# Cost-Effective Design Modification of a Sleeve Bearing with Large Bearing Clearance
## 3 Numerical Simulation
### Bearing Models
#### 3.3.2 Model of the TPJB  
As displayed in Figure 2, the hydropower machine is supported by three bearings, and one of them is the TB. The TB is a 12-shoe TPJB, and the schematics and technical details are illustrated in Figure 4 and Table 2, respectively. The bearing consists of 12 pads, each pivot positioned at \(\alpha=0^{\circ}\), \(30^{\circ}\), \(60^{\circ}\)... \(330^{\circ}\). The \(X\) and \(Y\) axes represent the global coordinate with the origin at the center of the bearing. On the other hand, the local coordinate is represented by the \(\xi\) and \(\eta\) axes, where the \(\xi\)-axis is always parallel to the line joining the bearing and journal centers. The location of the journal center from the bearing center is given as a function of the eccentricity angle (\(\alpha\)) and trajectory of the journal center (\(e_{j}\)).  
Figure 3: Schematic representation of a four-lobe bearing with grooves.  
The bearing was modeled as direct and cross-coupled stiffness and damping coefficients and included in the **K** and **C** matrices of Equation (1) for the rotordynamic simulations. The procedure that was used in this paper to determine the bearing coefficients closely resembles the method used in [24]. The bearing coefficients were precalculated by solving the fluid film lubrication model and represented using exponential functions as well as sine/cosine functions. Commercial software, RAPPID 3.22 [25], which employs a Navier-Stokes-based lubrication model, was used to calculate the bearing coefficients at eight predefined relative eccentricities, i.e., \(\epsilon=[0.01,0.1,0.2,\ldots,0.7]\). For each relative eccentricity, the coefficients were calculated for 25 different eccentricity angles (\(\alpha\)) in the interval \(\alpha\in[-15^{\circ},+15^{\circ}]\), and the maxima and minima coefficients were obtained. The bearing coefficients vary periodically between the maxima and minima coefficients as a function of \(\alpha\) (Figure 5a), which can be expressed by sine/cosine functions (Equations (10) and (11)). The maxima and minima coefficients were approximated with exponential equations, i.e., \(ae^{be}+ce^{de}\), where \(a\), \(b\), \(c\), and \(d\) are constants obtained from a curve-fitting. The small circles in Figure 5b represent the maxima and minima coefficients obtained from RAPPID, whereas the solid lines show the exponential curve-fitting using a MATLAB inbuilt function \(exp2\). The same procedure was applied for all stiffness and damping coefficients of the bearing.  
\[K_{ij}(\epsilon,\alpha,\Omega)=\frac{\overset{\sim}{K_{ij}}^{max}(\epsilon, \Omega)+\overset{\sim}{K_{ij}}^{min}(\epsilon,\Omega)}{2}\;sign\;\frac{ \overset{\sim}{K_{ij}}^{max}(\epsilon,\Omega)-\overset{\sim}{K  
Furthermore, the bearing coefficients in Equations (10) and (11) are given in the local coordinates. To be able to use them in Equation (1), they must be transformed into the fixed coordinates using Equations (12) and (13):  
\[\mathbf{K_{B}}=\mathbf{T^{T}K_{\beta}T} \tag{12}\]  
\[\mathbf{C_{B}}=\mathbf{T^{T}C_{\beta}T} \tag{13}\]  
where \(\mathbf{K_{\beta}}\) is the local bearing stiffness matrix, \(\mathbf{C_{\beta}}\) is the local bearing damping matrix, \(\mathbf{T}\) is the transformation matrix, and \(\mathbf{K_{B}}\) and \(\mathbf{C_{B}}\) are the bearing stiffness and damping matrices in the fixed coordinate, respectively.  
\[\mathbf{T}=\begin{bmatrix}\cos\left(\alpha\right)&\sin\left(\alpha\right)\\ -\sin\left(\alpha\right)&\cos\left(\alpha\right)\end{bmatrix}\]