# Cost-Effective Design Modification of a Sleeve Bearing with Large Bearing Clearance
## 3 Numerical Simulation
### Bearing Models
#### 3.3.1 The Fluid Film Force of the UGB and LGB  
For the rotordynamic simulation of the hydropower machine model defined in Equation (1), the bearing restoring forces (\(\mathbf{f_{b}}\)) of the UGB and LGB had to be calculated at each time step by solving the fluid film lubrication model. These bearing forces were modeled based on Reynolds' equation, which is a partial differential equation that describes the fluid film pressure in the gap between the journal and bearing surfaces (Equation (6)):  
\[\frac{1}{R^{2}}\frac{\partial}{\partial\psi}\left(h^{3}\frac{\partial p}{ \partial\psi}\right)+\frac{\partial}{\partial z}\left(h^{3}\frac{\partial p}{ \partial z}\right)=6\mu\Omega\frac{\partial h}{\partial\psi}+12\mu\frac{ \partial h}{\partial t} \tag{6}\]  
where \(p\) is the fluid film pressure, \(\mu\) is the lubricant's viscosity, \(h\) is the fluid film thickness, \(\Omega\) is the rotor speed, and \(\psi\) and \(z\) are the circumferential and axial axes, respectively. Figure 3 shows the schematics of the four-lobe bearing. The center of the lower left pad is displayed, with the segment radius equal to the sum of the journal's radius and half of the diametric pad clearance (\(R+0.5C_{p}\)). In the case of the sleeve bearing, the center of the pads coincides with the center of the bearing, and the diametric pad clearance (\(C_{p}\)) and bearing clearance (\(C_{b}\)) are equal. The location of the journal center is represented as a function of the eccentricity angle (\(\phi\)) and the trajectory of the journal center (\(e_{j,x}\) and \(e_{j,y}\)) at the \(j\)-node. The fluid film thickness, \(h(\psi)\), derived from geometrical relationships, is given in Equation (7) [21; 22]. Unlike the sleeve bearing, the equation for fluid film thickness of \begin{table}
\begin{tabular}{l c c c c} \hline \hline  & **UGB** & **LGB** & **TB** \\ \hline
**Geometry** & & & & \\ \hline Type & \begin{tabular}{c} Sleeve \\ bearing \\ \end{tabular} & \
\begin{tabular}{c} Sleeve \\ bearing \\ \end{tabular} &
\begin{tabular}{c} Four-lobe \\ bearing \\ \end{tabular} & TPJB \\ \hline Journal diameter (mm) & 1050 & 2450 & 2450 & 1100 \\ \hline Number of segments/pads & - & - & 4 & 12 \\ \hline Dia. bearing clearance (mm) & 0.6 & 1.6 & 0.45 & 0.35 \\ \hline Offset ratio (-) & 0.5 & 0.5 & 1 & 0.6 \\ \hline Preload ratio (-) & 0 & 0 & 0.718 & 0.65 \\ \hline Axial length (mm) & 345 & 185 & 185 & 180 \\ \hline Arc length (degree) & - & - & 85 & 18.5 \\ \hline
**Material** & & & & \\ \hline Surface pad material & & & & Babbitt \\ \hline Density (kg/m\({}^{3}\)) & & & & 7280 \\ \hline Radial thickness (mm) & 5 & 6 & 6 & 3 \\ \hline Base pad material & & & Steel \\ \hline Density (kg/m\({}^{3}\)) & & & & 7780 \\ \hline Radial thickness (mm) & 65 & 100 & 100 & 52 \\ \hline
**Lubricant Properties** & & & & \\ \hline Type & & & Turboway 68 \\ \hline Inlet oil temperature (\({}^{\circ}\)C) & & & 60 \\ \hline Oil supply pressure (MPa) & & & 0.1 \\ \hline Density (kg/m\({}^{3}\)) & & & 880 \\ \hline Viscosity at 40 \({}^{\circ}\)C (mPa-s) & & & 61 \\ \hline Viscosity at 100 \({}^{\circ}\)C (mPa-s) & & & 7.7 \\ \hline
**Bracket** & **UF** & **LF** & **TF** \\ \hline Mass (kg) & \(10\times 10^{3}\) & \(10\times 10^{3}\) & \(10\times 10^{3}\) \\ \hline Stiffness (N/m) & \(3\times 10^{8}\) & \(1.87\times 10^{9}\) & \(2.2\times 10^{9}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Technical specifications of the bearings. the four-lobe bearing includes an extra term (\(r_{p}cos(\psi-\psi_{c})\)) and requires treating every lobe separately.