Application of Auto-Regulative Sparse Variational Mode Decomposition in Mechanical Fault Diagnosis

Abstract

The variational mode decomposition (VMD) method has been widely applied in the field of mechanical fault diagnosis as an excellent non-recursive signal processing tool. The performance of VMD depends on its inherent prior parameters. Searching for the key parameters of VMD using intelligent optimization algorithms poses challenges for the internal essence and fitness function selection of intelligent optimization algorithm. Moreover, the computational complexity of optimization is high. Meanwhile, such methods are not competitive in evaluating orthogonality between intrinsic mode functions and the reconstruction error of the signal as a joint indictor for the termination of decomposition. Therefore, this paper proposes a new auto-regulative sparse variational mode decomposition method (ASparse–VMD) to achieve accurate feature extraction. The regularization term of the VMD handles sparsification by constructing an L₂-norm with a damping coefficient ε, and mode number K is set adaptively in a recursive manner to ensure appropriateness. The penalty parameter $\alpha$ is dynamically selected according to the number of modes and sampling frequency. The update step τ of the VMD algorithm is set using the signal-to-noise ratio to ensure the singleness and orthogonality of the modal components and suppress mode aliasing. The experimental results of the simulation signal and measured signal demonstrate the effectiveness of the proposed strategies for improving the inherent defects of VMD. Extensive comparisons with state-of-the-art methods show that the proposed algorithm is more effective and practical for hybrid feature extraction in mechanical faults.

1. Introduction

Mechanical vibration signals are an external manifestation of the interactions between various parts of a machine, reflecting important information about the state of the equipment. The strength, phase, frequency distribution, and some statistics are the eigenvalues of the machine state. The processing of mechanical vibration signals can be divided into time and frequency domains, and consists of two main processes: signal feature extraction and fault state identification. A mechanical vibration signal generally exhibits nonlinear and non-stationary features due to noise, structural deformation, velocity shift, and other factors [1]. Therefore, to accurately identify the fault characteristic information contained in a mechanical vibration signal, it is necessary to efficiently decompose and extract the characteristic modes. In recent years, commonly used signal decomposition methods include ensemble empirical mode decomposition (EMD) [2], synchrosqueezing wavelet transform (SWT) [3], synchrosqueezing S-transform (SSST) [4], local mean decomposition (LMD) [5], intrinsic time-scale decomposition (ITD) [6], intrinsic mode chirp decomposition [7], etc. These methods have been widely used in mechanical fault diagnosis with fruitful results. However, due to the limitations of the theoretical framework, the corresponding processing structure has some inherent flaws. EMD uses interpolation to decompose the signal, which is prone to end effects and modal aliasing. Although variants such as EEMD [8] and CEEMD [9] were proposed in subsequent studies, the structural defects of EMD cannot be fundamentally eliminated. As a modified method of the wavelet transform [10], SWT improves the time–frequency resolution of the energy signal. SSST neglects the instantaneous phase of the signal by more than an order of magnitude, leading to a bias in the estimation of the instantaneous frequency. LMD obtains a finite sum of independent AM/FM signals based on their characteristics, but still cannot avoid the end effect, over-envelope error, and under-envelope error. ITD combines the advantages of EMD and LMD with high time–frequency resolution, fast computation speed, and noise rejection. However, this method is prone to waveform distortion. Variational mode decomposition (VMD) is an adaptive signal processing method proposed on the basis of the submodes with limited bandwidth [11]. It is essentially different from the aforementioned methods and has been successfully applied in the field of mechanical fault diagnosis. VMD requires the manual setting of parameters such as the number of decomposition modes $K$, quadratic penalty term $\alpha$, initial center frequency, and other hyper parameters in advance. These inherent parameters have a critical impact on the decomposition results. Most importantly, there is a lack of criteria for measuring the results of decomposition.

To study the nature of VMD, many scholars have proposed a large number of optimization and improvement methods in recent years. Li et al. [12] introduced an adaptive selection principle for parameter $K$ in VMD using the mean instantaneous frequency. Jiang et al. [13] proposed a different coarse-to-fine VMD decomposition strategy that selects $K$ and $\alpha$ according to signal characteristics to obtain the optimal mode. Nazari et al. [14] put forward a novel signal decomposition method, termed successive variational mode decomposition (SVMD) that effectively solves the problem of mode number selection for VMD. In application, SVMD is prone to a residual signal without a strict mathematical definition. When the center frequencies of the two modes are particularly close, this method is not effective in distinguishing between them. Inspired by SVMD, a novel successive multivariate variational mode decomposition algorithm (SMVMD) was proposed in [15]. SMVMD can self-select the number of modes based on their signal characteristics and thus continuously extract joint or common modes, but the separation of modes with similar central frequencies remains to be solved. Based on the instantaneous linear mixture model, this method can continuously extract joint or common modes. Chen et al. [16] proposed self-tuning variational mode decomposition, in which the number of modes and penalty parameters are adaptively updated by the energy ratio and orthogonality between the modes in the frequency domain. This method offers good noise suppression and reduces mode aliasing and end effects. However, the validity of this method in the field of fault diagnosis has not been verified. Dey et al. employed a metric based on detrended fluctuation analysis to preset the mode number of VMD. In this study, $\alpha$ was based on a comparative selection from the values set by multiple tasks and not an adaptive value [17]. Zhong et al. applied the whale optimization algorithm (WOA) [18] to select the mode parameters [$K$, $\alpha$] of VMD to improve the accuracy and stability of the classifier for the residual injury detection of foreign object debris on airport runways [19]. Dibaj et al. proposed an end-to-end fault diagnosis method based on fine-tuned VMD. Data analysis of gearbox systems with compound faults was achieved by tuning the VMD mode frequency [20]. Nazari et al. [21] proposed a new variational mode extraction (VME) algorithm based on VMD. This approach improves VMD by predicting that the residual signal after extracting a particular mode has less energy at the mode’s central frequency. Aiming to solve the multi-fault diagnosis problem of rolling bearings, Li et al. [22] introduced an improved VME method to determine the proper penalty factor $\alpha$ based on the convergence characteristics of VMD. To evaluate the complex pulse fault components in the signal, a new syncretic impact index (SII) is combined with the artificial bee colony (ABC) algorithm [23] to select the optimal mode number and balancing parameter of VMD [24]. A cuckoo search algorithm [25] merged variational mode decomposition (CSA-VMD) was proposed in [26], and was adapted to optimize parameters to obtain decomposed multi-component signals. Wang et al. [27] proposed a multi-objective particle swarm optimization algorithm to optimize VMD parameters using dynamic symbolic entropy as the optimization index. In [28], correntropy was introduced to search for the optimal parameter $K$ of VMD, and the maximum correlation entropy criterion in information theory learning was used for the evaluation. Li et al. [29] constructed a genetic algorithm [30] to optimize the VMD combination parameters (mode number $K$ and penalty factor $\alpha$). Mao et al. explored an ensemble kurtosis index combining kurtosis and envelope spectrum kurtosis [31]. This index improves the number of modes and penalty parameters of the VMD algorithm. Bi et al. [32] proposed an approach to optimize the initial value of the modal center frequency. Zhang et al. [33] introduced a fast iterative variational mode decomposition to solve the problem of a traditional VMD parameter setting. Kumar et al. [34] developed a strategy using kernel mutual information as an evaluation function to find VMD optimization parameters for bearing defect identification.

