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Article

Step-Wise Parameter Adaptive FMD Incorporating Clustering Algorithm in Rolling Bearing Compound Fault Diagnosis

by
Shuai Xu
1,2,
Chao Zhang
1,2,*,
Jing Zhang
1,2,
Guiyi Liu
1,2,
Yangbiao Wu
1,2 and
Bing Ouyang
1,2
1
College of Mechanical Engineering, Inner Mongolia University of Science and Technology, Baotou 014010, China
2
Inner Mongolia Key Laboratory for Intelligent Diagnosis and Control of Electromechanical Systems, Baotou 014010, China
*
Author to whom correspondence should be addressed.
Symmetry 2024, 16(12), 1675; https://doi.org/10.3390/sym16121675
Submission received: 4 November 2024 / Revised: 9 December 2024 / Accepted: 16 December 2024 / Published: 18 December 2024
(This article belongs to the Section Engineering and Materials)
Browse Figures
Figure 1
<p>Filtering characteristics of FMD with different Hurst exponents: (<b>a</b>) <span class="html-italic">H</span> = 0.2; (<b>b</b>) <span class="html-italic">H</span> = 0.5; (<b>c</b>) <span class="html-italic">H</span> = 0.8; (<b>d</b>) <span class="html-italic">H</span> = 1.</p> "> Figure 2
<p>Filtering characteristics of FMD under different numbers of modes <span class="html-italic">n</span>: (<b>a</b>) <span class="html-italic">n</span> = 4; (<b>b</b>) <span class="html-italic">n</span> = 6.</p> "> Figure 3
<p>Filtering characteristics of FMD under different filter lengths <span class="html-italic">L</span>: (<b>a</b>) <span class="html-italic">L</span> = 60; (<b>b</b>) <span class="html-italic">L</span> = 40.</p> "> Figure 4
<p>Filtering characteristics of FMD under different numbers of segments <span class="html-italic">K</span>: (<b>a</b>) <span class="html-italic">K</span> = 10; (<b>b</b>) <span class="html-italic">K</span> = 7.</p> "> Figure 5
<p>The Robustness of Various Metrics under Different SNR: (<b>a</b>) Inner Race Fault; (<b>b</b>) Rolling Element Fault; (<b>c</b>) Outer Race Fault.</p> "> Figure 6
<p>Variation Rates under Different SNR: (<b>a</b>) Inner Race Fault Variation Rate; (<b>b</b>) Rolling Element Fault Variation Rate; (<b>c</b>) Outer Race Fault Variation Rate.</p> "> Figure 7
<p>Flowchart of composite fault diagnosis.</p> "> Figure 8
<p>Simulation signal processing results: (<b>a</b>) time-domain diagram of outer race fault; (<b>b</b>) time-domain diagram of inner race fault; (<b>c</b>) time-domain diagram of analog signal; (<b>d</b>) analog signal envelope diagram; (<b>e</b>) clustering renderings; (<b>f</b>) composite indicator iteration curve.</p> "> Figure 9
<p>Simulation signal decomposition: (<b>a</b>) Time-domain diagram of each IMF component; (<b>b</b>) Envelope plot of IMF components; (<b>c</b>) IMF1 envelope diagram; (<b>d</b>) IMF5 envelope diagram.</p> "> Figure 10
<p>Processing results of different indicators: (<b>a</b>) EE; (<b>b</b>) IE; (<b>c</b>) SE; (<b>d</b>) EnE.</p> "> Figure 11
<p>Results of different algorithms: (<b>a</b>) GWO; (<b>b</b>) PSO; (<b>c</b>) DE; (<b>d</b>) SSA.</p> "> Figure 12
<p>Comparison method processing results: (<b>a</b>) Simultaneous optimization of three parameters by WOA; (<b>b</b>) Time–Cost comparison.</p> "> Figure 13
<p>HZXT-DS-003 double-span rotor rolling bearing experimental bench and collection equipment.</p> "> Figure 14
<p>Experimental signal I processing results: (<b>a</b>) Time-domain diagram; (<b>b</b>) Envelope spectrum; (<b>c</b>) Clustering results; (<b>d</b>) Iterative curve.</p> "> Figure 15
<p>Experimental signal I decomposition: (<b>a</b>) Time-domain diagram of each IMF component; (<b>b</b>) Envelope plot of IMF components; (<b>c</b>) IMF time-domain diagram; (<b>d</b>) IMF envelope diagram.</p> "> Figure 16
<p>Comparison method processing results: (<b>a</b>) EE t; (<b>b</b>) IE; (<b>c</b>) SE; (<b>d</b>) EnE.</p> "> Figure 17
<p>Comparison method processing results: (<b>a</b>) GWO; (<b>b</b>) PSO; (<b>c</b>) DE; (<b>d</b>) SSA; (<b>e</b>) WOA optimizing three parameters simultaneously; (<b>f</b>) Time–cost comparison.</p> "> Figure 18
<p>Bearing data test bed, University of Paderborn, Germany.</p> "> Figure 19
<p>Experimental signal II processing results: (<b>a</b>) Time-domain diagram; (<b>b</b>) Envelope spectrum; (<b>c</b>) Clustering results; (<b>d</b>) Iterative curve.</p> "> Figure 20
<p>Experimental signal II decomposition: (<b>a</b>) Time-domain diagram of each IMF component; (<b>b</b>) Envelope plot of IMF components; (<b>c</b>) IMF1 envelope diagram; (<b>d</b>) IMF2 envelope diagram.</p> "> Figure 21
<p>Comparison method processing results: (<b>a</b>) EMD; (<b>b</b>) EEMD; (<b>c</b>) VMD; (<b>d</b>) EWT.</p> ">
Versions Notes

Abstract

:
Ideally, the vibration signal of a rolling bearing should be symmetrical. However, in practical operation, the vibration signals in both time and frequency domains often exhibit asymmetry due to factors such as load, speed, and wear. The relatively weak composite fault characteristics are easily masked. Although the Feature Modal Decomposition (FMD) method is outstanding in diagnosing composite faults in bearings, its effectiveness is easily constrained by parameter selection. To address this, this paper proposes a stepwise parameter adaptive FMD method combined with a clustering algorithm, specifically designed for diagnosing composite faults in rolling bearings. Firstly, this study employs the Density Peak Clustering algorithm to determine the number of modes n in the composite fault vibration signal. Subsequently, considering the signal spectral energy and modal characteristics, a new composite fault index is formulated, namely, the adaptive weighted frequency domain kurtosis-to-information entropy ratio, as the fitness function. The Whale Optimization Algorithm determines the filter length L and the number of segments K, thereby achieving step-wise signal decomposition. Through in-depth analysis of signal symmetry and asymmetry, simulation and experimental verification confirm the effectiveness of this method. Compared with four other index-optimized FMD methods and traditional techniques, this method significantly reduces the influence of parameters on FMD, is capable of separating the characteristic frequencies related to composite faults, and performs excellently in the diagnosis of composite faults in rolling bearings.

