Performance degradation assessment of rolling bearing under vibration signal monitoring based on optimized variational mode decomposition and improved fuzzy support vector data description

Vibration monitoring technology utilizes high-precision sensors to capture the vibration signals produced by mechanical equipment during operation. These signals carry crucial information about the machine's health status. Widely employed across industrial sectors, including rail transit, aviation, and petrochemical industries, this technology plays a vital role in ensuring equipment reliability and safety. It is also essential for preventive maintenance and fault diagnosis.^1,2 Sensors strategically positioned in key areas enable real-time monitoring of machine parameters such as displacement, velocity, and acceleration. These parameters can unveil underlying issues within the machine, including bearing damage, mechanical imbalance, or misalignment of shafts.³ The accuracy of vibration monitoring is directly influenced by the performance of sensors. Hence, it is imperative for sensors to possess high sensitivity and a broad frequency response range to detect subtle changes in machine operation effectively.^4,5 However, during the acquisition process, the vibration signal may encounter disturbances from environmental noise and complex dynamics within the machine. These disturbances can obscure the true vibration characteristics of the machine. To enhance monitoring accuracy and reliability in fault prediction, effective denoising and feature extraction from collected vibration signals are crucial. Advanced signal processing and feature extracting technology can be employed to improve the quality of sensor data, enabling more accurate diagnosis of the machine's health status.⁶ 

Rolling bearings stand as the most crucial and prevalent component within rotating machinery.⁷ In the event of an unforeseen failure, it can precipitate catastrophic accidents throughout the mechanical system, with the bearing's performance directly impacting operational reliability of the entire mechanical apparatus.⁸ Performance degradation assessment (PDA) analysis of components aids in organizing maintenance activities in a targeted manner, averting fault occurrences, and augmenting the operational reliability of the entire machine.⁹ Moreover, bearing degradation transpires progressively from a state of normalcy to eventual failure. If researchers can monitor this degradation progression and devise appropriate maintenance strategies to forestall bearing failures, it can substantially curtail production downtime and alleviate maintenance expenses.¹⁰ 

Monitoring the operating condition of bearings through vibration signals has emerged as a practical method for performance degradation assessment. However, the vibration signals collected by sensors often suffer from low purity due to the complex and variable external environment. These signals typically contain useful information for analysis as well as noise signals caused by harmonic interference. Directly extracting parameters from the collected original signal can lead to significant errors. Thus, the ongoing focus of scholars is on effectively removing noise from vibration signals and purifying key information as much as possible through continuous research efforts. Common methods used are wavelet transform,¹¹ wavelet packet transform,¹² and Hilbert transform.¹³ However, their poor adaptive ability, relying on experience to select the wavelet basis function, makes them less widely used than the prevailing adaptive processing methods at the application level. The common methods are empirical mode decomposition (EMD),¹⁴ ensemble empirical mode decomposition (EEMD),¹⁵ complete ensemble empirical mode decomposition (CEEMD),¹⁶ variational mode decomposition (VMD),¹⁷ etc. Since the VMD method can effectively suppress the end point effect and modal aliasing,¹⁸ this paper employs the VMD method to process the signal. However, determining the penalty factor $α$ and the number of modes K in VMD requires prior configuration, and for the problem of selecting the best intrinsic mode function (IMF) component after signal decomposition, the traditional methods usually use kurtosis,¹⁹ correlation coefficient,²⁰ envelope entropy,²¹ sample entropy,²² maximum information coefficient (MIC), and other indicators.²³ However, the above indicators used alone have their limitations. Under strong noise interference, there will also be selection errors, which reduces the adaptability of the diagnostic method. A technique is needed to ascertain the optimal parameter configuration and identify the most appropriate IMF component.

