Fault Diagnosis Method of Bearings Based on SCSSA-VMD-MCKD

Abstract

To tackle the issue of detecting early, subtle faults in rolling bearings in the presence of noise interference, the SCSSA-VMD-MCKD method is suggested. This method optimizes the Variational Mode Decomposition (VMD) and Maximum Correlated Kurtosis Deconvolution (MCKD) by integrating the sine-cosine and Cauchy Mutation Sparrow Search Algorithm (SCSSA). The approach aims to effectively diagnose faults in rolling bearings by leveraging the strengths of VMD and MCKD in noise reduction and highlighting fault frequencies. The method utilizes the SCSSA algorithm to autonomously search for parameters in both VMD and MCKD, using the EnvelopeCrest Factor $Ec$ as a fitness function. Firstly, SCSSA is employed to optimize the decomposition mode number $K$ and penalty factor $\alpha$ in the VMD algorithm. Secondly, the parameters in the MCKD algorithm are optimized, and MCKD is used for deconvolution to enhance the impulsive characteristics of the best modal component. Finally, the signal is further analyzed after deconvolution. The results demonstrate that this algorithm can effectively identify subtle fault signals in bearing signals and diagnose fault frequencies in noisy environments. The accuracy of fault diagnosis can reach nearly 99%.

1. Introduction

Rolling bearings are a vital component in large machinery, playing a critical role in enhancing the operational efficiency of mechanical equipment. The running state of the bearings directly affects the normal use of mechanical equipment [1]. In order to ensure that the bearing operates correctly, the various data signals of the bearings should be regularly detected and processed in real time. This places higher demands on the method used to diagnose bearing faults.

Early fault signals were very weak, non-linear and non-stationary. This made them difficult to diagnose in a noisy background. The essence of bearing fault diagnosis lies in identifying and interpreting fault states from rolling bearing vibration signals [2]. Early fault diagnosis methods encompass Wavelet Transform (WT) [3], Empirical Mode Decomposition (EMD) [4], Ensemble Empirical Mode Decomposition (EEMD) [5], etc. However, these all belong to recursive modal decomposition methods and have certain shortcomings. In the presence of strong noise, phenomena like modal aliasing and endpoint effects may occur, which can affect the diagnosis of fault signals. Dragomiretskiy et al. [6] introduced the Variational Mode Decomposition algorithm (VMD), which addresses the aforementioned issues. The VMD algorithm is capable of setting $K$ and $\alpha$ autonomously. Consequently, determining the optimal parameters represents a pivotal challenge for the VMD. Particle Swarm Optimization (PSO) was used to optimize the parameters of VMD by Wang Jie et al. [7]. Meanwhile, Yin Yantao et al. [8] and He Yong et al. [9] employed a Genetic Algorithm (GA) to optimize VMD. These studies further conducted envelope spectrum analysis on the optimized VMD. However, the algorithms exhibited low optimization efficiency. Wang Hengdi et al. [10] utilized the Beetle Antennae Search (BAS) algorithm to optimize VMD but encountered issues with insufficient local search capability.

To better eliminate interference and highlight fault characteristics, McDonald et al. suggested the Maximum Correlated Kurtosis Deconvolution (MCKD) algorithm [11]. By performing deconvolution operations to enhance the kurtosis of continuous impulse pulse signals, the MCKD algorithm accentuates the impulsive components within the signal, facilitating the extraction of frequency characteristics of fault signals. However, the MCKD algorithm similarly requires optimal parameter selection. Zhang Jun et al. [12] utilized the PSO algorithm to optimize MCKD but only optimized the $L$ and $T$ parameters, neglecting the effect of the displacement number $M$. Zeng Yaochuan et al. [13] employed the Grey Wolf Optimization (GWO) algorithm to optimize MCKD for extracting subtle fault features through envelope demodulation, but the convergence speed was found to be relatively slow. Wei Xiaopeng et al. [14], Liu Yingsong et al. [15] and Zichang Liu et al. [16], respectively, employed the Slime Mold Algorithm (SMA) and Sparrow Search Algorithm (SSA) [17] to optimize the parameters of MCKD, achieving adaptive selection and enhancing fault features. Nevertheless, these methodologies are susceptible to localized extremes during the latter stages of optimization. Based on the Sparrow Search Algorithm (SSA), Li Ailian et al. [18] proposed the Sparrow Search Algorithm with Cosine and Cauchy Mutation (SCSSA). This method solves the potential local optimal problem that may occur in SSA optimization, thereby enhancing the stability of the Sparrow algorithm.

