Research on the fault diagnosis method of drying roller based on VMD-RF

Abstract

The Variational Mode Decomposition (VMD) method is a signal processing technique commonly used for decomposing complex signals, especially for analyzing time-series signals (such as speech signals and vibration signals). It has been widely applied in signal diagnosis. A fault diagnosis method based on Variational Mode Decomposition (VMD) and Random Forest (RF) is proposed for the problem that early faults are difficult to diagnose when the drying roller is running with load. The method primarily applies VMD to decompose the raw vibration acceleration signal, breaking down the complex base frequency signal of the drying drum into a series of intrinsic mode functions in order to find the optimal frequency band decomposition. The decomposed modal signals are then synthesized to reconstruct the original signal. Finally, the method employs Random Forest, Decision Trees, and K-Nearest Neighbors for training. To eliminate random errors and uncertainties from the experiments, 20 repeated tests are conducted to establish a classification model. Vibration signals from three different models of drying drums were collected to further verify the generality and engineering application value of the VMD-RF-based fault diagnosis method. Experimental results show that the VMD-RF method has a higher overall recognition rate for early faults in rotary dryers, while also demonstrating good applicability to various models of rotary dryers.

Keywords

Variational mode decomposition random forest fault diagnosis drying roller

1. Introduction

Material drying technology has a wide range of applications in major industrial industries and fields, and the drying roller is primarily used in the mixing of materials. It is a key component in mixing equipment. Its main structure consists of the feed-end seat, discharge-end seat, drum frame, drum body structure, and drive motor. The drying material is moved by the rotational motion of the drum, coming into contact with the hot air or flame generated inside, which heats the material to form the dried product (Hou and Yang, 2019; Wang, 2021). As a large rotating machine, drying drums are highly susceptible to faults during operation due to factors such as processing precision, uneven material distribution, localized vibrations, and poor contact. The main causes of failure are the misalignment of the rollers, leading to axial displacement of the drum body, and poor contact between the drum rings. There are many mechanical fault diagnosis techniques. For instance, Li et al., (2024) have proposed SHO-mvMSloEn, which significantly improves efficiency in the nonlinear dynamic analysis of mechanical signals. At the same time, they extended the concept of dispersion entropy and its multi-scale form, and verified that the multi-scale version of EDisEn achieves the highest recognition rate in various situations due to its more accurate entropy estimation (Li et al., 2025), while requiring fewer features to reach the highest recognition rate.

Huang proposed an adaptive time-frequency analysis technique for processing nonlinear and non-stationary signals—Empirical Mode Decomposition (EMD) (Jin and Li, 2023: 888–892). EMD decomposes complex signals into a series of simple components with different frequency characteristics, known as Intrinsic Mode Functions (IMFs). It has been widely applied in the field of fault diagnosis (Chi et al., 2020: 9–16; Hu et al., 2019: 9–18), extracting fault-related feature information to help detect early faults or abnormal patterns. However, EMD still has some issues, such as endpoint effects, mode mixing, and limitations in adaptability. To address the problem of mode mixing in EMD, Wu and Huang proposed the Ensemble Empirical Mode Decomposition (EEMD) (Jia et al., 2020), but residual white noise remains in the decomposition process. Yeh and Huang et al. proposed Complementary Ensemble Empirical Mode Decomposition (CEEMD) (Chen et al., 2020), which involves adding random white noise of the same length as the original signal during the EMD decomposition process, and performing the decomposition multiple times. The overall mean of the decomposition results is taken as the final output, effectively suppressing mode mixing. Various methods derived from EMD can effectively perform multi-modal decomposition of the original signal, but they suffer from drawbacks such as high computational cost, lack of mathematical theoretical support, and the inability to manually select the required number of decomposition modes. In 2013, Gilles proposed the Empirical Wavelet Transform (EWT), a new method based on wavelet decomposition, with the goal of achieving adaptability similar to that of EMD in wavelet transform. EWT can extract the oscillation amplitude and frequency components of a signal, addressing some of the limitations of EMD. However, the scale-space function of EWT is susceptible to noise interference, leading to the emergence of invalid components. Additionally, excessive decomposition may occur during the decomposition process (Kong et al., 2024). Inspired by Huang and Gilles, Drago and others proposed the Variational Mode Decomposition (VMD) method (Li et al., 2020), which uses iterative search and optimization to find the best modal functions in the frequency domain, thereby reducing errors. VMD is supported by solid mathematical theory and performs well in terms of noise robustness, particularly for narrowband signals with non-overlapping frequency ranges. Wang employed VMD to extract features of rotor system rubbing faults (Wang et al., 2015), studied the equivalent filtering characteristics of VMD, and demonstrated that VMD is more effective than EWT, EMD, and EEMD in detecting multiple features caused by rubbing faults. Zhang et al., (2017) compared VMD and EMD in fault diagnosis of rolling bearings in multistage centrifugal pumps, and the simulation results indicated that VMD outperforms EMD in fault diagnosis. An and Zhang (2017) proved that VMD performs better than EMD and wavelet transform in diagnosing base looseness faults. VMD has also been combined with other methods for fault diagnosis of rotating machinery under harsh working conditions. However, in practical applications, the modulation capability of VMD heavily depends on the selection of two internal parameters: the secondary penalty factor $α$ and the number of decomposition layers $K$ . The different configurations of $α$ and $K$ directly affect the decomposition output and the final accuracy and efficiency of the decomposition, making the optimization of these parameters essential. Shuqin Liu et al. obtained the optimal solution for VMD through iterative computation, while Chao Zhang et al. improved VMD by optimizing it with Particle Swarm Optimization (PSO) to adaptively find the optimal parameter pair (Zhang and Huang, 2024). These adaptive time-frequency analysis methods have good applications in fault diagnosis, but there are still some issues. Despite VMD’s excellent noise robustness and strong mathematical theoretical support, its strong dependence on parameter selection and the complex optimization process remain challenges. Overall, it is crucial to select the appropriate time-frequency analysis method based on the specific application scenario and further improve the method.

