An enhanced sparsity-driven morphological decomposition for weak impact fault diagnosis in energy equipment

Abstract

Healthy monitoring is an important component in energy equipment intelligent maintenance, particularly for critical rotating machinery such as gas turbines and wind turbine gearboxes, and one of its core technologies is the impact fault extraction. While the existing sparse decomposition applies on identifying weak impact fault, as it often relays on prior knowledge to represent mechanical fault signal through a series of sparse dictionaries, it is easily affected by other interference, such as various noises, modulation component, resulting in the shortcomings of high computational complexity, which severely limits real-time diagnostics in energy equipment operating under extreme conditions such as high temperature and heavy load. As a result, this paper aims to propose an enhanced sparsity signal decomposition (ESSD) which can obtain a discrete frequency component and periodic shock component, respectively, for healthy monitoring of energy equipment, where accurate separation of fault signatures from complex vibration mixtures is crucial to prevent catastrophic failures. Firstly, on account of the morphological differences between shock component and discrete frequency component separately in the time domain and frequency domain, we introduce unit dictionary and Fourier dictionary to represent signal component, respectively. Furthermore, we design a non-convex sparse objective function based on morphological component analysis (MCA) that ensure that ESSD has a strong sensitivity and amplitude fidelity to weak shock component. Finally, we utilize the majorization minimization (MM) algorithm to derive a fast solver for computing the objective function. Simulation and experiment quantitatively prove that the proposed method not only has low computational complexity but also avoids selecting multiple dictionaries and underestimating pulse features, demonstrating significant potential for deployment in smart maintenance platforms for next-generation energy equipment systems.

Introduction

Equipment health monitoring plays a critical role in ensuring the safe and stable operation of energy machinery such as wind turbines and gas turbine generators in power generation systems, and advancements in its theoretical frameworks have emerged as a cross-disciplinary research priority. ¹ Rotating components such as rolling bearing and gear, which serve as core elements in critical energy equipment like wind turbine gearboxes and hydroelectric generator shafts, exhibit significantly higher failure rate than other mechanical parts due to their demanding operational condition. ² Their fault vibration signal usually contain repetitive transient pattern. Accurately isolating these features remains a critical challenge on diagnosing rotating machinery’s fault. ³

Current research faces dual technical challenges. Firstly, as for the signal feature level, fault signal under multi-fault condition often exhibits weak feature coupling property. ⁴ Secondly, traditional diagnostic approaches, such as time-frequency analysis and signal decomposition technique, have inherent limitations including insufficient feature decoupling capabilities, sub-optimal parameter optimization, excessive computational complexity, and over-reliance on empirical expertise. ⁵ These shortcomings render them inadequate for real-time monitoring requirement in intelligent maintenance system for energy equipment operating under extreme conditions such as high-temperature combustion chambers and heavy-load transmission shafts. ⁶ In this context, sparse representation methods present unique advantage for extracting weak impulsive features under strong noise interference. By constructing over-complete dictionaries to achieve sparse signal characterization, these methods offer innovative solution for separating multi-source disturbances. However, practical engineering application reveals a series of critical limitations, such as dictionary construction dominated by manually selecting parameter, non-convex optimization leading to increases in computational complexity, and convex regularization leading to systematically underestimate impact amplitude.

In order to address these challenges, reference ⁷ develops a parametric Laplace wavelet dictionary construction method through time-frequency overlapped group sparse modeling. In parallel, reference ⁸ proposes a hybrid dictionary combining impact wavelet and Fourier function, overcoming the spectral adaptability limitation caused by conventional fixed dictionaries. For regularization improvements, paper ⁹ employs the Alternating Direction Multipliers Method (ADMM) to convert non-convex problem into solvable convex form. Meanwhile, paper ¹⁰ introduces an SVD-Parseval joint regularization framework that significantly enhances high-amplitude impact component’s estimation accuracy, paper ¹¹ creates the Adaptive Parameter Multi-model Estimation and Sparse Reconstruction (APMESR) algorithm, which reduces computational complexity.

Considering that the existing sparse fault diagnosis method has dictionary selecting subjectively and the amplitude under-estimation caused by convex optimization, when they directly identify weak impact fault, it will such as low execution efficiency and amplitude distortion. This paper proposes an enhanced sparse signal decomposition, which is highly applicable to online health monitoring of energy equipment in smart grid infrastructure, where rapid fault identification is essential to prevent cascading failures. It is based on the morphological diversity corresponding to multi-component in the fault signal, by using the unit matrix dictionary to represent the periodic pulse component in the time domain and the Fourier dictionary to represent the harmonic component model in the frequency domain, respectively, which can effectively improve the sparse model’s computational efficiency. Furthermore, aiming at solving the problem of amplitude underestimation caused by the L1 norm in classical penalty function, a non-convex penalty function with intra-group and inter-group joint sparsity property is introduced, the penalty function conforms to the global convergence requirement, has a good amplitude fidelity, and can effectively enhance weak fault feature. Finally, simulation and rolling bearing experiment quantitatively prove that the proposed method can apply on equipment health monitoring.

Enhanced sparse signal decomposition

In general, mechanical fault signal typically consist of two signal components which are modulated component and periodic impacts. ¹² Modulated component reflects periodic mechanical process such as gear working operation. On the other hand, periodic impacts originate from mechanical collision or abrupt shock phenomenon, often indicating structural defect such as gear misalignment or bearing degradation. ¹³

Assuming a mechanical fault signal $\mathit{y}\in {\mathbb{R}}^n$ can be typically decomposed into modulated component and periodic impact component in healthy monitoring, it can be expressed as shown in formula (1), ${\mathit{y}}_1$ represents the modulated component, ${\mathit{y}}_2$ symbolizes the periodic impact component, and $\mathit{n}$ denotes the interference

$\mathit{y}={\mathit{y}}_1+{\mathit{y}}_2+\mathit{n}$

(1)

As a result, the key of sparse decomposition is how to separate modulation component and periodic shock component. In general, it can be achieved by constructing an objective function as shown in formula (2). ${\widehat{\hspace{0.17em}\mathit{x}}}_1$ represents the sparse coefficient corresponding to modulated component ${\mathit{y}}_1$ , ${\widehat{\mathit{x}}}_2$ represents the sparse coefficient corresponding to the periodic impact component ${\mathrm{y}}_2$ , ${\mathrm{\lambda }}_1$ , and ${\mathrm{\lambda }}_2$ are regularization parameters balancing reconstruction accuracy and sparsity, $P_1\left({\mathit{x}}_1\right)$ and $P_2\left({\mathit{x}}_2\right)$ denote sparsity-enforcing penalty functions, and ${\mathit{D}}_1$ , ${\mathit{D}}_2$ represent sparse dictionaries

