Abstract: Since bearings generate nonlinear vibrations when they fail, this paper uses a novel nonlinear dynamic method—sample entropy—to process the signals, extract features, and then uses a neural network to classify and predict the faults. The experimental results are good. Keywords: fault diagnosis, approximation entropy, sample entropy, neural network 1. Background Introduction Currently, the main fault diagnosis technologies for rolling bearings include vibration diagnosis technology, acoustic diagnosis technology, temperature diagnosis technology, oil film resistance diagnosis technology, and fiber optic diagnosis technology. Vibration analysis is frequently used for condition monitoring and fault diagnosis of rolling bearings because bearing vibration signals carry rich operational information and are highly sensitive to early-stage faults. During the fault occurrence process, the dynamic characteristics often exhibit complexity and nonlinearity, and the vibration signal also becomes non-stationary. With the development of nonlinear dynamics in the 1980s, some nonlinear dynamic methods have been applied in various fields, achieving good results in fault diagnosis. This paper uses an improved algorithm of approximate entropy—sample entropy—to process signals and extract features. Compared with other nonlinear dynamic methods (Lyapunov exponent, information entropy, correlation dimension, K-entropy), sample entropy, like approximate entropy, has the following characteristics: 1) It requires only a short amount of data to obtain relatively stable estimates; the required data points are approximately between 100 and 5000, typically around 1000. 2) It has good resistance to noise and interference, especially to occasional strong transient interference. 3) It can be used regardless of whether the signal is random or deterministic, and therefore can also be used for mixed signals composed of random and deterministic components. The approximate entropy value varies depending on the ratio of the two components. These advantages make approximate entropy and sample entropy analysis good tools for analyzing nonlinear time series, and they have achieved good results. Compared with approximate entropy, sample entropy has weaker dependence on data length and exhibits good consistency over a large numerical range. This project was proposed by the East China Petroleum Design Institute for bearing fault diagnosis in oilfield drilling. As a key component of oil drilling platforms, bearing failure can cause the entire equipment to stop production and even damage other components, greatly increasing maintenance time and causing serious economic losses. Therefore, it is important to develop a system for predictive fault diagnosis of the entire bearing system. This system can issue early warning signals before bearing failure occurs, allowing for early repair or replacement of bearings that are about to fail, thus shortening downtime and reducing maintenance costs, thereby minimizing drilling oil production losses. In addition, statistics show that 30% of actual field failures are caused by rolling bearings, making bearing fault diagnosis very important. 2. Algorithm Description We can see that m and r are two parameters set in ApEn and SampEn. For ApEn, Pincus suggests a value of r of 0.1-0.25SD, where SD is the standard deviation of the time series to be calculated. m should be 1 or 2. Lake et al. recommend using a standard autoregressive model to determine the m parameter of SampEn, or setting it to 1 or 2 as with ApEn, and determining r through the minimum relative error method. The value of m is also related to the sampling rate of the signal; different values ​​should be chosen for different sampling frequencies rather than consistently using a single m value. It should be noted that in the calculation of sample entropy, if the similarity tolerance r is too small, there will be few patterns satisfying the similarity condition; if r is too large, there will be too many patterns satisfying the similarity condition, resulting in a significant loss of detailed information in the time series. To avoid the influence of noise on the calculation results, r should be greater than the amplitude of significant noise. The meaning of sample entropy is similar to that of approximate entropy, both measuring the probability of a new pattern being generated in the time series when the dimension changes. The higher the probability of generating a new pattern, the more complex the sequence, and the larger the corresponding approximate entropy or sample entropy. Therefore, theoretically, approximate entropy and sample entropy can characterize the irregularity and complexity of a signal. To intuitively illustrate the meaning of sample entropy, the following shows the trends of approximate entropy and sample entropy of simulated white noise and frequency-modulated chirp signals as a function of r, with a data length of N=1000 and an embedding dimension of M=2. White noise is more complex than chirp signals, which should be reflected in the data comparison. [align=center] Figure 1: Approximate entropy of white noise and chirp signals Figure 2: Sample entropy of white noise and chirp signals[/align] As can be seen from the figures, the consistency of sample entropy is better than that of approximate entropy. When r < 0.15SD, the approximate entropy of chirp signals is larger than that of white noise, and when r > 0.15SD, it is smaller than that of white noise. Sample entropy consistently maintains this trend, so sample entropy provides a better analytical effect than approximate entropy. Under different sampling rate conditions, sample entropy also maintains good consistency, which is something that approximate entropy does not possess. 