## 2 Physical Model of a Jeffcott Rotor with Simultaneous Imbalance and Shaft Bowing Faults  
The schematic design involves a purely physics-based approach. This design, as elucidated in Figure 1a, provides a visual representation of the Jeffcott rotor systems consisting of an imbalance and shaft bowing with a rigid disk, supported by two simple bearings, while Figure 1b shows the disk's geometric relations. It is assumed that the shaft-bearing component possesses only stiffness \(K\) and light damping \(B\). Here, \(O\) stands for the center of the rotation axis, \(C\) stands for the disk's geometric center, \(e\) is the distance of the rotor's imbalance from the geometric center, \(\Omega\) represents the rotational speed, \(s\) represents the residual shaft bowing, \(m\) represents the imbalance mass, \(M\) represents the disk mass, \(\theta\) represents the shaft bowing angle, and \(\alpha\) represents the imbalance angle, both of which reference a key phasor (KP). Figure 1: Schematic diagram of a Jeffcott rotor (**a**) with simultaneous imbalance and shaft bowing faults and (**b**) the disk’s geometric relations. The rotor imbalance is commonly specified by a term of \(U=me\), with a unit of g-mm or kg-m. The acceleration of the imbalance mass at point \(A\) can be calculated based on fundamental dynamics, i.e. :  
\[\mathbf{a}_{A}=\ddot{u}\mathbf{i}+\ddot{v}\mathbf{j}-\Omega^{2}[e\cdot\cos( \Omega t+\alpha)\mathbf{i}+e\cdot\sin(\Omega t+\alpha)\mathbf{j}] \tag{1}\]  
where \(\mathbf{i}\) and \(\mathbf{j}\) respectively denote the unit vector in the X and Y directions. The inertia force resulting from the imbalanced mass \(m\) can be represented by:  
\[-m\mathbf{a}_{A}=[-m\ddot{u}+me\Omega^{2}(\cos\Omega t+\alpha)]\mathbf{i}+[-m \ddot{v}+me\Omega^{2}(\sin\Omega t+\alpha)]\mathbf{j} \tag{2}\]  
The equations of motion (EOMs) for the Jeffcott rotor with simultaneous imbalance and shaft bowing faults can be obtained by using [PERSON]'s second law:  
\[\Sigma F_{u}=M\ddot{u} \tag{3}\]  
\[\Sigma F_{v}=M\ddot{v} \tag{4}\]  
Subsequently, the EOMs subject to the imbalance and shaft bowing excitation in the X and Y directions are summarized as:  
\[(M+m)\ddot{u}+B_{x}\dot{u}+K_{x}u=K_{x}s\cos(\Omega t+\theta)+me\Omega^{2}\cos (\Omega t+\alpha) \tag{5}\]  
\[(M+m)\ddot{v}+B_{y}\dot{v}+K_{y}v=K_{y}s\sin(\Omega t+\theta)+me\Omega^{2}\sin (\Omega t+\alpha) \tag{6}\]  
or in terms of nominal vibration forms as:  
\[\ddot{u}+2\zeta_{x}\omega_{nx}\dot{u}+\omega_{nx}^{2}u=s\omega_{nx}^{2}\cos( \Omega t+\theta)+\frac{U}{M_{T}}\Omega^{2}\cos(\Omega t+\alpha) \tag{7}\]  
\[\ddot{v}+2\zeta_{y}\omega_{ny}\dot{v}+\omega_{ny}^{2}v=s\omega_{ny}^{2}\sin( \Omega t+\theta)+\frac{U}{M_{T}}\Omega^{2}\sin(\Omega t+\alpha) \tag{8}\]  
where \(M\), \(B\), and \(K\) respectively represent the mass, damping, and stiffness. Here, \(M_{T}=M+m\approx M\) denotes the total mass of the rotor, while \(\omega_{ni}=\sqrt{K_{i}/M_{T}}\) and \(\zeta_{i}=C_{i}/2\sqrt{K_{i}M_{T}}\), \(i=x,y\) represent the system's natural frequency and damping ratio in the X and Y directions, respectively. The right-hand terms in Equations (7) and (8) describe the excitation forces caused by shaft bowing and imbalances, respectively. The vibrational responses of \(u\) and \(v\) due to shaft bowing and imbalances can be separately solved and superimposed for the total responses. Nonetheless, identifying the rotor's imbalance (\(U\),\(\alpha\)) and shaft bowing (\(s\),\(\theta\)) from the total responses is a backward process involving nonlinear functions of \(\alpha\) and \(\theta\). Therefore, nonunique solutions exist, which causes difficulty in almost all cases of multi-fault diagnosis. Let us first provide the responses to these two types of faults separately and then introduce the idea of combining the physical modeling and machining learning techniques for multi-fault diagnosis.