Based on the analysis above, this paper proposes SCSSA-VMD-MCKD, a method for early fault diagnosis of rolling bearings, addressing the challenge of extracting weak early faults. This work’s primary contributions include the following:

(1)

The SCSSA algorithm was employed to achieve adaptive selection of VMD parameters, thus simplifying the issue.

(2)

The SCSSA algorithm was also utilized for parameter optimization in MCKD, enhancing computational efficiency and emphasizing the signal’s fault features.

(3)

The combination of optimized VMD and MCKD has been demonstrated to exhibit excellent efficacy. It has been successfully implemented in the field of bearing fault diagnosis.

The simulation analysis is conducted using MATLAB R2023a software, employing both simulated signals and the Case Western Reserve University (CWRU) rolling bearing dataset to represent inner and outer race faults, respectively. The SCSSA-VMD-MCKD method has shown its efficiency and accuracy, and its superiority is confirmed through comparative analysis.

The structure of the remaining article is as listed as follows:

(1)

Section 2 initially presents the fundamental principles of VMD, MCKD and SCSSA, accompanied by a flowchart depicting the SCSSA algorithm.

(2)

Section 3 details the specific process and flowchart of the SCSSA-VMD-MCKD method for fault diagnosis.

(3)

Section 4 validates the capability and performance of the SCSSA-VMD-MCKD method in extracting rolling bearing fault features using simulation signals.

(4)

Section 5 conducts experimental analysis on the CWRU bearing dataset based on simulated signals, verifying the potential application of this method in engineering.

(5)

Section 6 provides the conclusion of the article. The application prospects and developmental direction of this method are examined.

2. Description of Methods: SCSSA-VMD-MCKD

In this section, the fundamental principles and methodologies of VMD, MCKD and SCSSA are individually introduced.

2.1. Variational Mode Decomposition (VMD)

The VMD is a non-recursive signal processing method with adaptive properties, which decomposes the signal into multiple intrinsic mode functions (IMFs) through iterative search. The VMD algorithm tackles the issues posed by the EMD method, including the lack of mathematical foundation and problems with mode mixing [19]. It decomposes an actual signal $f(t)$ into $K$ intrinsic modes $u_k$ and accurately identifies both the center frequency and bandwidth of every mode component. To minimize the combined estimated bandwidth of all modalities, the constraint is that the sum of all modalities equals the total available bandwidth. The constraint conditions are as shown below:

$$\left\{\begin{array}{c}\underset{\left\{u_k\right\},\left\{w_k\right\}}{\min }\left\{{\displaystyle \sum _{\mathrm{k}=1}^{\mathrm{K}}{‖\partial \mathrm{t}\left[\left(\delta (t)+\frac{j}{\pi t}\right)*u_k(t)\right]e^{-j{w}_{k}t}‖}_2^2}\right\}\\ s.t.{\displaystyle \sum _{k=1}^K{u}_k=f(t)}\end{array}\right.$$

(1)

where $\delta (t)$ represents the unit impulse function, $u_k$ denotes the $K$ decomposed IMF components, and $w_k$ are the central frequencies of each component.

To solve the aforementioned model, we introduce Lagrange multipliers $\lambda$ and penalty factor $\alpha$. Converting a constrained variational problem into an unconstrained one involves using Lagrange multipliers [20]:

$$\begin{array}{cc}\hfill L\left(\left\{u_k\right\},\left\{w_k\right\},\lambda \right)& =\alpha {\displaystyle \sum _k{‖\partial t\left[\left(\delta (t)+\frac{j}{\pi t}\right)*u_k(t)\right]e^{-jwkt}‖}_2^2} \hfill \\ & \hspace{1em}+{‖f(t)-{\displaystyle \sum _k{u}_k(t)}‖}_2^2+〈\lambda (t),f(t)-{\displaystyle \sum _k{u}_k(t)}〉\hfill \end{array}$$