The various improvement methods investigated above effectively rectify some inherent problems of VMD. However, these methods do not take into account certain special situations. That is, the center frequencies of two or more single components in a multi-component signal are very close, but remain far away from those of other components. Single components with very close center frequencies will be regarded as a mono-components orthogonal to other items, resulting in spectrum aliasing. Another core problem is that the actual impact of update step $\tau$ on VMD is not considered. On the basis of evaluating the termination conditions for acquiring the original VMD submodes, an embedded binary tree decomposition mode of empirical VMD without preset parameters is proposed in the literature [35]. This approach thoroughly takes into account the problem of the proximity of the center frequencies of two or more single modes in a multi-component signal, and shows a satisfactory result. However, the binary tree mechanism also suffers from the drawback of a time-consuming decomposition procedure. This is especially the case when the number of modes is relatively large.

In this paper, we propose an empirical sparse VMD algorithm with an automatic adjustment mechanism based on the operational strategy of VMD. This mechanism ensures that the decomposition process of VMD is a fully adaptive process that can effectively avoid mode aliasing problems to some extent. Finally, the proposed ASparse–VMD is verified using a simulation and measured signals, and the experimental results effectively demonstrated the method’s effectiveness. The main contributions of the study are as follows: (1) The number of modes present in the signal is determined by the recursion formalism. (2) We analyze the distribution properties of the signal bands and derive a setting formula for the parameter $\alpha$. (3) According to the operation mechanism of the VMD updating step, the noise level of the signal is applied as the basis for the setting of $\tau$. This strategy alleviates the problem of mode aliasing at different noise levels. (4) On the basis of updating the Lagrange multiplier $\lambda$, the iteration operator $\gamma$ is introduced to update $\lambda$ twice, which makes the result of each internal iteration of VMD more accurate. (5) Different tolerance parameters are proposed for the VMD algorithm to ensure the accuracy of the number of modes and obtain better signal reconstruction accuracy.

The remaining sections of this paper are as follows. The mechanism of VMD is described in Section 2 which provides the theoretical derivation and algorithmic structure of empirical sparse VMD. Section 4 presents experimental verification via simulated signals. In Section 5, the effectiveness of the proposed method is verified by the measured signal. Finally, the conclusions are drawn in Section 6.

2. The Principle of VMD

Classical variational mode decomposition is an adaptive signal decomposition method based on the Wiener filter [36], Hilbert transform [37], and the modulation and demodulation theory. According to the number of modes in the signal sequence, the optimal center frequency and finite bandwidth of each mode are adaptively matched during the search and solution process. Through this process, the intrinsic mode function (IMF) can be effectively separated.

2.1. The Construction of a Variational Model

Assuming that the input signal $x(t)$ consists of several components with different central frequencies and limited bandwidth, the signal decomposition problem is formulated as a variational model for decomposition. The steps of this process are as follows [11].

Step 1. The unilateral spectrum $\left[\delta (t)+\frac{j}{\pi t}\right]\ast u_k(t)$ is obtained by performing the Hilbert transform of each mode $u_k(t)$.

Step 2. Frequency mixing is performed to modulate the frequency spectrum of each mode to the corresponding fundamental band $\left[\delta (t)+\frac{j}{\pi t}\right]\ast u_k(t)$.

Step 3. Construct the desired variational model

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

(1)

where $K$ is the mode number. $\left\{u_k\right\}=\{u_1,u_2,---,u_K\}$ and $\left\{{\omega }_k\right\}=\{{\omega }_1,{\omega }_2,---,{\omega }_K\}$ represent $K$ limited-band components and the corresponding band center of each mode, respectively. ${\displaystyle {\sum }_k:={\displaystyle {\sum }_{k=1}^K}}$ is the sum of all band components.

2.2. Solution of Variational Model

By introducing the quadratic penalty factor $\alpha$ and Lagrange multiplicative operator $\lambda (t)$, the constrained variational problem is transformed into an unconstrained variational issue. The augmented Lagrangian expression is:

$$\begin{array}{c}L\left(\left\{u_k\right\},\{{\omega }_k\},\lambda \right):=\alpha {{\displaystyle \sum _k‖\partial t\left[\left(\delta (t)+\frac{j}{\pi t}\right)\ast u_k(t)\right]e^{-j{\omega }_{k}t}‖}}_2^2\\ {+‖f(t)-{\displaystyle \sum _k{u}_k(t)}‖}_2^2+〈\lambda (t),f(t)-{\displaystyle \sum _k{u}_k(t)}〉\end{array}$$

(2)

Based on this equation, $\left\{u_k^{n+1}\right\}$, $\left\{{\omega }_k^{n+1}\right\}$, and ${\lambda }^{n+1}$ are alternately updated using the alternating direction method of multipliers (ADMM) to find the saddle points of the augmented Lagrangian expression. Further details about the VMD algorithm can be found in [11].

In the original VMD decomposition procedure, decomposition performance heavily depends on the influence of the algorithm’s prior parameters. Inappropriate parameter settings may lead to mode aliasing and mode duplication problems in mode extraction. The number of modes $K$, the second penalty parameter $\alpha$, the noise adjustment parameter $\tau$, and the update step size $\lambda$ are the four key preset parameters.