Support vector data description (SVDD) can establish a degradation assessment model using only normal state data when detecting outliers.²⁴ Lu et al.²⁵ introduced a performance degradation assessment model for crane rolling bearings, which integrates the particle swarm optimization (PSO) algorithm and the VMD. The model extracts fault features by optimizing parameters and uses SVDD to accurately evaluate performance degradation. Zhang et al.²⁶ proposed a fault diagnosis method of rotating machinery combing grasshopper optimization algorithm (GOA) optimized SVDD. However, the above studies mostly use artificial experience to determine the key parameters such as penalty factor C and kernel parameters $σ$ of the SVDD model to determine the performance, which cannot give full play to the best detection performance of the SVDD model. To address the issues, Fu et al.²⁷ proposed a semi-supervised SVDD model combining VMD dispersion entropy feature extraction and improved grey wolf optimization algorithm (AMGWO) optimization parameters. Tang²⁸ proposed an airport runway foreign object detection method based on bispectral features and SVDD; the essential parameters were optimized by a genetic simulated annealing algorithm (GSAA) to improve detection performance and reduce the false alarm rate. However, the above methods have many problems such as low diagnostic accuracy and slow diagnostic speed of the SVDD model. To mitigate the impact of outliers on the SVM classification model and enhance its generalization, Hsu and Lin²⁹ proposed a fuzzy support vector machine (FSVM). The FSVM distinguishes different contributions of samples by penalty coefficients and assigns smaller weights (fuzzy membership) to outliers when constructing classification models to reduce their influence on classification models. Following this idea, Qu et al.³⁰ introduced the fuzzification of sample points to enhance the model's generalization and proposed a fuzzy support vector data description (FSVDD) model. Currently, the hippopotamus optimization (HO) algorithm discovers optimal parameters by emulating the three common behaviors of hippos in the biological realm, exhibiting rapid convergence and robust stability.³¹ 

To address these challenges, this paper introduces a novel approach for assessing the performance degradation of rolling bearings through vibration signal monitoring. First, the vibration signal undergoes decomposition utilizing the optimized VMD method, and the IMF components with obvious performance degradation trends are selected according to the MIC method quantification. Then, the selected IMF components are reduced in dimensions to construct the maximum information degradation feature set (MIDFS) that characterizes bearing performance degradation. Finally, the FSVDD model optimized by the improved hippo optimization (IHO) algorithm is employed to evaluate the degradation state of rolling bearings based on the performance degradation indicator (DI) constructed in this paper. The proposed method is validated using two sets of life-test data of rolling bearings.

This paper is organized as follows. Section II introduces the basic theory of the proposed method. Section III describes the proposed PDA method. Section IV shows the application. The paper is concluded in Sec. V.

VMD is a signal processing method capable of adjusting the time-frequency scale: it decomposes the signal into multiple intrinsic mode functions (IMFs), which exhibit sparse characteristics, and each IMF component possesses its own center frequency and a limited bandwidth.³² The process of the VMD algorithm is as follows.

The Fourier transform frequency domain of the kth IMFs obtained by decomposition can be expressed as $u ^ k n + 1(ω)$⁠, where $f ^(ω)$⁠, $u ^ i(ω)$⁠, and $λ ^(ω)$ represent the Fourier transforms of $f(t)$⁠, $u i(t)$⁠, and $λ(t)$⁠, respectively.

When applying VMD for signal decomposition, its effectiveness is significantly influenced by its parameters $[K,α]$⁠. The selection of parameters in most studies heavily relies on expert experience, often leading to suboptimal signal processing results. This paper proposes using the AO algorithm to optimize the parameter combination of VMD to address the aforementioned issues.

The $[K,α]$ in VMD must be preset, and varying parameter combinations can yield vastly different decomposition results. Manual parameter tuning to achieve optimal results is challenging and can lead to issues such as end point effects, excessive decomposition, and modal aliasing. To address this challenge, this paper utilizes the AO algorithm to adaptively select parameter combinations for VMD.³⁴ 

The AO algorithm necessitates the definition of an appropriate fitness function to compute the fitness value during the optimization of VMD parameters. Parameters are updated by comparing these fitness values. Sample entropy is selected as the fitness function.³⁵ The magnitude of sample entropy indicates the complexity and regularity of the time series. Lower sample entropy suggests that the signal is more regular and predictable, while higher sample entropy indicates that the signal is more complex and random. In bearing PDA analysis, sample entropy can detect abnormal signal changes. The flow chart of the AO-VMD method is shown in Fig. 1.

