*2.1. DPCA Method*

To identify the state of machinery, the mainly three steps are as follows: acquisition of monitoring signals, feature extraction of monitoring signals, and pattern recognition and diagnosis of the state are carried out. For the traction machine state recognition technology, the extraction of state features is hard work, which directly affects the accuracy of state diagnosis and the reliability of early prediction.

The state parameters of the tractor during operation are hidden in the raw signals. Therefore, the extraction of the state parameters has become an important factor affecting the accuracy of the state identification. Based on the feature extraction, various signal processing techniques have been developed, which mainly involved time domain analysis, frequency domain analysis, time-frequency analysis, etc. [30]. Although the above methods can extract the features of vibration signals, the collected vibration signals usually contain background noise and unknown frequency interference. The amplitude demodulation process (also known as high frequency resonance, resonance demodulation or envelope analysis) separates low frequency from high frequency background noise [31]. In this paper, the DPCA method was adopted for feature extraction [28]. DPCA algorithm mainly includes: time-frequency analysis, principal component analysis and feature extraction.

(1) Time frequency analysis.

When the traction machine operates stably, its key modulation component is modulation signal. The single component modulation signal of the traction machine can be expressed in Equation (1), which is mainly composed of modulation signal and carrier signal.

$$\mathfrak{x}(t) = \mathfrak{x}\_m(t)\mathfrak{x}\_c(t) \tag{1}$$

where *x*(*t*) is the amplitude modulation signal of the traction machine, *xm*(*t*) is the modulation signal, and *xc*(*t*) is the carrier signal.

The time-frequency distribution of the monitoring signal can be expressed in Equation (2).

$$P\_X(f,t) = \int\_{-\infty}^{\infty} x\_{\mathfrak{m}}(\tau) x\_{\mathfrak{c}}(\tau) w(t-\tau) e^{-j2\pi f\tau} d\tau \tag{2}$$

where *PX* (*f, t*) is the time-frequency distribution function of the monitoring signal, and *w* (*t*) is the window function of the STFT.

The STFT of the modulation signal model of the traction machine can be approximated as follows, as shown in Equation (3).

$$\int\_{-\infty}^{\infty} \mathbf{x}\_m(\tau) \mathbf{x}\_\varepsilon w(t - \tau) e^{-j2\pi f \tau} d\tau \approx \mathbf{x}\_m(\tau) \int\_{-\infty}^{\infty} \mathbf{x}\_\varepsilon(\tau) w(t - \tau) e^{-j2\pi f \tau} d\tau \tag{3}$$

The time spectrum of the modulated signal is further simplified to obtain Equation (4).

$$P(f,t) \approx \underset{-\infty}{\propto} \ x\_m(t) \int\_{-\infty}^{\infty} \mathbf{x}\_c(\tau) w(t-\tau) e^{-j2\pi f \tau} d\tau = \mathbf{x}\_m(t) P\_C(f,t) \tag{4}$$

where *P* (*f, t*) represents the time-frequency distribution function of the detection signal, *PC* (*f, t*) represents the time-frequency distribution function of the carrier signal.

(2) Principal component analysis.

Principal component analysis is a classical data dimensionality reduction method, which is mainly realized by the following algorithms.

Firstly, the covariance matrix is solved. The matrix formula is shown in Equation (5).

$$P\_{\rm cov} = \text{cov}(P(t, f))\tag{5}$$

where *P*cov represents the covariance matrix of the time-frequency distribution matrix, cov() represents the covariance operator.

Secondly, there is eigenvalue decomposition. As shown in Equation (6).

$$[\mathbf{V}, \mathbf{U}] = \text{eig}(P\_{\text{cov}}) \tag{6}$$

where eig() represents the eigenvalue decomposition operator. **V**, **U** represent eigenvalue matrix and eigenvector matrix respectively.

Thirdly, eigenvalue selection. The order of the selected eigenvalue is determined by the maximum value of the difference spectrum, as shown in Equation (7).

$$k \ge \left. i \right|\_{\max} (\mathcal{S}\_i = (\lambda\_i - \lambda\_{i+1})) \tag{7}$$

where *k* represents the order of the selected eigenvalue, δ*<sup>i</sup>* represents the difference spectrum value.

Finally, principal component reconstruction. The corresponding principal component modulation signal PPC*i*(*t*) can be obtained, as shown in Equation (8).

$$\text{PPC}\_{i}(t) = P(t, f)u\_{i} \tag{8}$$

(3) Feature extraction.

The principal component analysis method can be used to obtain the principal component of the monitoring signal, which includes the low-frequency modulation component of the monitoring signal. The characteristic modulation frequency can be extracted by frequency analysis, as shown in Equation (9).

$$P\_i(f) = \int\_{-\infty}^{\infty} \text{PPC}\_i(t) e^{-j2\pi ft} dt\tag{9}$$

#### *2.2. Elevator Traction Machine Parameters*

The model of tractor selected in the experiment is GETM3.DM. The detailed parameters are listed in Table 1.