Random Forest is a powerful and flexible machine learning algorithm that makes predictions by constructing an ensemble of decision trees and combining their results. In this approach, the tests of each decision tree are independent of each other. A single decision tree is prone to overfitting the training data, while a Random Forest, composed of many independent decision trees, mitigates this issue by aggregating the classification results of each tree through statistical voting or weighted averaging to form the final classification outcome. This approach provides high classification accuracy and effectively avoids the problem of overfitting on the training data. Fault diagnosis methods based on Random Forest (Chen et al., 2021; Fan et al., 2020; Lin et al., 2024; Li et al., 2024) have already been successfully applied in the mechanical field.

In actual production processes, drying drums are heavy, with complex overall structures. The vibration signals generated under load are often accompanied by strong background noise and severe coupling, making it difficult to detect weak fault features in the signal, which complicates early fault diagnosis of the equipment. Traditional fault diagnosis methods for drying drums mainly rely on manual recognition, heavily depending on human experience, and their diagnostic effectiveness is poor, especially in identifying early faults (Cheng, 2019; Jiang and Shi, 2019; Xu, 2016). Although some scholars have used various methods to diagnose faults in equipment similar to drying drums (Bencheikh et al., 2020; Mei et al., 2022; Yuan, 2023), such as rotary kilns, the research on drying drums is relatively scarce due to their large size. For example, research on the GT10.0 type drying drum primarily addresses fault diagnosis of raw vibration signals that are nonlinear, non-stationary, and severely coupled, with weak fault features. A novel fault diagnosis method based on VMD-RF for intelligent fault diagnosis of drying drums has been proposed. By decomposing the complex vibration signals of the drying drum using Variational Mode Decomposition (VMD), clearer and more physically meaningful frequency-domain and time-domain features can be extracted. These features are then input into a machine learning model for classification, enabling fault prediction and the diagnosis of the health state of the drying drum equipment.

2. Diagnostic method design

2.1. Theoretical design of the fault diagnosis method

The fault diagnosis process based on VMD-RF is shown in Figure 1. The time-domain statistical feature formulas are listed in Table 1, and the frequency-domain statistical feature formulas are listed in Table 2. The specific experimental procedure is as follows:

Figure 1.

The flow chart of fault diagnosis based on VMD-RF.

Table 1.

Time-domain statistical characteristics.

