Novel Fault Detection Method for Rolling Bearings Based on Improved Variational Modal Decomposition Method

To enhance the precision of rolling bearing fault detection and lessen the likelihood of safety mishaps, this paper proposes a fault detection method grounded in improved variational mode decomposition. This technique initially employs the Northern Eagle Algorithm to determine the optimal parameter value for variational mode decomposition, subsequently decomposing the signal. This is followed by the utilization of the Spearman correlation coefficient to differentiate between the effective component and the noise-dominant component. Finally, the wavelet packet decomposition is adopted to filter noise and yield the Hilbert envelope spectrum of the de-noised signal to ascertain the bearing’s health status based on the extraction of characteristic fault frequencies and harmonics. The experimental findings illustrate that the enhanced variational mode decomposition technique not only escalates the inner ring signal’s signal-to-noise ratio from −10.844 dB to 8.4471 dB and the outer ring signal’s ratio from −4.5852 dB to 3.0997 dB but also reduces the error of outer ring fault detection from 3.14% to 0.37%, and improves the frequency of inner ring fault detection from a feature extraction inability to an error frequency of 0.45%.


I. INTRODUCTION
Rolling bearings are a critical component in mechanical systems, with their health condition directly influencing the normal functioning of the entire system.However, given the harsh working conditions they are subjected to, rolling bearings often see high failure rates.Therefore, the accurate extraction of fault characteristic frequencies and timely detection of their health status is crucial.The task of recording the vibration signals from rolling bearings is often met with substantial noise interference.Given their nonlinearity and non-stationarity, these challenges further complicate fault identification [1].To accurately extract fault characteristic The associate editor coordinating the review of this manuscript and approving it for publication was Julien Le Kernec .frequencies, efficiently filtering the noise from the vibration signal without compromising the effective information has emerged as a key focus of current research.
Currently, numerous universities and researchers are conducting extensive studies on noise removal from signals.For instance, in February 2020, Hua et al. introduced a Grasshopper optimization algorithm, which enhances Variational Mode Decomposition by integrating it with the Hausdorff distance and wavelet transform for denoising purposes.This denoising method outperformed traditional ones [2].In January 2021, Liu et al. put forward a denoising technique combining Variational Mode Decomposition with machine learning online optimization and interval threshold technology.This approach showed promising results in Lidar signal denoising [3].September 2014 saw Lahmiri proposing a mixed denoising method comprising Variational Mode Decomposition and Discrete Wavelet Transform, effectively eliminating additive Gaussian noise from electrocardiogram data [4].Moreover, in October 2021, Lu et al. suggested utilizing the ratio of mean to variance of permutation entropy as the fitness-function within the Sparrow Search Algorithm.The goal was to search for the optimal parameters for variational mode decomposition to decompose pipeline leakage signals [5].Li et al. proposed a mixed quadratic denoising algorithm combining variational mode decomposition and correlation coefficients in February 2018 to filter out noise in ship sound signal data [6].Wang et al. proposed the crow algorithm for optimizing variational mode decomposition in August 2021 and applied it to the field of microelectromechanical system gyroscopes.Noise separation and effective components were separated based on the sample entropy of each intrinsic mode component, and noise was filtered using smooth denoising [7].Wang et al. proposed a wavelet transform joint variational mode decomposition algorithm for LiDAR signals based on the sparrow search algorithm in September 2022, aiming to remove noise from LiDAR signals.Simulation experiments showed significant denoising effects [8].In September 2022, Hu Hongping and his team introduced a joint denoising algorithm based on wavelet threshold denoising, multiverse optimizer, and particle swarm optimization for variational mode decomposition.The algorithm aimed to eliminate noise in signals received by MEMS vector hydrophones.A series of simulation experiments demonstrated the superior performance of this algorithm compared to other similar ones [9].In February 2019, Yan Xiaoan et al. proposed the introduction of cuckoo algorithm optimized variational mode decomposition and optimal scale morphology slice bispectral analysis techniques in rolling bearing fault detection.The experimental results showed that cuckoo algorithm optimized variational mode decomposition can greatly suppress Gaussian noise, providing a new perspective for bearing fault detection [10].Hu et al. proposed a wavelet threshold denoising method based on variational mode decomposition and genetic algorithm optimization for bearing fault diagnosis in September 2022, which achieved good results [11].In July 2021, Ye Maoyou and his team proposed a process in which a multi-scale permutation entropy decomposition signal is based on variational mode decomposition and then reconstructed using the characteristic energy ratio criterion.To round off their proposal, a particle swarm optimization support vector machine model was constructed.This model can automatically identify different failure modes [12].In August 2022, Jin et al.Proposed a fault diagnosis method using the whale optimization algorithm to optimize the variational mode decomposition and the improved particle swarm optimization algorithm to optimize the least squares support vector machine, which improved the accuracy of bearing fault detection [13].Guo Junchao et al.Proposed a fault diagnosis scheme based on optimized wavelet packet de-noising and modulated signal bispectrum in August 2021.
Experiments show that this method has superior performance in extracting fault features of early defects on different bearing components [14].In April 2022, in order to improve the recognition performance of bearing fault signals, Li et al.Proposed the variational mode decomposition and center frequency extraction method optimized by genetic algorithm, which reduced the difficulty of fault feature extraction [15].Informed by the insights of the above literature and in the pursuit of more effective methods for bearing fault detection, this article introduces an innovative method for detecting faults in rolling bearings, which involves optimizing variational mode decomposition in combination with wavelet packet decomposition using the Northern Eagle Algorithm.This new approach uses the principle of minimum fuzzy entropy as the fitness function to globally search for optimal values of the key parameters of variational mode decomposition, enhancing its efficiency.It separates effective components from those dominated by noise by taking advantage of the fact that noise-dominated signals have less effective information and have lower correlation with the original signal.Here, the Spearman correlation coefficient is calculated.Once the noise-dominated components are separated, wavelet packet decomposition is applied to filter the noise and combine it with other components.Subsequently, the Hilbert envelope spectrum of the vibration signal is obtained.Based on whether the periodic impact signal of the fault frequency and its harmonics is extracted, it is determined whether there is a fault in the bearing.The experimental results outline that this detection method is excellent at filtering out signal noise, has a robust anti-interference ability, can make the fault frequency's impact component clearer, and ultimately improves fault detection accuracy.The experiment led to an increased inner ring signal signal-to-noise ratio of 19.2911 dB and an outer ring signal signal-to-noise ratio of 7.6849 dB.Additionally, outer ring fault detection error decreased by 2.77%.This provides valuable new insights for bearing fault detection.
The structure of this article is as follows: The first part introduces the research progress in the relevant fields of this article.The second part proposes a fault detection strategy based on an improved variational mode decomposition method.The third part conducts experiments on the signals of the inner and outer rings of rolling bearings to verify the effectiveness of the new method for fault detection of rolling bearings proposed in this paper.The fourth part compared several other methods and verified the superiority of the improved variational mode decomposition method in the field of fault detection.The fifth part provides experimental conclusions.