The MIC, denoted as C, quantifies the strength and non-linearity of the relationship between two variables. Its non-parametric nature and capability to detect non-linear relationships make it well-suited for analyzing complex datasets.²³ The MIC primarily utilizes mutual information and grid partitioning techniques, placing it within the realm of non-parametric exploratory statistics.

KPCA technology reduces the dimensionality of datasets while enhancing data interpretability.³⁷ It preserves most data features while mapping the nonlinearity of the input space to a high-dimensional feature space, effectively transforming it into a linear problem. Subsequently, the problem is addressed within the high-dimensional space.^38–40 

The kernel function in Eq. (7) can generally use the Gaussian kernel function, polynomial kernel function, and other kernel functions. Through its mapping, the inner product operation of data in the high-dimensional feature space can be obtained. At this point, the objective function of KPCA can be converted to

Time domain feature analysis refers to the statistical analysis of the vibration signal with time as the independent variable, which can reflect the amplitude change, fluctuation size, and energy distribution of the signal. Its parameters are mainly divided into dimension indicators and dimensionless indicators, reflecting the degree of failure in the process of equipment operation and the probability density of operating characteristics, respectively. These two types of indicators in a more unified form can better reflect the performance of the bearings in the process of operation changes. In this paper, a total of 15 kinds of the above two indicators are considered for analysis. The specific equation is shown in Table I.

Based on the time domain features and entropy energy ratio features, the degradation feature set W is constructed. After feature screening and dimensionality reduction, the MIDFS is constructed, which is recorded as H, and the construction method is shown in Fig. 2, where H is input into the improved hippopotamus optimization algorithm optimized fuzzy support vector data description (IHO-FSVDD) model for performance degradation assessment.

The primary concept behind the SVDD model is to establish a curve, surface, or closed hypersphere with a small radius to encompass the data as much as possible. This allows the majority of the described data to fall within the decision boundary, while abnormal data points are delineated outside the boundary. This enables the distinction between the two types of samples.

In the SVDD model, the kernel function is introduced to map the training data from the original sample space to a high-dimensional space without increasing the algorithm's complexity. This mapping allows for a more accurate description and division of the data. This enables a more accurate description and division of the data. It effectively addresses the issue of linear inseparability in the original space, allowing for precise characterization of the samples. The Gaussian kernel function is utilized in this paper.⁴¹ After introducing the kernel function, the SVDD model can generate a hypersphere model by describing the input sample. This model allows for the qualitative and quantitative determination of whether the sample being tested belongs to the interior of the hypersphere and the degree of deviation from the sphere. This determination is based on the relationship between the center distance of the sphere and its radius.

It can be seen from Eq. (21) that within the minimum hypersphere, the closer the sample is to the center of the sphere, the larger the assigned $s i$⁠, and the closer the sample is to the edge of the sphere, the smaller the assigned $s i$⁠. Outside the minimum hypersphere, the greater the distance of the sample from the center of the sphere, the larger the fuzzy coefficient is, indicating higher importance.

The determination of the kernel parameter $σ$ and the penalty parameter C in FSVDD is optimized by IHO. The main methods of IHO-FSVDD are described below.

Amiri³¹ drew inspiration from the inherent behavior of hippos and demonstrated an innovative meta-heuristic method, the hippopotamus optimization (HO) algorithm. The algorithm adopts a three-stage model in concept, which includes the location update of the hippo in the river or pond, the defense strategy, and the avoidance method for the predator, and is expressed by mathematical methods. The HO algorithm effectively avoids the trap of falling into local minima, making the global optimization search more effective.

In addition, random initialization may result in similar individuals, leading to a reduction in search space and limiting the algorithm's searchability, causing it to converge to local optimal solutions. Therefore, this paper introduces the logistic chaotic mapping method based on the HO algorithm to enhance the randomness and distribution of population initialization, thereby improving the algorithm's global search performance. This enhancement is denoted as the improved hippopotamus optimization (IHO) algorithm.

The overall process of the IHO-FSVDD algorithm is shown in Fig. 3.

In performance degradation assessment, FSVDD is used as a distance index model. First, the health feature sample is used to establish the boundary of the model. By calculating the distance from the test set data to the boundary, the class membership degree relative to the health status sample is determined.

