Fault Diagnosis of Rolling Bearings Based on Acoustic Signals in Strong Noise Environments

Abstract

Compared to vibration sensors, microphones offer several advantages, including non-contact detection, high sensitivity, low cost, and ease of installation. To address the challenges posed by the complex components and significant interference in rolling bearing sound signals, we proposed a fault diagnosis method for rolling bearing acoustic signals based on Secretary Bird Optimization Algorithm (SBOA)-optimized Feature Mode Decomposition (FMD). Initially, a microphone is utilized to collect sound data while the bearing operates, followed by the application of S-FMD (Secretary Bird Optimization Algorithm-optimized Feature Mode Decomposition) to decompose the sound signal and extract components that may contain fault information related to the bearing. The SBOA is employed to adaptively optimize four influencing parameters of FMD: mode number n, filter length L, frequency band cutting number K, and cycle period m. By minimizing envelope entropy as the objective function, we achieve FMD of the bearing sound signal with the assistance of the SBOA. Additionally, this paper introduces an Integrated Signal Evaluation Index (ISEI) to extract potential bearing failure characteristics from the filtered components. Simulation experiments and test results indicate that, compared to Empirical Mode Decomposition, Complementary Ensemble Empirical Mode Decomposition, fixed-parameter FMD, and adaptive variational mode decomposition methods, the proposed approach more effectively extracts weak characteristic information related to early faults in bearing sound signals.

1. Introduction

Mechanical failure poses a significant threat to the safe and reliable operation of large mechanical equipment, including wind power generation systems, aerospace engines, and high-end Computer Numerical Control (CNC) machine tools [1]. Rolling bearings are critical components for the effective functioning of rotating machinery [2], and their condition is crucial for the overall performance of the entire mechanical system. In the realm of fault diagnosis, signal decomposition technology is regarded as one of the most established and technically comprehensive methods. This technique can effectively suppress environmental noise and extraneous components in mechanical vibration signals, thereby facilitating the extraction of fault features [3,4]. In contrast to vibration signals, sound signals offer advantages such as non-contact monitoring and convenient remote assessment, making them valuable in specialized environments [5]. By performing time–frequency domain analysis on bearing sound signals, various bearing fault types can be identified. The diagnostic features derived from acoustic signals demonstrate greater accuracy than those obtained from vibration signals, allowing for a more effective differentiation between healthy and faulty conditions [6].

In practical measurements, the sound field is inherently complex and is frequently contaminated by noise signals produced by various sources, including reverberation, machinery, motors, rotating shafts, and human activity. This leads to a notably low signal-to-noise ratio for the acoustic fault signal [7,8]. Consequently, traditional signal processing methods often struggle to effectively separate and extract acoustic fault signals. As a result, the acoustic signal diagnosis methodology has been progressing at a slower pace compared to the vibration signal analysis techniques [9].

In recent years, the empirical mode decomposition (EMD) method proposed by Huang et al. [10] has emerged as an adaptive signal processing technique for decomposing non-stationary signals into multiple intrinsic mode functions (IMFs). This method has been successfully applied to fault feature extraction from bearing vibration signals. However, EMD is associated with several challenges, including modal aliasing, under-envelopment, over-envelopment, and boundary effects. To address these issues, ensemble empirical mode decomposition (EEMD) [11] has been proposed, which offers improved accuracy and efficiency compared to traditional EMD. Nonetheless, the integration process of EEMD can be time-consuming. To mitigate this limitation, the Complementary EEMD (CEEMD) [12] method has been introduced, which enhances decomposition efficiency by adding Gaussian white noise with opposite signs to the signal. CEEMD effectively alleviates modal aliasing and reconstruction errors. Nevertheless, not all IMFs contain useful information, making the selection of the most informative IMF a critical step. For instance, Sahu and Rai employed CEEMD to decompose bearing vibration signals into multiple IMFs and determined the grouping and selection of valuable modes based on correlation coefficients [13,14].

Drag et al. [15] proposed variational mode decomposition (VMD) based on empirical mode decomposition (EMD), which is a signal processing method with variable scaling. Unlike traditional recursive algorithms, VMD exhibits strong robustness and is supported by rigorous mathematical theory. The mode number K and the penalty factor α significantly influence the results. If K is set too high, it can result in over-decomposition (modal loss); conversely, if K is too low, it may lead to under-decomposition (modal aliasing). Additionally, improper configuration of the penalty factor can result in mode aliasing and loss, which adversely affects the extraction of important features. Wang [16] proposed an adaptive VMD method utilizing a bat algorithm search (BAS) optimization technique for the early fault diagnosis of rolling bearings. This method employs the kurtosis value as a metric, selecting the intrinsic mode function (IMF) with the highest kurtosis for analysis. Wang [17] also introduced a rolling bearing acoustic signal fault diagnosis method based on adaptive variational mode decomposition (AVMD) to enhance accuracy in noisy environments. Furthermore, Li [18] proposed a novel method based on frequency band entropy (FBE) for selecting sensitive intrinsic mode functions (IMFs) that contain variational mode decompositions (VMDs) rich in fault information. Subsequently, envelope power spectrum analysis is conducted on the selected IMFs to extract the fault characteristic frequencies of rolling bearings.