(2)

In the Alternating Direction Method of Multipliers (ADMM), we incrementally adjust $u_k$, $w_k$ and $\lambda$ to locate the saddle point of the augmented Lagrangian expression. In the frequency domain, the updates are as follows:

$${\widehat{u}}_k^{n+1}(w)=\frac{\widehat{f}(w)-{\displaystyle \sum _{i\ne k}{\widehat{u}}_k(w)+}\frac{\widehat{\lambda }(w)}{2}}{1+2\alpha {(w-w_k)}^2}$$

(3)

$$w_k^{n+1}=\frac{{\displaystyle \underset{0}{\overset{\infty }{\int }}w{\left|{\widehat{u}}_k(w)\right|}^{2}dw}}{{\displaystyle \underset{0}{\overset{\infty }{\int }}{\left|{\widehat{u}}_k(w)\right|}^{2}dw}}$$

(4)

where ${\widehat{u}}_k^{n+1}(w)$, $\widehat{f}(w)$ and $\widehat{\lambda }(w)$ correspond to the Fourier transforms of $w_k^{n+1}$, $f(t)$ and $\lambda (t)$, respectively.

The VMD method indeed has certain limitations. The selection of empirical parameters exerts considerable influence and instability in the results that might arise during the optimization process. Therefore, it is essential to pre-determine the value of $K$ and $\alpha$ in advance.

Indeed, parameter selection directly affects the decomposition effectiveness of VMD. In complex fault diagnosis scenarios, VMD might struggle to effectively identify and address intricate faults, especially in cases where fault features are very subtle. The straightforward fault diagnosis of VMD is not capable of achieving the optimal outcome. Consequently, an optimization algorithm is required to optimize it and enhance its fault characteristics in the context of strong interference.

2.2. Maximum Correlated Kurtosis Deconvolution (MCKD)

MCKD is designed to enhance the reproduction of the bearing signal’s periodic component, focusing on fault frequencies while simultaneously reducing noise frequencies. The principle of the MCKD algorithm is to utilize a series of finite impulse response (FIR) filters to process the signal such that the kurtosis of the filtered signal is maximized. By adjusting the parameters of the filters, MCKD can extract components from the signal that best match the filter response, thereby achieving signal deconvolution.

The definition of kurtosis is as follows [21]:

$$K_c(T)=\underset{f}{\max }\frac{{\displaystyle \sum _{i=1}^N({\displaystyle \prod _{m=0}^M{y}_i-mT{)}^2}}}{{({\displaystyle \sum _{i=1}^N}{y}_i^2)}^{M+1}}$$

(5)

In Equation (5), $T$ represents the period of the impulse signal; $M$ is the shift number of the FIR filter; $y_i$ denotes the impulse sequence of the periodic signal. The number of sampling points in the signal is N.

To find the optimal filter $df$, the above equation is differentiated:

$$\frac{d}{d{f}_k}{K}_c(T)=0,k=1,2,\cdot \cdot \cdot ,L$$

(6)

where $L$ is the parameter representing the length of the filter.

The combination coefficients of the obtained filter vector $f$ can be represented by the following matrix:

$$f=\frac{{‖y_n‖}^2}{2{‖\beta ‖}^2}{(X_0{X}_0^T)}^{-1}{\displaystyle \sum _{m=0}^M{X}_{mT}}{\alpha }_m$$

(7)

among which

$$\beta ={\left[\begin{array}{c}{y}_1{y}_{1-T}\cdots y_{1-MT}\\ y_2{y}_{2-T}\cdots y_{1-MT}\\ ⋮\\ y_1{y}_{1-T}\cdots y_{1-MT}\end{array}\right]}_{N\times 1}$$