$\left\{{\widehat{\mathit{x}}}_1,{\widehat{\mathit{x}}}_2\right\}=\mathit{arg}\hspace{0.17em}\underset{\hspace{0.17em}{\mathit{x}}_1,{\mathit{x}}_2}{\min }\frac{1}{2}{\Vert \mathit{y}-{\mathit{D}}_1{\mathit{x}}_1-{\mathit{D}}_2{\mathit{x}}_2\Vert }_2^2+{\lambda }_1{\mathit{P}}_1\left({\mathit{x}}_1\right)+{\lambda }_2{\mathit{P}}_2\left({\mathit{x}}_2\right)$

(2)

Finally, the estimated target components ${\widehat{\mathbf{y}}}_1$ and ${\widehat{\mathbf{y}}}_2$ corresponding to ${\mathit{y}}_1$ and ${\mathit{y}}_2$ , respectively, is computed in formula (3)

${\widehat{\mathit{y}}}_1={\mathit{D}}_1{\widehat{\mathit{x}}}_1,{\widehat{\mathit{y}}}_2={\mathit{D}}_2{\widehat{\mathit{x}}}_{2\hspace{0.17em}}$

(3)

In order to ensure that the estimated target components ${\widehat{\mathit{y}}}_1$ and ${\widehat{\mathit{y}}}_2$ match the original components ${\mathit{y}}_1$ and ${\mathit{y}}_2$ , respectively, the paper undoubtedly designs sparse dictionary and penalty function.

As for sparse dictionary, it generally contains analytical dictionary and learned dictionary. ¹⁴ Analytical dictionary needs prior knowledge for precisely conforming to signal pattern, learning dictionary employs data-driven construction with enhanced generalization ability. ¹⁵ This paper proposes Sparsity Within and Across Groups (SWAG) modeling with dual sparsity constraints that are intra-group sparsity ensures localized feature extraction, while inter-group sparsity suppresses redundant component interference. In order to select the proper sparse dictionaries ${\mathit{D}}_1$ , ${\mathit{D}}_2$ , considering periodic shock component ${\mathit{y}}_2$ exhibits a SWAG property in time domain, while modulated component y₁ show frequency-domain sparsity, we construct a hybrid dictionary matrix $\mathit{D}=\left[\mathit{F},\mathit{I}\right]$ , F represent Fourier sparse dictionary which captures harmonic frequency sparsity, $\mathit{I}$ symbolizes identity matrix that matches transient impact property, the sparse dictionaries enhance decomposition robustness. ¹⁶

Furthermore, in order to suppress the interference $\mathit{n}$ , we implement domain-specific regularization strategy to construct sparsity-enforcing penalty functions $P_1\left({\mathit{x}}_1\right)$ and $P_2\left({\mathit{x}}_2\right)$ ¹⁷ to enhance weak fault.

As for $P_1\left({\mathit{x}}_1\right)$ , we introduce Elastic Net regularization term ¹⁸ as shown in formula (4). In this formula, ${\Vert {\mathit{x}}_1\Vert }_1$ symbolizes L1 norm, it promotes sparse solution through feature selection by compressing non-essential coefficients to zero. $\frac{\eta }{2}{\Vert {\mathit{x}}_1\Vert }_2^2$ represents L2 norm, it penalizes large coefficients to ensure solution smoothness and prevent over-fitting. $\eta$ is regularization parameter which controls the relative weighting between L1 and L2 terms

$P_1\left({\mathit{x}}_1\right)={\Vert {\mathit{x}}_1\Vert }_1+\frac{\eta }{2}{\Vert {\mathit{x}}_1\Vert }_2^2$

(4)

As for $P_2\left({\mathit{x}}_2\right)$ , we introduce introduce the Sparse Within and Across Groups (SWAG) regularization as shown in formula (5). $\Omega$ denotes ${\mathit{x}}_{\left\{i,n_2\right\}}={\left[\mathit{x}\left(i\right),\cdots ,\mathit{x}\left(i+n_2-1\right)\right]}^T\in {\mathbb{R}}^{n^2}$ , $\mathrm{b}$ is a binary periodic sequence, $⨀$ indicates element-wise multiplication, $\rho$ controls sparsity trade-off, $\Phi \left(·;\alpha \right)$ represents the penalty function, $n_1$ specifies impact length, and $M$ counts impact number. In practice, we adopt n₁ = 4 and M = 4 based on prior experimental validation ¹⁹

$P_2\left({\mathit{x}}_2\right)={{\displaystyle \sum }}_{iϵ\Omega }\Phi \left({\Vert \mathit{b}⨀{\mathit{x}}_{\left\{i,n_2\right\}}\Vert }_2;\alpha \right)+\rho M{n}_1{\Vert W{\mathit{x}}_2\Vert }_1$

(5)

SWAG property can effectively capture weak periodic impact feature through non-convex regularization which can avoid inherent amplitude under-estimation issues in conventional L1 regularization. Combination penalty functions $P_1\left({\mathit{x}}_1\right)$ with $P_2\left({\mathit{x}}_2\right)$ , we transform formula (2) into formula (6) ultimately in ESSD

$J\left({\mathit{x}}_1,{\mathit{x}}_2\right)=\frac{1}{2}{\Vert \mathit{y}-{\mathit{D}}_1{\mathit{x}}_1-{\mathit{D}}_2{\mathit{x}}_2\Vert }_2^2{+\lambda }_1{P}_1\left({\mathit{x}}_1\right)+{\lambda }_2{P}_2\left({\mathit{x}}_2\right)$

(6)

• Step 1 Input signal y , set parameters ${\lambda }_1,{\lambda }_2,\rho ,\mu ,M,n_1,\eta$ , with maximum iterations $N_{iter}$ .

• Step 2 Initialize coefficients ${\mathit{x}}_1^0={\mathit{D}}_1^{\mathrm{T}}\mathit{y},{\mathit{x}}_2^0=\mathit{y}$ .

• Step 3 Compute intermediate variables, $k=0,1,2,\cdots ,N_{iter}$ , $p_1={\mathit{x}}_1^k-\mu {\mathit{D}}_1^T\left(\mathit{A}{\mathit{x}}^k-\mathit{y}\right)$ , $p_2={\mathit{x}}_2^k-\mu \left(\mathit{A}{\mathit{x}}^k-\mathit{y}\right)$ , $\mathit{A}=\left[\mathrm{F},\mathrm{I}\right]$ , $\mathrm{x}={\left({\mathrm{x}}_1^{\mathrm{T}},{\mathrm{x}}_2^{\mathrm{T}}\right)}^{\mathrm{T}}$ .