3. Sparse Variational Mode Decomposition Framework

3.1. Intrinsic Mode Function

Intrinsic mode function (IMF) is an amplitude modulation (AM-FM) signal that can reflect the intrinsic characteristics of the original representation. The IMF equation is as follows:

$$u_k(t)=a_k(t)\cos ({\varphi }_k(t))$$

(3)

where $u_k(t)$ represents the k-th components. $a_k(t)$ is the envelope of the signal, which is a non-negative value. ${\varphi }_k(t)$ is the phase of the signal, which is a non-decreasing function. Taking the derivative of ${\varphi }_k(t)$ gives the corresponding instantaneous frequency ${\omega }_k(t):=\frac{d}{dt}{\varphi }_k(t)$, which varies considerably more slowly than the phase function. Assuming that the spectrum of each IMF component is as close as possible to its central frequency, the limited bandwidth of the IMF is required to be sparse. Meanwhile, the orthogonality principle is further used for determination so that there is no frequency confusion between the obtained IMF components.

3.2. Wiener Filter

The observation signal $f_o(t)$ typically includes the original signal $f(t)$ and the observation noise signal $n(t)$:

$$f_o(t)=f(t)+n(t)$$

(4)

where $f(t)$ is an unknown signal that needs to be reconstructed, which is an ill-posed inverse problem. In [11], $n(t)$ represents an observation noise with tiny amplitude and adopts standard Tikhonov regularization to solve this challenge. The expression is as follows:

$$J(t)=\underset{f(t)}{min}\left\{{‖f(t)-f_o\left(t\right)‖}_2^2+\alpha {‖{\partial }_{t}f(t)‖}_2^2\right\}$$

(5)

Translating the above equation into the Fourier domain, it becomes:

$$J(\omega )=\underset{f(\omega )}{min}\left\{{‖f(\omega )-f_o\left(\omega \right)‖}_2^2+\alpha {‖(j\omega )f(\omega )‖}_2^2\right\}$$

(6)

where ${‖f(\omega )-f_o\left(\omega \right)‖}_2^2$ is the data-fitting term in the sense of $L_2$-norm, and ${‖(j\omega )f(\omega )‖}_2^2$ is the stabilizing penalty term. $\alpha$ indicates the regularization parameter, which plays a major role in balancing the fitting and penalty terms. Standard Tikhonov regularization enables a smooth reconstruction of the signal $f(\omega )$. However, in practical applications, the signal needs to be reconstructed sparsely. In order to sparsely reconstruct $f(\omega )$, the standard Tikhonov regularization method needs to be improved. The objective function of Tikhonov regularization can be transformed into the following form:

$$J(\omega )=\underset{f(\omega )}{min}\left\{{‖f(\omega )-f_o\left(\omega \right)‖}_2^2+\alpha {‖(j\omega )f(\omega )‖}_0^{}\right\}$$

(7)

where ${‖(j\omega )f(\omega )‖}_0^{}$ denotes the $L_0$-norm of the penalty term, that is, the number of nonzero vectors. However, this term is not distinguishable, and it is also unsolvable as a non-deterministic polynomial (NP). To solve this challenge, the following forms are adopted:

$$J(\omega )=\underset{f(\omega )}{min}\left\{{‖f(\omega )-f_o\left(\omega \right)‖}_2^2+\alpha {‖(j\omega )f(\omega )‖}_1^{}\right\}$$

(8)

where ${‖(j\omega )f(\omega )‖}_1^{}$ denotes the $L_1$-norm of the penalty term. The use of $L_1$-norm instead of $L_0$-norm can be effective in improving computational efficiency, but its penalty terms remain indistinguishable. Therefore, more improvements based on the $L_1$-norm are required as follows:

$$J(\omega )=\underset{f(\omega )}{min}\left\{{‖f(\omega )-f_o\left(\omega \right)‖}_2^2+\alpha {‖(j\omega )f(\omega )‖}_{2,\epsilon }^2\right\}$$

(9)

where ${‖(j\omega )f(\omega )‖}_{2,\epsilon }^2$ is the $L_2$-norm with the damping coefficient $\epsilon$, which is the sparse regularization term. The auxiliary parameter $\epsilon$ is a real vector and ${‖(j\omega )f(\omega )‖}_{2,\epsilon }^2={‖j(\omega +\epsilon )f(\omega )‖}_2^2$.

By solving the Euler–Lagrange equations of the above equation, we can obtain the following:

$$\widehat{f}(\omega )=\frac{{\widehat{f}}_o(\omega )}{1+\alpha {(\omega +\epsilon )}^2}$$

(10)

3.3. Sparse Variational Model

Assuming that the input signal $x(t)$ is composed of several components with different center frequencies, limited bandwidth and noise, the construction steps are as follows:

Step 1: Convert the analytic signal of each modal function $u_k(t)$ through Hilbert transform to obtain the unilateral spectrum.

Step 2: Perform frequency mixing to modulate the frequency spectrum of each mode to the corresponding fundamental frequency band.

Step 3: Calculate the square L² norm with the damping coefficient $\epsilon$ of the above demodulated signal and estimate the bandwidth of each modal signal. The optimal variational model is constructed by introducing constraints.

$$\left\{\begin{array}{l}\underset{\left\{u_k\right\},\left\{{\omega }_k\right\}}{\min }\left\{{\displaystyle \sum _{k=1}^K{‖{\partial }_t\left[\left(\delta (t)+\frac{j}{\pi t}\right)\ast u_k(t)\right]{\mathrm{e}}^{-j{\omega }_{k}t}{\mathrm{e}}^{-j\epsilon t}‖}_2^2}\right\}\\ s.t. {\displaystyle \sum _{k=1}^K{u}_k(t)=x(t)}\end{array}\right.$$

(11)

3.4. Parameters Selection of Model

The parameter values of variational mode decomposition have a large impact on the decomposition performance, and properly choosing parameter values is crucial to ensure decomposition performance. Here, the sparse variational model is chosen as follows.

(1) Choosing the total number of modes $K$. The performance of VMD is extremely sensitive to the value of $K$. The mechanism of constrained conditional self-superposition is used for adaptive selection, which effectively avoids the problem of decomposition validity caused by the empirical and accidental choice of the parameter K.