II. PROPOSED FAULT DETECTION STRATEGY A. VARIATIONAL MODAL DECOMPOSITION
Variational Modal Decomposition(VMD) is a new adaptive time-frequency analysis method proposed by Dragomiretskiy in 2014, which can suppress interference signals and avoid the loss of useful information [16].VMD has the advantages of self setting the number of modal decompositions to achieve effective separation of intrinsic modal components and frequency domain division of signals.The principle is: Assume that the original signal f (t) is decomposed into K eigenmodal IMF components, and the equation constraint to be satisfied by the K components is the original signal after superposition, and the inequality constraint is the minimum sum of the estimated bandwidths of each modal component.
In the above formula, δ t is the derivative symbol, δ(t) represents the Dirac function, K represents the number of IMFs, and u k (t) and ω k (t) represent modal classification and center frequency, j is an imaginary unit, f (t) is the input signal, s.t.represents the constraint condition.The quadratic penalty term α and the Lagrange multiplier λ are introduced to construct the augmented Lagrange function.
The alternating directional multiplier algorithm (ADMM) is used to calculate the optimal solution of the variational constraint model, and the solution process can be divided into inner and outer loops [16], and the formula for {u k }, {ω k }, {λ}.
In the third step, the second step is repeated, During each execution, one is automatically added until k = K , the Lagrange multiplier λ is updated according to equation (6).
Outer loop: when the modal component satisfies equation ( 6), the iteration is terminated and the modal decomposition is completed, if not, the result of the inner loop is used as the initial value to execute the inner loop again [16].

