# Bearing Fault Diagnosis Based on Vibration Envelope Spectral Characteristics
###### Abstract  
Deep learning methods based on neural network models have been widely applied to bearing fault classification. Although they can achieve high accuracy, they also come with significant complexity. Bearing faults often generate impact vibrations, which produce regular fault characteristic peaks on the envelope spectrum. This paper utilizes the differences in frequency and intensity of the envelope spectrum characteristic peaks under different bearing fault conditions as fault features. By combining these features with the simple and efficient Naive Bayes classifier for fault diagnosis, the algorithm complexity is reduced from the perspective of feature extraction and fault identification. The proposed method was validated using bearing fault data from the Case Western Reserve University (CWRU) dataset and the Machinery Fault Prediction Technology (MFPT) dataset. The results show that the method can classify bearing faults and achieve accurate diagnostic results. The average diagnostic accuracy for the four groups from these two datasets was 99.90% and 99.65%, respectively. The Naive Bayes classification algorithm was compared with classic algorithms in terms of classification accuracy and classification time. Additionally, the algorithm was compared with recent bearing fault diagnosis methods using the CWRU dataset in terms of algorithm complexity. The complexity of the proposed algorithm is only O(N(2280)), which is lower than that of other bearing fault diagnosis methods, where N represents the number of fault samples. This demonstrates that the method significantly reduces algorithm complexity while ensuring accuracy, improving diagnostic efficiency, enhancing the timeliness of real-time industrial bearing fault diagnosis, and reducing hardware setup and operating costs. b 20  
degree of non-stationarity and periodic impact characteristics. Therefore, in engineering practice, signal processing methods are often used to identify and extract fault characteristic components from the vibration signals of rolling bearings to achieve condition monitoring and fault diagnosis of the bearings. In practical applications, bearing fault diagnosis is mainly conducted using vibration signals. The process of rolling bearing fault diagnosis based on vibration signals can be divided into two steps: feature extraction and fault recognition. Since it is challenging to extract features directly from raw signals that can differentiate between different defective states and normal states of bearings, vibration analysis techniques, including time domain analysis, frequency domain analysis, and time-frequency domain analysis, are employed to address this issue [3]. Time domain analysis is based on analyzing the characteristics of a signal over time. We can reveal the signal's dynamic behavior and characteristics by observing features such as the waveform, amplitude, and frequency as the signal changes over time. Frequency-domain analysis transforms a signal into a series of discrete frequency components, making it easy to analyze. The signal is called the signal spectrum after transforming characteristic frequency components for fault analysis. Standard frequency domain analysis methods include peak spectrum analysis [4] and envelope spectrum analysis [5]. Time-frequency domain analysis methods simultaneously study the characteristics of signals in both the time and frequency domains. This enables a more comprehensive understanding of the dynamic properties of the signal and reveals its changing patterns in time and frequency. There are many time-frequency domain analysis methods, such as wavelet analysis [6; 7], Wagner-Ville distribution, and variational mode decomposition (VMD) [8; 9]. Fault recognition, also known as fault pattern recognition, involves extracting information containing equipment fault characteristics from historical fault data. A fault diagnosis model is established by training on historical feature information using machine learning, deep learning, and other methods.