(2) Choosing the quadratic penalty parameter $\alpha$. The distribution characteristics of the signal bands are analyzed and the value of $\alpha$ is derived using the following.

$$\alpha =\left[2\left(S-0.5\right)-K-\frac{f_s}{2}\right]\cdot D$$

(12)

where $K$ is the total number of modes expected to be included in the signal. $S=\left(\frac{1}{1+e^{-K}}\right)$ denotes the Sigmoid function. $f_s$ is the sampling frequency of the signal. D is the fine-tuning parameter. Clearly, $2\left(S-0.5\right)\in (0,1)$. The VMD method transforms the time-domain signal $x(t)$ into the Fourier domain $X(\omega )$, and constrains it to $\omega \in \left(-0.5,0.5\right)$. Since the range of the penalty parameter $\alpha$ is $\alpha \in {\Re }^{N+}$, $2\left(S-0.5\right)\in (0,1)$ is used to constrain $\omega$ to $\omega \in \left(0,1\right)$ which does not affect the width of $\omega$ in the frequency domain. In addition, $2\left(S-0.5\right)\cdot K\cdot \frac{f_s}{2}$ constrains the average bandwidth of K mode components to $2\left(S-0.5\right)\cdot \frac{f_s}{2}$. Given that each IMF is a single component, if the center frequency ${\omega }_c$ is too close between neighboring components in the signal $X(\omega )$, and the value of $\alpha$ is too small, the bandwidth of the modal components will be too large, resulting in modal aliasing. On the other hand, the modes in the signal $X(\omega )$ contain harmonic components. If the value of A is too high, the bandwidth of the mode component will be too narrow, and some of the harmonic components in the original component may be lost, leading to signal leakage. As a fine-tuning parameter, $D$ can adjust the degree of mode aliasing between IMF components. $D$ is a fine-tuning parameter. Appropriate bandwidth constraints can be obtained by dynamically adjusting the value of $D$, to avoid mode aliasing and reduce reconstruction errors.

(3) Selection of noise tolerance $\tau$.

The selection of $\tau$ is related to the noise level of the signal. In practical applications, the noise level of multi-component signals (i.e., mechanical vibration signal) is normally high. If $\tau$ is set to 0, closing the Lagrangian multiplier will essentially reduce Tikhonov regularization to a least-squares form. In other words, even if the noise level of the measured signal is slight, such noise may cause mode aliasing problems. The proposed sparse regularization term with the damping coefficient $\epsilon$ can still ensure Tikhonov regularization even if the Lagrangian multiplier is turned off. Here, the necessity of $\tau$ is illustrated by an example (the instance is derived from the original code of VMD) [38]. First, three single-frequency sinusoidal signals with different frequencies ($f_1$ = 2 Hz, $f_2$ = 24 Hz, and $f_3$ = 288 Hz) are constructed, as shown in Figure 1.

These three signals are linearly superimposed and the mixed signal is shown in Figure 2. In Figure 2a, the mixed signal does not contain noise, but the signal in Figure 2b contains interference with Noise = 0.2 × randn(•) (a normally distributed random number). Then, the signal in Figure 2b is decomposed using the original VMD.

The parameters are set as $K$ = 3 and $\alpha$ = 2000, and the decomposed result is shown in Figure 3. When $\tau$ = 0, it can be observed from Figure 3a that the first IMF and second IMF suffer from modal aliasing. The modal extraction results for $\tau$ = 0.2 are shown in Figure 3b. Here, there is no significant mode aliasing between the independent components. Consequently, the following conclusion can be drawn from the experimental results: when the signal contains noise, tuning $\tau$ can alleviate the aliasing problem to some extent. However, this effect is rarely mentioned in relevant studies. Considering that signal-to-noise (SNR) is a relatively simple and effective noise measurement tool, we use SNR as the basis for setting $\tau$. When the noise level in the signal is extremely low, and the SNR is large, the noise level can be normalized to 1. If the signal contains a large amount of noise, the SNR will be a negative number, in which case it will be normalized to 0. If the SNR of the signal to be decomposed is within [0, 1], it will be taken as the setting value of τ. In this way, the value can be selected to provide a theoretical basis and the convergence rate of the VMD algorithm can also be guaranteed.

(4) Iteration operator $\gamma$.

In this paper, an iterative operator $\gamma$ is introduced to update the Lagrange multiplier $\lambda$ twice. The mathematical formula of the iteration operator $\gamma$ is as follows:

$${\gamma }_{n+1}=\frac{\sqrt{4{\gamma }_n^2+1}+1}{2}$$

(13)

where ${\gamma }_{n+1}$ is the n-th iteration operator, and the initial value of ${\gamma }_{n+1}$ is 0, i.e., ${\gamma }_0$ = 0. Then, we use $\gamma$ to update $\lambda$ as follows:

$${\widehat{\lambda }}^{n+1}(\omega )\leftarrow {\widehat{\lambda }}^{n+1}(\omega )+\left(\frac{{\gamma }_n-1}{{\gamma }_{n+1}}\right)\left({\widehat{\lambda }}^{n+1}(\omega )-{\widehat{\lambda }}^n(\omega )\right)$$

(14)

where $\left(\frac{{\gamma }_n-1}{{\gamma }_{n+1}}\right)\in [0,1]$ is the iteration step. The iteration operator $\gamma$ is introduced to update $\lambda$ twice. Consequently, the updating process of $\lambda$ uses not only the current ${\widehat{\lambda }}^{n+1}(\omega )$, but also common information of the current ${\widehat{\lambda }}^{n+1}(\omega )$ and the previous iteration ${\widehat{\lambda }}^n(\omega )$.

(5) Choosing the convergence tolerance $\epsilon$. Based on the VMD algorithm, this paper proposes different convergence tolerance parameters, namely $\epsilon 1$, $\epsilon 2$, and $\epsilon 3$. Among them, $\epsilon 1$ controls the reconstruction accuracy, whereas $\epsilon 2$ and $\epsilon 3$ are applied to adjust the total number of modes $K$.

In this paper, we empirically improve the five key parameters of VMD. This strategy can effectively ensure the distinction of VMD decomposition items and reconstruction accuracy.

3.5. Sparse Variational Model

The constrained variational problem in Equation (11) is converted into a non-constrained variational problem, and the extended Lagrangian expression is as follows:

$$\begin{array}{cc}\hfill L\left(\left\{u_k\right\},\{{\omega }_k\},\lambda \right):& =\alpha {{\displaystyle \sum _k‖\partial t\left[\left(\delta (t)+\frac{j}{\pi t}\right)\ast u_k(t)\right]e^{-j{\omega }_{k}t}{e}^{-j\epsilon t}‖}}_2^2\hfill \\ & {+‖f(t)-{\displaystyle \sum _k{u}_k(t)}‖}_2^2+〈\lambda (t),f(t)-{\displaystyle \sum _k{u}_k(t)}〉\hfill \end{array}$$

(15)

The algorithm flow of the sparse variational model is outlined in Algorithm 1.