Assuming that the standard deviation of the health sample feature vector is $σ h$⁠, the radius of the failure threshold is $R+3 σ h$⁠. If $D≤R+3 σ h$⁠, it indicates that the rolling bearing runs normally; otherwise, it indicates that the rolling bearing is faulty.

This paper proposes a novel method for rolling bearing performance degradation assessment based on MIDFS construct and IHO-FSVDD under vibration signal monitoring, as shown in Fig. 4.

• Step 1: The vibration signal of the whole life cycle is collected according to the sensor arranged in the rolling bearing to be tested.

• Step 2: The AO algorithm is employed to optimize essential parameters in the VMD decomposition process to obtain the IMF component. Subsequently, the VMD decomposition of the bearing's life-cycle vibration signal is conducted using the optimized parameters. The MIC method is then applied to screen the IMF component containing the maximal degradation feature information.

• Step 3: Feature extraction establishes the maximum information degradation feature set (MIDFS). The feature extraction of the IMF component with high information content is carried out. Next, the feature set undergoes dimensionality reduction using the KPCA algorithm, resulting in the composition of the maximum information degradation feature set H.

• Step 4: Construct the IHO-FSVDD model and get the relevant parameters. The maximum information degradation feature set under the bearing health condition is selected as the training set sample; the IHO-FSVDD model is trained to obtain the FSVDD parameters.

• Step 5: Calculated DI to assess the performance degradation of rolling bearings. The whole life-cycle sample is input into the IHO-FSVDD model as the test set sample, and the distance D between the test sample and the center of the sphere is obtained. According to the calculation method of Eq. (24), the DI can be obtained, and then, the performance degradation assessment of rolling bearings can be carried out.

To comprehensively demonstrate the validity and general applicability of the proposed methodology, two experimental cases are analyzed in this section. The first case involves analyzing the bearing life-test data from the University of Cincinnati,⁴⁵ primarily aimed at verifying the effectiveness of the method in performance degradation assessment and early fault detection. The second case involves analyzing the dataset from XJTU-SY,⁴⁶ primarily focused on establishing the universality of the method in accelerated-life experiments.

The overall structure of the rolling bearing fatigue test rig is shown in Fig. 5. The motor drives the spindle to run at a speed of 2000 rpm through a belt drive, and the spindle is successively equipped with four double-row roller bearings of ZA-2115. The physical parameters of the test bearings are shown in Table II, and the outer ring fault frequency f₀ = 236.4 Hz can be calculated by using the speed and bearing structure parameters. Each bearing seat is equipped with a thermocouple and an acceleration sensor. The vibration signal of the bearings is sampled every 10 min, with a sampling frequency of 20 kHz. The fatigue experiment lasted 164 h, during which a total of 984 sets of data were collected. Each set of data contains four columns of data with a length of 20 480, which, in turn, correspond to the four bearings in Fig. 5. The data analyzed in this paper are the first column, and its vibration signal is shown in Fig. 6.

After the experiment, all bearings were disassembled, revealing severe outer ring failure. Based on the change in the amplitude, the life-cycle data of rolling bearings can be roughly categorized into three stages: the health state, degradation state, and failure state. However, quantitatively determining the initial point of fault occurrence is challenging. Therefore, it becomes necessary to extract fault features for analysis and diagnosis. In the calculation, the last two file data are outliers, which are excluded.

First, the AO algorithm is utilized to optimize the parameters of VMD to achieve the minimum sample entropy of the IMF after decomposition. Subsequently, the characteristic frequency of the IMF is refined. The maximum iteration during the experiment is set to 100 times, with detailed parameters provided in Table III.

To reflect the superiority of the optimization method, it is compared with the VMD algorithm based on the whale optimization algorithm (WOA-VMD)⁴⁷ and the VMD algorithm based on the particle swarm optimization algorithm (PSO-VMD).⁴⁸. The convergence of the algorithms is shown in Fig. 7.

As shown in Fig. 7, after eight iterations, the fitness function reaches the lowest value, and the algorithm converges. Currently, the fitness value is 0.16507, and the corresponding optimal parameter combination is $[ K , α]=[ 7 , 5847]$⁠. It shows that the AO-VMD algorithm has a fast convergence speed and strong spatial searchability. The time domain and frequency domain diagrams after decomposition are shown in Fig. 8.

