## 2 Materials and Methods
### The Mathematical Model of a Squeeze Film Damper
The pressure distribution in the full oil film is governed by the Reynolds equation, which is adapted to bilinear material and short dampers:  
\[\frac{\partial}{\partial Z}\left(\frac{1}{\eta_{C}}h^{3}p^{\prime}\right)=12 \dot{h}\text{ for }\dot{h}<0\text{ and }0\leq Z\leq Z_{C}, \tag{1}\]  
\[\frac{\partial}{\partial Z}\left[\frac{1}{\eta}\left(h^{3}p^{\prime}+3\dot{h}^{ 2}\tau_{y}+8\frac{\tau_{C}^{3}}{p^{\prime 2}}-12\frac{\tau_{y}\tau_{C}^{2}}{p^{ \prime 2}}\right)-\frac{8}{\eta_{C}}\frac{\tau_{C}^{3}}{p^{\prime 2}}\right]=12 \dot{h}\text{ for }\dot{h}<0\text{ and }Z>Z_{C}, \tag{2}\]  
\[Z_{C}=-\frac{\tau_{C}\dot{h}^{2}}{6\eta_{C}\dot{h}}, \tag{3}\]  
\[p^{\prime}_{C}=-\frac{2\tau_{C}}{h}. \tag{4}\]  
where; \(p\) is the pressure in the magnetorheological oil film, \(p^{\prime}\) stands for the pressure gradient in the axial direction, \(\eta\) is the oil viscosity if not affected by a magnetic field, \(\eta_{C}\) is viscosity in the oil core, \(\tau_{y}\) is the yielding shear stress, \(\tau_{C}\) is the shear stress at the core border, \(h\) is the thickness of the magnetorheological fluid film [37; 38], and \(Z\) is the axial coordinate defining the position in the oil film. From the physical point of view, \(Z_{C}\) and \(p^{\prime}_{C}\) are the axial coordinate and the pressure gradient in the axial direction at the core border, respectively. The governing equations are related to the coordinate system XYZ (Figure 5). Equation (1) describes the pressure profile in the high flow rate area, while Equation (2) represents the same in the core. More details on derivation and solution of Equations (1) and (2) can be found in [22]. Figure 4: The flow curve of the bilinear (biviscous) material. Figure 3.0 The coordinate system related to the damper. Components of the hydraulic force, by which the oil layer acts on the rotor journal, is calculated by integration of the pressure distribution around the circumference and along the length of the damper, considering different pressure profiles in the full oil film and cavitation areas. Figure 3.1  
Figure 3.2  
\(F_{mry}\), \(F_{mrz}\) are the y (horizontal) and z (vertical) components of the magnetorheological hydraulic force acting on the rotor journal, \(R_{D}\) is the radius of the damper gap, \(L_{D}\) is the damper length, and \(\varphi\) is the circumferential coordinate (Figure 5). Referring to Equation (2), the pressure in the full oil film depends on the yielding shear stress. Its magnitude can be approximated by a power function:  
Figure 3.3  
where; \(B\) is the magnetic induction, and \(k_{y}\) and \(n_{y}\) are the proportional and exponential constants, respectively. To determine magnetic induction of the magnetic flux generated by the electric coil in the damper gap, it is considered that the damper body consists of a set of meridian segments. Assuming that the magnetic reluctance is significant only in the gap between the rings and negligible in the ferromagnetic material of the damper housing, it holds for the magnetic induction in the damper clearance:  
Figure 3.4  
where; \(k_{B}\) is the design parameter, \(\mu_{0}\) is the vacuum permeability, \(\mu_{MR}\) is the relative permeability of the magnetorheological oil, and \(I\) is the applied current in the coil. The design parameter \(k_{B}\) depends on the number of the coil turns and efficiency of the transmission of the magnetic flux in the magnetic circuit. More details on its determination can be found in [39].