(8)

$$X_{mT}={\left(\begin{array}{ccccc}{\mathrm{X}}_{1-\mathrm{r}}& {\mathrm{X}}_{2-\mathrm{r}}& {\mathrm{X}}_{3-\mathrm{r}}& \cdot \cdot \cdot & {\mathrm{X}}_{\mathrm{N}-\mathrm{r}}\\ 0& {\mathrm{X}}_{1-\mathrm{r}}& {\mathrm{X}}_{2-\mathrm{r}}& \cdot \cdot \cdot & {\mathrm{X}}_{\mathrm{N}-1-\mathrm{r}}\\ 0& 0& {\mathrm{X}}_{1-\mathrm{r}}& \cdot \cdot \cdot & {\mathrm{X}}_{\mathrm{N}-2-\mathrm{r}}\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& 0& \cdot \cdot \cdot & {\mathrm{X}}_{\mathrm{N}-\mathrm{L}-\mathrm{r}+1}\end{array}\right)}_{L\times N}$$

(9)

$${\alpha }_m={\left(\begin{array}{c}{y}_{1-\mathrm{m}T}^{-1}(y_1^2{y}_{1-T}^2\dots y_{1-mT}^2)\\ y_{2-\mathrm{m}T}^{-1}(y_1^2{y}_{2-T}^2\dots y_{2-mT}^2)\\ ⋮\\ y_{N-\mathrm{m}T}^{-1}(y_1^2{y}_{N-T}^2\dots y_{N-mT}^2)\end{array}\right)}_{N\times 1}$$

(10)

The specific steps of MCKD can be referenced in the literature [22].

The MCKD method can be employed to identify impulsive elements within a signal, thereby overcoming the inherent limitations of the VMD algorithm in isolating critical fault indications. Nevertheless, this approach necessitates the fine-tuning of parameters to obtain the optimal outcomes.

2.3. SCSSA Algorithm

This article utilizes the Sparrow Search Algorithm with combined sine-cosine and Cauchy Variations (SCSSA) to optimize the parameters of VMD and MCKD, respectively.

The SCSSA algorithm builds upon the SSA for optimization purposes. It addresses the issues arising from the Sparrow Search Algorithm’s later stages, such as a decrease in population diversity and a lack of convergence accuracy due to falling into local optima.

The Sparrow Search Algorithm integrates a reverse learning strategy when operating in high-dimensional spaces:

$$x_{i,j}^{\ast }=\frac{l_j+u_j}{2}+\frac{l_j+u_j}{2k}+\frac{x_{i,j}}{k}$$

(11)

In Equation (11), $x_{i,j}$ represents the i-th sparrow in the j-th dimension $(i=1,2,\cdots ,D;j=1,2,\cdots ,N)$; $D$ represents the number of populations; $N$ is the dimensionality; $x_{i,j}^{\ast }$ is the mirrored reverse position of $x_{i,j}$; $l_j$ and $u_j$ represent the lower and upper bounds of the j-th dimension of the search space, respectively.

When the discoverer finds food at a local optimum in the sparrow population, many followers will gather there. This phenomenon can reduce population diversity and result in issues with local extremes. To address this phenomenon, the SCSSA algorithm introduces a sine-cosine strategy based on SSA. This strategy utilizes the oscillatory variations of sine and cosine functions for both global and local optimization, aiming to achieve the globally optimal value.

The scaling factor for step size in the fundamental sine-cosine algorithm is as follows:

$$r_1=a-\frac{at}{Ite{r}_{\max }}$$

(12)

The step-size scaling factor is updated as follows, where $a$ is a constant, $t$ represents the iteration count, and $Ite{r}_{\max }$ represents the maximum iteration limit, in conjunction with the diminishing strategy [23]:

$$r_1\prime =a*{\left(1-{\left(\frac{t}{Ite{r}_{\max }}\right)}^{\eta }\right)}^{1/\eta }$$

(13)

$\eta$ is the tuning coefficient, where $\eta$ ≥ 1. The new nonlinear decreasing search factor has a larger weight and slower decreasing speed in the early stage, which is conducive to improving the global optimality search ability; the lower weight factor in the subsequent period can enhance the algorithm’s advantage within regional development and accelerate the efficiency of achieving the optimal solution.

The SSA algorithm often considers the influence of the current position during the updating process, so a weighting factor $\omega$ is introduced to reduce the impact of the current position on the updating result. In the initial stages of optimization, a smaller $\omega$ reduces the effect of the seeking individual’s position update on the current position, thereby enhancing the algorithm’s global optimization performance. Conversely, in the later stages, a larger $\omega$ accelerates the convergence of the algorithm. It is recommended that the optimized performance be further optimized.