B. FAULT DETECTION BASED ON IMPROVED VARIATIONAL MODAL DECOMPOSITION
The decomposition effect of VMD depends on the choice of parameters α and K .The Northern Goshawk Algorithm (NGO) is an optimization algorithm proposed by MOHAM-MAD DEHGHANI et al. in 2022, the algorithm simulates the process of recognizing the prey, attacking the prey, and pursuing the prey after it escapes during the capture of prey by northern goshawks [17], have better global search capability with higher convergence accuracy, Moreover, fuzzy entropy can effectively adapt to nonlinear and nonsmooth fault signals [18].To address the weaknesses of the conventional VMD, this paper introduces a novel fault detection method based on Improved Variational Modal Decomposition (IVMD).This method integrates the advantages of three different approaches to enhance the precision of bearing failure detection.The method consists of two primary stages: The first stage involves the removal of perturbations from the signal.NGO is employed to determine the optimal values of the core parameters of the VMD.These optimal parameters are then utilised in the VMD to decompose the target signal.Capitalizing on the fact that noise-dominated Intrinsic Mode Functions (IMF) hold less valid information and correlate minimally with the original signal, this study separates these noise-dominant IMFs.The separation is based on each IMF's Spearman's correlation coefficient with the original signal.After filtering out the noise in them using Wavelet Packet Decomposition (WPD), these IMFs are superimposed with the remaining IMFs to successfully filter out noise interference in rolling bearing signals.In the second stage, fault detection is carried out.The Hilbert envelope spectra of rolling bearing vibration signals are obtained.Based on theories related to resonance mediation techniques, it is evident that bearings carrying damage will invariably generate periodic repetitive transient shocks when in operation [18].If the envelope spectrum of the rolling bearing vibration signal exhibits a shock signal of the fault frequency and its octave, it can be concluded that the bearing is flawed.Appropriate technicians should be alerted promptly to inspect and diagnose the precise type of fault, thereby eliminating potential safety risks.The steps to implement the IVMD method are as follows: Step 1: Configure the parameters for the NGO.The population size should be initialized at 20, and the maximum number of iterations should be set to 30.The search ranges for parameters K and α are set between [3,11] and [800,9000], respectively.
Step 2: Define the fitness function for NGO.This is done by using minimal fuzzy entropy as a fitness function for NGO.The calculation for both fuzzy entropy and the fitness function is as follows: Eq. ( 7) is the formula for fuzzy entropy, Eq. ( 8) is the fitness function of the NGO algorithm, where N denotes the number of IMFs and F i denotes the fuzzy entropy of the i-th IMF.
Step 3: Update the values of parameters K and α according to the minimum fuzzy entropy of the IMF decomposed at each iteration.
Step 4: Reiterate steps 1 through 3 until the number of iterations reaches the maximum set limit, and output the optimal parameters K and α.
Step 5: Decompose the signals to separate the noise dominant IMF.This is accomplished by decomposing the target signal according to the optimal values of parameters K and α, followed by calculation of the Spearman correlation coefficient between the IMF and the original signal.This step realizes the separation of effective IMF from noise-dominated IMF.
Step 6: Noise interference is removed.This is done by separating the noise dominant IMFs and utilizing WPD to filter out the noise within them.Once a small amount of valid information is extracted, it is superimposed onto the valid signal to obtain a pure signal with removed noise.
Step 7: Fault detection.This involves producing the Hilbert envelope spectrum of the signal derived from step 6.The presence of faults is then detected by observing if there are any periodic shock signals of the fault frequency and its multiples within its envelope spectrum.
Steps 1 through 7 are the fundamental stages of the new IVMD-based rolling bearing fault detection method proposed in this paper.A flowchart of these steps is illustrated in Figure 1.The superiority of this method will be confirmed in this paper both in terms of perturbation resistance and fault frequency extraction error.