Miao [19] proposed Feature Mode Decomposition (FMD), a novel adaptive decomposition method specifically designed for mechanical fault feature extraction. FMD updates the adaptive finite impulse response (FIR) filter bank using correlation kurtosis (CK), thereby overcoming the limitations of the variational mode decomposition (VMD) and Empirical Wavelet Transform (EWT) algorithms regarding filter shape and bandwidth. This enhancement allows for the extraction of more fault information, making FMD more advantageous than VMD in mechanical signal decomposition analysis, particularly in scenarios where prior knowledge of the failure cycle is lacking. Jia [20] introduced a fault feature extraction method that integrates an improved whale optimization algorithm (WOA) to optimize FMD. This approach refines the FMD parameter settings, employs information entropy as the fitness function, and selects the optimal intrinsic mode function (IMF) through kurtosis for fault feature extraction. Sumika [21] developed a bearing fault detection method based on adaptive feature mode decomposition (Adaptive FMD) and the health index (HI). Yan [22] proposed a fault diagnosis method based on adaptive multivariate eigenmode decomposition (AMFMD). The method addresses the challenge of fault information in mechanical equipment being easily obscured by significant noise, and it overcomes the limitation of potentially missing valuable features when relying solely on data from a single sensor. Yan [23] designed the signal period kurtosis-to-noise ratio (SCKNR) as the objective function for the adaptive parameter search of FMD. He selected the primary mode component based on the characteristic energy rate and computed its square envelope spectrum for fault identification. However, he did not consider the influence of the frequency band cutting number K on the decomposition results.

Li [24] proposed an adaptive feature mode decomposition (AFMD) method for analyzing noisy bearing vibration signals. The method optimizes parameters using a cuckoo search algorithm and employs the feature frequency ratio (FFR) to improve fault diagnosis accuracy and efficiency, particularly in the presence of strong noise levels (−15 dB). Meanwhile, Wen [25] introduced a novel method known as Graph Modeling Singular Values (GMSV), which successfully balances sensitivity to early faults with robustness against noise. This method employs a Gaussian distribution for fault detection. Nevertheless, in practical engineering scenarios, the dynamic characteristics of bearings may exhibit greater complexity, necessitating verification of the feasibility of this assumption.

The effectiveness of FMD is highly sensitive to key parameters: mode number n, filter length L, frequency band division K, and cycle period m [19]. A mode number n that is too small may miss critical fault information, while an overly large n can introduce noise and redundancy. Similarly, a short filter length L may cause inaccurate separation, whereas an excessively long L increases computational demands. A higher K enhances frequency resolution but also raises computational complexity. An insufficient iteration period m may hinder algorithm convergence, while an excessive period can lengthen calculation time.

Currently, most research [5,6] on the use of sound signals for bearing fault diagnosis is conducted in controlled, quiet laboratory environments. While sound-based diagnostic methods offer distinct advantages over vibration signal methods—such as non-contact measurement, flexibility in sensor placement, and lower costs for large-scale monitoring—these methods face challenges in noisy real-world environments. Sound signals are particularly susceptible to interference from environmental noise, exhibit complex spectral characteristics, and require extensive data processing, all of which can limit diagnostic accuracy. Furthermore, many FMD-based noise reduction methods [21,22,23] optimize only two of the four parameters, which may diminish their effectiveness in practical applications. To address this issue, this paper investigates the feasibility of sound-based bearing fault diagnosis in high-noise environments by optimizing all four FMD parameters to enhance the extraction of fault information. The Secretary Bird Optimization Algorithm (SBOA) [26] is employed to adaptively optimize the FMD parameters with the objective of minimizing envelope entropy. Subsequently, the ISEI metric introduced in this study is utilized to select the optimal pattern for extracting fault features. Simulation experiments and tests demonstrate that, compared to the EMD [10], CEEMD [11], fixed-parameter FMD [20], and adaptive VMD [16] methods, this approach is more effective in capturing subtle feature information related to early faults in bearing sound signals, exhibiting improved performance and robustness under noisy conditions.

The structure of this paper is organized as follows: the Section 1 presents a literature review and the relevant research background; the Section 2 discusses the relevant algorithms; the Section 3 elaborates on the proposed fault diagnosis method along with its technical details; the Section 4 details the simulation tests; and the Section 5 focuses on experimental design and result analysis. Finally, the conclusion summarizes the main contributions of this article.

2. Methodologies

2.1. FMD

FMD decomposes a signal into multiple sub-signals by utilizing an FIR filter bank and employs correlation kurtosis (CK) in conjunction with signal impulse and periodicity for deconvolution. The mode is selected based on the correlation coefficient (CC) of the sub-signals, discarding those with lower CK values until the number of retained sub-signals aligns with the expected number of modes. The specific implementation process is detailed as follows [19]:

Step 1: Input the original signal $x\left(t\right)$ and the parameters of FMD initialization: mode number n, filter length L, frequency band cutting number K, and cycle period m.

Step 2: Initialize the FIR filter bank through K Hanning windows, usually set in the range of 5 to 10, and initialize iteration i =1.

Step 3: Obtain the filtered signal (i.e., the decomposed modal component) through $u_k^i=x^{\ast }{f}_k^i$; $u_k^i$ represents the decomposed mode obtained from the k-th filter during the i-th iteration of the process, where i = 1, 2, 3, … represents the iteration index, indicating the current step in the iterative process, k = 1, 2, 3, … indicates the filter index, denoting the k-th filter in the filter bank, and * is the convolution operation.

Step 4: Update the filter coefficients using the original signal $x\left(t\right)$, the decomposed modal component $u_k^i,$ and the estimated fault period $T_k^i$, where the estimated fault period${ T}_k^i$ is determined by the autocorrelation spectrum of $u_k^i$. After the zero point, the moment corresponding to the local maximum value $R_k^i$ is reached and $i=i+1$ is set.

Step 5: Determine whether the current iteration reaches the maximum number of iterations. If it is not reached, return to step 3; otherwise, continue to step 6.