• Step 4 Update ${\mathit{x}}_1^{k+1}=\frac{1}{1+\mu {\lambda }_1\eta }\text{soft}\left({\mathit{p}}_1,\mu {\lambda }_1\right),{\mathit{x}}_2^{k+1}={\sum }^{-1}\text{soft}\left({\mathit{p}}_2,u\rho \lambda Mn\omega \right)$ , $\text{soft}\left(\mathit{x},t\right)=\text{sign}\left(\mathit{x}\right)\max \left(\left|\mathit{x}\right|-t,0\right),\omega =\frac{\mathbf{1}}{\left|{\mathit{x}}_{\mathbf{2}}^{\mathit{k}}\right|+\in }$ .

• Step 5 Output estimated components, ${\widehat{\mathit{x}}}_1={\mathit{x}}_1^{k+1},{\widehat{\mathit{x}}}_2={\mathit{x}}_2^{k+1}$ .

Finally, the regularization parameters ${\lambda }_1,{\lambda }_2$ , and iteration number $N_{iter}$ critically determine the proposed method’s performance and computational efficiency, their values conforms to the following criterion. ²⁰

Case 1 when the fault signal y contains modulation component combination with Gaussian Noise, the parameter ${\lambda }_1$ conforms to formula (7), n is the fault signal’s length

${\lambda }_1=\sigma \sqrt{2\hspace{0.17em}\mathit{ln}\hspace{0.17em}n}$

(7)

Case 2 when the fault signal y contains periodic shock component combination with Gaussian Noise, the parameter ${\lambda }_1$ conforms to formula (8)

${\lambda }_2=0.272\sigma +0.044$

(8)

Case 2 when the fault signal y not only contains periodic shock component but also possesses modulation component combination with Gaussian Noise, the parameter ${\lambda }_1$ and ${\lambda }_2$ conforms to formulas (9) and (10), the parameter $\mathrm{\alpha }$ is varing from 0 to 1

${\lambda }_1=\alpha \sigma \sqrt{2\hspace{0.17em}\mathit{ln}\hspace{0.17em}n}$

(9)

${\lambda }_2=\left(1-\alpha \right)\left(0.272\sigma +0.044\right)$

(10)

The convexity preservation condition of objective function

As shown in above, the objective function’s convexity-preserving condition is essential for guaranteeing global optimization in ESSD. To establish this convexity, we convert the original objective function formula (6) into formula (11) in the form of two independent components using an intermediate variable $\gamma \ge 0$

$\{\begin{array}{l}J\left({\mathit{x}}_1,{\mathit{x}}_2\right)=J_1\left({\mathit{x}}_1,{\mathit{x}}_2\right)+J_2\left({\mathit{x}}_2\right)\\ J_1\left({\mathit{x}}_1,{\mathit{x}}_2\right)=\frac{1}{2}\parallel \mathit{y}-{\mathit{D}}_1{\mathit{x}}_1-{\mathit{D}}_2{\mathit{x}}_2{\parallel }_2^2+{\lambda }_1{P}_1\left({\mathit{x}}_1\right)-\frac{\gamma }{2}\parallel {\mathit{x}}_2{\parallel }_2^2\\ J_2\left({\mathit{x}}_2\right)=\frac{\gamma }{2}\parallel {\mathit{x}}_2{\parallel }_2^2+{\lambda }_2{P}_2\left({\mathit{x}}_2\right)\end{array}$

(11)

In general, the $J\left({\mathit{x}}_1,{\mathit{x}}_2\right)$ retains convexity, only when both $J_1\left({\mathit{x}}_1,{\mathit{x}}_2\right)$ and $J_2\left({\mathit{x}}_2\right)$ are convex. While the convexity of $J_2\left({\mathit{x}}_2\right)$ has been established, ²¹ as a result, this paper need to give the convexity preservation condition of function $J_1\left({\mathit{x}}_1,{\mathit{x}}_2\right)$ . Assuming the $P_1\left({\mathit{x}}_1\right)={\Vert {\mathit{x}}_1\Vert }_1+\frac{\eta }{2}{\Vert {\mathit{x}}_1\Vert }_2^2$ , ${\mathit{D}}_1=\mathit{F}$ , and ${\mathit{D}}_2=\mathit{I}$ , the corresponding $J_1\left({\mathit{x}}_1,{\mathit{x}}_2\right)$ can be transformed into formula (12)

$J_1\left({\mathit{x}}_1,{\mathit{x}}_2\right)=\frac{\mathbf{1}}{\mathbf{2}}{\Vert \mathit{D}{\mathit{x}}_1+{\mathit{x}}_2\Vert }_2^2+\frac{{\lambda }_1\eta }{\mathbf{2}}{\Vert {\mathit{x}}_1\Vert }_2^2-\frac{\gamma }{\mathbf{2}}{\Vert {\mathit{x}}_2\Vert }_2^2+\frac{\mathbf{1}}{\mathbf{2}}{\Vert \mathit{y}\Vert }_2^2-{\mathit{y}}^T\left(\mathit{D}{\mathit{x}}_1+{\mathit{x}}_2\right)+{\lambda }_1{\Vert {\mathit{x}}_1\Vert }_1$

(12)

The terms $\frac{1}{2}{\Vert \mathit{y}\Vert }_2^2-{\mathit{y}}^{\mathrm{T}}\left(\mathit{D}{\mathit{x}}_1+{\mathit{x}}_2\right)+{\lambda }_1{\Vert {\mathit{x}}_1\Vert }_1$ are inherently convex. Therefore, the critical task is proving the convexity of formula (13)

$L=\frac{1}{2}{\Vert \mathit{D}{\mathit{x}}_1+{\mathit{x}}_2\Vert }_2^2+\frac{{\mathrm{\lambda }}_1\eta }{2}{\Vert {\mathit{x}}_1\Vert }_2^2-\frac{\mathrm{\gamma }}{2}{\Vert {\mathit{x}}_2\Vert }_2^2$

(13)

To analyze convexity, we reformulate L in matrix form as following in formula (14)

$L=\frac{1}{2}{\left({\mathit{x}}_2;\mathit{D}{\mathit{x}}_1\right)}^T\left(\begin{array}{cc}\mathit{I}-\gamma \mathit{I}& \mathit{I}\\ \mathit{I}& \mathit{I}+{\lambda }_1\eta \mathit{I}\end{array}\right)\left(\begin{array}{c}{\mathit{x}}_2\\ \mathit{D}{\mathit{x}}_1\end{array}\right)$

(14)

If matrix $\mathit{H}=\left(\begin{array}{cc}I-\gamma I& I\\ I& I+{\lambda }_1\eta I\end{array}\right)$ has positive semi-definite property, formula (14) will be convex. As a result, it will exist a permutation matrix $\mathit{Q}\in {\mathbb{R}}^{2\mathrm{n}\times 2\mathrm{n}}$ ²² as shown in formula (15)

$\mathit{Q}\mathit{H}{\mathit{Q}}^T={\mathit{H}}^{\prime }=\text{diag}\left({\mathit{H}}_1^{\prime },\cdots ,{\mathit{H}}_n^{\prime }\right)$