1. Introduction

Rolling bearings play a crucial role in rotating machinery and are an indispensable component of modern industrial systems [1]. As the operating time increases, bearings may become damaged and gradually develop into faults. If not diagnosed promptly, this could lead to the failure of the entire mechanical system [2]. Although the geometric structure of rolling bearings is typically symmetrical, during operation, the vibration signals often exhibit asymmetry in both the time and frequency domains due to factors such as load, speed, and wear. Through in-depth analysis of these signals in the time and frequency domains, it is possible to detect the asymmetric characteristics of abnormal signals. Overall, rolling bearing fault diagnosis enhances the detection and preventive capabilities of mechanical system faults by analyzing signal symmetry and asymmetry. Therefore, assessing the type of bearing faults through vibration signal analysis has become one of the core areas of fault diagnosis research [3].
Rolling bearings can develop various types of faults, including but not limited to inner race faults, outer race faults, ball faults, and cage faults. These faults may occur individually or simultaneously, forming compound faults. Compound fault signals typically contain multiple components, with fault characteristic frequencies being mixed and coupled, making diagnosis exceptionally complex [4]. Current fault diagnosis technologies for rolling bearings primarily focus on the identification of single faults. However, in practical engineering applications, due to the complexity of the transmission systems and the diversity of working environments, rolling bearings often exhibit compound faults. Consequently, accurately separating modal components with periodic impact characteristics from compound fault vibration signals has become a key issue in the diagnosis of rolling bearing compound faults [5].
Time-frequency analysis, as a common signal processing tool, is excellent in extracting sensitive fault features of non-smooth signals [6,7]. Compared with deep learning methods [8,9], time-frequency analysis not only provides more comprehensive details but also possesses stronger interpretability. Methods based on time-frequency analysis, such as Empirical Modal Decomposition (EMD) [10], Ensemble Empirical Modal Decomposition (EEMD) [11], and Local Mean Decomposition (LMD) [12], can automatically decompose a multi-component signal into a number of single-component signals without a priori knowledge by means of adaptive signal processing theory. These adaptive methods, leveraging the characteristics of multi-resolution analysis and filter banks, can identify fault features at different time scales, thereby providing more accurate and reliable diagnostic information. Therefore, they are widely used in the diagnosis of single faults in rolling bearings. However, they have certain limitations when dealing with composite faults, specifically being susceptible to background noise interference and failing to adequately consider the specific form of fault features due to their recursive and data-driven nature. This results in difficulties in effectively extracting representative fault features from complex signals.
Variational Mode Decomposition (VMD) [13] is an adaptive non-recursive decomposition method that can effectively process composite fault features. The decomposition performance of VMD is highly sensitive to parameter selection, where the number of decomposition layers influences the center frequency of the Wiener filter, and the penalty factor determines the bandwidth of the modal components [14]. To optimize the parameter selection of the VMD, some scholars have proposed improved methods. For example, in Li et al. [15], the kurtosis value of the envelope signal is used to determine the optimal number of modes, and the penalty factor is optimized accordingly. Yi et al. [16] guide VMD with the Power Spectral Density (PSD), adaptively determining VMD parameters, which are successfully used for determining the resonance frequency band in compound fault diagnosis. Although there has been extensive research on VMD parameter selection, using VMD without prior knowledge still presents challenges: The pulse and periodicity of the signal make it difficult to separate different components completely; decomposition performance is highly dependent on the choice of filter shape and bandwidth. Specifically, if the bandwidth is too large, different components and noise may be mixed into a single mode; if the bandwidth is too small, it may lead to an increase in redundant modes and the loss of important details [17,18]. In recent years, FMD [19] has received much attention as an emerging signal-processing method. Compared with the traditional EMD, EEMD, and VMD, FMD adopts an adaptive FIR filter, which fully takes into account the impulsive and periodic nature of the signal during the component separation process, without the need to preset the fault period as a priori knowledge. In addition, FMD combines the advantage of Correlated Kurtosis (CK) to make the Intrinsic Modal Function (IMF) separation more thorough. Experimental results show that FMD performs better in the decomposition analysis of mechanical signals.
However, there is currently limited research on the characteristics analysis and input parameter optimization of the FMD method. Some scholars have borrowed VMD parameter optimization to solve related problems in FMD. Common intelligent optimization algorithms, such as the Whale Optimization Algorithm (WOA) [20], particle swarm optimization (PSO) [21], Grey Wolf Optimizer (GWO) [22], Sparrow Search Algorithm (SSA) [23], and Differential Evolution (DE) [24], show potential in optimizing the parameter selection of FMD methods and improving their decomposition performance. For example, Wang et al. [25] utilized the periodic influence characteristics of fault signals and noise intensity to construct the ratio of sample entropy to integral kurtosis as the objective function for each step of the sparrow search algorithm, thereby achieving stepwise adaptive feature mode decomposition. Chauhan et al. [26] combine the Artificial Hummingbird Algorithm (AHA) to optimize the Sparsity Influence Metric Index (SIMI), adaptively optimizing FMD parameters for bearing defect detection. Yu et al. [27] optimized the FMD and MNAD parameters by combining adaptive FMD and Minimum Noise Amplitude Deconvolution (MNAD) with an improved chaos leader northern goshawk optimization algorithm (CLNGO) for composite fault diagnosis of bearings. In addition, Yan et al. [28] use the signal Cyclic Kurtosis Noise Ratio (SCKNR) as the objective function, combined with the Particle Swarm Optimization (PSO) algorithm to adaptively select the best parameters for FMD and to separate and extract the main modal components, successfully addressing the issues of non-stationarity and low signal-to-noise ratio in mechanical equipment vibration signals.
Although heuristic algorithms are suitable for parameter optimization problems [29], simultaneously optimizing three parameters has certain limitations, is unable to guarantee a globally optimal solution, and has significant result randomness. Therefore, this study adopts a stepwise optimization method for FMD decomposition. According to the FMD decomposition principle, the same fault component should be dispersed into the same mode. However, in the absence of prior knowledge, traditional methods can only preset the number of modes, which may lead to the same fault component being dispersed into different modes in practical applications, thereby affecting the accuracy of fault diagnosis. Therefore, it is crucial to reasonably determine the number of modes n in FMD.
Density Peak Clustering (DPC) [30] is a density-based clustering data processing method widely applied in the field of image processing. The advantages of this method include the ability to automatically identify cluster centers in data without the need to preset the number of clusters, making it easy to integrate with signal processing techniques. Wu et al. [31] used an improved Adaptive Noise Complete Ensemble Empirical Mode Decomposition (CEEMDAN) combined with DPC, accurately identifying fault modes through clustering algorithms and significantly enhancing the accuracy and efficiency of diagnosis. In addition, Zhang et al. [32] used DPC to identify the cluster centers of training samples and combined normalized clustering distance to evaluate the membership of test samples, integrating with Improved Variational Mode Decomposition (IVMD) to accurately identify fan bearing faults.
DPC applies to various data distributions and features, has high fault tolerance, and can be combined with a variety of signal processing techniques. Particularly, DPC can determine the number of source signals in compound fault vibration signals from a data processing perspective, i.e., the modal n, without prior knowledge of the source information. Therefore, this study proposes an innovative method that combines DPC with FMD. By utilizing DPC to identify high-density peak points to determine cluster centers, the method infers the number of modes n. This approach not only overcomes the limitations of traditional methods that require pre-setting the number of modes but also enhances the accuracy and robustness of other parameter optimizations.
In rolling bearing composite fault diagnosis, different types of faults cause vibration signals to exhibit unique characteristics. By utilizing the density kurtosis clustering algorithm, the center of clustering can be determined from the collected composite fault signals by finding the maximum point of local density. Subsequently, other data points are assigned to the nearest high-density point, thus automatically completing the clustering based on the density distribution of the data. Thus, the DPC algorithm is able to determine the number of source signals, i.e., the number of modes n, in the composite fault vibration signals from the data processing point of view without the need for a priori knowledge.
Based on the aforementioned research, this paper proposes a stepwise parameter adaptive FMD method integrated with DPC, aiming to optimize parameters to effectively identify the primary mode components in composite fault vibration signals, thereby achieving accurate diagnosis of composite faults in rolling bearings. Specifically, the DPC algorithm is incorporated to identify high-density peak points and determine the cluster centers, which in turn infers the number of modes n, overcoming the limitation of traditional methods that require predefined mode numbers. Furthermore, an adaptive weighted frequency domain kurtosis and entropy ratio (AKER) indicator is designed. This indicator not only exhibits high robustness to impulse noise but also demonstrates high sensitivity to periodic impulse signals. Finally, by utilizing the prey capture and spiral position update mechanisms of the WOA, the method efficiently and adaptively selects the optimal filter length L and the number of segments K for FMD, significantly enhancing the accuracy and robustness of FMD parameter optimization.
The remainder of this paper is structured as follows: Section 2 briefly outlines the fundamental principles and filtering characteristics of FMD, Section 3 delves into the construction methods of the new composite indicator and compares it with existing indicators, Section 4 provides a detailed explanation of the research framework, Section 5 validates the proposed method through simulation analysis and case studies, and Section 6 offers a thorough discussion of the obtained results.