III. CASE STUDY A. DATA SOURCE
To validate the efficacy of the rolling bearing fault detection method proposed in this paper, experimental data was gathered from the Case Western Reserve University of the United States.The measured signals were from a 6205-2RS JEM SKF deep groove ball bearing's drive end, specifically the inner ring vibration signal and outer ring vibration signal, with the first 3,000 data pieces used as experimental data.The specifics of this experimental data are outlined in Table 1.

B. FAULT CHARACTERISTIC FREQUENCY AND EVALUATION INDEX CALCULATION METHOD
In this paper, signal-to-noise ratio (SNR) and root-meansquare error (RMSE) are used as indicators for evaluating the effectiveness of bearing vibration signal denoising methods, the actual fault eigenfrequencies were extracted using Hilbert envelope mapping, SNR, RMSE and the theoretical fault eigenfrequencies of the inner and outer rings are calculated as: SNR in Eq. ( 9) represents the signal-to-noise ratio, P s and P n represent the power of the active signal and the power of the noise, respectively.RMSE in Eq. ( 10) represents the root-mean-square error.f i and f o denote the theoretical fault eigenfrequencies of the inner and outer rings, Z is the number of scrollers, F is the frequency of the rolling bearing, d is the diameter of the rolling element, D indicates the pitch diameter of the bearing raceway, and α is the contact angle.