Algorithm 1: Complete optimization of the Adaptive Sparse-VMD

Initialize:$\left\{{\widehat{u}}_k^1(\omega )\right\}$, $\left\{{\omega }_k^1\right\}$, ${\widehat{\lambda }}^1$, $\epsilon 1$, $\epsilon 2$, $\epsilon 3$, $k\leftarrow 1$, $n\leftarrow 0$, $D\leftarrow 0.1$ repeat: $k\leftarrow k+1$ repeat: $n\leftarrow n+1$ Update ${\widehat{u}}_k(\omega )$ for all $\omega \ge 0$: ${\widehat{u}}_k^{n+1}(\omega )\leftarrow \frac{\widehat{f}(\omega )-{\displaystyle {\sum }_{i<k}{\widehat{u}}_i^{n+1}(\omega )-{\displaystyle {\sum }_{i>k}{\widehat{u}}_i^n(\omega )}+\frac{{\widehat{\lambda }}^n(\omega )}{2}}}{1+2\alpha {(\omega +\epsilon -{\omega }_k^n)}^2}$ Update ${\omega }_k$: ${\omega }_k^{n+1}\leftarrow \frac{{\displaystyle {\int }_0^{\infty }\omega {\left|{\widehat{u}}_k^{n+1}(\omega )\right|}^{2}d\omega }}{{\displaystyle {\int }_0^{\infty }{\left|{\widehat{u}}_k^{n+1}(\omega )\right|}^{2}d\omega }}$ Dual ascent for all $\omega \ge 0$: ${\widehat{\lambda }}^{n+1}(\omega )\leftarrow {\widehat{\lambda }}^n(\omega )+(\tau +\frac{\epsilon }{n})\left(\widehat{f}(\omega )-{\displaystyle \sum _k{\widehat{u}}_k^{n+1}(\omega )}\right)$ Update $\lambda (\omega )$: ${\widehat{\lambda }}^{n+1}(\omega )\leftarrow {\widehat{\lambda }}^{n+1}(\omega )+\left(\frac{{\gamma }_n-1}{{\gamma }_{n+1}}\right)\left({\widehat{\lambda }}^{n+1}(\omega )-{\widehat{\lambda }}^n(\omega )\right)$ until convergence: ${{\displaystyle {\sum }_k‖{\widehat{u}}_k^{n+1}(\omega )-{\widehat{u}}_k^n(\omega )‖}}_2^2/{‖{\widehat{u}}_k^n(\omega )‖}_2^2<\epsilon 1$ If $‖u_{i<k}^{(n+1)*}(\omega )\cdot u_{(i+1)\le k}^{(n+1)*}(\omega )‖\ge \epsilon 2$ Update $D$: $D=D+0.1$ $k\leftarrow k-1$ until convergence: $D>5$ until convergence: ${\mu }_1=$$Max\left[\frac{{‖u_k^{(n+1)*}(\omega )\cdot u_{k-1}^{n+1}(\omega )‖}_2^2}{{\displaystyle {\sum }_k{‖u_k^{n+1}(\omega )‖}_2^2}}\right]\ge 5\epsilon 2$ and ${\mu }_2=$$Min\left[\frac{{‖u_k^{(n+1)*}(\omega )\cdot u_{k-1}^{n+1}(\omega )‖}_2^2}{{\displaystyle {\sum }_k{‖u_k^{n+1}(\omega )‖}_2^2}}\right]\le \epsilon 3$

The specific steps of the pseudo-code for solving the sparse variational model are as follows:

(1) Initialize the parameters of adaptive sparse-VMD, including $\left\{{\widehat{u}}_k^1(\omega )\right\}$, $\left\{{\omega }_k^1\right\}$, ε, ${\widehat{\lambda }}^1$, $\epsilon 1$, $\epsilon 2$, $\epsilon 3$, $k$, $n$, and $D$. The initial setting values of $\left\{{\widehat{u}}_k^1(\omega )\right\}$, $\left\{{\omega }_k^1\right\}$, ${\widehat{\lambda }}^1$, $\epsilon 1$, and $n$ are consistent with those of the standard VMD. The parameters $k$, $n$, and $D$ are set as 1, 0, and 0.1, respectively. Moreover, the parameters $\epsilon$, $\epsilon 1$, $\epsilon 2$, and $\epsilon 3$ are set according to the characteristics of the signal. $\tau$ is set according to the SNR, and $\epsilon$ can be set to 0.01.

(2) The initial signal decomposition parameters K = 2, $n$ = 1, and $D$ = 0.5. The first $\alpha$ value is obtained with Equation (12), and the first layer of the inner loop is executed to update $u_k$ according to $u_k^{n+1}=\arg \min L\left(\left\{u_{i<k}^{n+1}\right\},\left\{u_{i\ge k}^{n+1}\right\},\left\{{\omega }_i^n\right\},{\lambda }^n\right)$.

(3) Execute the second inner loop and update ${\omega }_k$ with ${\omega }_k^{n+1}=\arg \min L\left(\left\{u_i^{n+1}\right\},\left\{{\omega }_{i<k}^{n+1}\right\},\left\{{\omega }_{i\ge k}^n\right\},{\lambda }^n\right)$.

(4) Update parameter $\lambda$ in terms of ${\lambda }^{n+1}={\lambda }^n+\tau \left(f-{\displaystyle \sum _k{u}_k^{n+1}}\right)$ and ${\lambda }^{n+1}={\lambda }^{n+1}+\tau \left({\lambda }^{n+1}-{\lambda }^n\right)$.

(5) Run n = n + 1 until n = N (N is the maximum number of cycles) to end the loop. Or when ${{\displaystyle {\sum }_k‖u_k^{n+1}-u_k^n‖}}_2^2/{‖u_k^n‖}_2^2<\epsilon 1$, stop the iteration and calculate the orthogonality of the adjacent IMF components, that is $‖u_{i<k}^{(n+1)*}(\omega )\cdot u_{(i+1)\le k}^{(n+1)*}(\omega )‖$. If the orthogonality between adjacent IMF components is greater than the given threshold $\epsilon 2$, the fine-tuning parameter $D$ is adjusted by 0.1 to update the $\alpha$ value. Then, the number of IMF components k is set to k = k − 1 until the stopping condition is met or the orthogonality condition of the IMF component is satisfied and the inner second layer of the cycle is ended.

(6) Continue to execute the outer cycle. Set $k=k+1$ and repeat steps (3)–(5) until both ${\mu }_1\ge 5\epsilon 2$ and ${\mu }_2\le \epsilon 3$ are satisfied. After completing the entire iteration, the $k$ narrow-band IMF components can be obtained. The control indexes of the two cycles used in this step are ${\mu }_1$ and ${\mu }_2$. The indicator ${\mu }_2$ determines the orthogonality between IMF components, thereby ensuring that information aliasing between IMF components is minimal and meets a given threshold. However, this evaluation condition does not effectively reflect the single-component characteristics of the obtained IMFs. Consequently, the index ${\mu }_1$ is introduced to control the singleness of IMF components, but this index can lead to the modal replication phenomenon due to the small quantity of components. This index can then be merged using the orthogonal evaluation criterion ${\mu }_2$. By fusing these two constraints, the orthogonality between the IMF components and the monodromy of the IMF components can be efficiently guaranteed. The diagram of ASparse–VMD is shown in Figure 4.