Feature parameters	Formula	Feature parameters	Formula
Maximum value	$T_{1} = Max (x (i))$	Skewness	$T_{8} = \frac{\sum_{i = 1}^{N} {[x (i) - T 3]}^{3}}{N}$
Peak value	$T_{2} = Max (x (i)) - Min (x (i))$	Kurtosis	$T_{9} = \frac{\sum_{i = 1}^{N} {[x (i) - T 3]}^{4}}{N}$
Mean value	$T_{3} = \frac{1}{N} \sum_{i = 1}^{N} x (i)$	Form factor	$T_{10} = \frac{T_{7}}{\| T_{3} \|}$
Standard deviation	$T_{4} = \sqrt{\frac{1}{N - 1} \sum_{i = 1}^{N} {[x (i) - T 3]}^{2}}$	Peak factor	$T_{11} = \frac{T_{1}}{T_{7}}$
Variance	$T_{5} = \frac{1}{N - 1} \sum_{i = 1}^{N} {[x (i) - T 3]}^{2}$	Pulse factor	$T_{12} = \frac{T_{1}}{\| T_{3} \|}$
Minimum value	$T_{6} = Min (x (i))$	Margin factor	$T_{13} = \frac{T_{1}}{{(\frac{1}{N} \sum_{i = 1}^{N} \sqrt{\| x (i) \|})}^{2}}$
Root mean square value	$T_{7} = \sqrt{\frac{1}{N} \sum_{i = 1}^{N} {(\| x (i) \|)}^{2}}$	—	—

Table 2.

Frequency domain statistical characteristics.

Feature parameters	Formula	Feature parameters	Formula
Amplitude mean value	$F_{1} = \frac{1}{N} \sum_{i = 1}^{N} s (k)$	Amplitude skewness	$F_{6} = \frac{\sum_{i = 1}^{N} {(s (k) - F 1)}^{3}}{{(F 2)}^{3}}$
Amplitude RMS value	$F_{2} = \sqrt{\frac{\sum_{i = 1}^{N} {[s (k) - F 1]}^{2}}{N - 1}}$	Amplitude kurtosis	$F_{7} = \frac{\sum_{i = 1}^{N} {(s (k) - F 1)}^{4}}{{(F 2)}^{4}}$
Center frequency	$F_{3} = \frac{\sum_{i = 1}^{N} f_{k} s (k)}{\sum_{i = 1}^{N} s (k)}$	Frequency skewness	$F_{8} = \frac{\sum_{i = 1}^{N} {(f_{k} - F 3)}^{3} \cdot s (k)}{N \cdot {(F 5)}^{3}}$
Frequency STD	$F_{4} = \sqrt{\frac{\sum_{i = 1}^{N} {(f_{k} - F 3)}^{2} \cdot s (k)}{\sum_{i = 1}^{N} s (k)}}$	Frequency kurtosis	$F_{9} = \frac{\sum_{i = 1}^{N} {(f_{k} - F 3)}^{4} \cdot s (k)}{N \cdot {(F 5)}^{4}}$
Frequency RMS	$F_{5} = \sqrt{\frac{\sum_{i = 1}^{N} {f_{k}}^{2} s (k)}{\sum_{i = 1}^{N} s (k)}}$	Maximum magnitude	$F_{10} = Max (s (k))$

Under different operating conditions of the drying drum, vibration signal samples are collected. The vibration acceleration signal from a single sensor is decomposed into a finite number of IMF components using VMD. Time-domain and frequency-domain features are extracted from each IMF component. These features are then concatenated to form a high-dimensional feature vector, which is input into a machine learning model for training. The resulting trained model is used to diagnose the fault state of the drying drum.

2.2. Drying roller fault signal acquisition

Taking the GT10.0 type drying drum as the research object, the signal acquisition system of the drying drum consists of an NI data acquisition card, an IEPE piezoelectric accelerometer, an industrial computer, and a host computer. A material conveying platform is set up, and fault signal collection is carried out under an operating condition with a load of 11.5 t/h. The overall test platform is shown in Figure 2(a), the acquisition equipment used is shown in Figure 2(b), and the main signal acquisition parameters are listed in Table 3.

Figure 2.

Overall experiment platform (a) and Signal acquisition equipment (b).

Table 3.

Main acquisition parameters.