2. Feature Mode Decomposition

2.1. Feature Mode Decomposition Principle

FMD achieves the simultaneous adaptive selection of different mode components by constructing an FIR filter bank and regenerating the filter coefficients. The filtering process is inspired by the theory of inverse convolution, where the coefficients are iteratively updated to make the filtered signal gradually approach the inverse convolution objective function. The main steps of the FMD include (1) initialization of the adaptive FIR filter bank, (2) adaptive updating of the filter bank, and (3) mode selection.

2.1.1. Adaptive FIR Filter Bank

In order to initialize the Hanning window, the frequency band of the original signal is first divided into k bands, and the whole band is covered by a uniformly distributed FIR filter bank. Subsequently, the fault modes can be adaptively decomposed by filter updating and mode selection.
f k = f k 1 ⋯ f k l ⋯ f k L T f l = k ⋅ f s / 2 K f u = k + 1 ⋅ k ⋅ f s / 2 K
where f s is the sampling frequency of the original signal, f k is the kth filter with length L, f l and f u are the lower and upper cutoff frequencies of these filters, and L denotes the length of the filter.

2.1.2. Filter Update and Period Estimation

Although the initialized FIR filter bank provides initial denoising capabilities, it may lead to noise and interference being mixed into the candidate modes. Additionally, the same fault component might be dispersed across different modes, increasing the complexity of the analysis. To address these issues, FMD formulates the decomposition process as a constrained optimization problem, thereby achieving a more effective separation of fault modes.
arg max f k l C K M u k = ∑ n = 1 N ∏ m = 0 M u k n − m T s 2 / ∑ n = 1 N u k n 2 M + 1 s . t . u k n = ∑ l = 1 L f k l x n − l + 1
where u k n is the kth decomposition mode, T s is the input period measured using the number of samples, and M is the shift order.
An iterative eigenvalue decomposition algorithm is introduced to solve the constraint problem. The decomposition pattern can be represented in matrix form:
u k = X f k
u k = u k 1 ⋮ u k N − L + 1
X = x 1 ⋯ x L ⋮ ⋱ ⋮ x N − L + 1 ⋯ x N

2.1.3. Mode Selection

To avoid redundancy and repetition, the FIR filters are not updated throughout the entire processing period but are instead initialized with a Hanning window to provide a fuzzy direction. Subsequently, the fault information is accurately captured through period estimation and a refined update process. In FMD, the mode with the highest correlated kurtosis (CK) value is selected, and this mode is locked using the highest correlation coefficient (CC) value, while modes with lower CK values are discarded to retain as much fault information as possible. The formula for calculating the correlation coefficient matrix is as follows:
C C p q = ∑ n = 1 N u p n − u ¯ p u q n − u ¯ q ∑ n = 1 N u p n − u ¯ p 2 ∑ n = 1 N u q n − u ¯ q 2
where u p and u q are the mean of the plurality u ¯ p and u ¯ q , respectively.

2.2. FMD Filtering Characteristics

To investigate the filtering characteristics of FMD, numerical simulations using fractional Gaussian noise (FGN) are conducted, and the equivalent filtering characteristics of FMD are analyzed [33]. The autocorrelation sequence array of FGN with the Hurst index can be represented as:
x H ( j ) i i = 1 , 2 , 3 … , I , j = 1 , … , J
p H n = σ 2 2 n − 1 2 H − 2 n 2 H + n + 1 2 H
where σ is the standard deviation of the FGN series and H shows the Hurst index.
J sets of independent FGN sequences with I data points per noise sequence are generated. The modulus of FMD is set to n. FMD is performed on each independent FGN sequence to obtain the corresponding n modal components d n , H j k , For the nth mode component, the coefficient mean of the autocorrelation function can be expressed as:
p ^ n , H m = 1 J ∑ j = 1 J 1 I ∑ i = 1 I − m d n , H j i d n , H j i + m , m ≤ I − 1
Based on the above coefficient averaging of the mode components, the power spectral density of the nth mode component estimated through the Hamming window can be expressed as:
p ^ n , H f = ∑ m = − I + 1 I − 1 p ^ n , H m w m e − i 2 π f m , f ≤ 1 / 2
where w m means the Heming window function.
Thus, through the aforementioned steps, the power spectral densities of n modal components can be continuously obtained and plotted together in the frequency domain, serving as the final output result of the FMD filtering characteristics.
The modulus n, the filter length L, and the number of segments K of the FMD determine the decomposition effect to some extent. By adjusting these parameters, the effect of each parameter on the filtering characteristics of the FMD is explored. For the numerical simulation tests, a sequence of J = 500 randomly generated independent fractional Gaussian white noises with standard deviation σ = 1 and data length I = 5000 is generated. H denotes the parameter affecting the statistical properties and σ denotes the standard deviation. FMD decomposition is performed on each set of data to obtain 500 data sets for each mode, and then the power spectra of each mode are averaged. The averaged power spectra of all modes represent the equivalent filtering characteristics of FMD in the frequency domain.
In this study, two methods are employed to validate the impact of various parameters on the filtering characteristics of FMD. First, the number of FMD modes (n), filter length (L), and segment number (K) are fixed at {6, 40, 7}, while the Hurst index (H) of FGN is varied. When H is between 0 and 0.5, the sequence exhibits negative correlation; when H is between 0.5 and 1, the sequence exhibits positive correlation; and when H = 0.5, the sequence behaves as Gaussian white noise, which is uncorrelated and uniformly contains various frequency components. Therefore, to further investigate the filtering characteristics of FMD under different H values, H is set to 0.2, 0.5, 0.8, and 1, respectively. The decomposition results are shown in Figure 1. It is evident from Figure 1 that although the spectral density amplitudes of the modal components obtained by FMD differ under different Hurst indices H, their filtering characteristics are essentially the same, all exhibiting the properties of a bank of band-pass filters.
Secondly, H is fixed at 0.5, and the filtering characteristics of FMD are studied by adjusting the number of FMD modes (n), the filter length (L), and the number of segments (K). As shown in Figure 2, if the filter length (L) and the number of segments (K) are fixed and the number of modes (n) is set too small, the filter bandwidth becomes larger and passband ripples occur, which are not conducive to signal denoising. As shown in Figure 3, if the number of modes (n) and the number of segments (K) are fixed and the filter length (L) is set too large, the filtering characteristics of FMD also exhibit passband ripples. As shown in Figure 4, if the number of modes (n) and the filter length (L) are fixed and the number of segments (K) is set too large, the filtering characteristics of FMD will deteriorate. Therefore, from the above analysis, it can be concluded that the number of FMD modes (n), the filter length (L), and the number of segments (K) collectively influence the signal processing capabilities of FMD, and effective methods should be employed for their selection and optimization.