The weight factor $\omega$ is as follows:

$$\omega =\frac{e^{\frac{1}{Ite{r}_{\max }}}-1}{e-1}$$

(14)

The revised equation for the explorer positions is as stated below:

$$X_{i,j}^{i+1}=\left\{\begin{array}{c}\omega •X_{i,j}^t+r_1•\sin r_2+\left|r_3•X_{best}-X_{i,j}^t\right|,R_2<ST\\ \omega •X_{i,j}^t+r_1•\cos r_2+\left|r_3•X_{best}-X_{i,j}^t\right|,R_2\ge ST\end{array}\right.$$

(15)

In the formula, $r_2$ and $r_3$ are random numbers between 0 and 2π.

$$X_{i,j}^{t+1}=X_{best}(t)+cauchy(0,1)\oplus X_{best}(t)$$

(16)

The standard Cauchy distribution function, denoted as $cauchy(0,1)$, defines the one-dimensional Cauchy mutation function centered at the origin as follows:

$$f(x)=\frac{1}{\pi }(\frac{1}{1+x^2}),-\infty <x<\infty$$

(17)

Figure 1 is a flowchart of the SCSSA algorithm.

The detailed derivation of SCSSA can be found in Reference [17].

3. Methodology of Analysis: SCSSA-VMD-MCKD

The SCSSA-VMD-MCKD method adopted in this article addresses the shortcomings of SCSSA-VMD, which lacks sufficient fault information. It achieves the adaptive selection of VMD and MCKD parameters, enabling the extraction of weak fault characteristics in rolling bearings amidst strong noise interference. The envelope peak factor $Ec$ was selected as the optimization index to optimize the fitness functions of VMD and MCKD using SCSSA. $Ec$ is the dimensionless ratio of the peak envelope spectrum to the effective value, considering the impact of signal periodicity and intensity.

The definition of $Ec$ is

$$E\mathrm{c}=\frac{\max \left(X(z)\right)}{\sqrt{{\displaystyle \sum _{z}X{(z)}^2/Z}}}$$

(18)

A higher $Ec$ index indicates increased signal SNR and energy post-filtering [24]. The component with a higher $Ec$ value exhibits more pronounced periodic impact characteristics and more discernible fault characteristics.

The process of SCSSA-optimized VMD and MCKD is shown in Figure 2.

The specific process of fault diagnosis is as follows:

(1)

Initialize the SCSSA parameters with a population size of 10 and a maximum iteration count of 20. Apply SCSSA for the optimal parameters $K$ and $\alpha$, tuning for VMD. After multiple experiments, set the optimization range of $K$ to [3,10] and the optimization range of $\alpha$ to [300, 2500]. Obtain the optimal parameters [$K_0$, ${\alpha }_0$], employing the $E\mathrm{c}$ as the fitness function.

(2)

Decompose the fault signal using VMD to obtain $K_0$ IMF components. Compare the envelope peak factors of each component and choose the component with the highest value as the optimal IMF component when performing envelope spectrum analysis.

(3)

Select the optimized IMF component obtained from VMD as the input data for optimizing MCKD using SCSSA. Optimize the parameters $L$, $T$ and $M$, where the optimization range for $L$ is $L$ ⊆ [100, 1000]. The shift number $M$ exerts a significant influence on the number of peaks present in the signal following the completion of the deconvolution processing. It is generally chosen between 1 and 7. When $M$ exceeds 7, the precision decreases due to the iterative method exceeding the floating-point exponent range [25]. Therefore, set the optimization range for $M$ to $M$ ⊆ [1, 7]. $T=f_s/f_i$. The fitness function is also chosen as $Ec$ to obtain the optimal parameters [$L_0$, $T_0$, $M_0$].

(4)

The deconvolution operation for the MCKD is conducted using the parameters that were optimized in the previous step. The results are then subjected to envelope spectral analysis to identify the most prominent spectral lines within them.

(5)

A comparison must first be made between the theoretical fault frequency and the peak apparent spectral line, with a subsequent determination being made of the nature of the fault and the location of the occurrence. The objective of fault diagnosis can then be achieved by employing these findings.

