###### Abstract  
Bearings cause the most breakdowns in induction motors, which can result in significant economic losses. If faults in the bearings are not detected in time, they can cause the whole system to fail. System failures can lead to unexpected breakdowns, threats to worker safety, and huge economic losses. In this investigation, a new approach is proposed for fault diagnosis of bearings under variable low-speed conditions using a smooth sliding digital twin analysis of indirect acoustic emission (AE) signals. The proposed smooth sliding digital twin is designed based on the combination of the proposed autoregressive fuzzy Gauss-Laguerre bearing modeling approach and the proposed smooth sliding fuzzy observer. The proposed approach has four steps. The AE signals are resampled and the root mean square (RMS) feature is extracted from the AE signal in the first step. To estimate the resampled RMS bearing signal, a new smooth sliding digital twin is proposed in the second step. After that, the resampled RMS bearing residual signal is generated using the difference between the original and estimated signals. Next, a support vector machine (SVM) is proposed for crack detection and crack size identification. The effectiveness of this new approach is evaluated by AE signals provided by our lab's bearing dataset, where the benchmark dataset consists of one normal and seven abnormal conditions: ball, outer, inner, outer-ball, inner-ball, inner-outer, and inner-outer-ball. The results demonstrated that the average accuracies of the anomaly diagnosis and crack size identification of AE signals for the bearings used in this new smooth sliding digital twin are 97.75% and 97.78%, respectively. induction motor; bearing; acoustic emission signal; smooth sliding digital twin; autoregressive fuzzy Gauss-Laguerre; fuzzy logic technique; support vector machine; crack detection; crack size identification +
Footnote †: journal: Applied Sciences  
To design bearing fault diagnosis techniques, various approaches have been carried out that can be used in three main domains: the time domain, frequency domain, and time-frequency domain. Time and frequency domain analyses have the challenge of high feature dimensions. Time-frequency domain analysis, such as the short-time Fourier transform [4], Wigner Ville distribution [5], and wavelet packet transform [6], have been recommended to solve problems associated with the nonstationary and nonlinear nature of the bearings' signals. Moreover, digital twins are an efficient technique for anomaly diagnosis in various applications. These approaches allow for diverse and much more precise analyses by designing models/estimates of the original system [7]. In addition, various techniques can be introduced to design digital twins, such as model-based techniques, signal-based approaches, artificial intelligence-based algorithms, and hybrid schemes [8]. In signal-based approaches, signal processing algorithms can be used for anomaly diagnosis [8; 9]. The application of a signal-based algorithm for anomaly detection and identification was suggested in [10]. The application of artificial intelligence-based approaches for anomaly identification was proposed in [9]. These techniques have been used in several applications, such as bearing fault diagnosis and pipeline crack diagnosis [9]. In model-based approaches, mathematical-based fault modeling algorithms can be used for reliable and stable fault detection and identification. In reliable model-based fault diagnosis, the faults are identified using a small dataset; however, mathematical modeling of complex systems is challenging work [11]. Hybrid approaches can be designed based on a combination of the above techniques [12]. These techniques are able to reduce the weakness of each of the above methods and increase their strengths. Various researchers have implemented rigorous studies in the field of fault diagnosis for bearings using model-based approaches.