</sup> is displayed in Figure 13.

**Figure 13.** *F*<sup>13</sup> got based on EWT. (**a**) Time domain wave; (**b**) power spectrum density.

In Figure 13a, the wave of *F*<sup>13</sup> 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*<sup>13</sup> in Figure 13a is low, similar with that of the original loose slipper fault signal in Figure 12.

The spectrum of *F*<sup>13</sup> 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

IEWT is also applied to decompose the signal in [1 22] of the *coe*ffi*cient* range, where the power spectrum of the loose slipper fault signal is denoted as *Pcoe*ffi*cient*, the threshold value *THVA* is set as *coe*ffi*cient* × mean (*Pcoe*ffi*cient*), *coe*ffi*cient* is an integer, and mean (*Pcoe*ffi*cient*) is mean value of *Pcoe*ffi*cient*.

The mode number of each decomposition result is obtained based on the different coefficients, as displayed in Figure 14.

**Figure 14.** The distribution diagram of mode numbers based on loose slipper fault signal.

From Figure 14, the mode number decreases with the increase of *coe*ffi*cient* in all results. There are 36 modes in the case of *coe*ffi*cient* = 1, which means that there are 36 contiguous segments and 37 boundaries. All of the results have six modes in the case of *coe*ffi*cient* = 19–21. Thus, 36 modes mean that there is over-decomposition, and six modes indicate that there is mode mixing.

For the sake of obtaining the best decomposition result, *FER* of each mode is computed in each of all 22 results and the *FER* value of the highest-order mode is the biggest in each result. The *FER* value of each highest-order mode in all decompositions is displayed in Figure 15.

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

In the case of *coe*ffi*cient* ≥ 5, *FER* values change little and maintain at the maximum in Figure 15. Thus, the reasonable decomposition result is in [5 22].

In order to get the best decomposition result, it is necessary to figure out whether there ia over-decomposition and mode mixing in each result. *FERcoe*ffi*cient, max* of the highest-order mode is compared with *FERcoe*ffi*cient, 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 *FERcoe*ffi*cient, max* and *FERcoe*ffi*cient, secondmax* in each result is demonstrated in Figure 16.

**Figure 16.** The comparison distribution diagram of FER based on the loose slipper fault signal.

In the case of *coe*ffi*cient* = 8, it can be found that *Amax* = 46.89% in Figure 16, and it can be also seen that *F*<sup>11</sup> 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*<sup>11</sup> is 0.5644, and *FER*4*, max* corresponding to *F*<sup>4</sup> 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 *coe*ffi*cient* = 8. The best decomposition result of IEWT can be obtained, as displayed in Figures 17 and 18.

It can be seen from Figure 17 that periodic impact feature information of the highest-order mode *F*<sup>11</sup> is more obvious than that of *F*1–*F*10, and that of *F*<sup>11</sup> is high similar with that of the original loose slipper fault signal in Figure 12. The spectrum of *F*<sup>11</sup> 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*<sup>11</sup> contains the largest amount of fault feature information.

Compared with *F*<sup>13</sup> got by EWT in Figure 13a, the periodic impact feature information of *F*<sup>11</sup> got by IEWT is very obvious and regular in Figure 17a, and the amplitudes of *F*<sup>11</sup> are also much higher than those of *F*13. By comparing with *F*<sup>13</sup> got by EWT in Figure 13b, it can be seen that the amplitudes of *F*<sup>11</sup> 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.

It can be concluded from the above analysis that the right segment can be obtained according to Fourier power density spectrum, and then the effective mode number is obtained. Based on the mode number, the best decomposition of the loose slipper fault signal can be got by IEWT in the case of *coe*ffi*cient* = 8, and then the mode which contains the richest fault feature information can be selected based on *FER*. Moreover, it performs better than EWT.

**Figure 17.** *Cont.*

**Figure 17.** *Cont.*

**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.** *Cont.*

**Figure 18.** *Cont.*

**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.

#### **4. Conclusions**

When the hydraulic pump works, it is often faced with of high pressure and high speed working conditions. The vibration of the hydraulic pump is usually caused by mechanical and fluid impact, and the vibration is intensified if it is broken, and thus the fault vibration signal is contaminated by a lot of noises. The Fourier amplitude spectrum is sensitive to the noises, and the segment is got based the above spectrum of the contaminated fault signal in EWT, and thus the signal is decomposed in the wrong way.

Aiming to resolve the shortcomings of EWT, an improved method IEWT is proposed, and IEWT replaced the Fourier amplitude spectrum of EWT with power spectrum in acquiring the segment, and thus the bad influence of the interference on the segment acquirement is much reduced. Based on the right segment, the loose slipper fault signal can be decomposed by IEWT in the best way, and the mode that contains the most amount of the fault feature information can be selected based on *FER*. Therefore, mode-mixing and over-decomposition can be eliminated as much as possible, and IEWT performs much better than EWT.

**Author Contributions:** Conceptualization, Z.Z., Z.W., Y.Z., and S.T.; Methodology, Z.Z., and Y.Z.; Investigation, Z.Z., Y.Z., and S.T.; Writing-Original Draft Preparation, Z.Z., and Y.Z.; Writing-Review and Editing, Z.W., Z.Z., Y.Z., S.T. and B.W.

**Funding:** This research was funded by the Startup Foundation for the Doctors of North China University of Science and Technology (0088/28412499), the Cultivation Foundation of North China University of Science and Technology (JP201505), the National Natural Science Foundation of China (51505124, 51805214), and the China Postdoctoral Science Foundation (2019M651722).

**Conflicts of Interest:** The authors declare no conflict of interest.
