## 3 Rotor-Bearing Model  
### Five-Pad Tilting Pad Journal Bearing (TPJB)  
Figure 3 represents the schematics of the five-pad TPJB model employed in this study. The hydrodynamic pressure of the lubricant film on each pad, \(p\), is able to be estimated using the Reynolds equation for iso-viscous, incompressible, and laminar flow conditions as follows:  
\[\frac{\partial}{\partial\theta}\bigg{(}\frac{h^{3}}{12\mu_{\
u}}\frac{\partial p }{\partial\theta}\bigg{)}+\frac{\partial}{\partial z}\bigg{(}\frac{h^{3}}{12 \mu_{\
u}}\frac{\partial p}{\partial z}\bigg{)}=\frac{R\omega}{2}\frac{ \partial h}{\partial\theta}+\frac{\partial h}{\partial t} \tag{3}\]  
where \(z\) and \(\theta\) are the axial and angular positions of the bearing, respectively; \(\mu_{\
u}\) is the dynamic viscosity of the lubricant oil; \(\omega\) is the angular velocity of the journal; and \(h\) is the film thickness on the pad at the {\(z\), \(\theta\)} position:  
\[ \begin{array}{c}h(\theta,z)=C_{p}^{(j)}-\bigg{(}e_{x}-e_{z}\delta_{roll} \cos(\theta_{p}^{(j)})\bigg{)}\cos(\theta)-\bigg{(}e_{y}-e_{z}\delta_{roll} \sin(\theta_{p}^{(j)})\bigg{)}\sin(\theta)\\ -\bigg{(}C_{p}-C_{b}\bigg{)}\cos(\theta-\theta_{p}^{(j)})-\delta_{pitch}R \sin(\theta-\theta_{p}^{(j)})\end{array}  \tag{4}\]  
where \(e\) is the vector that designates the current position in the reference frame {x, y, z}; \(\theta_{p}\) is the pivot position in the \(\theta\) axis; \(\delta\) is the tilt angle of the pad; and \(C_{b}\) and \(C_{p}\) are the clearances of the bearing and pad, respectively. Superscript (\(j\)) denotes the \(j\)th pad, and subscripts \(x,y,z,pitch\) and \(roll\) represent the relevant axes and motions. The solution of the Reynolds equation was calculated by applying a finite element model, as shown in Figure 4. It consisted of a three-node simplex, triangular-type mesh created on the overall fluid film layer on the pad. The fluid repelling force between the pad and shaft was obtained by integrating the pressure throughout the mesh for each pad as follows:  
\[\left\{ \begin{array}{c}F_{x}^{(j)}\\ F_{y}^{(j)}\end{array} \right\}=\int_{0}^{L}\int_{\theta_{p}^{(j)}}^{\theta_{p}^ {(j)}}p^{(j)}(\theta,z)\bigg{\{} \begin{array}{c}-\cos(\theta)\\ -\sin(\theta)\end{array} \bigg{\}}d\theta dz \tag{5}\] \begin{table}
\begin{tabular}{c c c c c c} \hline  & & & & **Viscosity (c5i)** & \\ \cline{2-5}  & **40**\({}^{\circ}\)**C** & **100**\({}^{\circ}\)**C** & **55**\({}^{\circ}\)**C (at Inlet)** & **Specific Gravity at 15**\({}^{\circ}\)**C** & **Viscosity Index** \\ \hline
**ISO-VG22** & 22 & 4.3 & 13.8 & 0.850 & 98 \\ \hline \end{tabular}
\end{table} 
Table 2: Lubricant condition parameters for the SCM. Figure 2: Stribeck curve for the friction coefficient as a function of pad tilting speed on its spherical pivot. where \(F^{(j)}\) is the hydrodynamic force on the \(j\)th pad, \(L\) is the bearing length, \(\theta_{B}^{(j)}\) is the beginning angle of the \(j\)th pad, and \(\theta_{E}^{(j)}\) is the end angle of the \(j\)th pad. The boundary conditions was incorporated with an assumption of ambient pressure at the bearing ends (\(p_{\text{amb}}=p|_{z=\pm\frac{1}{2}}\)) and supply pressure at the leading and trailing edges (\(p_{\text{sup}}=p|_{\theta=\theta B}^{(j)},\theta_{E}^{(j)}\)). In addition, Reynolds cavitation boundary conditions were applied in the calculation. The integrated pad pressure functions as the normal force, which induces the friction moment in Equation (2). The overall hydrodynamic force from entire pads to the journal can be calculated as:  
\[\left\{ \begin{array}{c}F_{x}\\ F_{y}\end{array} \right\}=\sum_{j=1}^{N_{p}}\left\{ \begin{array}{c}F_{x}^{(j) }\\ F_{y}^{(j)}\end{array} \right\} \tag{6}\]  
The moment on the pad about its pivot owing to the pressure distribution can be expressed as:  
\[