Acquisition parameters	Parameter value
Sensor sensitivity	500mv/g
Sensor resolution	0.0004 m/ $s^{2}$
Sampling frequency	2048 Hz

As a large rotating machine, the drying drum is highly prone to failures during operation due to factors such as processing accuracy, uneven internal material distribution, local vibrations, and poor contact. Based on engineering experience, the main causes of failure in drying drums are typically due to drum body movement caused by roller misalignment and poor contact of the roller rings. By manually adjusting the roller positions, several fault conditions of the drying drum are obtained, including a roller displacement of 5 mm on one side, 5 mm on both sides, 3 mm on the left side, and 6 mm on the right side, and bearing seat height adjustment at the roller position. Signal acquisition is performed under these fault conditions on the overall test platform. Given that the position adjustment is made at the roller, accelerometers are installed at the roller supports. The structure of the drying drum is shown in Figure 3(a), and the positions of the accelerometers are shown in Figure 3(b).

Figure 3.

GT10.0 drying roller (a) and Acceleration sensors placement (b).

2.3. VMD key parameter selection

The variational mode decomposition (VMD) can be viewed as an implementation of an adaptive optimal Wiener filter bank. It uses iterative search and optimization to find the best mode functions in the frequency domain. This method leverages the narrowband characteristics of signal components, allowing it to adaptively select the corresponding frequency bands. It is assumed that the original signal is decomposed into several variational mode components, each of which is a narrowband signal centered around its respective central frequency. A constraint model is established based on the narrowband condition of each component (Fan et al., 2021):

{\begin{cases} \min_{{u_{k}}, {ω_{k}}} {\sum_{k} {‖ \partial_{t} [(δ (t) + \frac{j}{π t}) * u_{k} (t)] e^{- j ω_{k} t} ‖}_{2}^{2} \\ s . t . \sum_{k = 1}^{K} u_{k} = f (t) \end{cases}

(1)

In the equation,

{u_{k}}

represents the k-th mode function,

{ω_{k}}

denotes the central frequency of the k-th mode,

δ (t)

is the Dirac function, and

*

is the convolution sign. Additionally, penalty parameters are introduced to control the bandwidth and smoothness of the modes, and Lagrange multipliers are used to constrain the signal reconstruction. These are employed to solve the optimal constraint model.

Step 1: Perform spectrum estimation on the original signal to obtain the initial spectral components. Initialize ${{\hat{u}}_{k}^{1}}$ 、 ${{\hat{ω}}_{k}^{1}}$ 、Lagrange multiplier ${\hat{λ}}^{1}$ 、number of iterations $n = 0$ .

Step 2: Through an iterative optimization process, continuously adjust the amplitude and phase of each frequency component to minimize the residual energy between the signal and the modal function. The principle of variational methods is applied, and the modal function is updated by solving a constrained optimization problem. Number of iterations $n = n + 1$ perform optimization process.

Step 3: To control the number and smoothness of the modal functions, a regularization term is introduced to constrain the optimization problem. Update ${u_{k}}$ and ${ω_{k}}$ according to equation (2) and equation (3).

{\hat{u}}_{k}^{n + 1} (ω) = (\hat{x} (ω) - \sum_{i \neq k} {\hat{u}}_{i} (ω) + \frac{\hat{λ} (ω)}{2}) \frac{1}{1 + 2 α {(ω - ω_{k})}^{2}}

(2)

{\hat{ω}}_{k}^{n + 1} = \frac{\int_{0}^{\infty} ω | {\hat{u}}_{k} (ω) | d ω}{\int_{0}^{\infty} {| {\hat{u}}_{k} (ω) |}^{2} d ω}

(3)

Step 4:Update $λ$ according to equation (4).

{\hat{λ}}^{n + 1} (ω) = {\hat{λ}}^{n} (ω) + τ (\hat{x} (ω) - \sum_{k} {\hat{u}}_{k}^{n + 1} (ω))

(4)

Step 5: Repeat steps 1 to 4. The above process is repeated, and the iterative procedure continues until the stopping criterion in equation (5) is met. Once the stopping condition is satisfied, the iteration halts, and the loop terminates, yielding a set of intrinsic mode functions (IMFs) with distinct frequency characteristics.