It is evident from the figure that the characteristic frequency distribution of the signal is uniform after decomposition, with no modal aliasing phenomenon observed. However, the temporary absence of observable bearing performance degradation necessitates further feature extraction to explore the deeper information of the signal.

Reference 49 highlights that the RMS value of the vibration signal can directly indicate the vibration energy of the component and reflect the fault occurrence over time. Therefore, the RMS value is utilized as the reference for the degradation characteristics of the component, and the RMS value is shown in Fig. 9.

Observe Figs. 8 and 9: qualitative analysis is carried out from the data in the figure. Among the time domain signals of different IMF components, the amplitude changes in the IMF1 component and the IMF5–IMF7 component cannot effectively characterize the running state of the bearing. In the frequency domain signals of different IMF components, the frequency domain distribution from high frequency to low frequency is reasonable, and there is no modal aliasing phenomenon, indicating that VMD decomposition has a good decomposition effect; by analyzing the RMS value of each IMF component, it can be concluded that IMF1 and IMF5–IMF7 are not sensitive to the early vibration energy change: IMF1 component belongs to the high-frequency component, and the obvious fault energy change is identified only at 7000 min, so does the low-frequency component. The low-frequency component IMF7 can identify the fault energy change only at 9000 min, while the IMF2–IMF4 component belongs to the intermediate frequency component, which can sensitively identify the energy change in the vibration signal, effectively reflect the degradation trend of fault characteristics, and better reflect the early fault generation time and the fault development stage.

To select the most suitable IMF component for analysis and verify the conclusion of qualitative analysis, the correlation analysis between the original signal and the decomposed IMF component is carried out. Because the MIC algorithm is more suitable for calculating nonlinear variables, and the concept of information theory and probability is introduced, it is more suitable for the research problem in this paper than the correlation coefficient method. The MIC coefficients between different IMF components and the RMS value of the original signal are calculated, and the calculation results are shown in Fig. 10.

By analyzing the results in Fig. 10, the maximum MIC value of the IMF2 component is 0.98666, indicating that the component retains the most fault energy information. Therefore, the IMF2 component is selected as the research object, and the degradation feature set W is extracted and constructed.

According to the time domain features and entropy energy ratio feature of the whole life extracted from Table I and Eq. (9), the degradation feature set W is formed, as shown in Fig. 11.

Given that the RMS value effectively captures the performance degradation trend of rolling bearings, this paper selects features with higher MIC coefficients for further analysis. The calculated MIC values are shown in Fig. 12, revealing that features such as square root amplitude, standard deviation, absolute mean amplitude, kurtosis factor, and entropy energy ratio have MIC coefficients greater than 0.9. The meaning of each feature is shown in Table IV.

Conversely, other features exhibit unclear overall trends and significant noise, potentially rendering monitoring metrics insensitive to initial faults and leading to misjudgments.

KPCA is applied to diminish the dimensionality of high-dimensional features with MIC coefficients higher than 0.9. Specifically, the Gaussian kernel function is designated as the kernel function for this endeavor.⁵⁰ As delineated in Table V, the contribution rates of the initial five principal components are depicted. Notably, the contribution rates of the foremost three principal components amount to 91.55%, 7.25%, and 1.11%, respectively. It is discerned that the initial principal component encapsulates more than 90% of the variance, thereby encompassing comprehensive degradation information pertaining to the bearing. This proportion notably exceeds that of the residual principal components. Consequently, the initial principal component is extracted to aptly portray the performance degradation trajectory of the bearing.

The first three principal components are shown in Fig. 13, and the first principal component is taken as the maximum information degradation feature set of the bearing, which is denoted as $H=[ P 1]$⁠, and H is input to the performance degradation assessment model.

For the purpose of bolstering evaluation precision, this study incorporates the IHO algorithm to optimize the parameters of the kernel function denoted as $σ$ and the penalty factor C, within the framework of the FSVDD model. The initial parameter configurations of the algorithm are elucidated in Table VI.