3. A New Indicator Suitable for Compound Faults

3.1. Construction of the Compound Fault Indicator

The traditional kurtosis indicator may not effectively highlight the important features of complex signals, especially those containing a large amount of noise or non-stationary components. In contrast, Adaptive Weighted Frequency Domain Kurtosis (AWFK) significantly enhances the sensitivity to target features by applying weighting to the frequency components of the signal. AWFK is primarily used to describe the concentration of spectral energy and is particularly suitable for detecting impulse components in weak signals. Its calculation formula is similar to that of time-domain kurtosis, but the introduction of an adaptive weighting mechanism allows for better recognition and extraction of fault features in the frequency domain, thereby improving the accuracy and reliability of fault diagnosis. The specific calculation formula is as follows:
K a w n f d = ∑ k = 1 N E ω n f · U n f − μ ω , n 4 σ ω , n 4
where ω n f is the corresponding weight, U n f is the amplitude of the nth mode in the frequency domain, μ ω , n is the mean of the weighted frequency domain amplitude of the nth mode, and σ ω , n is the standard deviation of the weighted frequency domain amplitude of the nth mode.
Information entropy can effectively quantify the uncertainty and complexity of vibration signals, aiding in the identification of signal characteristics. When abnormal changes occur in a bearing, the statistical properties of the vibration signal may undergo significant alterations, leading to changes in the information entropy. Generally, the higher the uncertainty, the greater the value of the information entropy.
H X = − ∑ i = 1 n p i log 2 p i
In this paper, considering the spectral energy in the signal and the signal characteristics, we construct a comprehensive evaluation index of decomposition effect–adaptive weighted frequency domain kurtosis to information entropy ratio (AKER).
C I = K H
When the spectral energy of the IMF is larger and the information entropy is smaller, the value of the composite indicator CI is greater (CI stands for AKER). Therefore, the maximum value of CI can be used as the fitness function, which is expressed as:
f i t n e s s = max 1 ∼ k C I

3.2. Response of Various Metrics to Faults Under Different SNR

Excellent evaluation metrics are frequently employed as objective functions in optimization algorithms to guide the selection and optimization process of parameters. For example, Envelope Entropy (EE) [34] can effectively quantify the complexity and uncertainty of a signal, as shown in Equation (15); the higher the envelope entropy, the more complex the signal. Information Entropy (IE) [35] is used to measure the uncertainty and information content of data. When the information entropy is minimized, it typically indicates that the system is in its most certain or ordered state, meaning the uncertainty in the signal is minimized, as shown in Equation (16). Sample Entropy (SE) [36] is a metric that quantifies the complexity of time series data by calculating the negative logarithmic probability of similar patterns in the signal, thus measuring the system’s complexity and degree of chaos. As shown in Equation (17), the larger the sample entropy, the more complex or disordered the signal. Energy Entropy (EnE) [37] measures the distribution of energy within a signal by calculating the proportion of energy across different frequency components relative to the total energy, thereby quantifying the signal’s complexity and uncertainty, as shown in Equation (18). These evaluation metrics have been widely applied in vibration signal analysis and fault diagnosis research, effectively assisting optimization algorithms in parameter selection and model optimization, and thus enhancing the accuracy and robustness of fault diagnosis.
H e = − ∑ i = 1 n p i log ( p i )
H ( X ) = − ∑ i = 1 n x i log ( x i )
S a m p E N ( m , r , N ) = − ln ( A / B )
E s = − ∑ i = 1 n N i log ( N i )
The comparative data in this study utilize the bearing fault dataset from Case Western Reserve University, aiming to systematically analyze the variation characteristics of AKER, EE, IE, SE, and EnE under different fault types and SNR, and to calculate the change rates of these metrics. By comparing the metric variations of different fault types under various noise levels, the effectiveness of these metrics in signal feature analysis and fault diagnosis is evaluated (e.g., outer race fault f b p f o , inner race fault f b p f i , rolling element fault f b s f ). The specific experimental parameters are shown in Table 1.
Firstly, using the real fault signal data, AKER, EE, IE, SE and EnE are calculated. Subsequently, noise is added to the real fault signal data and the above experimental steps are repeated. Under the conditions of different signal-to-noise ratios, the sensitivity and robustness of the new indicators under different fault characteristics are verified by analyzing the trend and rate of change of the indicators. In theory, the value of AKER tends to decrease with the increase in noise intensity, while the four indicators, namely, EE, IE, SE, and EnE, tend to increase when the noise intensity increases.
To visually present the processing results of each index, the values of each index are marked in Figure 5 using bar charts of different colors and specifically analyzed as follows: In the inner race fault, when the SNR is −5 and −10, the IE value shows a decreasing trend with the increase in noise intensity. In the rolling element fault, when the SNR is −3 and −8, both EE and IE show a decrease with the increase in noise intensity. In the outer race fault, the IE shows a similar decreasing trend when the SNR is −10. The above results indicate that these entropy metrics are not robust enough to effectively cope with the effect of noise on fault feature recognition under strong noise interference.
Next, as shown in Equation (19), the rate of change of the metrics for the outer race fault under different SNR is calculated to analyze the sensitivity of each metric to periodic pulses. Then, the fault type is changed and the same experimental procedure is repeated. If the rate of change is positive and increases with the increase in noise level, this indicates that the metric shows an increasing trend under noise interference, which means that the metric is more sensitive to periodic pulses.
β = A i − A A × 100 %
where A is the value of the indicator when no noise is added, and Ai is the value of the indicator under different SNR.
By Figure 6 to intuitively demonstrate the variation rates of various metrics under different fault types and SNR, clear trends in how the metrics change with SNR can be observed. When comparing the variation rates of AKER, EE, IE, SE, and EnE, it can be noted that the variation rate of AKER is not only high but also stable. This indicates that AKER is the most sensitive to periodic impulses, effectively detecting subtle changes under specific conditions and thereby providing higher fault diagnosis accuracy.

4. Composite Fault Diagnosis Method

4.1. Principle and Process of Density Peak Clustering

Density Peak Clustering (DPC) is a density-based clustering algorithm that primarily identifies cluster centers by detecting high-density peak points. The algorithm defines cluster centers based on two key characteristics: first, their local density is higher than that of other points in their neighborhood; and second, the distance between them and points with higher density is relatively large. The DPC algorithm can recognize potential cluster centers, allowing the number of clusters to naturally emerge during the clustering process without the need for pre-specification. Furthermore, because the definition of cluster centers considers both density and distance, low-density outliers are automatically excluded, thus not interfering with the clustering results. This feature of the DPC algorithm enables it to effectively identify clusters regardless of their shape or the dimensionality of the embedding space, demonstrating strong generalization and robustness.
  • The clustering process is divided into two main steps: local density estimation and minimum distance calculation. Using the two metrics of local density and minimum distance, the cluster centers in the data can be effectively identified, while other data points are assigned to the corresponding clusters based on their similarity to the cluster centers.
The local density ρ i is defined as follows:
ρ i = ∑ j χ d i j − d c
χ x =       1       i f       x < 0 0       i f       o t h e r
where d i j represents the distance between point i and point j, and d c is the cutoff distance.
The high local density point distance δ i , which is defined as:
δ i = min j : ρ j > ρ i d i j
δ i   is the closest distance to object i among all objects with local densities higher than object i . At the extreme, for the one object with the highest density, we set δ = m a x d i j ; only those points whose densities are local or global maxima will have distances to high local density points that are much larger than normal.
2.
Cluster centers are selected based on two key metrics: local density ρ i and high local density distance δ i . Points with large local density ρ i and large high local density distance δ i are identified as cluster centers. On the contrary, points with larger high local density distance δ i but smaller local density ρ i are considered anomalies. Once the clustering centers are identified, the other data points are classified based on their distance from the nearest clustering center, or based on a density accessibility approach.