3. Data Analysis The data in this experiment were obtained by artificially damaging the bearings to simulate the two main faults: spalling and cracking. Three sensors were installed in the experiment: measuring point 1 was located radially above the rear side (without a motor), measuring point 2 was located radially above the front side (with a motor), and measuring point 3 was located transversely to the right side (with a motor). Therefore, the data for each condition was 3-channel. The sampling frequency for bearing 1 was 51.2 kHz. The sampling rate for bearing 2 was 128 kHz, and the sample entropy parameters were m=2, r=0.2, and N=2048. Each data point in Table 1 is the average of 100 data points. [align=center]Table 1[/align] The data in Table 1 shows that the bearings have different sample entropy values ​​under different failure modes. The sample entropy value is lowest under normal operating conditions, highest under fatigue spalling conditions, and falls between these two values ​​when cracks occur. The significant difference in values ​​between bearing 1 and bearing 2 under the same failure mode is mainly due to their different sampling rates, and is also related to differences caused by processing and usage. When a bearing rotates, the rolling elements roll on the inner and outer raceways. Because the forces acting on the rolling elements vary at different positions, and the number of rolling elements also differs, these structural characteristics of the bearing cause variations in load-bearing stiffness, resulting in bearing vibration. When the bearing speed and load are constant, this vibration is deterministic. Wavy surfaces and roughness left during bearing assembly machining generate alternating excitation forces that cause the bearing system to vibrate. Although these excitations are mostly periodic, the actual constituent factors are very complex, and there are no specific relationships between the factors. Furthermore, the vibrations of the test motor, the working bearing, and other mechanical components on the testing machine also exhibit significant randomness in their excitation forces, containing multiple frequency components. This is why the sample entropy is relatively high even under normal operating conditions. If the rolling surface of the bearing has defects such as fatigue spalling or indentations, alternating excitation forces will occur when the rolling bearing rotates on these damaged surfaces. Because rolling surface defects are irregular, the resulting excitation forces are also random, containing multiple frequency components. Generally, the faster the shaft rotates, the higher the vibration frequency caused by surface damage. The smaller sample entropy in the cracked state compared to the peeled state might be due to the shallower crack depth. Only when the crack is larger does its nonlinear effect on the system response become apparent. 4. Neural Network Classification Neural networks are a branch of artificial intelligence that has developed rapidly in recent years and has been widely applied in various fields due to the following advantages: 1) It is a large-scale complex system, providing a large number of adjustable variables. 2) It implements parallel processing mechanisms, thus providing high-speed processing capabilities. 3) Its connection strength is variable, making the network topology highly flexible and thus possessing strong adaptive capabilities. 4) The characteristics (input and output) of artificial neural networks are nonlinear; therefore, artificial neural networks are a class of large-scale nonlinear systems, which provides the potential for system self-organization and collaboration. For specific application problems, different scholars have proposed many different new models and algorithms, some even proposing combinations with other disciplines such as nonlinear dynamics or wavelets to achieve good results. This paper uses the most widely used BP network. When there are many sensor inputs, using neural networks for classification is very convenient. In this experiment, the input signal had 3 channels, and the experimental data showed that the results were quite satisfactory. The three-layer BP network used here consists of an input layer, a hidden layer, an output layer, and connections between inter-layer neurons. Since three sensors were used, the neural network structure is three inputs and two outputs. The outputs are represented by three states: 00, 01, and 10, representing normal, crack, and peeling, respectively. The structure of the neural network is shown in Figure 3. The activation function f is the Sigmoid function. 1000 sets of data were used, each set representing the sample entropy values ​​of the three sensors under various operating conditions. The software used was NeuroShell 2. 20% of all data was randomly selected as test data, 20% as validation data, and the remaining 60% as training data. The average error was 0.002, the learning rate was 0.5, and the initial weights were 0.1. After training, the neural network achieved recognition rates of 94%, 89%, and 90% for normal, crack, and peeling, respectively. 5. Conclusion Since nonlinear vibration signals are generated when rolling bearings fail, nonlinear dynamics method is used for analysis to extract feature quantities under different conditions. Finally, the classification and prediction capabilities of neural networks are used. The experimental results prove that this method is effective, especially when the fault type is more complex and there are more input variables, this method can show its superiority. References: [1] Yang Fusheng, Liao Wangcai. Approximate entropy: a complex measure suitable for short data. Chinese Journal of Medical Devices, 1997, 21(5): 283-286. [2] Xinnian Chen, Irence C. Solomon Comparison of the use of Approximate Entropy and Sample Entropy: Application to Neural Respiratory Signal IEEE Engineering in Medicine and Biology 27th Annual Conference Shanghai china, September 1-4 2005. 4212-4215. [3] Wang Guofeng, Zhou Yiwu, Guo Jinguang. Early fault diagnosis method for bearing spalling. Journal of Dalian Maritime University, August 2002, 101-104. [4] Zhao Jiangang. Exploration of the application of artificial neural network in real-time hydrological correction. Sichuan University, 2002, 09:20. [5] Zou Minghu. Research on intelligent fault diagnosis system of radar printed circuit board based on fuzzy neural network. Microcomputer Information, 2004, No.12. [6] Gu Yunhui. Research on fault diagnosis method based on automatic measurement and control system. Microcomputer Information, 2005, No.19.