Step 6: Calculate the correlation coefficient between every two modal components, construct a K × K correlation matrix CC (K × K), select the two modal components with the largest correlation coefficient, pass the estimated failure period $T_k^i,$ calculate their correlation kurtosis (CK), and then select the modal component with the larger correlation kurtosis (CK) as the decomposed modal component and set K = K − 1.

Step 7: Determine whether the current number of modes meets the specified threshold, n. If it does, the decomposition process will conclude, and the results will be presented. If it does not, return to step 2 and continue the previous operations. The specific FMD process is illustrated in Figure 1.

2.2. SBOA

The Secretary Bird Optimization Algorithm (SBOA) [26] is a new meta-heuristic optimization algorithm with the advantages of robustness, high efficiency, and fast convergence. It was inspired by the hunting behavior of the secretary bird. The algorithm steps are as follows:

Firstly, initialize the secretary bird’s position using the following formula:

$$X_{i,j}=l{b}_j+r\ast \left(u{b}_j-l{b}_j\right)$$

(1)

where $X_{i,j}$ represents the value of the ith secretary bird in the jth dimension, $l{b}_j$ and $u{b}_j$ are the lower and upper bounds of the jth dimension, respectively, and r is a random number within the range of [0, 1].

The SBOA represents the secretary bird group by the following formula:

$$X={\left[\begin{array}{cccccc}{x}_{1,1}& x_{1,2}& \cdots & x_{1,j}& \cdots & x_{1,Dim}\\ x_{2,1}& x_{2,2}& \cdots & x_{2,j}& \cdots & x_{2,Dim}\\ ⋮& ⋮& \ddots & ⋮& \ddots & ⋮\\ x_{i,1}& x_{i,2}& \cdots & x_{i,j}& \cdots & x_{i,Dim}\\ ⋮& ⋮& \ddots & ⋮& \ddots & ⋮\\ x_{N,1}& x_{N,2}& \cdots & x_{N,j}& \cdots & x_{N,Dim}\end{array}\right]}_{N\times Dim}$$

(2)

where N is the number of secretary birds and $Dim$ is the dimension of the problem variable, which is equal to 4 in this paper. The SBOA is used to optimize the FMD parameters: the number of modes n, the filter length L, the number of frequency band cuts K, and the cycle period m. Each secretary bird represents a parameter combination, and the objective function value is given by the following formula:

$$F={\left[\begin{array}{c}{F}_1\\ ⋮\\ F_{\mathrm{i}}\\ ⋮\\ F_{\mathrm{N}}\end{array}\right]}_{N\times 1}={\left[\begin{array}{c}F\left(X_1\right)\\ ⋮\\ F\left(X_i\right)\\ ⋮\\ F\left(X_N\right)\end{array}\right]}_{N\times 1}$$

(3)

where F is the objective function value vector.

In this paper, the envelope entropy value is used as the objective function. By comparing the objective function values, the secretary bird with the smallest envelope entropy value is selected as the best candidate solution. The position and objective function value of the secretary bird are updated in each iteration, so the best candidate solution needs to be re-determined in each iteration.

The test function described by Formulas (4)–(7) is employed to compare the performance of the Snake Optimization Algorithm (SO) [27], Whale Optimization Algorithm (WOA) [28], and Secretary Bird Optimization Algorithm (SBOA). Figure 2 illustrates the convergence curve of the average efficiency value for the selected function obtained from the SBOA. This algorithm demonstrates a rapid convergence speed, strong stability, and high robustness.

$$\begin{array}{c}{F}_1\left(x\right)={\displaystyle \sum _{i=1}^n} x_i^2\end{array}$$

(4)

$$\begin{array}{c}{ F}_2\left(x\right)={\displaystyle \sum _{i=1}^n}\mid x_i\mid +\underset{x\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}{\displaystyle \prod _{i=1}^n}\mid x_i\mid \end{array}$$

(5)

$$\begin{array}{c}{F}_3\left(x\right)={\displaystyle \sum _{i=1}^n}i{x}_i^2+\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{o}\mathrm{m} (0,1)\end{array}$$

(6)

$$\begin{array}{c}{F}_4\left(x\right)=-20\exp \left(-\left({\displaystyle \frac{1}{5}}\right)\sqrt{{\displaystyle \frac{1}{n{\displaystyle {\sum }_{i=1}^n} x_i^2}}}\right)\end{array}$$

(7)

3. SBOA-Optimized FMD

The SBOA uses the envelope entropy value of the reconstructed signal after FMD as the fitness function to optimize the parameters of FMD. At the same time, this paper proposes the Integrated Signal Evaluation Index (ISEI) as the selection criterion for the best mode. The detailed steps are as follows:

(1)

Initialize the population: The maximum number of SBOA iterations P is set to 70 and the population size Q is set to 6. P candidate solutions are randomly generated, and each candidate solution represents an eigenmode selection scheme. Combined with the influence of FMD parameters on the noise reduction effect in the literature, this paper sets the range of the mode number n as 3 to 20, the filter length L as 10 to 300, the frequency band cutting number K as 5 to 40, and the cycle period m as 5 to 30, where the frequency band cutting number K ≥ the mode number n.

(2)

Fitness calculation: The signal is decomposed according to the FMD parameters corresponding to the candidate solution, the envelope entropy value of the reconstructed signal is calculated, and $F\left(x\right)$ is constructed as the fitness function as the criterion for stopping the SBOA.

