## 2 System Equations of Motion  
The Laval/Jeffcott rotor is a fundamental model in rotordynamics, comprising a flexible shaft with a centrally located rigid disk (Figure 1). Despite its simplicity, this model effectively captures critical dynamic behaviors such as critical speeds, unbalance response, and whirling [15]. As a foundational element in the field, it serves as a basis for understanding more complex rotor systems and is extensively employed in both theoretical and practical engineering applications.  
Assuming the rotor disk has no impact on the stiffness of the weightless shaft, the lateral bending stiffness at the midpoint of a uniformly supported beam with consistent properties can be expressed as follows:  
\[k=\frac{48 EI}{L^{3}} \tag{1}\]  
where: \(E\) is the Young's modulus of the shaft,  
\(L\) is the shaft length between the bearings,  
\(I\) is the shaft area moment of inertia.  
For the off-centered disk, or disk in the arbitrary position on shaft:  
\[k=\frac{3 EIL}{c^{2}d^{2}} \tag{2}\]  
where: \(c\) and \(d\) are the distances from the left and right bearing, respectively.  
A small amount of viscous damping, represented by the constant cs, affects the disk's lateral movement. This damping results from a combination of factors including shaft flexibility, fluid resistance within the machine, and bearing friction. By applying [PERSON]'s laws to the rotor disk, assuming the shaft has negligible mass, we can derive equations describing the disk's lateral motion in both the horizontal (z) and vertical (y) directions.  
Figure 1: Single mass Jeffcott/Laval rotor on flexible bearings.  
These equations, and their inputs visualized in Figures 1 and 2, account for the disk's inertia and the forces produced by the shaft's deformation. They can be written as:  
\[m\ddot{z}_{T}=-b\dot{z}_{H}-kz_{H} \tag{3a}\] \[m\ddot{y}_{T}=-b\dot{y}_{H}-ky_{H} \tag{3b}\]  
where: \(m\) is the disk mass,  
\(b\) is the damping coefficient,  
\(k\) is the stiffness coefficient,  
\(z_{H}\), \(y_{H}\), \(z_{T}\), \(y_{T}\) are the co-ordinates of the geometric center and mass center.  
The positional co-ordinates of the disk's mass center can be mathematically represented in relation to its geometric center [16]. These co-ordinates are dynamically influenced by the rotor's rotational angle, which varies over time.  
\[z_{T}=z_{H}+\varepsilon\cos(\omega t) \tag{4a}\] \[y_{T}=y_{H}+\varepsilon\sin(\omega t) \tag{4b}\]  
where: \(\varepsilon\) is the eccentricity,  
\(\omega\) is the angular velocity,  
\(t\) is time.  
By inserting the second-order derivatives derived from Equation (4a,b) into Equation (3a,b), the equations of motion for the Foppl/Jeffcott rotor can be reformulated in terms of the disk's geometric center [17; 18].  
\[m\ddot{z}_{H}+b\dot{z}_{H}+kz_{H}=m\varepsilon\omega^{2}\cos(\omega t) \tag{5a}\] \[m\ddot{y}_{H}+b\dot{y}_{H}+ky_{H}=m\varepsilon\omega^{2}\sin(\omega t) \tag{5b}\]  
It is important to note that this model simplifies the rotor system by excluding gyroscopic effects. This simplification is due to assumptions of infinitely stiff bearings and a rigid disk without tilt. Consequently, the shaft is fixed and always aligned with the bearing centerline.  
Figure 2: Disk of a single mass Jeffcott/Laval rotor.