A total of 500 sets of data, characterized by a healthy operational period ranging from 1000 to 5000 min, are chosen for calculating H. These datasets are input into the IHO-FSVDD model for training purposes. Subsequently, the entire dataset is input into the trained IHO-FSVDD model for testing. The optimal parameter set and evaluation accuracy are presented in Table VII, and the DI value is calculated by Eq. (24), and the degradation evaluation result is shown in Fig. 14.

According to the information depicted in Fig. 14, the DI is roughly divided into four sections on the time axis: normal state—[0,5320] min, early degradation—[5330,7020] min, severe degradation—[7030,9090] min, and failure state—[9100,9820] min. During the period of 7030–9100 min, the DI of the bearing suddenly increases and then decreases, and repeatedly experiences fluctuations. The reason is that the wear degree of the rolling bearing fault point gradually increases with the running time, resulting in a larger impact signal. Then, the fault is smoothed again, resulting in a reduced vibration impact; with the increased running time, the wear degree of the worn place is increased again. After 9100 min, the DI exhibits chaotic fluctuations, indicating severe bearing failure.

To verify the above experimental results, samples of 5320, 5330, 6050, 7030, 9100, and 9580 min were selected for envelope spectrum analysis, as shown in Fig. 15.

As shown in Fig. 15(a), the frequency of faults in the outer ring of the sample at 5320 min cannot be clearly observed, suggesting that the sample remains in a healthy state at this juncture. Moving on to Fig. 15(b) at 5330 min, a weak characteristic frequency of failure can be observed at 1-time frequency, suggesting the onset of early failure in the outer ring of the rolling bearing. Subsequently, in Fig. 15(c) at 6050 min, the 1-, 2-, and 3-time frequency components become clearly observable, indicating the bearing's entry into an accelerated degradation stage. At 7030 min, as shown in Fig. 15(d), the fault characteristic frequency of the signal is highlighted, with the 2-, 3-, and 4-time frequency components clearly observable, signifying that the bearing has entered a severe degradation stage. Moving further to Fig. 15(e) at 9100 min, almost only the fault eigenfrequency and its times frequency are left in the envelope spectrum, indicating the bearing's entry into the failure stage. Finally, as depicted in Fig. 15(f) at 9580 min, due to cracks in the outer ring of the bearing, in addition to the characteristic frequency of the bearing failure, a decrease in the amplitude of the 2-, 3-, and 4-time frequency components can be observed, marking the complete failure of the bearing.

To validate the superiority of the proposed method, a comparative study is undertaken utilizing the control variable method. In the first set of comparison experiments, in order to compare the performance of different models, the input parameter of all three comparison models is the maximum information degradation feature set H. The three degradation evaluation models are the FSVDD model (with a specified penalty factor C of 1 and a Gaussian kernel function of 180), the IPSO-FSVDD model and the ISSA-FSVDD model. In the Improved Sparrow Search Algorithm (ISSA) model, the optimization parameters are the Gaussian kernel function $σ$ and the penalty factor C. The population set is 100, and the maximum iteration is 200; the results are shown in Fig. 16.

As shown in Fig. 16, the state assessment results of all three performance degradation assessment models produce a certain degree of misclassification after inputting H. The specific results are shown in Table VIII.

Analyzing Fig. 16 and Table VIII reveals that the ISSA-FSVDD method exhibits the highest assessment accuracy. All three models utilizing H as an input variable for performance degradation assessment successfully identified early failures at 5330 min. However, both the FSVDD model and the IPSO-FSVDD model displayed two outliers before 5300 min at 134 and 471 min, respectively, with the DI value of the IPSO-FSVDD model smaller than that of the FSVDD model in these two instances. Moreover, the ISSA-FSVDD model encountered one outlier before 5300 at 134 min. This observation indicates that different assessment models employing H as an input variable can accurately predict early failure times, albeit with some outliers and false fluctuations. However, the accuracy of the performance degradation assessment model introduced in this paper surpasses that of these methods, as discerned from comparative analysis. Thus, the superiority of the IHO-FSVDD model posited in this paper is unequivocal.