(3)

Update location: According to the parameter search and position+n update process outlined in the literature [26], the exploration phase employs a differential evolution strategy to introduce population diversity. This phase combines Brownian motion and Lévy flight strategies [26] to expand the search range, assisting the algorithm in overcoming local optimality. In the utilization phase, the SBOA mimics the secretary bird’s behavior of evading natural predators through two strategies: camouflage and rapid movement. The camouflage strategy is implemented using a dynamic perturbation factor ${\left(1-T_t\right)}^2$, which enables the search agents to disperse more effectively within the solution space, helping to avoid local optima. The rapid movement strategy, on the other hand, adjusts the position and velocity of the search agents to simulate the secretary bird’s swift movement, thus improving the efficiency of the search and accelerating convergence. Specifically, the rapid movement strategy incorporates a dynamic perturbation factor ${\left(1-T_t\right)}^2$, which allows the search agents to move quickly across the solution space, facilitating a faster approach to the global optimal solution. Additionally, the search direction and step size are dynamically adjusted to guide the algorithm towards efficiently approaching the global optimal solution, thereby ensuring accurate determination of the FMD parameters during the optimization process.

(4)

Individuals with better fitness values (FMD combined solutions) are selected according to the criterion of the minimum envelope entropy value, individuals with poor fitness are replaced, and the optimal solution is retained to ensure that the algorithm converges to the global optimum.

(5)

Iteration: The fitness calculation, position update, local search and selection, and replacement steps are repeated until the termination condition is met (the maximum number of iterations P is reached).

(6)

Output the optimal solution: When the algorithm terminates, the optimal eigenmode combination solution and FMD parameters are output.

Mode Selection Criteria: ISEI

As a common indicator to characterize signal impact characteristics, kurtosis is easily interfered with by a small number of large-value abnormal data or random impacts. In this regard, an Integrated Signal Evaluation Index (ISEI) weighted by kurtosis combined with the harmonic-to-noise ratio is proposed as the Secretary Bird Optimization Algorithm-optimized Feature Mode Decomposition (SFMD) criteria for selecting the best mode after the signal.

(1)

Normalization: Kurtosis formula.

$$\begin{array}{c}K={\displaystyle \frac{\frac{1}{N}{\displaystyle {\sum }_{i=1}^N}{\left(x_i-\mu \right)}^4}{{\left(\frac{1}{N}{\displaystyle {\sum }_{i=1}^N}{\left(x_i-\mu \right)}^2\right)}^2}}\end{array}$$

(8)

where N is the length of the bearing signal, $x_i$ is the $i$th data point, and $\mu$ is the mean value of the signal.

The formula for the harmonic noise ratio (HNR) is as follows

$$\begin{array}{c}HNR={\displaystyle \frac{{\displaystyle {\sum }_{i=1}^n}{A}_{i\times FCO}}{n\times 3{\sigma }_{\mathrm{n}\mathrm{o}\mathrm{i}\mathrm{s}\mathrm{e}}}}\end{array}$$

(9)

where $A_{i\times FCO}$ represents the amplitude of the $i$th harmonic frequency component, $FCO$ represents the fundamental frequency (fault characteristic frequency), $n$ is the number of harmonic frequency components, ${\sigma }_{\mathrm{n}\mathrm{o}\mathrm{i}\mathrm{s}\mathrm{e}\mathrm{d}\mathrm{d}\mathrm{d}\mathrm{l}}$ is the standard deviation of the noise. The peak search algorithm is used to look for peaks at the bearing’s fault frequency. A high HNR value indicates that harmonic components are more significant relative to noise.

The normalized formula of kurtosis K is

$$\begin{array}{c}{Z}_K={\displaystyle \frac{K-{\mu }_K}{{\sigma }_K}}\end{array}$$

(10)

where ${\mu }_K$ is the mean value of kurtosis and ${\sigma }_K$ is the standard deviation of kurtosis.

The normalized formula of the harmonic noise ratio RFN is

$$\begin{array}{c}{Z}_{RFN}={\displaystyle \frac{RFN-{\mu }_{HNR}}{{\sigma }_{RFN}}}\end{array}$$

(11)

where ${\mu }_{HFN}$ is the mean value of the harmonic-to-noise ratio and ${\sigma }_{RFN}$ is the standard deviation of the harmonic-to-noise ratio.

(2)

Weighted Integrated Signal Evaluation Index (ISEI).

$$\begin{array}{c}ISEI={\mathsf{\delta }}_1\cdot \left|Z_K\right|+{\mathsf{\delta }}_2\cdot \left|Z_{RFN}\right|\end{array}$$

(12)

where ${\mathsf{\delta }}_1$ and ${\mathsf{\delta }}_2$ are weights. According to the literature [23], the weights are 0.3 and 0.7, respectively.

As a criterion for mode selection, the ISEI takes into account the periodicity and impact resistance of the signal and can more comprehensively reflect the fault characteristics of the bearing signal. The modal component with the largest ISEI index is selected for further analysis. Subsequently, the Hilbert transformation [29] is applied to calculate its envelope spectrum, which is then used to extract bearing fault characteristics. The fault diagnosis flowchart is shown in Figure 3.

After denoising the collected bearing signals, the modal components containing the most significant fault information are extracted. As early bearing fault signals are predominantly distributed in high-frequency bands, traditional Fourier transform methods often struggle to provide accurate analyses. The resonant frequency of the sensor modulates the bearing fault information into a high-frequency band, necessitating the use of the Hilbert transform to demodulate this high-frequency information into a lower frequency range. The specific procedure involves applying the Hilbert transform to the optimal modal component to obtain the analytical signal, from which the envelope curve is extracted. By taking the modulus of this curve, the analytical function is derived. Subsequently, the FFT Fourier transform is applied to the analytical function to generate the envelope spectrum, enabling the clear identification of fault characteristic frequencies and their multiples, thus facilitating the precise diagnosis of bearing faults. Utilizing the bearing parameters and the fault frequency calculation formula, the theoretical bearing fault characteristic frequency can be computed. A comparative analysis of the envelope spectrum against the corresponding fault frequency and its multiples is then conducted. If a peak in the envelope spectrum aligns with the theoretical fault frequency and its multiples, it confirms the presence of a corresponding fault in the bearing.