4. Simulation Analysis

To verify the efficacy of the SCSSA algorithm in optimizing VMD and MCKD, a rolling bearing fault model is employed to simulate impact signals. Gaussian white noise is introduced to emulate faint inner race fault signals that are typically obscured by noise in real operational settings. This paper refers to the rolling bearing fault model in Reference [11] for analysis. The simulation fault signal is as follows:

$$\left\{\begin{array}{l}h(t)=\exp (-Bt)\sin (2\pi f_n\mathrm{t})\\ A_{\kappa }=A_0\sin (2\pi f_{\mathrm{r}}\mathrm{t})+1\\ y(t)={\displaystyle \sum _{\mathrm{K}}{A}_{\kappa }}h(t-k{T}_1-{\tau }_k)\\ x(t)=y(t)+n(t)\end{array}\right.$$

(19)

To set the fault environment more accurately, we set the parameters as follows: $A_0$= 0.5; the attenuation coefficient $B$ is 800; the rotational frequency $f_{\mathrm{r}}$ is 25 Hz; the resonance frequency $f_n$ is 4000 Hz; the inner race fault frequency $f_{\mathrm{i}}$ is 120 Hz; $n(t)$ stands for Gaussian white noise, and the SNR was set to −16 dB; ${\tau }_k$ A is a minor fluctuation with a mean of 0 and a standard deviation of 0.05% of $f_{\mathrm{r}}$, which follows a normal distribution; the sampling frequency $f_s$ is 12,800 Hz; and there are a total of 8192 points in the sampling array ($N$).

Figure 3 displays the fault simulation signal, and Figure 4 depicts the time-domain waveform post-addition of noise. The envelope spectrum is presented in Figure 5. The addition of noise to a signal results in a substantial reduction in the visibility of prominent frequencies within the envelope spectrum, thereby significantly hindering the identification of these frequencies.

In the conventional analytical approach, it is challenging to discern the fault frequency following the introduction of noise, which impedes the extraction of fault information and introduces significant limitations.

During the signal analysis process, the SCSSA algorithm was used for optimization of $K$ and $\alpha$. The optimization range for $K$ was set to [3, 10], and for $\alpha$, it was set to [100, 2500]. Iteration was capped at 20, and the population size was set to 10. The best optimization result for VMD obtained was [2372, 10]. To compare the advantages of the SCSSA algorithm in parameter optimization, the results of VMD optimized by the SCSSA algorithm are compared with those optimized by the SSA and PSO algorithms based on convergence speed and envelope peak factor. The parameter settings of the SSA and PSO algorithms are the same as those of the SCSSA algorithms mentioned above in Section 3. Under the condition that $Ec$ is also used as a fitness function, the time of each optimization of SCSSA, SSA and PSO functions is 6.378 s, 6.312 s and 6.286 s, respectively, indicating that the three functions have similar spatial complexity.

Figure 6 displays optimization curves for the SCSSA, PSO and SSA algorithms. The $Ec$ values of the SCSSA, SSA and PSO algorithms are $E{c}_1$ = 5.9705, $E{c}_2$ = 5.9469 and $E{c}_3$ = 5.9619, respectively. The SCSSA algorithm achieves the highest envelope peak factor after optimization, and it also demonstrates faster convergence speed compared to the SSA and PSO algorithms. The results indicate that the SCSSA algorithm is demonstrably more efficacious than the SSA and PSO algorithms regarding both the speed of solution acquisition and the accuracy of the optimization process.

After optimizing with the SCSSA algorithm, the best parameters are inputted into the VMD, setting $K$ to 10 and $\alpha$ to 2372. Upon decomposing with VMD, the time-domain waveforms of each IMF are depicted in Figure 7, and the envelope peak factors $Ec$ are listed in

Table 1. The values of each component Ec after VMD decomposition.

Component	IMF1	IMF2	IMF3	IMF4	IMF5	IMF6	IMF7	IMF8	IMF9	IMF10
Ec	3.63	3.80	3.94	4.55	4.08	5.10	5.62	4.05	3.52	4.51

. It can be observed that the optimal IMF component is $K_0$ = 7. Further envelope spectrum analysis is conducted on IMF 7. As shown in Figure 8, the envelope spectrum at this point exhibits only two prominent frequency lines, with other frequencies being less significant. The prominent frequencies are approximately around 120 Hz, which is near the fault frequency.