**Table 1.** Parameters of elevator traction machine.


#### *2.3. Equipment Selection*

The test system was used for the state identification research of elevator traction machine, as shown in Figure 1. The instruments included vibration acceleration sensor, data acquisition instrument, computer, and other auxiliary instruments. The acceleration sensor was fixed to the traction machine, and the vibration signals collected by the sensor were transmitted to the data acquisition instrument.

**Figure 1.** Flow chart of traction machine vibration signal acquisition and analysis.

#### *2.4. Test Conditions*

In order to realize the recognition of different states of the traction machine, the test conditions involved in this paper are listed in Table 2.

**Table 2.** Test conditions.


#### **3. Case Analysis**

*3.1. Analysis of Influence of Elevator Running Speed on Main Engine Vibration*

To compare features extracted by the different signal analysis methods, FFT was used to transform the time domain signal into spectrum domain. Their peaks value of the spectrum under different conditions are recorded in Tables 3 and 4. Under the working condition of 1 m/s, the frequency spectrum, time-frequency spectrum, and DPCA result are shown in Figures 2–4, respectively. Under the working condition of 2.4 m/s, the frequency spectrum, time-frequency spectrum, and DPCA result are shown in Figures 5–7, respectively.

With the comparation of the FFT results, it can be found that operating speed has an impact on the amplitude of the vibration spectrums. The amplitude of each frequency under low-speed operation (1.0 m/s) was lower than the amplitude of each frequency under normal operation (2.4 m/s), as shown in Figures 2 and 5. In general, the peak frequency in vibration spectrum will increase with acceleration of the traction machine.

In the time-frequency spectrum shown in Figure 3, it can be found that the prominent frequency of the elevator machine at the operating speed of 1 m/s is approximately 25 Hz. While the prominent vibration frequency under normal speed (2.4 m/s) is approximately 50 Hz, as shown in Figure 6. This characteristic is positively related to the operating speed of the traction machine, which can be regard as the main feature of elevator vibration. The vibration level of the traction machine can be evaluated by the amplitude change of this frequency band in the time-frequency spectrum.

Through comparative analysis, the DCPA results shown in Figures 4 and 7a. The k refers to the serial number of principle frequency bands selected by Equation (7). The modulated frequency of elevator traction machine is indicated by *f* m, which can be found in each modulation spectrum. It can be observed that when *k* = 1, the amplitude modulation difference of *f* <sup>m</sup> under two working conditions is 2.58 times. This value is approximate to the speed ratio under two working conditions. Therefore, the mechanism of vibration level-up caused by the operating speed-up is the increase of modulation effect in principle frequency bands.

As a result, for the working conditions with obvious differences, such as the influence of different operating speeds of the elevator on the vibration signal of the traction machine, the difference between the two states could be obtained by analyzing the frequency-domain diagram through the FFT. The time-frequency diagram and the demodulation diagram can more clearly highlight the difference and complete the identification of the state of the traction machine.




**Table 4.** Frequency domain peak value of vibration response in different running directions of elevator.

**Figure 2.** Vibration spectrum diagram of main engine at elevator running speed of 1 m/s. (**a**) Motor with up-drive condition. (**b**) Motor with down-drive condition.

**Figure 3.** Time-frequency diagram of main engine vibration at elevator running speed of 1 m/s. (**a**) Motor with up-drive condition. (**b**) Motor with down-drive condition.

**Figure 4.** Vibration demodulation diagram of main engine when elevator speed is 1 m/s.

**Figure 5.** Vibration spectrum of main engine in different running directions of elevator (2.4 m/s). (**a**) Motor with up-drive condition. (**b**) Motor with down-drive condition.

**Figure 6.** Time frequency diagram of main machine vibration in different operating directions of elevator (2.4 m/s). (**a**) Motor with up-drive condition. (**b**) Motor with down-drive condition.

**Figure 7.** Vibration demodulation diagram under different operating conditions (2.4 m/s). (**a**) Motor with up-drive condition. (**b**) Motor with down-drive condition.

## *3.2. Analysis on the Influence of Elevator Running Direction on the Vibration of Main Engine*

The spectrum analysis about different running direction is shown in Figure 5 and Table 4. It can be found that the vibration response was largest at 100.8 Hz under up-drive operation, while the largest vibration response under down-drive operation was at 48.1 Hz.

When the frequency was 48.1 Hz, the peak value of the downlink is much larger than that of the uplink, which was twice that of the downlink. As the frequency was 144.5 Hz, the peak value of the uplink was much larger than that of the downlink, which was three times that of the downlink. Except for 48.1 Hz and 144.5 Hz, the characteristic frequency of the most obvious peak under the two working conditions of the elevator was basically unchanged, and the height of the main peak slightly changed.