4.2. Whale Optimization Algorithm

The Whale Optimization Algorithm (WOA) is inspired by the feeding behavior of humpback whales. The algorithm initializes a group of individuals representing potential solutions in the solution space by simulating the humpback whale’s behavior of rounding up and searching for prey. The optimal solution is gradually approximated by continuously updating individual positions and fitness values. The core behaviors include rounding up, capturing, and searching for prey.
  • Initialization: The expression to initialize the position of the whale population is as follows:
X i = l b + r a n d × u b − l b
2.
Surrounding prey: Whales update their position by surrounding prey, and the expression is as follows:
X t + 1 = X ∗ t − A D
D = C X ∗ t − X t
where t denotes the current number of iterations, D denotes the distance of variables of the solution in the problem, A and C denote the coefficients, X * ( t ) denotes the current optimal whale position, and X ( t ) denotes the position of the whale in the current number of iterations.
3.
Capture of prey: The whale captures prey utilizing a spiral motion with the following expression:
X t + 1 = X ∗ t − D p e b l cos 2 π l
where D p shows the size of the distance between the whale and the prey, b is a constant that indicates the shape of the spiral, and l is a (−1, 1) random number.
In order to spiral encircle the prey, the whale also needs to continuously contract the encirclement, so it is necessary to set probabilities Pi and 1 − Pi for the contraction encirclement mechanism and the spiral model in order to dynamically update the position of the whale. The expressions are as follows:
X t + 1 = X ∗ t − A D                                   p < P i X ∗ t − D p e b l cos 2 π l       p ≥ P i
By adjusting the probability p , the parameter α , and the fluctuation range A , the whale updates its position in different ways and gradually approaches to more optimal solution.
4.
Searching for prey: The goal of searching for prey is to find a better solution, i.e., the global optimal solution. The expression is as follows:
X t + 1 = X r a n d − A D
D = C X r a n d − X t
where X r a n d is a random whale position.
When A > 1, an individual whale is randomly selected to update the positions of other whales, thus increasing the randomness of exploration. Through this process, the search ability of the algorithm is enhanced, making the WOA more likely to find the global optimal solution.

4.3. Bearing Composite Fault Diagnosis Procedure

  • Step 1. Data Collection: Acquire vibration signals using accelerometers.
  • Step 2. Determination of FMD Modal Quantity n: Analyze the distance distribution and local density of the signals using the Density Peak Clustering (DPC) algorithm to identify cluster centers and infer the modal quantity n for FMD. The specific steps are as follows:
First, calculate the number of sampling points for each fault characteristic cycle and select a portion of the cycle lengths. Then, analyze the fault signals using the Short-Time Fourier Transform (STFT) to obtain the time-frequency matrix.
Next, convert this time-frequency matrix into a two-dimensional point set and compute the local density and distance for each point. By selecting points with higher local density and greater distance as cluster centers, perform the clustering assignment.
Finally, map the clustering results back to the time-frequency matrix for visualization.
  • Step 3. Construction of the Objective Function: Integrate the spectral energy and modal characteristics in the signal to construct a composite fault indicator.
  • Step 4. Determination of Filter Length and Segments: Evaluate the decomposition characteristics of the composite indicator using the Whale Optimization Algorithm (WOA) and optimize the filter length and number of segments through iterative processes. The specific steps are as follows:
(1) Parameter initialization:
Set the number of individuals in the whale population to n = 30 and limit the maximum number of iterations of the algorithm to 20. The FMD parameters to be optimized include filter length L; the search range is set to {30, 150}. The number of segments K, the search range is set to {n, 12}. For each combination of L and K, initial values are randomly generated to form a diverse initial population distribution.
(2) Execute the whale optimization algorithm:
Initialization of whale positions: Randomly initialize the positions of each individual whale that correspond to different combinations of FMD parameters (L and K).
Adaptation evaluation: Using the objective function constructed in step 3, calculate the adaptation value of each whale individual under the corresponding parameter configuration.
Position update:
Surrounding prey: In each iteration, the position of each whale individual is adjusted according to its distance to the global optimal solution to simulate the behavior of the whale population gradually approaching the prey.
Capture prey: According to the current optimal solution (i.e., the best fitness value), the search strategy is quickly adjusted to narrow the search area in order to accelerate the convergence process.
Searching for prey: In order to prevent the algorithm from falling into a local optimum, the whales will randomly explore new locations to maintain some global search capability.
Iteration: The fitness evaluation and position update are repeated until the maximum number of iterations is reached or the fitness value converges.
(3) Filter out the best parameters:
At the end of the algorithm iterations, select the individual whale with the highest objective function value whose position corresponds to the optimal FMD parameter combination (L and K).
  • Step 5. FMD Decomposition: Based on the optimized parameters determined in Steps 2 and 4, use FMD to decompose the original vibration signal into multiple IMF components.
  • Step 6. Fault diagnosis: According to the composite fault indicator AKER, select the IMF components that are most sensitive to the fault characteristics. Perform envelope analysis on the selected IMF components to extract their envelope spectra, and conduct a detailed analysis of the characteristic frequencies and their harmonics within the envelope spectra to accurately identify the composite fault patterns of the bearing.
The specific fault flow chart is shown in Figure 7.

5. Experimental Analysis

5.1. Analyzing the Simulated Signals

5.1.1. Bearing Composite Fault Simulation Model

In order to verify the feasibility of this research method, this paper uses a composite fault signal composed of two single fault simulation signals to test the performance of the proposed method. The analog signal expressions are as follows:
  x ( 1 ) = E 1 . × sin ( 2 π × f n 1 × t ) x ( 2 ) = E 2 . × sin ( 2 π × f n 2 × t ) x = x ( 1 ) + x ( 2 ) + n ( t )
where the rotational speed f r of the bearing is 30 Hz, the characteristic frequency f b p f o of the outer race fault is 60 Hz, the characteristic frequency f b s f of the rolling element fault is 142 Hz, f n 1 is the intrinsic frequency of the outer ring, f n 2 is the resonance frequency of the bearing structure, E 1 is the attenuation signal generated by a specific function to simulate the vibration characteristics of the outer race fault, and E 2 is the attenuation signal that characterizes the rolling element fault, which is generated by a function and multiplied with the modulation signal to reflect the complexity of the signal. The composite fault simulation signal x is generated by superimposing the outer race fault signal, the rolling element fault signal, and Gaussian white noise.