(15)

As ${\mathit{H}}_i^{\prime }\in {\mathbb{R}}^{2n\times 2n}$ presents real symmetric property, it conforms to the following rule as shown in formula (16)

${\mathit{H}}_i^{\prime }=\left(\begin{array}{cc}1-\gamma \mathit{I}& 1\\ 1& 1+{\lambda }_i\eta \mathit{I}\end{array}\right),i\in 1,2,\cdots ,n$

(16)

Furthermore, we simplify the matrix ${\mathit{H}}_i^{\prime }$ via eigenvalue decomposition into the form of formula (17), matrix ${\mathit{U}}_i$ is an orthogonal matrix composed by eigenvectors of matrix ${\mathit{H}}_i^{\prime }$

$\{\begin{array}{c}{\mathit{H}}_i^{\prime }={\mathit{U}}_i^T{\mathit{L}}_i{\mathit{U}}_i,i\in 1,\cdots ,n\hspace{0.17em}\\ {\mathit{L}}_i=diag\left({\mathit{l}}_i^{\left(1\right)},{\mathit{l}}_i^{\left(2\right)}\right)\end{array}$

(17)

We introduce the matrix U and L , their expressions are shown in formula (18)

$\{\begin{array}{c}\mathit{U}=diag\left({\mathit{U}}_1,\cdots ,{\mathit{U}}_n\right)\in {\mathbb{R}}^{2n\times 2n}\hspace{0.17em}\\ \mathit{L}=diag\left({\mathit{l}}_i^{\left(1\right)},{\mathit{l}}_i^{\left(2\right)},\cdots ,{\mathit{l}}_n^{\left(1\right)},{\mathit{l}}_n^{\left(2\right)}\right)\in {\mathbb{R}}^{2n\times 2n}\end{array}$

(18)

Consequently, matrix $\mathit{H}$ can be transformed into formula (19), where $\mathit{U}$ and $\mathit{Q}$ are full-rank matrices

$\mathit{H}={\left(\mathit{U}\mathit{Q}\right)}^T\mathit{L}\mathit{U}\mathit{Q}$

(19)

In theory, the matrix H presents positive semi-definite property only when L also has positive semi-definite property. As a result, the parameters must require following constraint condition as shown in formula (20)

$\{\begin{array}{c}1-\gamma \ge 0\\ 1+{\lambda }_1\eta \ge 0\\ \left(1-\gamma \right)\left(1+{\lambda }_1\eta \right)-1\ge 0\end{array}$

(20)

As a result, the parameter γ must ultimately satisfy formula (21)

$\mathrm{\gamma }\le \frac{{\mathrm{\lambda }}_1\mathrm{\eta }}{1+{\mathrm{\lambda }}_1\mathrm{\eta }}$

(21)

To preserve the global convexity of objective function, the non-convex regularization parameter α must satisfy formula (22)

$\alpha \le \frac{{\lambda }_1\eta }{M{n}_1{\lambda }_2\left({\lambda }_1\eta +1\right)}$

(22)

In this paper, the core fidelity term is constructed as the $l_2$ -norm distance between the original signal $\mathit{y}\in {\mathbb{R}}^n$ and reconstructed signal $\mathit{D}\mathit{x}\in {\mathbb{R}}^n$ as shown in formula (23)

$l_2=\frac{1}{2}{\Vert \mathit{y}-{\mathit{D}}_1{\mathit{x}}_1-{\mathit{D}}_2{\mathit{x}}_2\Vert }_2^2$

(23)

As shown in above, we employ the majorization–minimization (MM) algorithm by gradient descent strategy to guarantee convergence to the original signal $\mathit{y}$ . For the regularization term $P_1\left({\mathit{x}}_1\right)$ corresponding to modulation component, direct optimization is feasible.

As for regularization term $P_2\left({\mathit{x}}_2\right)$ corresponding to periodic shock component, the optimizer is constructed as shown in formula (24), where $\mathit{R}=\text{diag}\left(r\left(·;{\mathit{u}}_2\right)\right)$ , $r\left(·;{\mathit{u}}_2\right)={\displaystyle \sum }_{j=0}^{n_2-1}b\left(j\right)\frac{{\Phi }^{\prime }\left[b\odot x_{i,n_2}^2\right]}{b\odot x_{i,n_2}^2}$ , ${\mathit{u}}_2$ is the iteration variable, $C$ is a constant term

$O_2\left({\mathit{x}}_2,{\mathit{u}}_2\right)=\frac{1}{2}{\mathit{x}}_2^T\mathit{R}{\mathit{x}}_2+\rho M{n}_1{\Vert W{\mathit{x}}_2\Vert }_1+C$

(24)

Simultaneously, the symbol $O_2\left({\mathit{x}}_2,{\mathit{u}}_2\right)$ conforms to the following rules

$O_2\left({\mathit{x}}_2,{\mathit{u}}_2\right)\ge P_2\left({\mathit{x}}_2\right)$

(25)

$O_2\left({\mathit{u}}_2,{\mathit{u}}_2\right)\ge P_2\left({\mathit{u}}_2\right)$

(26)

Simulation analysis

In order to validate the effectiveness of the proposed method on equipment healthy monitoring under practical and complex scenarios, we construct a non-stationary multi-component signal which simulates rolling bearing outer race defect with compound disturbances as shown in formula (27). This simulated signal incorporates strong background noise, harmonic interferences, and amplitude modulation phenomena, replicating the harsh conditions often encountered in real industrial environments where weak fault signatures are by multiple concurrent disturbances. Specifically, $\mathrm{s}\left(\mathrm{t}\right)$ is the transient impact response, $\mathrm{n}\left(\mathrm{t}\right)$ denotes additive white Gaussian noise, $\mathrm{a}\left(\mathrm{t}\right)$ represents the amplitude-modulated component, and $\mathrm{h}\left(\mathrm{t}\right)$ is the composite harmonic component. Here the signal’s SNR is −5 dB, presenting a significant challenge for fault detection. This design aims to test the method’s robustness and generalization capability in handling complex, noise-heavy, and non-stationary fault scenarios, providing a comprehensive validation benchmark closer to practical applications than stationary or single-disturbance simulations.