According to the analysis of Figure 6, the frequency (rotation speed) of the elevator gradually increased from the start to a certain state and then remains stable. After a cycle of operation, the frequency gradually decreased. Comparing (a) and (b) in Figure 6, it can be found that the peak value of the uplink was greater than that of the downlink at 145 Hz, while the difference in other frequency bands were not significant.

From the time-frequency spectrum shown in Figure 6, it can only be concluded that the difference between the two working conditions was the most significant at the frequency of 145 Hz. However, it was not enough to support the identification of the elevator's up and down conditions. Therefore, based on the spectrum analysis of STFT, the modulation signal in the vibration signal of the traction machine was extracted by the PCA technology.

The vibration demodulation diagram of the main engine under different operating conditions of the elevator in Figure 7 was analyzed. When *k* = 1, there was little difference between the up-working condition and the down-working condition; when *k* = 2, the frequency modulation of one *f* m is generated more in the upstream working condition than in the downstream working condition; when *k* = 3, a *f* m frequency modulation is generated in the downstream working condition more than in the upstream working condition; and when *k* = 4, the uplink has a frequency modulation of *f* m and the downlink has a frequency modulation of 2*f* m.

For the up and down working conditions of the elevator, the influence on the traction machine was not obvious. Only the frequency domain diagram and the time-frequency diagram cannot accurately distinguish the two working conditions. Therefore, the demodulation method was used to separate the signal from the raw signal, highlight the weak state characteristic signal, and distinguish the states of different working conditions.

#### *3.3. Analysis of Influence of Elevator Load on Main Engine Vibration*

Taking the behavior under the elevator as an example, the measured time domain diagram was transformed into a spectrum diagram by FFT, as shown in Figure 8.

**Figure 8.** Vibration spectrum diagram of elevator main engine with different loads. (**a**) Motor running under 140 kg load condition. (**b**) Motor running under 325 kg load condition.

From the Figures 5b and 8 and Tables 4 and 5, it was hard to identify different working conditions only by using the frequency domain diagram. Then, the time-frequency spectrums obtained by STFT was analyzed, as shown in Figure 9.


**Figure 9.** Vibration time-frequency diagram of down main machine under different loads of elevator. (**a**) Motor running under no-load condition. (**b**) Motor running under 140 kg load condition. (**c**) Motor running under 325 kg load condition.

However, it was also difficult to distinguish the working conditions of different loads by time-frequency spectrums. Therefore, based on the spectrum obtained by STFT, the modulation signal in the vibration signal of the traction machine was extracted by the DPCA method.

From the vibration demodulation diagram of the elevator main engine in Figure 10, it can be found that the frequency modulation of 2*f* <sup>m</sup> was less than that of the other two working conditions when *k* = 1, and the frequency modulation of 2*f* <sup>m</sup> was less than that of the other two directions when *k* = 2. When the load was 325 kg and *k* = 3, the frequency modulation of *f* <sup>m</sup> was less than that of other working conditions. In addition, the amplitude corresponding to the common frequency modulation in the three working conditions increases with the increase of the load.

**Figure 10.** Vibration demodulation diagram of down main machine under different loads of elevator. (**a**) Motor running under no-load condition. (**b**) Motor running under 140 kg load condition. (**c**) Motor running under 325 kg load condition.

According to the working conditions of different loads of the elevator, the influence on the traction machine is not obvious. It was difficult to distinguish the two working conditions only through the frequency-domain diagram and the time-frequency diagram. Therefore, the demodulation method is used to separate the signal from the raw signal, highlight the weak state characteristic signal, and distinguish the states of different working conditions.

#### **4. Conclusions**

In this paper, the application of the DCPA method provides an alternative way to realize fast and effective condition monitoring of a traction machine, which could be extended to detect other background interference and typical faults. The conclusion is as follows:


**Author Contributions:** Conceptualization, D.L. and J.Y.; methodology, D.L.; software, D.L.; validation, D.L., J.Y. and Y.L.; formal analysis, D.L.; investigation, D.L.; resources, D.L.; data curation, D.L.; writing original draft preparation, D.L.; writing—review and editing, D.L.; visualization, D.L.; supervision, D.L.; project administration, D.L.; funding acquisition, J.Y. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by Open Research Project of the State Key Laboratory of Industrial Control Technology grant number [No. ICT2022B13]. The APC was funded by the Science and Technology Project of the State Administration for Market Regulation [No. 2021MK140].

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Data will be made available on request.

**Acknowledgments:** This work was supported by the Open Research Project of the State Key Laboratory of Industrial Control Technology, Zhejiang University, China (No. ICT2022B13) and the Science and Technology Project of the State Administration for Market Regulation (No. 2021MK140).

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

#### **References**