C. ROLLING BEARING INNER RING FAILURE DETECTION
Proposed in this study, particularly for identifying inner ring faults, Gaussian white noise of 20 dB was added to the vibration signals from the inner ring.Subsequently, the SNR of the vibration signal from the faulty inner ring, after the addition of noise, resulted in −10.844 dB while the RMSE was 1.0226.By contrasting the SNR and RMSE of the signals before and after noise removal, the fault detection method's anti-disturbance ability was ascertained.Also, the fidelity of the fault detection was validated by comparing the spectra of signals before and after disturbance removal, and by evaluating the discrepancy in the extraction of fault frequency.Fig 2 illustrates the time-domain signal of the vibration signal from the inner ring for a duration of 0-0.2s after the addition of Gaussian white noise.
Figure 2 clearly illustrates that the vibration signal from the inner ring, after the addition of noise, manifests conspicuous spikes, burrs, and shocks, leading to substantial noise interference.In order to accurately extract the fault frequency, it's imperative to first filter out the noise within the vibration signal from the inner ring, an essential step in reducing errors in fault feature frequency extraction.In order to validate the effectiveness and superiority of the NGOs, using the Whale Algorithm (WOA), Gray Wolf Algorithm (GWO), African Vulture (AVOA), Dung Beetle Optimization Algorithm (DBO), Pelican Optimization Algorithm (POA), and Sparrow Search Algorithm (SSA) as a control.The convergence curve is shown in Figure 3.
Figure 3 presents the convergence curves of the seven optimization algorithms when applied to the vibration signals of the inner ring with added Gaussian white noise.As depicted in Figure 3, the NGO algorithm exhibits the quickest convergence rate, achieving convergence as early as the 2nd iteration.This process not only displays the highest convergence accuracy, but also results in an optimal  fitness value of 0.1767.The search results derived from the optimization of the seven algorithms are documented in Table 2.
The optimal parameter decomposition of the inner ring fault signal for the VMD is designated as [10,8990].Postdecomposition, each IMF plots in both time and frequency domains are illustrated in Figure 4.
Figure 4 presents various plots.Subplot (a) demonstrates the time-domain view of the decomposed vibration signal from the inner ring for 10 IMFs, while subplot (b) delineates the frequency domain view of the 10 IMFs from the aforementioned vibration signal.Analysis of Figure 4 indicates that there's no modal aliasing occurring across all IMFs.Subsequently, the Spearman correlation coefficient is computed between each IMF and the fault signal of the inner ring, the results of which are tabulated in Table 3. Imposing a threshold value of 0.2, IMFs with correlation coefficients below this value are considered predominantly noise-driven Intrinsic Mode Functions due to either a weak correlation or no correlation at all with the signal.
Reviewing the data in Table 3, it is clear that the correlation coefficients of IMF1, IMF9, and IMF10 fall below the threshold of 0.2, implying that noise predominantly influences these IMFs.Conversely, IMFs ranging from IMF2 to IMF8 are confirmed to be effective IMFs.
WPD relies on the selection of wavelet basis functions.The denoising impact of IVMD, when implemented with eight wavelet basis functions such as Bior5.5,Coif5, and Dmey, can be observed in Table 4.  Table 4 demonstrates that IVMD provides the most effective denoising results when using Bior5.5 as the wavelet basis function.The denoised inner ring fault vibration signal presents the highest SNR of 8.4471 dB and the minimal RMSE of 0.1108.Compared to the pre-denoising inner ring vibration signal, the SNR has improved by 19.29 dB, and the RMSE has advanced by 0.91. Figure 6 shows a time-domain comparison of the signal before and after the perturbation extraction.
Figures 5 and 6 illustrate that when utilizing IVMD for the fault detection of inner ring vibration signals, satisfactory results have been achieved in terms of anti-disturbance, most of the noise within the signal has been effectively filtered out [19].In summary, these findings highlight that the method proposed in this paper can efficiently eliminate noise in the vibration signal of the inner ring.Moreover, it demonstrates strong anti-interference capability in detecting faults in the inner ring.
The Hilbert envelope spectrum of the inner ring vibration signal, without noise removal, is displayed in Figure 7, while Figure 8 showcases the Hilbert envelope spectrum of the denoised signal.The theoretical fault eigenfrequency of the inner ring, as per the formula, calculates to 159.93 Hz.The  Hilbert envelope spectrum is then utilized to ascertain if the inner ring is malfunctioning.
The Hilbert envelope spectra of the inner circle signals within the frequency range of 0-1000 Hz, before and after the perturbation removal, are intercepted and analyzed.In the envelope spectrum of the pre-denoising signal, the shock signal spectral lines of the fault frequency and its octaves are drowned by interference spectral lines, these are numerous in all low-frequency bands, resulting in a low signal-tonoise ratio.This prevents the effective extraction of the actual fault characteristic frequency shock signal, rendering it impossible to determine whether a rolling bearing has an inner ring failure.It is evident that signal perturbations can significantly impact bearing fault detection [14].Upon analyzing Figure 8, it is noticeable that the denoised signal can clearly extract the shock components of the actual fault frequencies: f i = 159.2Hz, 2f i = 318.41Hz, 3f i = 477.61Hz, 4f i = 636.8Hz, 5f i = 796 Hz, and 6f i = 955.2Hz.The deviation from the theoretical failure frequency is e = 0.45%, with the fault frequency and amplitude at the octave frequency exhibiting a periodic trend of decreasing and then increasing, in line with octave multiplicity [20].From this, it can be deduced that there's a failure in the inner ring of the bearing, necessitating immediate shutdown for maintenance.The exact location and the type of the fault will be determined by a professional, ensuring safety hazards are eliminated before resuming operation.
In conclusion, the IVMD technique, even in strong interference environments, can extract the fault frequency with minor error, accurately detecting potential issues in the inner ring.This discovery is of considerable practical application value.