\sum_{k} \frac{{‖ {\hat{u}}_{k}^{n + 1} - {\hat{u}}_{k}^{n} ‖}_{2}^{2}}{{‖ {\hat{u}}_{k}^{n} ‖}_{2}^{2}} < ε

(5)

3. Experimental analysis

3.1. Selection of key parameters for VMD

In the VMD algorithm, the signal decomposition result is highly dependent on the number of modes $K$ and the penalty parameter $α$ . The value of $K$ affects the number of IMF components, while the penalty parameter $α$ influences the bandwidth of these components. Studies (Sadoughi and Hu, 2019) have found that when $K$ is chosen too small, the original signal is decomposed into fewer and coarser frequency bands, leading to the loss of detailed information in the signal. On the other hand, when $K$ is too large, it results in more modes, making the decomposition overly complex and leading to frequency aliasing. A smaller $α$ value allows modes to have wider bandwidths, while a larger $α$ value reduces the bandwidth of the spectrum, making each mode smoother. However, if the bandwidth of each component is too small, incomplete signal decomposition may occur, resulting in the loss of some signal information. Therefore, the penalty parameter $α$ is typically controlled within the range of 1000 to 3000. By adjusting the penalty parameter $α$ , this method can be made more universally adaptable (Fan et al., 2021; Zhou et al., 2020).

For VMD decomposition, vibration acceleration signals under normal operating conditions were selected, and the central frequencies for different values of

K

were calculated, as shown in Table 4. The correlation coefficients between the modes for the same value of

K

were also computed, as shown in Table 5. As seen in Table 4, when

K

exceeds 5, the central frequency of IMF5 is 791.14 Hz, and the central frequency of IMF6 is 880.24 Hz. The difference in their central frequencies is small. Furthermore, according to the data in Table 5, when

K

exceeds 5, the correlation coefficients R12 and R23 show significantly larger differences compared to other correlation coefficients, indicating local correlations between the modes, which leads to mode aliasing. Therefore, the optimal choice for the number of modes is 5, and the penalty parameter should be set to 2000. The VMD decomposition results are shown in Figure 4.

Table 4.

Center frequency under different values.

K	Center frequency/Hz
K	IMF1	IMF2	IMF3	IMF4	IMF5	IMF6
2	98.066	570.108	—	—	—	—
3	71.626	197.355	876.301	—	—	—
4	70.888	195.072	535.975	876.545	—	—
5	70.272	192.934	466.987	598.809	876.801	—
6	65.703	169.135	282.439	532.822	791.140	880.244

Table 5.

Inter-modal correlation coefficients under different values.

K	R12	R23	R34	R45	R56
2	0.0137	—	—	—	—
3	0.0981	0.0034	—	—	—
4	0.0979	0.0209	0.0155	—	—
5	0.0978	0.0263	0.0988	0.0218	—
6	0.1086	0.1144	0.0385	0.0230	0.0587

Figure 4.

Vmd result.

3.2. GT10.0 loaded drying roller fault diagnosis

Vibration signals were collected from the GT10.0-type drying drum under normal and early fault conditions, along with four other operating states, totaling five states. The sampling time was 180 seconds, with data recorded every second, each group containing 2048 data points. For each state, 180 groups of data were collected, amounting to a total of 900 groups. Each experiment was repeated 20 times to reduce random errors and variability, and the average identification accuracy was recorded. The specific experimental procedure is as follows:

Step 1: Perform VMD decomposition on the collected raw vibration acceleration signals under different operating conditions of the drying drum.

Step 2:After each training sample undergoes VMD decomposition, the original training signal is decomposed into five different modal components. For each modal component, 13 time-domain statistical features and 10 frequency-domain statistical features are extracted. Combine these features to form a high-dimensional feature vector.

Step 3:These features are input into three machine learning models: Random Forest (RF), Decision Tree (DT), and k-Nearest Neighbors (K-NN) for training. To eliminate random errors and variability.

Step 4:Each experiment is repeated 20 times, and a classification model is established. Use the validation dataset samples to verify the fault diagnosis performance of the obtained model.

The comparison of fault recognition results is shown in Figure 5. The fault diagnosis method based on VMD-RF achieved the highest identification rate using both time-domain index and frequency-domain index, and the comprehensive identification rate reached 99.45%.

Figure 5.

Drying roller fault identification results.

From the identified standard deviation data as shown in Table 6, the fault diagnosis method based on VMD-RF demonstrates a smaller standard deviation for both time-domain and frequency-domain indicators compared to the other two methods, indicating more stable performance.

Table 6.

Standard deviation of recognition results.