In the second group of comparison experiments, the degradation performance assessment models all adopt the IHO-FSVDD model. The degradation indicators take the square root amplitude, absolute average amplitude, and standard deviation extracted from the original signals as inputs to the IHO-FSVDD model. These three degradation indicators were chosen because they can characterize vibration signals in terms of energy and dispersion. The results are shown in Fig. 17.

As shown in Fig. 17, when different degradation indicators are input into the IHO-FSVDD model, all three performance degradation assessment models exhibit a certain degree of assessment outliers during the early period, as demonstrated in Table IX.

Analyzing Fig. 17 and Table VII reveals that different degradation indicators exhibited numerous instances of outliers and spurious fluctuations before 5330 min. Among them, the square root amplitude value degradation indicator displayed the fewest outliers and spurious fluctuations, whereas the standard deviation value exhibited the highest. However, in terms of the overall trend, all three degradation indicators demonstrated significant changes in the DI value at 5330 min. From these findings, it is evident that the bearing experienced outer ring failure at 5330 min. However, relying solely on a single indicator for performance degradation assessment may lead to misleading condition monitoring and generate unnecessary maintenance costs. Therefore, it is not advisable to use a single indicator for performance state assessment.

The localized zoom assessments of each comparative method disclose that the degradation indicator (DI) values obtained through the proposed method in this paper manifest the most consistent trend during the healthy operational periods, in contrast to the comparatively more erratic fluctuations observed in the DI values derived from alternative methods. Additionally, the proposed method showcases superior robustness and accuracy in DI computation. Furthermore, envelope spectrum analysis corroborates the precise identification of the initial failure time by the proposed method, affording ample warning for equipment maintenance. These observations underscore the proposed method's sensitivity to rolling bearing performance degradation across its lifespan, alongside its adeptness in mitigating outliers and incidental fluctuations, thus furnishing a more precise depiction of rolling bearing degradation throughout its operational lifespan.

To substantiate the superiority of this method, the XJTU-SY bearing dataset is subjected to analysis. The test data are from the joint laboratory of Xi ‘an Jiaotong University and Changxing Shengyang Technology Co., Ltd.⁵¹  Figure 18 is the structure of the experimental platform, which is mainly composed of an AC motor, motor speed controller, support shaft, two support bearings, and a hydraulic loading system. The test bench is generally used for accelerated degradation tests of bearings.

The original vibration signal is acquired by employing an acceleration sensor positioned on the outer ring of the test bearing. Concurrently, a radial load is imposed to expedite the degradation process of the bearing. The sampling frequency of the bearing vibration signal stands at 25.6 kHz, with a sampling duration of 1.28 s. Each sampling interval spans 1 min, and individual sampling data comprise 32 768 data points. This paper focuses on evaluating the performance degradation of bearing 2_1. A total of 491 sets of samples are collected under the selected working conditions, and the vibration signals are shown in Fig. 19. Bearing 2_1 has an inner ring fault after the experiment, and the characteristic frequency of the inner ring fault is calculated to be f₁ = 196.67 Hz. The physical parameters of the test bearings are shown in Table X.

The performance degradation of bearing 2_1 is evaluated according to the method proposed in this paper, and the results are shown in Fig. 20.

Based on the contents of Fig. 20, it can be concluded that the performance degradation metrics are roughly divided into four segments on the time axis, which are the normal state—[0,452] min, early degradation—[453,456] min, severe degradation—[457,476] min, and failure state—[477,491] min.

To verify the above experimental results, the samples of 452 and 453 min were selected for envelope spectrum analysis, as shown in Fig. 21.

As shown in Fig. 21(a), the failure frequency within the inner ring of the sample at 452 min is not distinctly discernible, suggesting that the sample remains in a healthy state at this juncture. Conversely, in Fig. 21(b), precisely at 453 min, a faint characteristic frequency of the failure emerges at 1x and 2x frequencies, indicating a weak early failure within the inner ring of the rolling bearing.

To prove the superiority of the proposed method in this paper, comparative experiments are conducted based on the comparative experimental criteria in Sec. IV A 4, and the results are shown in Figs. 22 and 23.

Based on the experimental results in Fig. 22, it can be concluded that all three optimization models can accurately predict the weak early failure appearance time at 453 min, and the absence of spurious fluctuations and outliers proves the superiority of the maximum information degradation feature set H proposed in this paper.