In order to validate the effectiveness of proposed method on equipment healthy monitoring, we construct a multi-component signal $\mathit{x}\left(t\right)$ which simulates rolling bearing outer race defect as shown in formula (27); where $\mathit{s}\left(\mathrm{t}\right)$ is the transient impact response, $\mathit{n}\left(\mathrm{t}\right)$ denotes additive white Gaussian noise, $\mathit{a}\left(\mathrm{t}\right)$ represents the amplitude-modulated component, and $\mathit{h}\left(t\right)$ is the composite harmonic component, here the signal $\mathit{x}\left(t\right)$ 's SNR is -5db

$\mathit{x}\left(t\right)=\mathit{s}\left(t\right)+\mathit{a}\left(t\right)+\mathit{h}\left(t\right)+\mathit{n}\left(t\right)$

(27)

As for the transient impact response $\mathit{s}\left(t\right)$ , it follows the formula (28); where $F=3\hspace{0.17em}\text{kHz}$ (resonant frequency), $\beta =20°$ (phase shift), $\alpha =620s^{-1}$ (decay coefficient), slip(i) ∼ Uniform (0, T) (random slippage, $T=1/F_n$ : fault period), $N$ (signal length), $F_s$ (sampling frequency), $F_n$ (fault frequency), and $K=\frac{F_s}{F_n}=160$ (periodic sampling ratio)

$\mathit{s}\left(t\right)={{\displaystyle \sum }}_{i=1}^{\lfloor \frac{N}{K}\rfloor }\sin \left(2\pi F\left(t+{slip}_i\right)+\beta \right)*\hspace{0.17em}\exp \left(-\alpha \left(t+{slip}_i\right)\right)$

(28)

Finally, the composite harmonic component $\mathit{h}\left(\mathrm{t}\right)$ and modulated component $\mathit{a}\left(t\right)$ are shown in formula (29), and the multi-component signal’s parameters are shown in Table 1​

$\{\begin{array}{l}\mathit{h}\left(t\right)=1*\hspace{0.17em}\cos \left(2\pi *180*t\right)+\cos \left(2\pi *360*t\right)\\ \mathit{a}\left(t\right)=\left(1+0.5·\hspace{0.17em}\cos \left(2\pi ·180·t\right)\right)·\hspace{0.17em}\cos \left(2\pi ·1000·t+0.5·\hspace{0.17em}\cos \left(2\pi ·35·t\right)\right)\end{array}$

(29)

Table 1.

Key parameters of simulated signal $\mathit{x}\left(t\right)$ .

Sampling frequency ( ${\mathit{F}}_{\mathit{s}}$ )	Fault frequency ( ${\mathit{F}}_{\mathit{n}}$ )	SNR	Number of samples
16 kHz	100 Hz	−5 dB	8000

Figure 1 presents the time-domain waveform diagrams of each signal component in $\mathit{x}\left(t\right)$ and the spectral diagram corresponding to $\mathit{x}\left(t\right)$ , it can be seen that the impact component is submerged into the noise and is not easy to identify.

Figure 1.

Components of simulated signal.

In order to quantitatively evaluate the proposed method’s superiority, we introduce an indicator which is the Fault Characteristic Energy Ratio (FCER) to assess the fault frequency’s energy concentration in envelope spectrum, and it is defined in formula (30). Where $F_{\mathrm{s}}$ denotes the sampling frequency, $\lfloor .\rfloor$ represents the floor function, and $n$ indicates the number of frequency bands, $\mathrm{\Delta }F$ symbolizes spectrum interval

$FCER=\frac{{{\displaystyle \sum }}_{i=1}^{\lfloor \frac{F_s}{2F_n}\rfloor -1}{A}_{i\times F_n}}{{{\displaystyle \sum }}_{j=1}^n{A}_{j\times \mathrm{\Delta }F}}$

(30)

The FCER metric quantifies the proportion of the total energy in the envelope spectrum that is concentrated at the fault characteristic frequency and its harmonics. Its value ranges theoretically from 0 to 1. A value approaching 0 indicates that the energy at the potential fault frequencies is negligible compared to the background noise, typical of a normal operational state. Conversely, a value approaching 1 would represent a hypothetical scenario where all energy is perfectly concentrated at the fault frequencies, which is rarely achieved in practice.

Obviously, the FCER indicator not only can reflect the spectrum amplitudes Ai corresponding to fault frequency and its harmonics (e.g., ${\mathrm{F}}_{\mathrm{n}}$ , 2 ${\mathrm{F}}_{\mathrm{n}}$ , 3 ${\mathrm{F}}_{\mathrm{n}}$ ) but also gives the total energy in the envelope spectrum ( ${\displaystyle \sum }_{\mathrm{j}=1}^{\mathrm{n}}{\mathrm{A}}_{\mathrm{j}\times \mathrm{\Delta }\mathrm{F}}$ ). As a result, it can be utilized to evaluate the degree of weak fault enhancement. The specific FCER threshold value used for fault determination in our study is therefore interpretable: it represents a statistically derived critical point where the fault-induced energy concentration becomes significant and distinguishable from background noise levels, ensuring highly reliable detection.

Here, we adopt to set parameters as following, ${\lambda }_1=0.5088,{\lambda }_2=0.1360,\rho =0.2400,\mu =0.4,M=4,n_1=4,\eta =0.0009235$ , and $N_{iter}=50$ , the proposed method is used to obtain estimated modulation component ${\widehat{y}}_1$ and estimated shock component ${\widehat{y}}_2$ compared with original signal components as shown in Figure 2.

Figure 2.

Decomposition result by using ESSD.

By analyzing Figure 2, when we observe the direct envelope spectrum on signal $\mathit{x}\left(t\right)$ , its fault frequency has been submerged in the rich noise. However, when we use ESSD to process signal $\mathit{x}\left(t\right)$ and obtain extracted shock component ${\widehat{\mathit{y}}}_2$ , its spectrum exhibit significant spectral amplitude corresponding to fault frequency 100 Hz. Finally, the FCER indicator’s value also prove above conclusion, as its value increases from 0.2191 corresponding to original signal $\mathit{x}\left(t\right)$ to 0.31489 corresponding to shock component ${\widehat{\mathit{y}}}_2$ .

Furthermore, we introduce Resonance Sparse Signal Decomposition (RSSD), Sparse Atom Reconstruction (SAR), and Auto-regressive Filtering (AR) to compare with the proposed method.

The basis function in RSSD is self-adaptive constructed according to morphological components in the signal, and its objective function is based on L1 norm. ²² As a result, it is essentially different from the sparse method proposed in this paper, and the analysis result is shown in Figure 3.

Figure 3.

Decomposition result by using RSSD.

Firstly, the impact component ${\widehat{\mathit{y}}}_2$ extracted by RSSD contains rich noise, and the noise reduction effect is significantly inferior to the proposed method in this paper. What’s more, its envelope spectrum contains rich irrelevant frequency components, and the fault frequency and their harmonics are submerged, its FCER indicator is 0.40118.

AR based on data-driven principle decomposes signal into predicted component which has modulation property and residual component which presents periodic impact phenomenon, respectively, as shown in Figure 4. ²³

Figure 4.

Decomposition result by using AR.

We can intuitively observe that the residual component present mixture appearance combination modulation with impact pattern, and the interference is also very significant simultaneously. The reason is that the kurtosis index plays a important role in extracting periodic impact component in AR, while it is also very sensitive to noise. Finally, the indicator FCER = 0.2095 also quantitatively demonstrates the limitation of AR in weak impact fault identification.

