# Fault Diagnosis of Rolling Bearings Based on Acoustic Signals in Strong Noise Environments
## 3 SBOA-Optimized FMD
### 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 Algorithmoptimized Feature Mode Decomposition (SFMD) criteria for selecting the best mode after the signal. (1) Normalization: Kurtosis formula. \[K=\frac{\frac{1}{N}\sum_{i=1}^{N}(x_{i}-\mu)^{4}}{\left(\frac{1}{N}\sum_{i=1}^{N }(x_{i}-\mu)^{2}\right)^{2}} \tag{8}\]  
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  
\[HNR=\frac{\sum_{i=1}^{n}A_{i\times FC0}}{n\times 3\sigma_{\text{noise}}} \tag{9}\]  
where \(A_{i\times FC0}\) represents the amplitude of the \(i\)th harmonic frequency component, \(FC0\) represents the fundamental frequency (fault characteristic frequency), \(n\) is the number of harmonic frequency components, \(\sigma_{\text{noisedddl}}\) 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  
\[Z_{K}=\frac{K-\mu_{K}}{\sigma_{K}} \tag{10}\]  
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  
\[Z_{RFN}=\frac{RFN-\mu_{HNR}}{\sigma_{\text{{\tiny{RFN}}}}} \tag{11}\]  
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). \[ISEI=\delta_{1}\cdot|Z_{K}|+\delta_{2}\cdot|Z_{RFN}| \tag{12}\]  
where \(\delta_{1}\) and \(\delta_{2}\) 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.