4. Simulation Verification Analysis

By constructing a simulation signal of a fault in a rolling bearing [20], the feasibility of the SFMD method is verified. The simulation signal expression is:

$$\begin{array}{c}\left\{\begin{array}{l}x\left(t\right)=s\left(t\right)+n\left(t\right)={\displaystyle \sum _i}{A}_{i}h\left(t-iT\right)+n\left(t\right)\\ A_i=1+A_0\mathrm{c}\mathrm{o}\mathrm{s}\left(2\pi f_{r}t\right)\\ h\left(t\right)=\mathrm{e}\mathrm{x}\mathrm{p}\left(-Ct\right)\mathrm{c}\mathrm{o}\mathrm{s}\left(2\pi f_{n}t\right)\end{array}\right.\end{array}$$

(13)

where $s\left(t\right)$ is the periodic impact component;

The amplitude $A_0$ is 0.3;

The rotation frequency $f_r$ is 30 Hz;

The attenuation coefficient $C$ is 700;

The resonance frequency $f_n$ is 4 khz;

The inner ring fault characteristic frequency is $f_i=1/T=120$ Hz;

$n\left(t\right)$ is Gaussian white noise; the signal-to-noise ratio of the noise-containing signal is −13 dB (calculation formula SNR $=20{\mathrm{l}\mathrm{o}\mathrm{g}}_{10}\left(v_s/v_n\right)$; $v_s$ and $v_n$ are the effective values of the impact component and noise component, respectively); the sampling frequency $f_s$ is 16 kHz; and the number of analysis points is 4096 points.

As shown in Figure 4, fault information cannot be identified in the time domain diagram, spectrogram, or envelope spectrum of the simulated signal after adding noise.

Using the proposed method for signal denoising processing, the convergence curve of the SBOA obtained based on the minimum envelope entropy value is shown in Figure 5. The iterations converged after 13 times, and the optimal parameter combination was [250, 5, 5, 10].

FMD optimized by the SBOA is S-FMD, and its decomposed components are F-IMF. VMD optimized by CPO is C-VMD, and its decomposed components are V-IMF. Figure 6 shows the decomposed time domain diagram after optimizing FMD. Figure 7 shows the kurtosis, harmonic noise ratio, and ISEI of each F-IMF.

As shown in Figure 7, F-IMF4 has the largest kurtosis value. As shown in Figure 6, F-IMF4 (Figure 6d) has a large kurtosis because it contains three abnormal shocks, while RFN and ISEI indicate that F-IMF3 contains the most faults. An envelope spectrum analysis is performed on the F-IMF decomposed by each S-FMD. The envelope spectrum of F-IMF3 contains the most fault information.

The envelope spectrum of F-IMF3 in Figure 8 clearly shows the first eight harmonic components of the inner raceway fault frequency $f_i$ 120 Hz and the rotation frequency $f_r$ (30 Hz). This signifies a substantial reduction in the strong noise component added to the simulated signal at −13 dB, effectively mitigating noise interference and enhancing the extraction of fault information. This shows that using the kurtosis value as the selection criterion for the best mode is not applicable, but the ISEI comprehensive index is effective as the selection criterion. In addition, the RFN value of F-IMF3 is 3.31 dB, which is 16.31 dB higher than the −13 dB signal-to-noise ratio of the original simulation signal.

The simulation signal is denoised separately using four methods including adaptive C-VMD. Based on the parameter settings in the literature [19], the fixed FMD parameters are set as follows: the filter length L is 40, the frequency band cutting number K is 5, the mode number n is 4, and the cycle period m is 15. The other methods include empirical mode decomposition (EMD) and complete ensemble empirical mode decomposition (CEEMD). Each method decomposes the simulated signal and calculates its RFN, kurtosis, and ISEI as illustrated in Figure 9.

The noise tolerance τ and convergence error limit ε for VMD are set to the values reported in the literature [16], specifically $\tau =0$ and $\epsilon =1\times {10}^{-7}$. The convergence curve of the CPO algorithm, derived from the minimum envelope entropy value, is shown in Figure 10. After 18 iterations, the envelope entropy value reaches its minimum, yielding the parameter combination K = 12 and α = 1217. The decomposed modal spectrum is shown in Figure 11, demonstrating that the VMD optimized by CPO effectively mitigates modal aliasing and endpoint effects. Each frequency band is concentrated around the center frequency, indicating a more thorough decomposition. The optimal component following C-VMD decomposition was selected based on the ISEI value, identifying V-IMF4 for envelope analysis. As shown in Figure 12, only $f_i$ is visible, with its harmonic component being subtle. Under conditions of strong noise, the ability to extract fault information from the bearing remains limited.

The noise tolerance τ and convergence error limit ε for VMD are set to the values reported in the literature [16], specifically, $\tau =0$ and $\epsilon =1\times {10}^{-7}$. The convergence curve of the CPO algorithm, derived from the minimum envelope entropy value, is shown in Figure 10. After 18 iterations, the envelope entropy value reaches its minimum, yielding the parameter combination K = 12 and α = 1217.

The decomposed modal spectrum is shown in Figure 11, demonstrating that VMD optimized by CPO effectively mitigates modal aliasing and endpoint effects. Each frequency band is concentrated around the center frequency, indicating a more thorough decomposition. The optimal component following C-VMD was selected based on the ISEI value, identifying V-IMF4 for envelope analysis.