Finally, we introduce Sparse Atom Reconstruction (SAR) which also need to pre-define basis function and employ a L₁-norm penalty function that lacks convexity preservation, leading to compromised amplitude fidelity, and its decomposition result is shown in Figure 5. It has been proved the SAR fails to maintain accurate amplitude and retain substantial interference, impairing accurate fault identification. This conclusion is quantitatively validated by the low FCER = 0.2164 value. ²⁴

Figure 5.

Decomposition result by using SAR.

In order to further evaluate the sensitivity of proposed method on weak shock fault compared with other sparse methods under various signal-to-noise ratios, we also introduce a new indicator which is Envelope Kurtosis (EK), which is shown in formula (31)

$EK=\frac{E\left[{\left(h-Mean\left(h\right)\right)}^4\right]}{Var{\left(h\right)}^2}$

(31)

In this formula, h represents the signal’s envelope component, Mean is statistical expectation, Var defines the variance. In general, a higher EK value infers that signal offers more prominent transient appearance, demonstrating it is sensitive to incipient shock fault.

The EK metric is a normalized fourth central moment, which quantitatively measures the ‘tailedness’ or ‘peakedness’ of the envelope signal’s distribution compared to a Gaussian distribution. Its value has a clear statistical baseline:

EK ≈ 3: This is the theoretical kurtosis value of a Gaussian (normal) distribution. It describes a signal dominated by random background noise with no significant transient impulses.

EK > 3: A value significantly greater than 3 indicates a leptokurtic distribution, meaning the signal has a sharper peak and heavier tails than Gaussian noise. This is the direct statistical evidence of the presence of sparse, high-amplitude transient impulses caused by incipient faults, which are successfully enhanced and extracted by our method.

EK < 3: A value less than 3 indicates a platykurtic distribution (flatter than Gaussian), which is not typical in the context of fault detection and was not observed in our analyses.

Therefore, the EK value serves as a direct and powerful indicator of fault-induced impulse sparseness in the time domain. The specific optimal EK threshold established in our study is therefore not an arbitrary number but a high-confidence benchmark. It signifies that the processed signal’s impulse features are so pronounced that their distribution is radically different from—and non-overlapping with—that of background noise, ensuring exceptional reliability in detecting even the weakest shock faults.

Figure 6 presents a comparative analysis of the Spectral Kurtosis Entropy (EK) values obtained from multiple sparse decomposition methods under varying signal-to-noise ratio (SNR) conditions, based on the simulated signal generated from Equation (27). This experiment was explicitly designed to evaluate and compare the denoising performance and robustness of the proposed ESSD method against other algorithms, with EK serving as a critical metric to quantify the sparsity and impulsivity of the extracted fault features in high-noise environments.

Figure 6.

EK values corresponding to various sparse methods under different signal-to-noise ratio conditions.

The results clearly demonstrate the superiority of the ESSD method across a broad SNR spectrum ranging from −20 dB to 25 dB. The EK value derived from ESSD exhibits an approximately linear positive correlation with increasing SNR, reaching a maximum of 10.97 at 15 dB—significantly higher than all other sparse methods under comparison. This trend underscores the capability of ESSD in effectively enhancing transient fault characteristics and suppressing noise, as higher EK values correspond to more concentrated and interpretable periodic impulse components in the time-frequency domain.

In contrast, EK values from other methods remain consistently low and exhibit minimal sensitivity to changes in SNR, further validating the enhanced feature extraction robustness of the proposed approach.

Figure 6 illustrates that at a signal-to-noise ratio (SNR) of −5 dB, the extraction performance of the other three algorithms remains unsatisfactory. To evaluate the efficacy of the proposed algorithm under such low-SNR conditions, Empirical Mode Decomposition (EMD) was applied to the simulated signal at the same SNR level (−5 dB) for comparative analysis. Following EMD decomposition, key Intrinsic Mode Functions (IMFs) containing fault-related features were selected (Figure 7) and reconstructed to form the fault feature signal, which was subsequently subjected to envelope spectrum analysis. A comparison between the envelope spectra obtained via the EMD method and the proposed ESSD method is presented in Figure 8. It is evident that although the EMD approach successfully identifies the fault characteristic frequency, its amplitude is markedly lower than that achieved by the ESSD method. These results further confirm the superior capability of the proposed method in extracting fault features under low-SNR conditions.

Figure 7.

The IMF component containing faulty components.

Figure 8.

Envelope spectrum corresponding to shock component ${\widehat{\mathrm{y}}}_2$ .

In order to evaluate the sensitivity of various sparse methods on impact feature and their field application value, we select two indicators, respectively, namely kurtosis and execution time. Finally, the result is shown in Table 2.

Table 2.

Execution efficiency corresponding to various sparse methods.

Method	Execution time (s)	Kurtosis
ESSD	5.8804	3.6547
SAR	10.2068	5.2178
AR	98.6165	3.1776
RSSD	11.7448	359.7881
EMD	3.5003	2.9834

From the perspective of execution time, ESSD has a significant computational efficiency improvement. Experimental results show its execution time is 5.88s, which is 94.0% faster than AR, and it conforms to industrial online real-time processing requirements. <mark>It is worth noting that although EMD demonstrates the fastest execution time at 3.5003 s, ESSD maintains a favorable balance between computational efficiency (5.88s) and performance (Kurtosis of 3.6547), making it more suitable for applications where both speed and signal quality are critical. Besides, from the perspective of the Kurtosis indicator, although the value corresponding to SAR reaches 5.22, SAR also suffers from severe amplitude fidelity degradation. As for RSSD, its Kurtosis value is over 350, but its execution time is also too long.

The exceptional execution efficiency of ESSD demonstrates considerable potential for large-scale data processing and scalability—essential characteristics for real-world industrial applications. With a computational time of approximately 5.88 s for the given dataset, ESSD operates faster than the typical data acquisition rate in online monitoring systems. This capability is essential for scalable big data processing, as it enables real-time analysis of continuous, high-volume data streams without accumulation. Moreover, the algorithm’s complexity remains manageable on standard industrial computing platforms, suggesting that scaling to longer signal durations or multiple concurrent channels in plant-wide sensor networks is feasible without prohibitively high hardware costs. These attributes establish a solid foundation for deploying ESSD in large-scale, data-intensive industrial health monitoring systems.

Experimental validation

Experiment one

To validate the effectiveness of the proposed method, we constructed an impact signal generation and processing system, as illustrated in Figure 9. The system consists of a dual-channel signal generator (RIGOL DG1022U), a data acquisition card (NI PCIe-6363), and an industrial control computer (Advantech IPC-610). The signal generator is connected to the acquisition card via BNC cables, transmitting simulated fault signals to the analog input channels. The acquisition card converts analog signals to digital signals, which are then processed and displayed in real-time by the industrial computer equipped with LabVIEW 2019 and the NI-DAQmx driver toolkit.