We decompose the analog signal by EMD, VMD and EEMD, respectively. Figure 9, Figure 10 and Figure 11 depict the spectrum.

As can be seen from the above figures, both EMD and EEMD produce frequency aliasing, but VMD has a relatively good decomposition effect and can effectively extract local fault information, so VMD is more suitable for signal feature extraction.

After deconvolution, the optimal IMF components decomposed by VMD are input into MCKD to optimize parameters. With MCKD optimization, the parameters are set as follows: $L$ is 644, $T$ is 96, and $M$ is 1. These parameters were optimized within specified ranges: $L$ from 100 to 1000, $T$ from 85 to 142 and $M$ from 1 to 7. The optimization process used a population size of 10 and was limited to a maximum of 20 iterations. After sampling at 12.8 kHz, the optimized parameters [$L$, $T$, $M$] were determined to be [644, 96, 1] for use in the MCKD algorithm.

The fitness function still selects the peak envelope spectral factor $Ec$. Figure 12 shows the fitness curve of MCKD optimized by SCSSA, with the envelope peak factor $Ec$ being 10.3188. It is evident from the graph that convergence occurred around the 16th generation of optimization. The envelope spectrum of the best IMF component after deconvolution is illustrated in Figure 13.

Comparing the envelope spectrum before and after MCKD deconvolution, it is evident that the fault feature has undergone a notable enhancement. After deconvolution, the fault characteristic frequency $f_i$ and its harmonics are all visible, including the second, third and fourth harmonics, indicating accurate extraction of the characteristic frequencies. This demonstrates that the SCSSA algorithm optimization of MCKD greatly amplifies the impact components of fault signals, making it highly applicable in fault diagnosis. The accuracy of fault frequency and frequency doubling is shown in the following

Table 2. Failure frequency and accuracy of frequency doubling.

Theoretical Frequency/Hz	True Frequency/Hz	Accuracy Rate
120	120.312	99.7%
240	239.062	99.6%
360	359.375	99.8%
480	478.125	99.6%
600	598.438	99.7%

, with an accuracy of up to 99%.

To verify the stability of SCSSA algorithm optimization, repeated experiments were conducted, and the results of parameter optimization are shown in

Table 3. Parameters of several SCSSA tests.

Number of Tests	1st	2nd	3rd	4th	5th	$\mathit{E}(\mathit{x})$	$\mathit{S}$
$\alpha$	2372	2448	2500	2406	2318	2408.8	62.387
$K$	10	9	9	10	10	9.6	0.489
$L$	644	446	482	378	252	440.4	128.53
$T$	96	101	90	99	104	98	4.775
$M$	1	1	1	1	1	1	0

.

The formulas for mean and standard deviation are as follows:

$$E(x)=\frac{X_1+X_2+\cdots +X_n}{n}$$

(20)

$$s=\sqrt{\frac{{(X_1-E(x))}^2+{(X_2-E(x))}^2+\cdots +{(X_n-E(x))}^2}{n}}$$

(21)

where $E(x)$, $S$ and $n$ stand for mean, standard deviation and sample size, respectively. It can be observed from the table that the variation range of parameter optimization results is not large, indicating that the algorithm has stability.

5. Experimental Signal Analysis

To gain further insight into the efficacy of the SCSSA-optimized VMD and MCKD methods employed in this study, the outer ring bearing data of the CWRU dataset was selected for optimization. Multiple experiments were conducted to validate this approach. The bearing data used are listed in

Table 4. Bearing parameters.

Bearing Model	Fault Diameter/mm	Motor Speed r/min	Sampling Frequency/kHz	Load/hp
SKF6205	0.1778	1797	12	0

.

The theoretical fault frequency of the outer race of the bearing can be calculated by the following formula:

$$f_o=\frac{r}{60}\times \frac{1}{2}\times n\times \left(1-\frac{d}{D}\times \cos \alpha \right)$$

(22)

where $r$ is the shaft rotation speed; $n$ represents the number of rolling elements; $d$ represents the diameter of the rolling elements; $D$ is the pitch diameter of the rolling elements; $\alpha$ represents the contact angle of the pivot, which is set to 0° in this article. Following the completion of the requisite calculations, it was determined that the theoretical fault frequency of the outer race is 107.3 Hz. To increase contrast, Gaussian white noise with a level of −16 dB was added, simulating weak fault signals hidden by noise in the bearing. Figure 14 depicts the time-domain waveform, and Figure 15 shows the envelope spectrum of the fault signal.