According to the experimental results in Fig. 23, it can be concluded that all three different degradation features exhibit outliers and false fluctuations in the health stage. Additionally, they all show significant fluctuations in DI values at the beginning of the experiment, attributed to the break-in period during the early operation stage of the bearing. In terms of performance degradation assessment, it is notable that, in contrast to the method proposed in this paper, the DI values associated with the three degradation features exhibit no alteration, post the 453 min mark. This indicates that the three degradation features cannot effectively characterize the degradation stage of the bearings. The method proposed in this paper can significantly characterize the performance degradation trend after 453 min, which is of great significance in engineering practice, especially in maintenance decisions. Although the traditional time domain features used in accelerated-life experiments can effectively assess the time of the first failure, their robustness and interpretability are significantly inferior to the method proposed in this paper.

High-precision sensor monitoring of vibration signals throughout the full life cycle and subsequent performance degradation assessment offers a means to mitigate the risk of component failure, but the degradation indicators in the traditional input assessment model, the determination of which has not been adaptive decomposition to extract degradation characteristics, have low tolerance difference constants and false fluctuations as disadvantages. To solve this problem, this paper proposes a rolling bearing degradation performance assessment method based on MIDFS and IHO-FSVDD under vibration monitoring. Through experimentation, the effectiveness of the method is validated, leading to the following conclusions:

1. The VMD decomposition of vibration signals holds paramount importance in the assessment of rolling bearing performance degradation. It proficiently mitigates the interference of signal-degraded features arising from extraneous frequency components and identifies the IMF component harboring the most pertinent degraded information. Specifically, the high-frequency IMF component and the low-frequency IMF component of the vibration signal carry less degradation information than the IMF mid-frequency component, and the MIC algorithm can be used to determine the portion of the IMF component that carries the richest degradation information.

2. A novel method for constructing the maximum information degradation feature set (MIDFS) is introduced: this method entails determining the adaptive maximum information degradation feature set through the adaptive decomposition of vibration signals, leveraging the MIC method and the KPCA method for constructing the degradation features. This approach obviates the need for manual parameter configuration, such as weights and thresholds, thereby facilitating a more objective determination of the bearing degradation stage compared to the direct utilization of vibration signals for constructing the degradation feature set. Consequently, the quality of input for the PDA model is enhanced, culminating in improved accuracy in assessing bearing degradation.

3. The proposed IHO-FSVDD method optimizes key parameters in the FSVDD model, enabling effective early failure identification and performance degradation assessment of rolling bearings through training on early-stage health samples. This method surpasses other comparative approaches by mitigating the effects of outliers and spurious fluctuations, resulting in a more accurate characterization of bearing degradation stages.

4. The efficacy of the proposed method is verified through the utilization of both Cincinnati full life-cycle bearing data and XJTU-SY bearing data. The proposed method was evaluated with 99.62% accuracy on the IMS life-test bearing dataset. In the first set of comparison experiments, an improvement of 1.06% was achieved compared to the ISSA-FSVDD method, which had the highest accuracy, and in the second set of comparison experiments, an improvement of 3.79% was achieved, compared to the square root amplitude degradation metric, which had the highest accuracy. Multiple comparative experiments involving different performance degradation assessment models and degradation indicators demonstrate that the model proposed in this paper outperforms many mainstream methods. It exhibits superior accuracy and degradation trend determination compared to the comparative methods.

This work was supported by the Basic Research Project of Liaoning Provincial Department of Education, China (Grant No. LJKMZ20222192), the Liaoning Provincial Department of Education Science and Technology Innovation Characteristic Program (Grant No. JYTMS20230002), and the National Science Foundation for Young Scientists of China (Grant No. 62001079).

The authors have no conflicts to disclose.

Chaoqun Hu: Conceptualization (equal); Investigation (equal); Methodology (equal); Resources (equal); Software (equal); Writing – original draft (equal). Zhe Chen: Validation (equal). Yonghua Li: Supervision (equal); Writing – review & editing (equal). Xuejiao Yin: Validation (equal).

Data sharing is not applicable to this article as no new data were created or analyzed in this study.