The standard deviation	Random forest	Decision tree	KNN
The time domain	0.1510	0.2751	0.5176
The frequency domain	0.1504	0.2370	0.5762
Comprehensive	0.1507	0.2561	0.5469

3.3. Multi-model drying roller fault diagnosis

Vibration signals of three different models of drying roller GT160F.0, GT320E4.0, and GT400CF.0, were collected to further verify the generality and engineering application value of the fault diagnosis method based on VMD-RF. According to the method described in Chapter 1, determine the number of variational modal decompositions

K = 5

, the penalty parameter

α = 2000

.The fault diagnosis method based on VMD-RF using time-domain indicators for three different drying roller achieved high recognition accuracy with a combined recognition rate of 97%; Only the GT160F.0 drying roller achieved 99.17% identification accuracy using the frequency-domain indicators diagnostic method, while the identification rate of the other two rollers was around 80%. The experiment results are shown in Table 7.

Table 7.

Multi-model drying roller identification results.

Model	Recognition accuracy %
Model	Time-domain indicators	Frequency-domain indicators
GT160F.0	97.50	99.17
GT320E4.0	94.79	81.77
GT400CF.0	97.22	79.17

4. Conclusion

Fault diagnosis methods for drying drums are typically limited to manually identifying severe faults in the later stages, with early faults often going unnoticed. These methods heavily rely on the experience of human experts, and the diagnostic effectiveness is suboptimal. This paper proposes a method for decomposing the fault diagnosis signals of a drying drum using Variational Mode Decomposition (VMD), obtaining a series of mutually orthogonal intrinsic mode functions, analyzing the different frequency components of the signal, and combining the Random Forest (RF) algorithm to find the optimal solution.

Through field testing and practical verification, the following conclusions have been drawn:

(1) The approach of using load-bearing signals has high research value, as it more closely reflects real operating conditions.

(2) The early fault signals of the drying drum are analyzed using the VMD (Variational Mode Decomposition) method, which makes it easier to capture potential fault characteristics and enhances the ability to understand and analyze the operational state of the equipment in more detail.

(3) For early faults caused by roller position misalignment in the drying drum, the proposed VMD-RF-based (VMD-Random Forest) diagnostic method achieves the highest recognition accuracy, with a comprehensive recognition rate of 99.45%. This indicates that the VMD-RF method is highly effective and accurate in distinguishing different types of faults, demonstrating its superiority in fault diagnosis.

(4) The VMD-RF-based fault diagnosis method was applied to multiple different models of drying drums. The time-domain and frequency-domain indicators obtained showed high recognition accuracy, with a comprehensive recognition rate of 97.50%. Through these experimental analyses, this indicates that the method has high versatility and promising engineering application prospects in the fault diagnosis of drying drums.

In summary, the proposed VMD-RF method demonstrates great potential in fault diagnosis for drying drums, especially in load-bearing conditions, where traditional methods may struggle. The approach can be extended to other rotating machinery, providing a robust and efficient way to detect early faults and improve maintenance practices across industrial equipment.

Footnotes

Author’s note

This manuscript has not been published or presented elsewhere in part or in entirety, and is not under consideration by another journal.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research is supported by “National Natural Science Foundation of China (No. 52275530)”, “Fujian Provincial Natural Science Foundation (NO. 2020J01068)”.

ORCID iD

Haiyan Jia

Data Availability Statement

The data that support the findings of this study are available from the corresponding author, [Wei Fan], upon reasonable request.*

References

Zhang

(2017) Pedestal looseness fault diagnosis in a rotating machine based on variational mode decomposition. Proceedings of the Institution of Mechanical Engineers - Part C: Journal of Mechanical Engineering Science 231(13): 2493–2502.

Bencheikh

Harkat

Kouadri

, et al. (2020) New reduced kernel PCA for fault detection and diagnosis in cement rotary kiln. Chemometrics and Intelligent Laboratory Systems 204: 104091.

Chen

Peng

Zhou

(2020) Signal decomposition and its application in mechanical fault diagnosis: a review. Journal of Mechanical Engineering 56(17): 91–107.

Chen

, et al. (2021) Vibration time-domain feature-based fault diagnosis method for marine rolling bearings. Machine Tool and Hydraulic Journal 49(14): 193–200.

Cheng

(2019) Operation stability inspection and adjustment method of drying roller. Internal Combustion Engine and Accessories Journal 09: 68–69.