5.1.2. Comparison of Fault Diagnosis Results with the Performance of Other Methods

The analog signals as demonstrated in Figure 8a,b correspond to the outer race fault and rolling element fault, respectively. Figure 8c,d show the time-domain waveform and envelope spectrum analysis of the composite fault signal, respectively. Although the outer race fault eigenfrequency f b p f o and its harmonic components can be effectively extracted from the envelope Spectrum of Figure 8d, the rolling element fault eigenfrequency f b s f cannot be identified. This indicates the limitation of the direct spectral analysis method in dealing with periodic impulse fault eigenfrequencies that are swamped by noise or other disturbances.
The simulated signals constructed above are analyzed using the method proposed in this paper. Firstly, the DPC algorithm is used to analyze the distance distribution and local density of the signal, as shown in Figure 8e, to determine the clustering center and infer the FMD modal number of 6. Then, the composite metrics are computed by using the whale algorithm to assess the decomposition characteristics, and the filter length and the number of segments are iteratively optimized. The optimization results are shown in Figure 8f.
Next, the optimal combination parameters {6,32,8} are input to decompose the simulated signal into six modal components. The results of the time-domain waveform and envelope spectrogram analyses are shown in Figure 9a,b. The IMF1 component and IMF5 component are analyzed in detail, and the results are shown in Figure 9c,d. The outer race fault characteristic frequency f b p f o and harmonic components can be observed in Figure 9c, whereas the rolling element fault characteristic frequency f b s f , which is drowned out by noise and other interferences, can be extracted efficiently in Figure 9d. This suggests that the proposed method can effectively detect the composite fault characteristic information of the bearing.
To further verify the sensitivity of AKER to compound faults, AKER is compared with four classical indicators. The processing results with four classical indicators as the objective function are shown in Figure 10.
The analysis results from Figure 10 show that when dealing with each IMF component, the use of the four classical indicators as the objective function can only identify the characteristic frequencies f b p f o of the outer race fault and their harmonics, but cannot effectively extract the characteristic frequencies of the rolling element faults that are submerged by noise and other interferences f b s f . This suggests that these classical indicators are not suitable for the diagnosis of composite faults in rolling bearings.
In order to fully evaluate the performance of the method in this paper, this study compares it with four other optimization algorithms. Figure 11 shows the results of optimizing the objective function AKER using these four optimization algorithms.
In addition, Figure 12 demonstrates the results of simultaneously optimizing three parameters in FMD using the whale optimization algorithm.
The analysis results of Figure 11a–d and Figure 12a show that the five comparative methods, which include optimizing the objective function AKER using four other optimization algorithms and optimizing the three parameters of FMD using the WOA, can all effectively extract the fault characteristic frequency f b s f of rolling elements submerged by noise interference. However, the time–cost comparison result shown in Figure 12b indicates that the method proposed in this paper requires the shortest amount of time during the optimization process, demonstrating the highest optimization efficiency. This result not only verifies the effectiveness of the proposed method in composite fault characteristic extraction but also highlights its significant superiority and higher practical value in actual applications.

5.2. Experimental Signal Analysis

5.2.1. Case 1: Double-Span Rotor Rolling Bearing Experimental Bench Data Set

To validate the performance of the proposed method under actual operating conditions, a set of rolling bearing experimental data was analyzed. This dataset was obtained from the HZXT-DS-003 (Produced by Wuxi Houde Instrument Co., Ltd. in Jiangsu, China) type double-span rotor rolling bearing test bench at the School of Mechanical Engineering, Inner Mongolia University of Science and Technology, as shown in Figure 13.
The test bench primarily consists of a drive motor, a speed torque meter, a radial loader, and the fault bearing to be tested. During the experiment, the bearing model at the drive motor end is 6205-2RS (Produced by Wuxi Houde Instrument Co., Ltd. in Jiangsu, China), and the vibration signals are collected using accelerometers installed in both the vertical and horizontal directions of the bearing. To simulate the fault characteristics of the bearing, electric spark technology is employed to create cracks with a diameter of 0.6 mm on both the outer and inner rings of the bearing. Additionally, pits with a diameter of approximately 1 mm and a depth of about 2 mm are formed on the rolling elements by cutting to simulate surface damage on the rolling elements.
This experimental study examined the data from a rolling bearing with combined inner and outer race faults at a rotational speed of 1750 r/min. As detailed in Table 2, the parameters of the rolling bearing are listed comprehensively. Based on these parameters, the calculated outer race fault characteristic frequency f b p f o is 104.56 Hz, and the inner race fault characteristic frequency f b p f i is 157.94 Hz; the specific results are presented in Table 3. The detailed information on the signal acquisition equipment is provided in Table 4.
In order to more accurately simulate the operation of bearings under low signal-to-noise ratio conditions in real working conditions, Gaussian white noise with a signal-to-noise ratio of 5 dB is introduced into the data signal in this study. Figure 14a,b show the waveform characteristics of the vibration signals in the time domain and envelope spectrum after the addition of noise, respectively. By analyzing the envelope spectrum in Figure 14b, although the fault characteristic frequency f b p f i of the inner ring of the bearing can be successfully extracted, the fault characteristic frequency f b p f o of the outer ring fails to appear. This result indicates that in a low signal-to-noise ratio environment, the fault characteristic frequencies with low amplitude are easily masked by the background noise, and it is difficult to effectively identify these weak fault characteristic frequencies that are submerged by the noise interference by directly analyzing the spectrum of the experimental signals.
The method adopted in this study is applied to the experimental bench vibration data shown in Figure 14a. Firstly, the distance distribution and local density of the signals are analyzed by the DPC algorithm, the cluster centers are identified, and the FMD modal number n = 7 is inferred, as shown in Figure 14c. Subsequently, the WOA is used to compute the composite metrics, evaluate the decomposition characteristics, and iteratively optimize the length and number of segments of the filter, with the iterative process shown in Figure 14d. Finally, the best combination of parameters {7,73,10} is input into the FMD to decompose the signal into seven modal components. Figure 15a,b show the time-domain waveforms of the filtered modal components and their envelope spectra, respectively. In Figure 15d, the inner race fault characteristic frequency f b p f i and its harmonic components can be clearly observed, and it is inferred that the experimental bearing may have an inner race fault. In addition, the outer race fault characteristic frequency f b p f o and its harmonic components under noise and other disturbances can be effectively identified, indicating that an outer race fault does exist in the experimental bearing. These results are consistent with the actual situation of the faulty bearing, which verifies the effectiveness of this method in detecting the composite faults of rolling bearings.
However, according to the results of the current study, the performance of the method is not yet satisfactory in a low signal-to-noise ratio environment, failing to fully realize the effective separation of the composite fault eigenfrequencies in different modal components, and there still exists the phenomenon of overlapping of multiple fault features in the middle. This indicates that the robustness and accuracy of the method under low signal-to-noise ratio conditions still need to be further improved and optimized.
In order to further verify the sensitivity of AKER to compound faults, the experimental signals are compared and analyzed with four classical indicators as the objective function, and the processing results are shown in Figure 16.
From the analysis results in Figure 16, it can be seen that the corresponding envelope spectra of the modal components screened after processing with the four classical indicators as the objective function are unable to effectively extract the outer race fault eigenfrequency f b p f o and its harmonic components, which are submerged by noise and other disturbances. This suggests that these metrics are not suitable for rolling bearing composite fault diagnosis.
It is compared with four other optimization algorithms. Figure 17a–d illustrate the results of optimizing the objective function AKER using these four optimization algorithms. Figure 17e shows the results of simultaneously optimizing the three parameters in FMD using the whale optimization algorithm.
The analysis results of Figure 17a–d show that the optimization of the objective function AKER using the other four optimization algorithms are all effective in extracting the outer-ring fault eigenfrequency f b p f o that is overwhelmed by the noise interference, although there is a significant difference in the magnitude extracted by each algorithm. It is noteworthy that the envelope spectrum results in Figure 17a,b,d are consistent, which may be attributed to the fact that these optimization algorithms are prone to falling into the local optimal solution during the optimization process, which leads to similar feature frequency extraction results. However, the results in Figure 17e show that the simultaneous optimization of the three parameters fails to successfully extract the outer race fault eigenfrequency f b p f o , indicating that the optimization effect of this method has a large degree of randomness and uncertainty. Finally, it is further verified by the time–cost comparison in Figure 17f that the method proposed in this paper takes the shortest time and produces the most accurate results in the optimization process.