D. ROLLING BEARING OUTER RING FAULT DETECTION
a bid to further demonstrate the of the IVMD method in fault detection enhance the efficiency and accuracy of this technique, we undertook additional steps.We introduced 20 dB of Gaussian white noise to the outer ring vibration signal, thus simulating environmental disturbances.The resulting SNR of the outer ring vibration signal post-noise addition stood at −4.5852 dB with a RMSE of 0.994.We proceeded with the same IVMD method for fault detection.The time-domain signal patterns of the outer ring vibration letter, from the original signal spanning 0-0.2s postaddition of Gaussian white noise, are depicted in Figure 9.
Fig. 9 illustrates the 0-0.2s time-domain plot of the outer ring vibration signal before and after the implementation of noise.It's evident from Figure 9 that there's considerable noise disruption in the outer ring signal after the addition of noise.To address this, seven optimization algorithms were employed to find the optimal core parameters of VMD on the faulty signals in the outer ring.The iteration curve is exhibited in Figure 10.Of all these methods, NGO has proven to be the most accurate in terms of convergence, with an optimal fitness value of 0.1911.It also has the fastest convergence rate, reaching convergence within the 5th iteration.The optimal results yielded by the seven optimization algorithms on the vibration signals of the outer ring are presented in Table 5.Additionally, when taking [10,8257] as the optimal parameter, the decomposition of the outer ring vibration signal is depicted in Figure 11.
Figure 11 presents the time and frequency domain diagrams of each IMF of the outer ring vibration signal.Subplot (a) in Figure 11 exhibits the time-domain plot for 10 IMFs of the outer ring vibration signal.In contrast, subfigure (b) illustrates the frequency-domain plot of the 10 IMFs of the same vibration signal -it's noteworthy, that no modal aliasing arose at the center frequencies of all IMF components.The Spearman's correlation coefficients of each IMF corresponding to the outer ring vibration signal are tabulated in Table 6, where the correlation coefficients of IMF6 and IMF7 are greater than 0.2.These are counted as effective IMFs, while the others are dominated by noise.Table 7 demonstrates the denoising effect of IVMD with different wavelet basis functions.
Drawing from Table 7, it can be observed that IVMD exhibits the best denoising performance when the ''sym7'' wavelet basis function is used.Post-denoising, the outer ring vibration signal ends up having the highest SNR at 3.0997 dB, and the smallest RMSE at 0.3799.Compared with pre-denoising signals, the SNR sees an improvement of 7.68 dB, and the RMSE records a reduction of 0.61.The time-domain diagrams of the signals pre-and post-IVMD processing are sketched out in Figures 12 and 13.
As visually depicted in Figures 12 and 13, the effectiveness of the IVMD denoising implementation is confirmed on the outer ring vibration signals, where it is seen to effectively filter out most of the noise interference.A comparative analysis of the Hilbert envelope spectra of the outer ring vibration signals, pre-and post-denoising, endorses the precision of this fault detection method specifically for outer ring fault detection.
Figuress 14 and 15 exhibit the Hilbert envelope spectra within the frequency range of 0-1000 Hz, both before and after the removal of perturbations from the outer ring signal.The characteristic frequency of the outer ring fault, calculated by Equation, equates to 105.87 Hz.However, the characteristic frequency diagnosed in the envelope spectrum of the inner circle signal before denoising is 102.54Hz, presenting a substantial error of 3.14% about the theoretical fault eigenfrequency.Moreover, excessive interference spectral   warranting immediate shutdown for maintenance to rectify the fault.To sum up, IVMD proves efficient in filtering noise interference in the outer ring signal and accurately detects the outer ring fault.

IV. EXPERIMENTAL COMPARISON
In a bid to validate the effectiveness, superiority, and practical worth of IVMD in noise filtering and fault detection in realworld engineering, we referenced literatures [21] and [23] robust denoising capability.SVMD, MVMD, and REMD, in contrast, exhibit limitations in their capacity to effectively remove perturbations from the signal.Fault detection is subsequently realized by deriving the Hilbert envelope spectrum of the denoised signal.A comparative presentation of the denoising and fault detection performance of IVMD versus the aforestated three techniques is encapsulated in Table 8 and 9.
As per the data presented in Table 8 and table 9, it becomes evident that IVMD exhibits the most effective denoising capabilities on inner and outer ring vibration signals.The processed signal boasts the highest SNR and the minimum RMSE.An examination of the time-domain plot of the signal, paired with the fault detection error, reveals that IVMD provides an excellent removal of noise interference from the signal.Furthermore, it fully restores the peaks and valleys state with high fidelity and solid robustness.Conversely, methods such as SVMD, MVMD, and REMD show lesser efficacy in suppressing noise within the signal.The processed signal demonstrates a negative SNR, implying that noise remains a dominant element in the signal.When combined with the fault detection error of the denoised signal, it can be observed that the IVMD denoised signal for fault detection bears a small error, thereby indicating that IVMD can more precisely diagnose whether a fault has occurred.In stark contrast, other fault detection methods record considerable errors during the extraction of fault characteristic frequencies and fail to accurately determine the health condition of the bearings.

