# Order-Based Identification of Bearing Defects under Variable Speed Condition
## 1 Introduction  
Machines in the 4.0 industry are becoming progressively complex, making their vibration signals analysis a challenging task, even for vibration diagnostic experts [1]. Bearings are recognized as critical components in industrial machinery. Thus, mastering their vibration behavior can significantly enhance the diagnostic process. The vibration signal of a defective bearing operating under stationary conditions can be considered as an amplitude modulated waveform [2]. This makes the envelope analysis (EA) one of the most effective methods for bearings health monitoring in stationary conditions [3]. At a constant operating speed, the impulses of bearing defects occur periodically with a known frequency, namely the bearing defect characteristic frequencies. Here, the diagnostic expert relies on the frequency spectrum of the signal being diagnosed to check the existence of the defect frequencies. Three EA-based features: SPRO, SPRI, and SPRR associated with the three common bearing defects (outer race, inner race, and ball defect) have been applied successfully in the literature and proven their effectiveness in the characterization of localized bearing defects in stationary conditions [4, 5]. In a real application, the operating speed of rotating machines is usually variable. The vibration impulses resulting from a localized bearing defect are therefore no longer periodic. EA as well as any other method based on the assumption of a constant rotational speed are therefore non-applicable. Order tracking methods are considered to be among the most powerful algorithms used to eliminate the effect of speed fluctuation [6]. The latter converts the non-stationary signals in the time domain into stationary ones in the angular domain [5],making Fourier transform-based analysis techniques applicable again. Order tracking methods are divided into two main types: with or without speed measuring devices, such as encoders and tachometers. It is important to note that in the case of available speed measurement, order tracking is usually performed with fairly high accuracy [2]. Despite their effectiveness in characterizing bearing defects in non-stationary operations, the use of order tracking methods is still reserved for diagnostic experts. A specialist can easily track the differences between the envelope order spectra corresponding to different bearing defects, which can be a difficult task for inexperienced operators. Machine learning algorithms (MLA) are increasingly being used as an effective solution for processing vibration signals [7]. [PERSON] et al. updated the unsupervised classifier Optics (Ordering Points to Identify the clustering structure) to provide real-time detection of bearing defects based on vibration monitoring. [PERSON] et al. [8] used two supervised machine learning classifiers: Multi kernel support vector machines (MSVM) and K-nearest neighbors (KNN), to determine the severity of a bearing outer race defect in a shaft bearing system. In many other studies [9; 10; 11], several kinds of MLAs are also applied to automate the detection and/or the characterization of the severity of bearing defects. However, there is no literature that deals with the automatic identification of bearing defects at variable speeds. This work aims to automate the identification of bearing defects under variable speed conditions. Vibration data collected from real bearings with different health states operating under variable speed are considered to validate the proposed approach. Based on an order analysis technique, the three features associated with bearing defects: SPRO, SPRI, and SPRR are updated to perform with non-stationary signals. These are extracted from the considered vibration signals and used to build a predictive classification model for the MLA-classifier: MSVM [8]. The constructed classification model has proven to be effective in the identification of bearing defects under variable speed conditions, confirming the performance of the proposed features. The paper is organized as follows.