5.2.2. Case 2: Germany University of Paderborn Bearing Dataset

To validate the effectiveness of the FMD method in separating the characteristic components of composite bearing faults based on the impulsiveness and periodicity of the signals, a set of comparative experimental data is introduced in this paper. The data are derived from the bearing dataset of the University of Paderborn, Germany. The experimental rig is shown in Figure 18 and consists of an electric motor, torque measurement shaft, rolling bearing test module, flywheel, and load motor. The bearings used in the data are all model 6203, and the vibration signals are collected by mounting ball bearings with different damage types in the bearing test module.
In this study, the inner ring and outer ring composite failure data with number N15_M01_F10_KB23 are used, as shown in Table 5, and the rolling bearing data parameters of Case II are listed in detail. Based on these parameters, the outer race fault characteristic frequency f b p f o is calculated to be 76.76 Hz, the inner race fault characteristic frequency f b p f i is 123.24 Hz, and the specific results are shown in Table 6.
Figure 19a,b show the time-domain waveforms and envelope spectra of the vibration signals collected by the experimental bench, respectively. In the envelope spectrum of Figure 19b, the characteristic frequencies of faults in the inner and outer rings of the bearing can be extracted. The experimental bench vibration data shown in Figure 19a are processed using the method in this paper. Firstly, the distance distribution and local density of the signals are analyzed by the DPC algorithm, which identifies the cluster centers and infers an FMD modal number of 5, as shown in Figure 19c. Next, the WOA is used to calculate the composite metrics, evaluate the decomposition properties, and iteratively optimize the length and number of segments of the filter, with the iterative process shown in Figure 19d.
Finally, the best combination of parameters {5,150,9} is input into the FMD to decompose the signal into five modal components. Figure 20a,b show the envelope spectra of the modal components after processing by this method. The outer race fault eigenfrequency f b p f o and its harmonic components can be extracted from Figure 20c, whereas the inner race fault eigenfrequency f b p f i can be effectively identified in Figure 20d, which results in the separation of the composite fault components.
In addition, this paper compares the performance of four traditional fault diagnosis methods in processing vibration signal data, including EMD, EEMD, VMD, and EWT. To present the results succinctly, the first five IMF components are intercepted for each method. Figure 21 illustrates the analysis results of these four methods. From the envelope spectra of the various IMF components, it can be observed that none of these methods effectively separates the fault characteristic frequencies of the outer and inner rings. In some cases, there is a significant reduction in the amplitude of certain inner race fault characteristic frequencies during the decomposition process. This indicates that these methods are not suitable for the diagnosis of compound-bearing faults.

6. Discussion

The method proposed in this paper has made significant progress in the field of composite fault diagnosis of rolling bearings, especially in processing composite fault signals, which demonstrates unique advantages. By fusing the DPC method and the AKER evaluation index of composite fault features, this paper proposes a step-by-step adaptive feature modal decomposition method, which innovates and improves the existing techniques in several aspects.
First, the introduction of the DPC method significantly improves the flexibility and computational efficiency of signal decomposition. Traditional feature mode decomposition methods usually require a preset number of modes, which not only increases the dependence on a priori knowledge but also has large uncertainty in practical applications. On the other hand, the DPC method can automatically identify the clustering center in the data and determine the number of modes in the signal from the perspective of data processing, which reduces the dependence on optimization parameters. This is especially important for dealing with complex and variable composite fault signals, which can adapt to different types of fault signals and improve the universality of the method.
Secondly, the proposed AKER further improves the fault feature extraction performance and noise immunity. AKER combines adaptive weighted frequency domain kurtosis and the information entropy ratio, which is not only highly robust to impulsive noise but also exhibits high sensitivity to periodic impulses. Compared with traditional evaluation metrics such as envelope entropy, sample entropy, information entropy, and energy entropy, AKER performs better in fault pulse extraction performance and noise immunity. This innovation provides strong support for the accurate diagnosis of composite fault signals and helps to improve the accuracy and reliability of fault diagnosis.
Finally, the stepwise adaptive feature modal decomposition method automatically identifies the number of modes by DPC and searches for the optimal parameter combinations by using AKER as the objective function and the WOA, which realizes the stepwise adaptive signal decomposition of parameters. This method is not only verified in analog and experimental signal analysis but also shows significant advantages in practical applications. Through the comparative analysis from multiple perspectives, this paper fully proves the effectiveness and applicability of the method in bearing composite fault diagnosis.
Although the method proposed in this paper has shown significant advantages in dealing with composite faults, it still has some limitations in the separation effect of composite fault signals in low-signal-to-noise-ratio environments. This is mainly because the noise in the low-signal-to-noise-ratio environment will have a greater impact on the accuracy of signal decomposition, thus reducing the accuracy of fault diagnosis. Future research will focus on improving the performance of the method in low-SNR environments to ensure that it maintains high accuracy and reliability in more complex and demanding application scenarios. In addition, the research will aim to extend the application of the method by successfully applying it to other rotating machines, such as gearboxes and motors, to further validate its general applicability. Meanwhile, exploring the combination with deep learning techniques aims to further enhance the accuracy and robustness of fault diagnosis through a data-driven approach, providing more diversified solutions and technical support for compound fault diagnosis.

Author Contributions

Writing—original draft preparation, S.X.; project administration, C.Z.; resources, J.Z.; data curation, G.L.; modifications, B.O. and Y.W. All authors have read and agreed to the published version of the manuscript.

Funding

This work was supported by the National Natural Science Foundation of China (grant number: 52365014), the Central Government Guiding Local Science and Technology Development Fund Project (grant number: 2022ZY0221), Key R&D Achievement Transformation Plan Project in Inner Mongolia Autonomous Region (grant number: 2023YFSW0003), and the Fundamental Research Funds for Inner Mongolia University of Science and Technology (grant number: 2024YXXS045).

Data Availability Statement

The rolling bearing dataset was obtained from the HZXT-DS-003 Double Span Double Rotor Rolling Bearing Test Stand at the School of Mechanical Engineering, Inner Mongolia University of Science and Technology.

Conflicts of Interest

The authors declare no conflicts of interest.