To illustrate the need for the orthogonality and singleness of IMF components, the three harmonic signals ($f_1$ = 2 Hz, $f_2$ = 24 Hz, and $f_3$ = 288 Hz) in Figure 1a are used as an additional illustration. These three signals are linearly superimposed as mixed signals to be decomposed, and the results are shown in Figure 5.

The key parameters of VMD are preset as $K$ = 2, $\alpha$ = 2000 and $\tau$ = 0, respectively. Then, the signal in Figure 5a is decomposed, and the result is shown in Figure 6. According to the decomposition results, the first IMF1 component contains two frequency characteristics ($f_1$ and $f_2$), while the second IMF2 component only contains $f_3$. Obviously, IMF1 is not a mono-component signal. At this time, the least squares mutual information (LSMI) between these two IMF components is 1.0 × 10⁻⁶ (close to 0), indicating that the two components are approximately orthogonal.

Reconstruction is performed for both IMF components, and the new signal is shown in Figure 7a. The distinctiveness between the reconstructed signal and the original mixed signal is very small, with a reconstruction error of only 1.0 × 10⁻¹⁰. These indicators satisfy the termination of the decomposition condition, but the resulting number of modes is two, while the actual number of components is three. As shown by the experiment, using the orthogonality between the modal components and the reconstruction error of the signal as an evaluation metric for the performance of the decomposition provides an incomplete result. Thus, it is critical to introduce another evaluation parameter to measure the monodromy of the IMF component itself. According to the decomposition strategy of the VMD algorithm, if the decomposition of a single IMF continues, the IMF will replicate itself. Essentially, the signal features are self-replicating and the total energy is invariant. The demonstration results are shown in Figure 8. Finally, the replicated modes can be merged by judging the correlation and the correct decomposition result can be obtained.

4. Numerical Validation

The main purpose of this section is to evaluate the performance difference between our proposed ASparse-VMD and other widely used signal-processing methods, as well as the advantage of ASparse-VMD using simulated signals. To verify the effectiveness of the proposed method, two simulation cases are used to analyze and validate the aforementioned methods.

4.1. Simulation Verification (Case 1)

Considering the nonlinear and non-stationary nature of faulty signals, it is difficult to qualitatively explain their nonlinearity and non-stationary nature using the measured signals. Here, the simulation signal in Equation (16) is used for illustration [26]. The waveform of this signal is shown in Figure 9. The mathematical definition of the simulation signal is as follows:

$$\left\{\begin{array}{l}x(t)=x_1(t)+x_2(t)+x_3(t)+n(t)\\ x_1(t)=1\times \sin (2\pi f_{1}t)\\ x_2(t)=0.25\times \sin (2\pi f_{2}t)\\ x_3(t)=0.06\times \sin (2\pi f_{3}t)\end{array}\right.$$

(16)

where $x(t)$ is a mixture consisting of four simulated signal components, namely $x_1(t)$, $x_2(t)$, $x_3(t)$, and $n(t)$. In particular, $x_1(t)$, $x_2(t)$, and $x_3(t)$ are three sinusoidal signals with different center frequencies: $f_1$ = 2 Hz, $f_2$ = 24 Hz, and $f_3$ = 288 Hz, respectively, and $n(t)$ is Gaussian white noise.

The simulated signal for $n(t)$ = 0 is shown in Figure 9. Here, Figure 9a represents the signals of three components, Figure 9b denotes the composite signal, and Figure 9c is the corresponding spectrum. First, CEEMDAN, LMD, ITD, SVMD, and ASparse-VMD are used to evaluate the decomposition performance of the signal, and the results are shown in Figure 10.

Figure 10a denotes the independent components obtained by CEEMDAN decomposition and the corresponding spectrum. Based on the time-domain waveforms of the components, the last IMF component clearly suffers from mode aliasing problems, as evidenced by the corresponding spectra. Figure 10b shows the decomposition results under the LMD method. From the obtained number of components, it can be seen that the LMD effectively decomposes three-product function (PF) components, which are consistent with the original number of components. However, these components appear heavily distorted, and the corresponding spectra also show the appearance of clutter. The results of ITD decomposition are shown in Figure 10c. Here, ITD effectively obtains three proper rotation components (PRC), but there is some mode aliasing between the components. Figure 10d presents the results of SVMD decomposition. The SVMD parameters are set as $\alpha$ = 20,000, $\tau$ = 0 and $stopc$ = 4. For the time-domain signal shown in Figure 10, a total of four signal components were obtained. The third IMF component is a residual component with no physical definition, while the modal aliasing obviously appears in the first component. The two peaks of the frequency ($f_1$ and $f_2$) are clearly visible in the spectrum. The result of ASparse-VMD is shown in Figure 10e, where the preset parameters are $\tau$ = SNR = 1, $\epsilon$ = 0, $\epsilon 2$ = 1.0 × 10⁻¹, and $\epsilon 3$ = 1.0 × 10⁻². The ASparse-VMD method efficiently obtains three single-signal components whose respective spectra exhibit perfect discriminative properties. The comparative experiments show that the proposed ASparse-VMD algorithm inherits the advantages of VMD with higher resolution feature extraction.

The simulated signal for when $n(t)$ = 0.2 × randn(•) (normally distributed random numbers) is shown in Figure 11. Here, Figure 11a presents the three original independent signals and noise components, and Figure 11b is their synthetic time-domain waveforms.

