# Research on a Nonlinear Dynamic Incipient Fault Detection Method for Rolling Bearings
## 1 Introduction  
The rolling bearing is one of the most commonly used components in rotating machinery and equipment [1, 2]. However, the running state of the rolling bearings are easily affected by shock and vibrations, and incomplete statistics show that about 30% of rotating machinery failures are caused by bearing failures [3, 4]. The failures of rolling bearings often develop from the normal stage to the incipient stage, then enter the stage of repeated failure, and finally reach the stage of complete breakdown. The serious faults do not occur in an instant, but has a process of gradual deterioration [5, 6, 7]. Therefore, it is of great significance to study the incipient fault detection of rolling bearings, find the incipient signal characteristics of faults, and eliminate the safety hazards in time when the fault has not developed toward a serious degree [8, 9]. The fault detection methods mainly include model-driven and data-driven methods. Some scholars have established rolling bearings degradation models by analyzing the prior physical knowledge of rolling bearings [10, 11], or completed the incipient fault detection of rolling bearings by analyzing the characteristic frequency of failure of rolling bearings [12, 13]. However, the model-based method requires an accurate prior physical model of rolling bearings. The motion and internal stress of rolling bearings are complicated, which causes the decline of accuracy and generalization. The data-driven method is used to finding incipient failures by analyzing the collected data, and is more suitablefor incipient failure detection of current large-scale industrial processes. The deep learning theory brings opportunities to solve the feature extraction problems in rolling bearings fault detection with its powerful feature extraction capabilities [14; 15; 16; 17]. By continuously increasing the abstraction of various representations, rich and differentiated features can be obtained and studied. However, the computational intensity of the complicated model becomes stronger, which will lead to a decrease in the training and detecting speed of the system in a large network structure. The method based on signal processing is also a common detection method for incipient faults [18; 19; 20]. For example, the signal feature processing based on the wavelet variation method [21] and empirical mode decomposition method [22] has been generalized successfully in many applications. However, the incipient fault features are usually weak and buried in the noises, and the noise in the entire signal makes the normal state similar to the actual incipient fault, thus affecting the effect of fault detection. Many scholars have proposed many methods for fault detection, but incipient fault detection still mainly focuses on the following challenges [23; 24; 25]: 1. The incipient failure characteristics are weak, and it is difficult to extract the fault characteristics. 2. The vibration signal is a kind of non-linear and non-stationary signal, which brings difficulty to extract the rolling bear fault weak feature 3. The collected signals can be easily affected by the noise and transmission loss, which makes it difficult to extract signal characteristics. Therefore, the weak fault feature extraction is the key technology for incipient fault detection of rolling bearings. Literature [26] proposed an incipient fault detection method based on the Deep Principal Component Analysis (Deep PCA). This method is used to the principle of deep decomposition to accurately divide the data to extract weak fault features and detect incipient faults. However, Deep PCA is a linear stationary transformation method, when it is used for a monitoring non-linear time-varying system, its performance is greatly reduced since it does not fully take account of the characteristics of the actual system. In order to solve this problem, many non-linear time-varying fault detection methods have been proposed [27; 28; 29; 30].