As shown in Figure 12, only $f_i$ is visible, with its harmonic component being subtle. Under conditions of strong noise, the ability to extract fault information from the bearing remains limited. The fixed-parameter FMD, EMD, and CEEMD methods denoise the simulated signal to obtain the best components according to the ISEI value, subsequently performing envelope spectrum analysis. As shown in Figure 13, Figure 14 and Figure 15, the envelope spectra of the best components were selected after fixed-parameter FMD, EMD, and CEEMD denoised the simulated signal, respectively. No obvious fault characteristics were detected; however, the effective information was obscured by strong noise. The fault information is buried within this noise, and fixed-parameter FMD methods demonstrate limited capabilities in fault extraction when compared to S-FMD and C-VMD, which yield more significant results. While the envelope spectrum of the optimal component is derived from the EMD and CEEMD methods, following the denoising of the simulated signal, can indicate the fault characteristic frequency of $f_i$, there is no discernible frequency doubling. This frequency doubling is entirely obscured by strong noise, which can easily lead to the misclassification of fault types. Furthermore, the fault extraction capabilities of the EMD and CEEMD methods for simulated signals are inherently constrained.

The proposed methods, i.e., S-FMD, C-VMD, EMD, CEEMD, and fixed-parameter FMD, presented in this article were employed to denoise the bearing inner raceway simulation signal after the introduction of noise. The RNF and ISEI obtained by the two methods in the figure have the same change trend, that is, the ISEI is as feasible as the RFN in selecting the best components. Each method yields multiple intrinsic mode function (IMF) components, from which the optimal components are selected for further analysis. The relative fault number (RFN) and Integrated Signal Energy Index (ISEI) are then calculated for these optimal components. Figure 16 illustrates the RFN and ISEI values of the optimal components obtained by each method. The figure indicates that the S-FMD method proposed in this article achieves the highest RFN and ISEI values, thereby retaining the most fault information. This finding aligns with the envelope spectrum derived from the proposed method, which distinctly reveals the fault characteristic frequency ($f_i$) along with multiple harmonics. Furthermore, it demonstrates that the fixed-parameter FMD method exhibits superior capabilities in extracting noise effects and fault features compared to C-VMD, EMD, and CEEMD. This suggests that the proposed method is effective for conducting bearing fault diagnosis in environments with strong noise, thereby affirming its feasibility for diagnosing faults based on rolling bearing sound signals. Additionally, it is noteworthy from Figure 16 that the adaptive C-VMD method shows higher RFN and ISEI values than EMD, CEEMD, and fixed-parameter FMD, indicating a better noise reduction effect compared to these three methods. Consequently, this article compares C-VMD, which demonstrates improved noise reduction effects, with the proposed S-FMD method for analyzing the measured sound signals.

Relying solely on one set of parameters may impose certain limitations. To address this, we introduced two additional simulated experiments with Gaussian white noise levels of −11 dB and −9 dB, respectively, to mitigate deviations and provide a more comprehensive evaluation of the fault detection performance of the S-FMD method under varying noise conditions. Figure 17 and Figure 18 illustrate the time domain diagrams corresponding to SNRs of −9 dB and −11 dB, while Figure 19 and Figure 20 present the envelope spectrum following noise reduction processing using this method, clearly revealing the inner ring fault. The presence of the seventh harmonic frequency indicates that this method effectively identifies faults across different SNRs of the simulation signal.

5. Fault Diagnosis of Measured Bearing Sound Signals

5.1. Test Bearings and Test Equipment

The test object is the 7205 angular contact ball bearing, as shown in Figure 21. Its geometric dimensions and fault characteristic frequency are shown in

Table 1. The geometry and the fault characteristic frequencies of bearing 7205.

Geometric Dimensions	Size	Unit
$\mathrm{Outer} \mathrm{raceway} \mathrm{diameter}, do$	52	mm
$\mathrm{Inner} \mathrm{raceway} \mathrm{diameter}, d_i$	25	mm
$\mathrm{Rolling} \mathrm{element} \mathrm{diameter}, d$	7.938	mm
$\mathrm{Number} \mathrm{of} \mathrm{rolling} \mathrm{elements}, z$	13	pieces
$\mathrm{Pitch} \mathrm{circle} \mathrm{diameter}, D$	38.5	mm
$\mathrm{Contact} \mathrm{angle}, \theta$	40°	(°)
$\mathrm{Spindle} \mathrm{rotation} \mathrm{frequency}, f_S$/Hz	200	Hz
Fault characteristic frequency	Value	Unit
$\mathrm{Inner} \mathrm{raceway} \mathrm{fault} \mathrm{characteristic} \mathrm{frequency}, f_i$	1505	Hz
$\mathrm{Outer} \mathrm{raceway} \mathrm{fault} \mathrm{characteristic} \mathrm{frequency}, f_o$	1094	Hz
$\mathrm{Rolling} \mathrm{element} \mathrm{failure} \mathrm{characteristic} \mathrm{frequency}, f_b$	945	Hz

.

The test device is a certain type of test bench from Schaeffler Trading (Shanghai, China) Co., Ltd.—Shanghai R&D Center, as shown in Figure 22. As shown in Figure 23, the red box is the position of the sound sensor, the radial position of the test bearing, and the radial data from the test bearing. The sound signal is picked up by a sound sensor composed of an AWA14423 microphone and an AWA14604 preamplifier. The data are sampled by NI acquisition card 9234. The sampling frequency and number of sampling points are both set to 51.2 kHz. The size and appearance of the microphone are shown in Figure 24. The sensor performance indicators are shown in

Table 2. Microphone performance indicators.

