## 3 Rotor-Bearing Model
### Rigid Jeffcott Rotor-TPJB Model  
As can be seen in Figure 5, a \"Jeffcott\"-type symmetric rigid rotor supported on the five-pad TPJB was employed as a mechanical model to investigate the rotordynamic bifurcation analysis owing to  
Figure 4: The finite element model of five-pad TPJB and typical pressure distributions on the pads.  
Figure 3: Five-pad tilting pad journal bearing (TPJB) (load on pad) diagram and the respective coordinates.  
the pad-pivot friction. The governing equations of motion for the rotor and TPJB pads in the system can be written as:  
\[\begin{array}{l}m_{j}\ddot{x}=2F_{x}+W_{dx}+W_{sx}\\ m_{j}\ddot{y}=2F_{y}+W_{dy}+W_{sy}\\ I_{p}^{(j)}\ddot{\delta}_{pitch}^{(j)}=M_{p,pitch}^{(j)}+M_{f,pitch}^{(j)}\\ I_{p}^{(j)}\ddot{\delta}_{roll}^{(j)}=M_{p,roll}^{(j)}+M_{f,roll}^{(j)}\end{array} \tag{8}\]  
where \(m_{j}\) and \(I_{p}\) are the rotor mass and pad moment of inertia, respectively; and \(W_{\rm s}\) and \(W_{\rm d}\) are the static and dynamic loads on the rotor, respectively. \(\ddot{x}\) and \(\ddot{y}\) are second derivatives of rotor translational motion in \(x\) and \(y\) direction respectively. \(\ddot{\delta}_{pitch}\) and \(\ddot{\delta}_{roll}\) are second derivatives of pad pitch and roll motion respectively. Generally, the static load is the weight of the rotor or a side load, and the dynamic force can be unbalance force due to the mass eccentricity on the disc. The separate moments of inertia equations were written for each pad (\(j\)). The numerical values of the disc, bearing, pad parameters, and pressure boundary condition have been provided in Table 3. In particular, the amount of unbalance on disc, bearing diameter to length ratio, pad preload, pad offset, and pivot radius are the major design parameters in the TPJB performance [15; 16]. It is noteworthy that the present rotor model is assumed to have a rigid shaft and symmetrical geometry. This implies that the two journals have the same attitude and no roll motion in the y-z plane. Hence, the second equation in Equation (7) and fourth equation in Equation (8) may be negligible in the numerical study.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Disc Parameter** & & **Value** & **Unit** \\ \hline
**Mass** & (\(M_{\rm J}\)/2) & 509.8 & kg \\
**Amount of Imbalance on Disc \({}^{*}\)** & (\(e_{\rm imb}\)) & 0.0-0.2 C\({}_{\rm b}\) & - \\
**Operation Speed Range \({}^{*}\)** & - & 0-25 & kpm \\ \hline
**Bearing Parameter** & & **Value** & **Unit** \\ \hline
**Bearing Diameter** & (\(D\)) & 100 & mm \\
**Bearing Length \({}^{*}\)** & (\(L\)) & 50, 75, 100 & mm \\
**Bearing Clearance** & (\(C_{\rm b}\)) & 0.1 & mm \\
**Bearing Load** & (\(W\)) & 5 & kN \\
**Lubricant Ambient Pressure** & (\(P_{\rm sup}\)) & 0 & Pa \\
**Lubricant Supply Pressure** & (\(P_{\rm amb}\)) & 0 & Pa \\ \hline
**Pad Parameter** & & **Value** & **Unit** \\ \hline
**Number of Pads (arc Length)** & - & 5 (60 deg. load on & \\  & & pad) & - \\
**Preload \({}^{*}\)** & (\(m_{p}\)) & 1/2, 2/3, 3/4 & - \\
**Offset \({}^{*}\)** & (\(x_{\rm p}\)/\(x\)) & 0.5, 0.55, 0.6 & - \\
**Pad Clearance** & (\(C_{\rm p}\)) & 0.2 & mm \\
**Pivot Radius \({}^{*}\)** & (\(R_{\rm pvt}\)) & 10, 15, 20 & mm \\ \hline \hline \end{tabular}  
* Parametric range for numerical study.  
\end{table}
Table 3: Disc, bearing, and pad parameters of the tilting pad journal bearing (TPJB)-rigid rotor system.  
Figure 5: The TPJB-rigid Jeffcott rotor system.