# Cost-Effective Design Modification of a Sleeve Bearing with Large Bearing Clearance
## 4 Results and Discussion
### Simulation Results  
_Rotor over-speeding condition without UMP_: The Campbell diagram and stability map were plotted for rotor speeds up to 3 \(\times\)\(\Omega_{0}\). An unbalance response simulation was first conducted at the nominal speed for both models (MOB and MMB), and the maximum amplitudes of the trajectory of the journal were obtained at each bearing location. Based on those values, the stiffness and damping coefficients of each bearing were computed at the prescribed relative eccentricity ratio for various rotor speeds, and subsequently,  
Figure 5: (**a**) The local direct stiffness (\(K_{\xi\xi}\)) as a function of \(\alpha\) for 60% relative eccentricity (\(\epsilon\)). (**b**) The maxima and minima coefficients of \(K_{\xi\xi}\) as a function of \(\epsilon\). the modal parameters of the system were identified. Figures 6 and 7 show the Campbell diagram and stability map of the models of the MOB and MMB, respectively. The first seven modes of the system are plotted as a function of the normalized rotor speed. Figure 8 displays the corresponding mode shapes plotted at \(1\times\Omega_{0}\). For each plot in Figures 6 and 7, the simulation results from the full and reduced models are overlaid and compared. The maximum deviation of the modal parameters due to model reduction (\(11-\text{Reduced/Full}\) \(\times\) 100%) was less than 1%, and thus, the reduced model can be considered as an accurate estimate of the full model. The first mode of both models, which were the sub-synchronous modes, demonstrated high damping values, exceeding 70%. For the MOB, the synchronous line (1X) crossed the second mode at 2.05 \(\times\)\(\Omega_{0}\), which is below the runaway speed. Furthermore, the second critical speed crossing the third mode at 2.25 \(\times\)\(\Omega_{0}\) is considerably close to the runaway speed (\(\Omega_{\text{r}}=2.23\times\Omega_{0}\)). For the MMB, however, the damped natural frequencies of the second and third modes were high enough to surpass the 1X-synchronous line with a significant margin; subsequently, no critical speed was found within the frequency range. The synchronous line crossed the fourth mode at 2.29 \(\times\)\(\Omega_{0}\), which exceeds the runaway speed. Figure 8: The mode shape of the first seven modes plotted at \(1\times\Omega_{0}\). The mode shapes are enclosed in color-coded boxes, where each color corresponds to specific lines in Figures 6 and 7. Figure 7: (**a**) Campbell diagram and (**b**) damping ratio of the MMB. The solid lines represent the results from the reduced model, while the circles show those from the full model. The color-coded lines correspond to a distinct mode shape shown in Figure 8.  
Figure 9: The simulated trajectory of the journal center at the UGB, LGB, and TB locations. Furthermore, the damped natural frequency and damping ratio of the two systems at the nominal rotor speed were calculated and compared. For illustration purposes, the change in dynamics with the bearing clearance was investigated by gradually changing the values of \(C_{b}\) from 1.6 mm (original LGB) to 0.45 mm (modified LGB). The unbalance response simulations were performed, and Figure 10 displays the maximum trajectory of the journal center as a function of \(C_{b}\). Note that the first points (\(C_{b}\) = 1.6 mm) refer to the simulation results for the MOB. Furthermore, the vertical axis of the right-hand side of each figure shows the percentile ratio of the response with respect to the reference (MOB). The magnitude of the trajectory of the journal decreases with the bearing clearance. The stiffness and damping coefficients of each bearing were calculated based on prescribed eccentricities (Figure 10), and the modal parameters of the system were identified. In Tables 3 and 4, the normalized damped natural frequencies and the damping ratios of the first seven modes are presented for models with different \(C_{b}\) values of the LGB, respectively.