Chi

Yang

Jiao

, et al. (2020) Pseudo-fault feature identification method for rolling bearings based on EMD-DCS. Journal of Vibration and Shock 9(9): 9–16.

Fan

Liu

, et al.(2020) Rolling bearing fault characterization method based on wavelet packet transform and random forest. Mechanical Design and Manufacture Journal 10: 59-63.

Fan

Wang

, et al. (2021) Linear vibrating screen excitation force unbalance fault diagnosis based on VMD-RQA. Vibration and Shock Journal 40(18): 25–32.

Hou

Yang

(2019) Analysis and control of energy saving and consumption reduction of sand and gravel aggregate drying equipment. China Equipment Engineering Journal 14: 60–61.

10.

Chen

, et al. (2019) Fault diagnosis method of planetary gearbox based on empirical mode decomposition and deep convolutional neural network. Journal of Mechanical Engineering 55(7): 9–18.

11.

Jia

Dong

, et al. (2020) A novel denoising method for vibration signal of hob spindle based on EEMD and grey theory. Measurement 169: 108490.

12.

Jiang

Shi

(2019) Analysis culation and improvement of fracture and shaft. Construction Machinery Journal 50(2): 38–42.

13.

Jin

(2023) Theoretical framework for a succinct empirical mode decomposition. IEEE Signal Processing Letters 30: 888–892.

14.

Kong

Chen

Yang

, et al. (2024) Harmonic and interharmonic detection based on adaptive wavelet and improved EWT. Zhejiang Electric Power Journal 43(11): 97–105.

15.

Liu

, et al. (2020) An optimized VMD method and its applications in bearing fault diagnosis. Measurement 166: 108185.

16.

Mao

, et al. (2024a) Research on turbine vibration fault diagnosis based on improved random forest algorithm. Zhejiang Electric Power Journal 43(09): 107–116.

17.

Tang

Jiao

, et al. (2024b) Optimized multivariate multiscale slope entropy for nonlinear dynamic analysis of mechanical signals. Chaos, Solitons & Fractals: The interdisciplinary journal of Nonlinear Science, and Nonequilibrium and Complex Phenomena Journal 179: 114436. Epub ahead of print December 2023. DOI: 10.1016/j.chaos.2023.114436.

18.

, et al. (2025) Extended dispersion entropy and its multiscale versions: methodology and application. Communications in Nonlinear Science and Numerical Simulation Journal 141: 108497. Epub ahead of print 13 December 2024. DOI: 10.1016/j.cnsns.2024.108497.

19.

Lin

Cao

Yang

, et al. (2024) Fault diagnosis of ship ballast water system based on random forest. Ship Engineering Journal 46(11): 92–97.

20.

Mei

Tang

, et al. (2022) Research on fault diagnosis of cement production line rotary kiln bearings based on hybrid GA-BP. Shandong Industrial Technology Journal 03: 47–53.

21.

Sadoughi

(2019) Physics-based convolutional neural network for fault diagnosis of rolling element bearings. IEEE Sensors Journal 19(11): 4181–4192.

22.

Wang

(2021) Structural analysis and advantages of double-drum asphalt mixture mixing. Construction Machinery Journal 07: 34–37.

23.

Wang

Markert

Xiang

, et al. (2015) Research on variational mode decomposition and its application in detecting rub-impact fault of the rotorsystem. Mechanical Systems and Signal Processing 60-61: 243–251.

24.

(2016) Severe vibration overhaul treatment of drum dryer. Equipment management and maintenance Journal 05: 26–27.

25.

Yuan

(2023) Fault Diagnosis of Cement Rotary Kiln Based on Multivariate Data-Driven Approach. MA thesis. China: Qingdao University of Science and Technology.

26.

Zhang

Huang

(2024) Denoising method of parameter optimization VMD combined with ImprovedWavelet packet threshold. Noise and Vibration Control Journal 44(5): 128–132.

27.

Zhang

Jiang

Feng

(2017) Research on variational mode decomposition in rolling bearings fault diagnosis of the multistage centrifugal pump. Mechanical Systems and Signal Processing 93: 460–493.

28.

Zhou

Tang

(2020) Wind turbine gearbox unbalance fault feature extraction based on Improved VMD-based. Vibration and Shock Journal 39(05): 170–176.