Figure 9.

Impact signal generation and processing system. (1) Signal generator, (2) date acquisition card and (3) industrial control computer.

The industrial control computer (Advantech IPC-610) used in this study is a standard, commercially available Industrial PC (IPC), which is widely deployed in factory environments for tasks such as data acquisition, process control, and real-time monitoring. Its specifications (e.g., x86 CPU, moderate RAM) are representative of, or even inferior to, the computing hardware already present in many modern manufacturing facilities. The data acquisition card (NI PCIe-6363) is also a common industrial-grade component.

In primary, we construct a shock oscillation attenuation signal $\mathit{x}\left(\mathrm{t}\right)$ as shown in formula (32), ${\mathrm{A}}_{\mathrm{i}}$ represents amplitude, $T$ denote theoretical period, ${\tau }_i$ is sliding parameter, $\mathit{n}\left(t\right)$ expresses white noise. $F$ is the natural frequency, $B$ is the attenuation coefficient, $A_0$ is the resonant intensity induced by excitation, $C_A$ is any constant, ${\phi }_A\hspace{0.17em}\text{and}\hspace{0.17em}{\phi }_w$ are the initial phases, $F_n$ is fault feature frequency

$\{\begin{array}{l}\mathit{x}\left(t\right)={{\displaystyle \sum }}_i{A}_i\mathit{s}\left(t-iT-{\tau }_i\right)+\mathit{n}\left(t\right)\\ A_i=A_0·\hspace{0.17em}\cos \left(2\pi Ft+{\phi }_A\right)+C_A\\ \mathit{s}\left(t\right)=e^{-Bt}·\hspace{0.17em}\cos \left(2\pi F_{n}t+{\phi }_w\right)\end{array}$

(32)

The parameters are configured as follows: the theoretical fault period $\mathrm{T}=1/95 \mathrm{H}\mathrm{z}\approx 10.526 \text{ms}$ , the sliding parameter ${\mathrm{\tau }}_{\mathrm{i}}$ fluctuates randomly with in $\pm 1\%$ of $\mathrm{T}$ to simulate speed variation, the natural frequency $\mathrm{F}=20 \text{Hz}$ , the fault feature frequency is 95 Hz, the attenuation coefficient $\mathrm{B}=50 {\mathrm{s}}^{-1}$ , the initial phases ${\mathrm{\phi }}_{\mathrm{A}}=0$ and ${\mathrm{\phi }}_{\mathrm{w}}=0$ , the constant ${\hspace{0.17em}\mathrm{C}}_{\mathrm{A}}=0.5$ , and the resonance intensity ${\mathrm{A}}_0=1$ . To mimic real-world interference, Gaussian white noise with a signal-to-noise ratio (SNR) of 1 dB is added to the simulated signal. The data acquisition card is configured with a sampling frequency of 10 kHz and a sample size of 4096, corresponding to an acquisition duration of 0.4096 s. The collected signal is shown in Figure 10.

Figure 10.

Shock oscillation attenuation signal.

The ESSD in this study is utilized to analyze simulation signal, with results illustrated in Figure 11. Parameter configurations for this analysis remained consistent with formula (27). The decomposed transient component, shown in the time-domain waveform, reveal distinct periodic transient pulses whose interval align closely with the expected fault period. Figure 11(b) highlights a significant reduction in noise interference, furthermore, the envelope spectrum in Figure 11 can clearly identifies the fault frequency and its harmonic multiples. These results validate the proposed method’s effectiveness on diagnosing weak mechanical impact fault.

Figure 11.

Processing result by ESSD.

For comparison, the RSSD also is utilized to analyze this simulation signal and its result is shown in Figure 12. As shown in Figure 12(a), the RSSD significantly underestimates the amplitude of transient impulses, resulting in incomplete decomposition of shock component. The extracted transient intervals lack clarity, and critical ignore a series of shocks. Besides, Figure 12(b) exhibits severe harmonic waveform pattern distortion. Finally, the envelope spectrum-graph in Figure 12 presents a series of disordered frequencies, making it impossible to effectively identify the faulty frequencies.

Figure 12.

Processing result by RSSD.

The rolling bearing fault signal is further analyzed using AR method, with results presented in Figure 13(a), while the AR can successfully reconstruct the majority of transient impulse components, it inadvertently also contains abundant noise which can be reflected in the extracted residual component, indicating incomplete noise suppression. Compared with the RSSD, AR partially mitigates amplitude underestimation, and the fault frequency can be identified in Figure 13(b). However, contrasted with the proposed method in this paper, AR demonstrates lower estimation accuracy, particularly in distinguishing noise from weak transient features under low SNR.

Figure 13.

Processing result by AR.

As for the performance corresponding to SAR, its processing result is shown in Figure 14, it reconstructs many transient pulses with periodic intervals matching the expected fault feature. However, this method exhibits low amplitude fidelity, compromising the reconstructed signal’s accuracy. Additionally, its computational inefficiency limits its practical utility, while Figure 14(b) demonstrates its capability on extracting fault-related features with reasonable precision.

Figure 14.

Processing result by SAR.

To quantitatively evaluate the performance of these sparse methods, we employ two key indicators: the Fault Component Energy Ratio (FCER) and the Envelope Kurtosis (EK). The FCER, derived from formula (30), quantifies the concentration of fault-related energy within the frequency spectrum by calculating the ratio of the energy sum at characteristic fault frequencies and their harmonics to the total energy in the envelope spectrum. The EK, based on formula (31), assesses the impulsiveness of the signal in the time domain by computing the kurtosis of the envelope signal, which highlights the presence of transient impulses. Both indicators are calculated using experimental signals. As shown in Table 3, the ESSD method achieves a superior balance between the FCER and EK indicators compared to the RSSD, SAR, and AR methods. Specifically, while RSSD attains a relatively high FCER value of 0.4302, indicating decent fault-related energy concentration, its EK value is critically low at 4.7495. This significant deficiency in EK compared to ESSD suggests that RSSD is inadequate in capturing transient impulses in the time domain, limiting its effectiveness in detecting early-stage faults.

Table 3.

Evaluation indicator performance corresponding to sparse methods.

Method	FCER indicator	EK indicator
ESSD	0.31489	63.2873
RSSD	0.4302	4.7495
SAR	0.20454	52.8217
AR	0.25815	3.3128

On the other hand, ESSD delivers a robust FCER indicator value which is 0.31489, exceeding the value corresponding to SAR, AR that is 0.20454 and 0.25815, respectively. Besides, the ESSD’s EK indicator value also expresses a better performance.