V. CONCLUSION
In this study, we introduce and experimentally validate a novel, optimized approach to rolling bearing fault detection.Our method primarily focuses on the vibration signals of the inner and outer rings, recorded from the drive end of the 6205-2RS JEM SKF deep groove ball bearing at Case Western Reserve University in the United States.
A comparison of different optimization methods such as SSA, NGO, WOA, GWO, AVOA, DBO, and POA has shown NGO as the fastest and most accurate, thus making it the best choice for optimizing VMD.The novel approach also employs the Spearman correlation coefficient to differentiate the effective IMF from the noise-dominated IMF.This involves utilizing the WPD to eliminate noise interference within the noise-dominated signal, followed by the reconstruction of the effective signal to achieve denoised, pure signal.
Experimental data supports the effectiveness of this approach, particularly in filtering out noise within the signal-a critical component for accurate bearing fault detection.After noise filtering, the Hilbert envelope spectrum from the signal enabled the extraction of the real bearing fault characteristic frequency.In comparing four different noise filtering methods-IVMD, SVMD, MVMD, and REMD, IVMD has proven to be the most effective at removing noise interference.By minimizing the residual noise and maximizing denoising, we could reduce errors during fault detection via the Hilbert envelope spectrum, improving the accuracy of bearing failure detection.
We believe that this new method of bearing failure detection, with its enhanced accuracy and noise filtering ability, promises significant contributions towards advancements in the field.Preliminary investigations indicate a potential for major impacts, including improved bearing lifespan and performance analysis.While the adoption and practical application of this method will bring specific benefits, we acknowledge that further research and improvements can help optimize its efficiency and utility.To that end, we continue exploring possibilities for further enhancement of this novel technique.In so doing, we aim to contribute meaningfully to the ongoing expansion of this critical area of inquiry.

FIGURE 1 .
FIGURE 1. Bearing vibration signal processing method flow chart.

FIGURE 2 .
FIGURE 2. Time domain diagram of the inner ring vibration signal before and after adding disturbance to the signal 0-0.2s.

FIGURE 3 .
FIGURE 3. Convergence curve of the optimization algorithm on the inner ring vibration signal.

FIGURE 4 .
FIGURE 4. Time frequency domain diagrams of each IMF of the inner ring fault signal.

FIGURE 6 .
FIGURE 6.Time domain diagram of 0-0.2s before and after denoising of inner ring vibration signal.

FIGURE 7 .
FIGURE 7. Envelope spectrum of inner ring vibration signal with perturbation.

FIGURE 8 .
FIGURE 8. Envelope spectrum of inner ring vibration signal without perturbation.

FIGURE 9 .
FIGURE 9. Time domain diagram of outer ring vibration signal 0-0.2s before and after adding noise.

FIGURE 10 .
FIGURE 10.Convergence curve of optimization algorithm on outer ring vibration signal.

FIGURE 11 .
FIGURE 11.Time frequency domain diagram of each IMF of the outer ring fault signal.

FIGURE 13 .
FIGURE 13.Time domain comparison of 0-0.2s before and after denoising of outer ring vibration signal.

TABLE 2 .
Optimization Algorithm Optimization Search Results.

TABLE 3 .
IMF Spearman correlation coefficient for each inner ring vibration signal.

TABLE 4 .
Comparison of the denoising effect of eight wavelet basis functions.

TABLE 5 .
Outer ring fault signal optimal parameter table.

TABLE 6 .
IMF Spearman correlation coefficient for each outer ring vibration signal.

TABLE 7 .
Comparison of the denoising effect of 8 wavelet basis functions of outer ring vibration signal.

TABLE 8 .
Time domain diagram of signal after SVMD denoising.andouterringvibration signals are visually represented in Figures 16 to 18.A comparative analysis between the signals post-IVMD denoising and the signals post-SVMD, MVMD, and REMD denoising, it becomes apparent that IVMD possesses a FIGURE 17.Time domain diagram of signal after MVMD denoising.FIGURE 18.Time domain diagram of signal after REMD denoising.Comparison of the effect of the inner ring fault detection method.

TABLE 9 .
Comparison of the effect of the outer ring fault detection method.