Similar to the above experiment, the signal is used to evaluate the decomposition performance of CEEMDAN, LMD, ITD, SVMD, and ASparse-VMD methods. Figure 12a shows the results of CEEMDAN decomposition, including the time domain waveform of the obtained components and corresponding spectrums. The CEEMDAN decomposition results in six signal components. The spectra of three IFM components (the third, fourth and fifth) correspond to the $f_3$ item in the original signal, indicating mode duplication. The frequency components $f_1$ and $f_2$ appear in the spectrum of the last IMF, indicating that it is under-decomposition and causing significant mode aliasing. Figure 12b presents the PF results of LMD decomposition. Here, six components are obtained by LMD decomposition, but they have serious distortion. By observing the corresponding spectra, it can be found that the second, third, and fourth PF components contain the same frequency feature $f_3$ simultaneously, indicating the occurrence of modal replication. In addition, the fifth and sixth PFs contain signal characteristic frequencies $f_1$ and $f_2$, respectively, indicating that these two modes can be correctly identified. In general, LMD is able to correctly obtain the individual components of the mixed signal, but there is apparent over-decomposition. As shown in Figure 12c, the ITD decomposition results in four PRCs, but specific signal features cannot be identified from their time-domain waveforms. From the corresponding spectra, it can be observed that the first three PRC components produce mode duplication ($f_3$ appears in all mode) and the last PRC component produces mode aliasing ($f_2$ and $f_3$). Worst of all, the feature $f_1$ in the raw signal is not found in all obtained modes, indicating severe distortion in the decomposition. The decomposition result of SVMD for parameters of $\alpha$ = 30,000, $\tau$ = 0, and $stopc$ = 4 are shown in Figure 12d. As in the time-domain waveform, SVMD resolves four signal components, but the first IMF component experiences a clear mode aliasing problem ($f_1$ and $f_2$). The second IMF corresponds to feature $f_3$, indicating that the characterization of this component is correct. In addition, the frequency spectra corresponding to the third and the fourth IMF components present trend items that should not appear. The result obtained by the decomposition is quite different from the real components. Figure 12e is the result of ASparse-VMD decomposition. When SNR < 0, the set parameters are $\tau$ = 0, $\epsilon$ = 0.01, $\epsilon 2$ = 1.0 × 10⁻¹ and $\epsilon 3$ = 1.0 × 10⁻². At this time, ASparse-VMD decomposes eight signal components. The first, second, and third components are consistent with the original signal features ($f_1$, $f_3$, and $f_3$), and no over-decomposition or under-decomposition occurs in adjacent components. Additional, In the spectra corresponding to these three components, there are fewer interference elements. The remaining five components are noise items. There is no mode duplication and mode aliasing problem during the whole decomposition process. The experimental results also verify the effectiveness and superiority of ASparse-VMD.

4.2. Simulation Verification (Case 2)

In order to verify the validity of the proposed ASparse-VMD method, in this section, a typical simulated fault excitation signal of a rolling bearing is employed for analysis. The simulated signal $x(t)$ is composed of a harmonic signals $x_1(t)$ and $x_2(t)$, a frequency modulation (FM) signal $x_3(t)$, a periodic impulse signal $x_4(t)$, a random interference pulse $x_5(t)$, and additive white Gaussian noise $n(t)$, as described in Equations (17) and (18). The signal frequencies are $f_1$ = 111 Hz, $f_2$ = 223 Hz, $f_3$ = 50 Hz, and $f_4$ = 85 Hz, respectively. The simulation signal is shown in Figure 13.

$$\left\{\begin{array}{l}x(t)=x_1(t)+x_2(t)+x_3(t)+x_4(t)+x_5(t)+n(t)\\ x_1(t)=0.2\times \sin (2\pi f_{1}t)\\ x_2(t)=0.2\times \sin (2\pi f_{2}t)\\ x_3(t)=0.2\times \sin (2\pi f_3{t}^2)\\ x_4(t)={\displaystyle {\sum }_i{A}_{i}h(t-iT-{\nu }_i)}\\ x_5(t)=rand(·)\times x_4(t)\\ n(t)=0.2\times randn(·)\end{array}\right.$$

(17)

$$\left\{\begin{array}{l}h(t)=e^{-Ct}\sin (2\pi f_{n}t)\\ A_i=1+A_0\sin (2\pi f_{r}t)\end{array}\right.$$

(18)

In this section, we further analyze the signal in case 2 using the CEEMDAN, LMD, ITD, SVMD, and ASparse-VMD methods. Figure 14a presents the results of CEEMDAN decomposition. The CEEMDAN method yields 11 signal components. Two dominant modes, $f_1$ and $f_2$, appear in the IMF3 component, indicating aliasing. The same mode appears between adjacent components (such as $f_2$ in IMF2 and IMF3, $f_1$ in IMF3 and IMF4), indicating mode replication. The FM signal $x_3(t)$ in the original signal is decomposed into three IMF components (IMF 5, 6, and 7) in different frequency bands. As shown in Figure 14b, LMD obtains seven signal components. Apparently, modal aliasing occurs between PF1 and PF3 components. The original FM signal appears in three different frequency bands (PF3–PF5), and some distortion occurs. The results of SVMD decomposition are shown in Figure 14d, where the decomposition parameters are set as $\alpha$ = 30,000, $\tau$ = 0 and $stopc$ = 4. SVMD decomposes 11 signal components, and the first three IMF components (IMF1-IMF3), respectively, correspond to $x_3(t)$, $x_1(t)$ and, $x_2(t)$ in the original signal. However, the periodic pulse component in the original signal is decomposed into seven IMF components of different frequency bands (IMF4–IMF11), and mixed with random noise. When the set parameters are $\epsilon$ = 0.01, $\epsilon 2$ = 1.0 × 10⁻¹. and $\epsilon 3$ = 1.0 × 10⁻²., ASparse-VMD divides the mixed signal into five modes. Figure 14e presents the result of ASparse-VMD decomposition. Among these components, the modes IMF1–IMF3 correspond to the FM signal $x_3(t)$, harmonic signal $x_1(t)$, and harmonic signal $x_2(t)$, respectively. IMF4 corresponds to the periodic pulse signal $x_4(t)$, but contains slight noise (random pulse signal $x_5(t)$). Here, IMF5 represents the residual random noise n(t). Finally, all IMFs are reconstructed and the reconstruction error is 2 × 10⁻².. Additional, Figure 14e shows the reconstructed signal and its corresponding spectrum. As can be seen from the experiments, the ASparse-VMD can efficiently extract each individual signal component of the original signal. This result further demonstrates the effectiveness and superiority of ASparse-VMD.

5. Experimental Results

The aforementioned experimental results were used to qualitatively analyze and verify the effectiveness, superiority and robustness of ASVMD from the perspective of theory. This section further verifies the practical application of the project.

5.1. Introduction to the Experimental Platform

The bearing accelerated life test platform used in this work was designed and manufactured by the joint laboratory of mechanical equipment health testing (XJTU-SY bearing datasets) [39]. As shown in Figure 15, the platform includes an AC motor, motor speed controller, rotating shaft, support bearing, hydraulic loading system, test bearing, and other connecting parts. The test bearing is a LDK UER204 rolling bearing, which is detailed in Figure 16. The relevant parameters are shown in

Table 1. Bearing parameters of LDK UER204.

