The purpose of this paper is to investigate some qualitative properties of solutions of nonlinear fractional retarded Volterra integro-differential equations (FrRIDEs) with Caputo fractional derivatives. These properties include uniform stability, asymptotic stability, Mittag–Leffer stability and boundedness. The presented results are proved by defining an appropriate Lyapunov function and applying the Lyapunov–Razumikhin method (LRM). Hence, some results that are available in the literature are improved for the FrRIDEs and obtained under weaker conditions via the advantage of the LRM. In order to illustrate the results, two examples are provided.

Distributed bearing faults are highly common in industrial applications and display unpredictable vibration patterns impeding their detection. These faults stem from issues such as lubrication deficiencies, contamination, electrical erosion, roughness of the bearing surface, or the propagation of localized faults. This study aims to detect distributed bearing faults by utilizing a multi-sensory approach consisting of current, accelerometer, and fluxgate sensors. A novel 2D deep learning framework is proposed, leveraging signals from six distinct sources, including three-axis vibration signals, stray magnetic flux signal, and two phase-current signals. Data is collected from 3 hp and 10 hp induction motors at 50 operational points, spanning 10 speed levels and 5 torque levels. These six signals are transformed into matrices and combined to create a comprehensive matrix that provides an overall depiction of the bearing condition. The proposed deep learning architecture employs a 2D convolutional model, which takes 2D images as input and determines the bearing status. To evaluate the system's robustness, the data is divided into training and testing sets. The proposed model demonstrates remarkable effectiveness in detecting distributed bearing faults, achieving an impressive accuracy rate of 99.92%. Furthermore, a comprehensive comparison is provided, highlighting the impact of using various sets of inputs as sources for the deep learning model on the accuracy rate for each set. Through the analysis of the obtained results, a clear conclusion can be drawn: the model performs at its best when all six input sources are utilized.

Distributed bearing faults are the most common ones in industry and create random vibration patterns, which make their detection difficult. They are caused by lubrication issues, contamination issues, electrical erosion, bearing roughness, or the spread of a local fault. This research mainly focuses on the distrusted bearing faults diagnosis using a multi-sensory kit. For this purpose, a novel deep-learning framework is proposed to detect these faults using 3 axis vibrations and one stray magnetic flux signal. The data is collected at 50 operating points, i.e., 10 speed and 5 torque levels. The proposed architecture benefits from a multi-input pipeline consisting of time-frame signals and extracted features. A feature-rich architecture is proposed combining convolutional and high-level information. Although a deep learning structure coherently learns from the features through convolutional and LSTM layers, 20 predefined features sampled from each instance are also fed into the network to improve accuracy. The robustness of the overall system is validated with train/test split data. Deep learning results are compared with two more classification algorithms, SVM and XGBoost. The high accuracy of the proposed model demonstrates the superiority of the deep learning architecture for distributed bearing fault detection.