Parameter	Value	Unit
Sensitivity, mV/Pa	50	mV/Pa
Dynamic range, dB	20~142	dB
Frequency range, Hz	10~20,000	Hz
$\mathrm{Size}, \varphi$ (mm)	12.7	mm
Frequency response characteristics	free field

.

In this experiment, the relationship between the placement of the sensor and the bearing under test significantly impacts the fault detection results. As illustrated in the structural diagram of the test bench in Figure 25, the test bearings are symmetrically distributed on either side of the support bearings. The dimensions of the tested bearings are as follows: the outer raceway diameter ($do$) is equal to 52 mm, the inner raceway diameter ($d_i$) is equal to 25 mm, the rolling element diameter ($d$) is equal to 7.938 mm, the number of rolling elements ($z$) is equal to 13, the pitch circle diameter ($D$) is equal to 38.5 mm, and the contact angle ($\theta$) is equal to 40°. In this paper, a disc spring is used for loading, with an axial load of 5 KN and a radial load of 2 KN. The drive mechanism is belt-driven. Bearings located near the belt side experience relatively large radial loads, and the fault characteristic signals are more pronounced in these locations. Consequently, the sound sensor is positioned at the bearing near the belt end to collect radial data, thereby enhancing the accuracy of fault characteristic signal capture. However, the experimental environment is characterized by considerable noise interference, which can lead to rapid attenuation of the sound signal during propagation. If the sound sensor is positioned too far from the test bench, the signal strength may be significantly diminished, and crucial fault information from the bearing may be obscured by noise, thereby compromising the accuracy of fault diagnosis. Conversely, if the sensor is placed too close, the vibrations and shaking of the bench during operation could potentially damage the front end of the contacting sound sensor, adversely affecting both the performance of the sensor and the stability of signal collection.

To effectively address these challenges, extensive trials and experiments led to the conclusion that the sound sensor should be positioned as close as possible to the radial end of the bench’s belt (as shown in Figure 26). This placement minimizes the signal propagation path and reduces noise interference. Additionally, 3D printing technology was employed to create a specialized shell (as shown in Figure 27) to secure the sound sensor, ensuring its stability throughout the experiment and preventing displacement or damage caused by bench vibrations. The shell shown in Figure 27 is made of polyamide (nylon). The principal dimensions of the shell are as follows: the total height is 36 mm, the inlet diameter is 16 mm, and the outlet diameter is 52 mm. Additionally, the cylindrical portion of the shell has a length of 20 mm and a wall thickness of 2 mm. The interior of the shell is filled with efficient noise-absorbing material for sealing, while the exterior is further reinforced with clay, providing dual protection. This design effectively diminishes external noise interference with sound signals, thereby ensuring the integrity and stability of sound signal transmission. Numerous experiments and tests [30,31] have demonstrated that these measures significantly enhance the accuracy and reliability of fault detection, establishing a robust data foundation for the early diagnosis of bearing faults. In this study, the primary noise sources identified were the vibrations of the bench and interference from surrounding equipment. To maintain the consistency of fault characteristic signals, the working conditions of the bearing (such as rotational speed and load) were kept constant during the experiments, ensuring that the sound sensor’s performance remained stable and capable of accurately capturing sound signals.

5.2. Early Failure of Inner Raceway

Figure 28 shows the sound signal of an early failure of the bearing inner raceway. Since the sound signal is extremely susceptible to interference and the interference of strong noise from various bench operations in the test environment, it is difficult to find periodic pulse characteristics from the picture, and the bearing cannot confirm the health status.

The method proposed in this article and the adaptive VMD method were respectively used to perform noise reduction processing on the sound signal, as shown in Figure 28. The convergence curves of the SBOA and the CPO algorithm obtained based on the minimum envelope entropy value are shown in Figure 29 and Figure 30. After 30 iterations of the SBOA, FMD reached the optimal combination. The filter length L is 34, the frequency band cutting number K is 14, the mode number n is 5, and the cycle period m is 34. CPO was iterated 21 times, and VMD reached the optimal combination. The number of decomposition modes K is 5, and the penalty factor α is 4985. The comprehensive index ISEI of each mode of the decomposition results is shown in

Table 3. ISEI values of the decomposed modes via parameter-optimized FMD-VMD (inner raceway fault).

F-IMF	IMF1	IMF2	IMF3	IMF4
ISEI	1.28	1.60	2.49	1.19
V-IMF	IMF1	IMF2	IMF3	IMF4
ISEI	0.89	0.70	0.97	1.55

.

As shown in Table 3, the F-IMF3 component in S-FMD has the largest ISEI value and contains the most fault information. The envelope spectrum analysis of F-IMF3 is shown in Figure 31. Two peaks with prominent amplitude energy are observed, corresponding to the inner raceway fault characteristic frequency $f_i$ and its harmonic components, respectively. Although the signal noise floor is high and there are serious interference components near low frequencies, the proposed method can still effectively extract bearing inner raceway fault information, showing its strong anti-interference ability. The ISEI value of the V-IMF4 component in C-VMD is the largest. The envelope of this component is shown in Figure 32. The fault frequency is not intuitive, and$f_i$ and its harmonic components cannot be observed.

5.3. Early Failure of Outer Raceway

Figure 33 shows the early fault sound signal of the bearing outer raceway. The analysis idea is the same as that of the bearing inner raceway. For noise reduction processing, the SBOA and CPO convergence curves obtained based on the minimum envelope entropy value are shown in Figure 34 and Figure 35, respectively. After 47 iterations of the SBOA, FMD reached the optimal combination. The filter length L is 240, the frequency band cutting number K is 7, the mode number n is 7, and the cycle period m is 35. CPO was iterated 26 times, and VMD reached the optimal combination. The number of decomposition modes K is 5, and the penalty factor α is 5784. The ISEI values of each mode are shown in

Table 4. ISEI values of the decomposed modes via parameter-optimized FMD-VMD (outer raceway fault).