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Figure 1. Filtering characteristics of FMD with different Hurst exponents: (a) H = 0.2; (b) H = 0.5; (c) H = 0.8; (d) H = 1.
Figure 1. Filtering characteristics of FMD with different Hurst exponents: (a) H = 0.2; (b) H = 0.5; (c) H = 0.8; (d) H = 1.
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Figure 2. Filtering characteristics of FMD under different numbers of modes n: (a) n = 4; (b) n = 6.
Figure 2. Filtering characteristics of FMD under different numbers of modes n: (a) n = 4; (b) n = 6.
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Figure 3. Filtering characteristics of FMD under different filter lengths L: (a) L = 60; (b) L = 40.
Figure 3. Filtering characteristics of FMD under different filter lengths L: (a) L = 60; (b) L = 40.
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Figure 4. Filtering characteristics of FMD under different numbers of segments K: (a) K = 10; (b) K = 7.
Figure 4. Filtering characteristics of FMD under different numbers of segments K: (a) K = 10; (b) K = 7.
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Figure 5. The Robustness of Various Metrics under Different SNR: (a) Inner Race Fault; (b) Rolling Element Fault; (c) Outer Race Fault.
Figure 5. The Robustness of Various Metrics under Different SNR: (a) Inner Race Fault; (b) Rolling Element Fault; (c) Outer Race Fault.
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Figure 6. Variation Rates under Different SNR: (a) Inner Race Fault Variation Rate; (b) Rolling Element Fault Variation Rate; (c) Outer Race Fault Variation Rate.
Figure 6. Variation Rates under Different SNR: (a) Inner Race Fault Variation Rate; (b) Rolling Element Fault Variation Rate; (c) Outer Race Fault Variation Rate.
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Figure 7. Flowchart of composite fault diagnosis.
Figure 7. Flowchart of composite fault diagnosis.
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Figure 8. Simulation signal processing results: (a) time-domain diagram of outer race fault; (b) time-domain diagram of inner race fault; (c) time-domain diagram of analog signal; (d) analog signal envelope diagram; (e) clustering renderings; (f) composite indicator iteration curve.
Figure 8. Simulation signal processing results: (a) time-domain diagram of outer race fault; (b) time-domain diagram of inner race fault; (c) time-domain diagram of analog signal; (d) analog signal envelope diagram; (e) clustering renderings; (f) composite indicator iteration curve.
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Figure 9. Simulation signal decomposition: (a) Time-domain diagram of each IMF component; (b) Envelope plot of IMF components; (c) IMF1 envelope diagram; (d) IMF5 envelope diagram.
Figure 9. Simulation signal decomposition: (a) Time-domain diagram of each IMF component; (b) Envelope plot of IMF components; (c) IMF1 envelope diagram; (d) IMF5 envelope diagram.
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Figure 10. Processing results of different indicators: (a) EE; (b) IE; (c) SE; (d) EnE.
Figure 10. Processing results of different indicators: (a) EE; (b) IE; (c) SE; (d) EnE.
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Figure 11. Results of different algorithms: (a) GWO; (b) PSO; (c) DE; (d) SSA.
Figure 11. Results of different algorithms: (a) GWO; (b) PSO; (c) DE; (d) SSA.
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Figure 12. Comparison method processing results: (a) Simultaneous optimization of three parameters by WOA; (b) Time–Cost comparison.
Figure 12. Comparison method processing results: (a) Simultaneous optimization of three parameters by WOA; (b) Time–Cost comparison.
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Figure 13. HZXT-DS-003 double-span rotor rolling bearing experimental bench and collection equipment.
Figure 13. HZXT-DS-003 double-span rotor rolling bearing experimental bench and collection equipment.
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Figure 14. Experimental signal I processing results: (a) Time-domain diagram; (b) Envelope spectrum; (c) Clustering results; (d) Iterative curve.
Figure 14. Experimental signal I processing results: (a) Time-domain diagram; (b) Envelope spectrum; (c) Clustering results; (d) Iterative curve.
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Figure 15. Experimental signal I decomposition: (a) Time-domain diagram of each IMF component; (b) Envelope plot of IMF components; (c) IMF time-domain diagram; (d) IMF envelope diagram.
Figure 15. Experimental signal I decomposition: (a) Time-domain diagram of each IMF component; (b) Envelope plot of IMF components; (c) IMF time-domain diagram; (d) IMF envelope diagram.
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Figure 16. Comparison method processing results: (a) EE t; (b) IE; (c) SE; (d) EnE.
Figure 16. Comparison method processing results: (a) EE t; (b) IE; (c) SE; (d) EnE.
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Figure 17. Comparison method processing results: (a) GWO; (b) PSO; (c) DE; (d) SSA; (e) WOA optimizing three parameters simultaneously; (f) Time–cost comparison.
Figure 17. Comparison method processing results: (a) GWO; (b) PSO; (c) DE; (d) SSA; (e) WOA optimizing three parameters simultaneously; (f) Time–cost comparison.
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Figure 18. Bearing data test bed, University of Paderborn, Germany.
Figure 18. Bearing data test bed, University of Paderborn, Germany.
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Figure 19. Experimental signal II processing results: (a) Time-domain diagram; (b) Envelope spectrum; (c) Clustering results; (d) Iterative curve.
Figure 19. Experimental signal II processing results: (a) Time-domain diagram; (b) Envelope spectrum; (c) Clustering results; (d) Iterative curve.
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Figure 20. Experimental signal II decomposition: (a) Time-domain diagram of each IMF component; (b) Envelope plot of IMF components; (c) IMF1 envelope diagram; (d) IMF2 envelope diagram.
Figure 20. Experimental signal II decomposition: (a) Time-domain diagram of each IMF component; (b) Envelope plot of IMF components; (c) IMF1 envelope diagram; (d) IMF2 envelope diagram.
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Figure 21. Comparison method processing results: (a) EMD; (b) EEMD; (c) VMD; (d) EWT.
Figure 21. Comparison method processing results: (a) EMD; (b) EEMD; (c) VMD; (d) EWT.
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Table 1. Data parameters.
Table 1. Data parameters.
NameParameters
Rolling bearing typeSKF6205
Shaft speed1797 r/min
Sampling frequency12 KHz
Inner race fault data105 .mat
Rolling element fault data108 .mat
Outer race fault data113 .mat
Table 2. Rolling bearing parameters of case I.
Table 2. Rolling bearing parameters of case I.
NameParameters
Rolling bearing type6205-2RS
Contact angle0°
Bearing pitch diameter39.04 mm
Ball diameter7.94 mm
Number of balls9
Shaft speed1750 r/min
Sampling frequency12 KHz
Number of sampling points12,000
Table 3. Fault characteristic frequency calculation.
Table 3. Fault characteristic frequency calculation.
NameParameters
Outer race fault characteristic frequency f b p f o 104.56 Hz
Inner race fault eigenfrequency f b p f i 157.94 Hz
Characteristic frequency of rolling element faults f b s f 137.48 Hz
Table 4. Configuration of relevant experimental equipment.
Table 4. Configuration of relevant experimental equipment.
NameParametersManufacturer
Acceleration sensorsPCB352C33American PCB Company
NI chassisCDAQ-9184American NI Company
Data acquisition cardsNI9234American NI Company
Table 5. Rolling bearing parameters of case II.
Table 5. Rolling bearing parameters of case II.
NameParameters
Rolling bearing type6203
Contact angle0°
Bearing pitch diameter29.05 mm
Ball diameter6.75 mm
Number of balls8
Shaft speed1500 r/min
Sampling frequency64 KHz
Number of sampling points60,000
Table 6. Calculation of fault characteristic frequency.
Table 6. Calculation of fault characteristic frequency.
NameParameters
Outer race fault characteristic frequency f b p f o 76.76 Hz
Inner race fault characteristic frequency f b p f i 123.24 Hz
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MDPI and ACS Style

Xu, S.; Zhang, C.; Zhang, J.; Liu, G.; Wu, Y.; Ouyang, B. Step-Wise Parameter Adaptive FMD Incorporating Clustering Algorithm in Rolling Bearing Compound Fault Diagnosis. Symmetry 2024, 16, 1675. https://doi.org/10.3390/sym16121675

AMA Style

Xu S, Zhang C, Zhang J, Liu G, Wu Y, Ouyang B. Step-Wise Parameter Adaptive FMD Incorporating Clustering Algorithm in Rolling Bearing Compound Fault Diagnosis. Symmetry. 2024; 16(12):1675. https://doi.org/10.3390/sym16121675

Chicago/Turabian Style

Xu, Shuai, Chao Zhang, Jing Zhang, Guiyi Liu, Yangbiao Wu, and Bing Ouyang. 2024. "Step-Wise Parameter Adaptive FMD Incorporating Clustering Algorithm in Rolling Bearing Compound Fault Diagnosis" Symmetry 16, no. 12: 1675. https://doi.org/10.3390/sym16121675

APA Style

Xu, S., Zhang, C., Zhang, J., Liu, G., Wu, Y., & Ouyang, B. (2024). Step-Wise Parameter Adaptive FMD Incorporating Clustering Algorithm in Rolling Bearing Compound Fault Diagnosis. Symmetry, 16(12), 1675. https://doi.org/10.3390/sym16121675

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