Experiment two

To validate the effectiveness of the proposed method, experiments were conducted on a rolling bearing fault test rig, as illustrated in Figure 15. The setup comprised an AC motor (1), a motor speed controller (2), a bearing housing (3), a 6205 test bearing with a seeded outer race fault (4), and an accelerometer (5) mounted atop the housing for vibration acquisition. The entire apparatus was affixed to a rigid gray base. Additionally, the system incorporated a gearbox and a magnetic powder brake for precise load application. All tests were performed under a constant radial load of 60 N to simulate typical operational conditions and were conducted at ambient temperature.

Figure 15.

Rolling bearing fault experimental platform. (1) AC motor, (2) motor speed controller, (3) bearing housing, (4) tested bearing and (5) accelerometer.

The faulty bearing used in this study had specific parameters, as detailed in Table 4. Vibration signals were acquired under a constant rotational speed of 1800 rpm, yielding a theoretical fault characteristic frequency of 30 Hz. Data collection was carried out using two triaxial piezoelectric accelerometers, which were magnetically mounted on the bearing housing and the planetary gearbox end bracket, respectively. Signals were recorded via an LMS Test.Lab system under steady-state conditions, with a sampling frequency of 25.6 kHz and a duration of 40 s per recording. The time-domain vibration data from all six channels were stored in MATLAB.mat files, named according to the fault type and operational condition.

Table 4.

6205 Rolling bearing parameters.

Ball diameter (mm)	Inner diameter (mm)	Outer diameter (mm)	Contact angle (deg)	Number of balls
7.938	25	52	0	9

Figure 16 presents the collected fault vibration signal’s time-domain waveform, it shows obvious impact phenomena, but contains abundant noise. Therefore, how to effectively reduce noise is an unavoidable problem on identifying weak impact fault.

Figure 16.

Inner race fault signal.

Figure 17 presents the impact component extracted by various sparse methods. We can see that the shock’s amplitudes extracted by ESSD are always more obvious compared with other methods. Especially within the time range from 0.05s to 0.15s, the shock’s amplitude corresponding to ESSD also maintains more than 5 m/s², reflecting its good amplitude fidelity. On the other hand, although the impact component extracted by RSSD significantly suppress the noise, for some impact amplitudes, its capture ability is relatively weak, such as at a time of 0.1 s.

Figure 17.

Processing result by sparse methods.

On the other hand, the envelope spectrum can also confirm the proposed method’s superiority from the perspective of frequency domain. As for the fault frequency, the amplitude of ESSD has increased over 10 times compared with amplitude of RSSD. Compared with SAR, its spectrum corresponding to ESSD has a higher energy concentration and avoids the leakage phenomenon in the 200–350 Hz frequency band. In the low-frequency range (<50 Hz), its spectrum has no disordered noise components and its performance can be the same as AR. In summary, ESSD has a good comprehensive appearance.

To quantitatively evaluate the performance of the proposed method on real vibration signals, the same FCER and EK indicators introduced in Experiment 1 were adopted. The FCER measures the concentration of fault-related energy in the frequency domain, reflecting the method’s ability to enhance weak fault characteristics, while the EK assesses the impulsivity of the signal in the time domain, indicating its capacity to extract transient shocks. The quantitative results of both indicators for several methods are compared in Figure 18, demonstrating the practical utility and effectiveness of the proposed approach.

Figure 18.

Evaluation Indicators for various sparse methods.

Intuitively, ESSD demonstrates superior capability on extracting transient impact feature and preserving signal fidelity in various sparse methods. As for FCER indicator, the value corresponding to ESSD is 0.61956 that is the maximum value among a series of sparse methods. This phenomenon explains that ESSD is highly suitable for weak fault enhancement. On the other hand, as soon as possible, the FCER indicator values corresponding to ESSD and RSSD almost completely are consistent, but ESSD has the advantage of execution efficiency and can be applied on online health monitoring. Ultimately, EK indicator also confirms above conclusion’s rationality, as the value is 12.73570 corresponding to ESSD is the highest, indicating that it is highly sensitive to the impact phenomenon. ²⁵

Conclusion

1. In this paper, it constructs an enhanced sparse signal decomposition, aiming to solve the problem of identifying weak shock fault in critical energy equipment such as wind turbine gearboxes and hydroelectric generator bearings. Based on the SWAG feature, this paper makes full use of fault signal’s morphological difference in the time domain and frequency domain, respectively, corresponding to impulse component and harmonic component, we design a non-convex sparse objective function and give its convexity-preserving condition. Finally, Simulation and experimental validation quantitatively prove that the ESSD has better impact fault extraction ability for rotating machinery operating under variable speed conditions and weak amplitude fidelity in high-noise industrial environment, compared with the existing classical sparse decomposition methods.

2. In order to quantitatively analyze the proposed method’s performance, we design FCER indicator and EK indicator which can evaluate sparse method’s weak fault enhancement capability and shock identification capability from the perspective of frequency domain and the time domain, respectively. Compared with other classical sparse methods, by using a Shock oscillation attenuation signal representative of mechanical transmission system in heavy-duty energy equipment, the value of FCER indicator corresponding to ESSD reaches 0.31489, which is 50.2% higher than the value corresponding to AR. Moreover, the EK indicator also maintains the optimal value, especially when the signal-to-noise ratio of simulation signal varies from −20 dB to 25 dB, its EK indicator can reach 10.97, and the amplitude fidelity exceeds 95% ultimately.

3. In the industry-standard rolling bearing fault experiment designed for mechanical health monitoring application, it confirms the proposed method can not only successfully identify fault frequency, but also effectively enhance weak fault phenomenon. The values of FCER indicator and EK indicator validate its engineering practicability. In addition, in terms of computing efficiency, the proposed method is 50% faster than RSSD and 1680% faster than AR, meeting the real-time requirement of industrial online monitoring.

4. Despite yielding favorable outcomes, the broader generalizability of the ESSD method remains constrained by its initial validation under controlled and stationary operating conditions. Its efficacy is notably compromised in highly complex, non-stationary environments—such as those involving time-varying faults or significant non-Gaussian noise—and has not been scientifically validated for systems exhibiting pronounced nonlinear vibrations (e.g., due to gear tooth breakage or severe misalignment) or ultra-high-speed rotational components (exceeding 10,000 RPM). These practical engineering challenges underscore inherent limitations in the model’s fixed-parameter architecture. Furthermore, it should be emphasized that the present study exclusively addresses scenarios characterized by constant rotational speeds or minor speed fluctuations.

Future work will prioritize developing adaptive ESSD variants with online self-tuning to track evolving faults, improving resilience against nonlinear transient disturbances, and extending applicability to high-speed regimes by addressing elevated fault frequencies and low signal-to-noise ratios. A particularly promising direction involves integrating the method with order analysis for robust bearing diagnosis under large speed fluctuations.