F-IMF	IMF1	IMF2	IMF3	IMF4	IMF5	IMF6	IMF7
ISEI	1.53	2.34	5.02	1.20	2.17	2.58	4.08
V-IMF	IMF1	IMF2	IMF3	IMF4	IMF5
ISEI	1.48	1.46	1.40	2.12	1.98

.

As shown in Table 4, F-IMF3 in S-FMD has the largest ISEI value and contains the most fault information. The envelope analysis of F-IMF3 is shown in Figure 36, and three peaks with prominent amplitude energy can be clearly seen, respectively corresponding to the outer raceway fault frequency $f_o$ and harmonic components. The interference frequency of 25 Hz appears in the form of side frequency bands at $f_o$ and ${2f}_o$, indicating that the noise interference is serious, but the impact of environmental noise is still significant. The ISEI of V-IMF4 in C-VMD is the largest. Its analysis is shown in Figure 37. The fault frequency is not intuitive, and the harmonic component of $f_o$is not obvious.

5.4. Early Failure of Rolling Elements

The sound signal of the early weak fault of the bearing rolling element was selected for analysis, as shown in Figure 38, and the noise reduction process was performed. The SBOA and CPO convergence curves obtained based on the minimum envelope entropy value are shown in Figure 39 and Figure 40. After 26 iterations of the SBOA, FMD reached the optimal combination. The filter length L is 120, the frequency band cutting number K is 7, the mode number n is 7, and the cycle period m is 15. CPO was iterated 15 times, and VMD reached the optimal combination. The number of decomposition modes K is 7, and the penalty factor α is 9697. The ISEI values of each mode are shown in

Table 5. ISEI values of the decomposed modes via parameter-optimized FMD-VMD (rolling element fault).

F-IMF	IMF1	IMF2	IMF3	IMF4	IMF	IMF6	IMF7
ISEI	3.93	5.24	2.65	2.34	2.13	1.78	2.12
V-IMF	IMF1	IMF2	IMF3	IMF4	IMF5	IMF6	IMF7
ISEI	1.96	2.74	1.24	1.10	1.15	2.41	1.30

.

As shown in Table 5, F-IMF2 in S-FMD has the largest ISEI value and contains the most fault information. F-IMF2 was selected for envelope analysis, as shown in Figure 41. From the envelope spectrum, four prominent amplitude energies can be clearly observed. The peak values respectively correspond to the bearing rolling element failure frequency $f_b$ and the harmonic component, indicating that the bearing rolling element is faulty. It is observed from the envelope spectrum that there is a strong noise floor below 500 Hz, indicating that it is low-frequency interference noise. At the same time, there are sidebands of low-frequency interference noise near the $f_b$, 2$f_b$, and 3$f_b$ fault frequencies. The ISEI value of V-IMF2 in C-VMD is the largest. This analysis is shown in Figure 42. The fault frequency is not intuitive, and $f_b$ and the harmonic components cannot be seen.

For further comparative analysis, the proposed methods S-FMD, C-VMD, EMD, CEEMD, and fixed-parameter FMD were used to perform noise reduction and decomposition on the simulated inner raceway fault signal and the measured sound signal of the inner raceway, outer raceway, and rolling element faults, and their ISEI values were selected respectively. The largest component was compared. The ISEI value is a comprehensive index obtained by merging kurtosis and the harmonic signal-to-noise ratio. Its size represents the degree of noise reduction effect of the bearing sound signal and the ratio of the bearing’s fault characteristic frequency to the background noise energy. Different The ISEI values of the optimal components obtained by this method are shown in Figure 43.

Different methods were applied for noise reduction to simulated inner raceway faults, measured inner raceway faults, measured outer raceway faults, and rolling element raceway faults. The component with the most fault-relevant information was then selected to calculate its ISEI. As illustrated in Figure 43, the ISEI value of the optimal component decomposed using the method proposed in this article is significantly higher than that of the best components obtained through the four methods: C-VMD, EMD, CEEMD, and fixed-parameter FMD. This finding demonstrates that the proposed method exhibits superior noise reduction and fault extraction capabilities in practical applications, as well as enhanced fault identification abilities, particularly in diagnosing weak faults.

6. Conclusions

Compared with vibration sensors, microphones have the advantages of non-contact detection, high sensitivity, low cost, and easy installation. In order to solve the problem of complex components and serious interference of rolling bearing sound signals, a rolling bearing acoustic signal fault diagnosis method based on the Secretary Bird Optimization Algorithm and SBOA optimization (Feature Mode Decomposition, FMD) was proposed. The method was verified through simulation and test signals. The following conclusions were obtained:

• This paper proposes a method for bearing fault diagnosis based on sound signals. Compared with vibration signals, sound signals have the advantages of non-contact, economical price, and convenient remote monitoring. However, sound signals are extremely susceptible to noise interference, so noise reduction processing of sound signals is crucial.
• The rolling bearing sound signal fault diagnosis method based on parameter-adaptive FMD studied in this article preprocesses the original sound signal through parameter-adaptive FMD, which can reduce the impact of noise and other interference components on diagnosis and then obtain the envelope spectrum for fault diagnosis.
• Simulation and experimental analysis show that compared with the fixed parameter FMD and parameter adaptive VMD methods, the method studied in this paper can effectively reduce noise interference, accurately extract fault information from the bearing sound signal, and realize bearing fault diagnosis.