The optimization steps and the range of optimization for $K$ and $\alpha$ are as mentioned earlier. Based on experimental experience, the iteration count was capped at 20, with a population of 10. The optimization result was [$K_0$, ${\alpha }_0$] = [2500, 8], with an envelope peak factor $Ec$ = 10.3246. The fitness curve is illustrated in Figure 16, demonstrating that the SCSSA algorithm converged in the fifth optimization iteration.

Using the optimal parameters [$K_0$, ${\alpha }_0$], the VMD decomposition was performed, resulting in the IMF components shown in

Table 5. The values of each component Ec after VMD decomposition.

Component	IMF1	IMF2	IMF3	IMF4	IMF5	IMF6	IMF7	IMF8
Ec	3.73	3.59	3.46	4.97	10.32	5.97	3.82	3.59

. The best IMF component is $K_0$ = 5. Figure 17 shows the envelope spectrum of the optimal IMF component, which is characterized by a prominent frequency of around 107 Hz.

The optimal IMF component was used as input data for MCKD, and the SCSSA algorithm was employed to further optimize MCKD. The optimization ranges for $L$, $T$ and $M$, $M$ remained the same as before. Eventually, the best parameters for $L$, $T$ and $M$ were determined to be [169, 112, 1], with an envelope peak factor $Ec$ = 6.577. The fitness curve for MCKD is depicted in Figure 18 and Figure 19 and shows the envelope spectrum after deconvolution of the optimal IMF component.

The peak of the deconvolved envelope spectrum is close to 107 Hz, which is congruent with the calculated value of the fault characteristic frequency. This demonstrates that the fault frequency can be identified. The frequency of outer ring failures can be accurately determined, as well as the harmonics. The optimization of MCKD using the SCSSA method effectively enhances the impulse components in the signal, thus accomplishing the objectives of fault diagnosis. The following

Table 6. Failure frequency and accuracy of frequency doubling.

Theoretical Frequency/Hz	True Frequency/Hz	Accuracy Rate
107.3	108	99.3%
214.6	216	99.3%
321.9	324	99.3%
429.2	429	100%
536.5	543	98.8%

shows the accuracy of frequency doubling, indicating that the diagnostic effect can reach more than 98.8%.

6. Conclusions from the Research

This paper proposes a novel feature extraction method based on SCSSA-VMD-MCKD for bearing failure diagnosis. It provides theoretical and technical support for the extraction of fault features in rolling bearings, with promising engineering application prospects. This method aims to extract weak fault characteristics of bearings in the context of complex noises. Through experimental analysis, the following conclusions were reached:

(1) By utilizing $Ec$ as the fitness function to optimize VMD, adaptive selection of VMD parameters is achieved. After obtaining the optimal parameters, the best modal components are selected through VMD decomposition with these parameters.

(2) After optimizing the best modal components using SCSSA, MCKD is applied with $Ec$ as the fitness function for the deconvolution operation. Subsequent envelope spectrum analysis significantly reduces interference, effectively highlighting impact components distorted by noise interference. The recognition accuracy of the real signal can reach nearly 99%.

(3) The optimization curve of the SCSSA algorithm exhibits faster convergence and higher accuracy compared to PSO, SSA and other algorithms. The SCSSA-VMD-MCKD method effectively enables fault diagnosis despite strong background noise.

The SCSSA-VMD-MCKD method has not only driven technological advancements in data-driven scientific research but also continues to enhance productivity, reduce costs and provide crucial support for ongoing improvements and innovations in industrial applications. Its development and application hold significant implications for the progress of modern industry and science. However, the algorithm still faces limitations and areas for improvement, such as optimizing parameter selection and adjustment and enhancing convergence speed and stability to accommodate more complex external interferences. Addressing multi-objective optimization problems and achieving multi-input multi-output solutions are directions for further research and development of the method.