Specification	Value
Diameter of inner race (mm)	29.3
Diameter of outer race (mm)	39.8
Bearing center diameter (mm)	34.55
Basic rated dynamic load (N)	12,820
The ball diameter (mm)	7.92
Ball number	8
Contact angle (°)	9
Basic rated static load (KN)	6.65

.

5.2. Experimental Data

In order to obtain the real-time state of the bearing under the failure condition, two unidirectional acceleration sensors are fixed to the horizontal and vertical directions of the test bearing through the magnetic base to collect vibration signals. In the test, the sampling frequency is 25.6 kHz, the sampling interval is 1 min, and the sampling time is 1.28 s. The sampling diagram is shown in Figure 17.

5.3. Bearing Failure Form and Vibration Signal Analysis

Figure 18 illustrates typical failure types of bearings [40]. The fault causes of the tested bearings include inner race wear, cage fracture, outer race wear, and outer race crack. Under the test conditions of a rotational speed of 2400 R/min and a radial force of 10 kN, vibration signal samples with the four faults were simultaneously collected. The time-domain signals obtained by the horizontal sensor are presented in Figure 19. The corresponding fault characteristic frequencies of the four faults are shown in

Table 2. Fault characteristic parameters.

Speed (r/min)/Hz	Fault Frequency (Hz)
2400/40	Inner race ($f_{inner}$)	Cage ($f_{cage}$)	Outer race ($f_{outer}$)	Rolling ball ($f_{ball}$)	Rotation frequency ($f_r$)
	196.9	15.4	123.3	165.3	40

.

In this section, the CEEMDAN, LMD, ITD, SVMD, and ASPARSE-VMD methods are used to process the measured bearing signals obtained in Figure 19, and the results are shown in Figure 20, Figure 21, Figure 22, Figure 23 and Figure 24. Figure 20 presents the results of CEEMDAN decomposition, which obtained eleven independent IMF components. Among them, a cage fault feature appears in IMF4 and the spectrum of the IMF1 contains the fault characteristic frequencies of the inner ring, outer ring, and rolling body of the bearing. This result indicates the occurrence of serious mode aliasing and, more importantly, the appearance of significant redundant interference in the corresponding spectra.

Figure 21 presents the results of LMD decomposition, which was used to decompose the signal into seven PF components. In the first three PFs, inner ring, outer ring, and rolling element fault features are observable. However, some types of fault features are mixed together (such as inner ring and outer ring faults), and appear simultaneously in multiple components (i.e., PF1 and PF3).

As shown in Figure 22, the ITD method decomposes the signal into eight PRC components. The PRC1 component spectrum contains the fault characteristic of the bearing inner ring, and PRC5 indicates the fault feature of the bearing outer ring. Nevertheless, the relative amplitudes of the two spectral lines are weak and are not prominent under noise items. In terms of other components, the effective information on other faults is not effectively demonstrated.

Figure 23 shows the results of SVMD decomposition, yielding 27 IMF elements. In the decomposition process, the inherent hyper-parameters are selected as $\alpha$ = 30,000, $\tau$ = 0, and $stopc$ = 4. The IMF2 component contains the fault signal of the bearing cage, IMF9 contains the fault characteristic of the bearing inner ring, and the fault characteristic frequency of the bearing rolling ball exists in the IMF19 component. However, the amplitudes of the required signals are relatively small. The fault characteristics of the bearing outer ring appear in IMFs 24–27, indicating that modal replication (over-decomposition) occurs in the decomposition process. Although SVMD can effectively detect the hybrid fault characteristic frequencies of bearings, this method has many invalid components and suffers from the modal replication problem.

In the proposed Asparse-VMD decomposition process, the parameters are set as $\epsilon 1$ = 1.0 × 10⁻², $\epsilon 2$ = 1.0 × 10⁻¹., and $\epsilon 3$ = 1.0 × 10⁻². It can be seen from the component information in Figure 24 that the spectrum of IMF3 contains the fault characteristic frequency of the bearing cage, IMF8 extracts the fault signatures of the bearing rolling body, and the independent component IMF9 detects the fault sensitive information of the inner ring. In the spectrum of IMF10, the fault characteristics of the bearing outer ring (including the fundamental frequency and double frequency) are clearly identified. In particular, the rotation frequency $f_{\mathrm{rotary}}$ of the bearing (39.3 Hz; the theoretical value is 40 Hz) is extracted using the IMF1 component, which cannot be obtained by the state-of-the-art CEEMDAN, LMD, ITD, and SVMD methods. Compared to the decomposition effect of SVMD, the number of modes is more sparse, and the anti-aliasing ability and resolution are better. The analysis process in this section fully verifies the effectiveness and superiority of ASparse-VMD in the practical application of multi-faults engineering. The above experimental results also fully illustrate the robustness of the proposed method.

6. Discussion and Conclusions

In this paper, we proposed an ASparse–VMD method to study the feature extraction of shock vibration signals from multi-fault rotating machinery by deeply evaluating the decomposition principle of VMD and the influence of key parameters. The regularization procedure of VMD was improved by adding a damping parameter, giving the VMD better sparsity. By making this change, the parameters ($K$ and $\alpha$) of the VMD were automatically adjusted via new update rules and convergence conditions. Then, the influence of $\tau$ on VMD was analyzed, and the SNR was used as the basis for setting the $\tau$ value. This approach additionally improved the mode aliasing problem and the mode resolution. Moreover, ASparse–VMD has lower computational complexity and versatility. The ASparse–VMD improved decomposition performance of WMD and with superior robustness. The experimental results showed that the proposed method is more advantageous in signal processing.

However, ASparse-VMD also suffers from some problems. (1) The update rules and convergence conditions for the parameters have some theoretical basis, but also retain empirical properties. (2) The common impacts of all embedding parameters of VMD on performance, such as the number of inner iterations, was not sufficiently considered. (3) Information sources for monitoring the operational status of mechanical devices show a tendency to diversify and develop in high dimensions, and the proposed Asparse-VMD is not suitable for processing such signals. (4) ASparse-VMD is essentially an iterative process that requires parameter updates and stopping rules to be satisfied, and further optimization is required for application scenarios with high real-time requirements. Consequently, the theoretical framework of VMD needs to be improved in the future or the proposed algorithm needs to be better implemented. Especially for real-time applications, it is urgent to solve the parameter iteration search problem and simplify the implementation steps of the method.