## 1 Introduction
[34] employed a transfer matrix technique to forecast the imbalance response of a Jeffcott rotor subjected to bowing. The rotor was affixed to various fluid film bearings, and the outcomes were subsequently contrasted with prior experimental investigations. [PERSON] et al. [35] investigated the impact of residual shaft bowing on the dynamic response of a simply supported single disk rotor. Their study systematically examined the interplay of factors such as disk skewing, mass imbalances, and the positioning of the disk between the bearings. [PERSON] et al. [36] scrutinized a warped Jeffcott rotor model across diverse bowing scenarios, revealing instances of self-balancing and phase jumping. [PERSON] et al. [37] employed an ANN and wavelet transform to categorize a rotor system that experienced both shaft bowing and an imbalance. This was achieved by monitoring the vibrations in the transverse and axial directions. [PERSON] et al. [38] recently presented research where they used a combination of a WTS (wavelet transform spectrum) with LSTM (long short-term memory) and an SVM (support vector machine) to detect imbalances and shaft bowing in rotor systems. When there are multiple faults, the vibrations caused by these faults are coupled with each other in a linear or nonlinear manner. As a result, the observed vibration signals are quite complicated, making it challenging to detect each fault using traditional methods. Even though model-based approaches have been used to diagnose and anticipate rotor defects for many years, there are drawbacks, including the need for lengthy models and a lack of self-adaptability to aging machinery. Therefore, with the aid of modern signal processing, hybrid methods that make use of AI techniques and physical modeling have appeared. Regarding the commonly named hybrid approaches, in light of their combination types, they can be divided into three categories--serial combinations (method fusion) [39], parallel combinations (output fusion) [40], and mixed combinations. Physical model may be derived and employed along with these hybrid methods, although mainly just to generate the residual in the parity equation. The present research, however, proposes a novel combination of physics-based and ML techniques, in which a physical model generates synthetic data for ML training. An exemplar in this domain is the work by [PERSON] et al. [41; 42], who adeptly introduced the concept of a hybrid approach by incorporating a neural network (NN) alongside physical modeling for imbalance identification. [PERSON] al. [43] used a hybrid method for fault prognosis based on a physical model, data clustering, and the geolocation principle to predict the remaining useful life (RUL) for wind turbine systems. Recently, [PERSON] et al. [44] reviewed hybrid approaches for FDD that combine data-driven analyses with physics-based and knowledge-based models to overcome a lack of data and increase the FDD accuracy. Moreover, [PERSON] et al. [45] presented a fault diagnosis and prognosis method based on a hybrid approach that combines structural and data-driven techniques. The objective of this study is to create a hybrid approach that is applicable for the diagnosis of multiple faults in rotor-bearing systems. The essence of this innovative method is as follows:  
1. We will establish a physical and mathematical model of a Jeffcott rotor-bearing system containing simultaneous imbalance and shaft bowing faults and identify the model parameters, including the imbalance and shaft bowing characteristics, from calculations and experimental tests. 2. After parameter identification, the physical model will be used to generate sufficient sets of simulated data for ANN-supervised training, which will help to produce a more reliable model. A trained ANN can be integrated into a Jeffcott rotor monitoring system for the online diagnosis of imbalance and shaft bowing fault components using simulated and experimental data from Jeffcott rotor experiments. The remainder of this paper's structure is as follows.