# Numerical Investigations on the Blade Tip Clearance Excitation Forces in an Unshrouded Turbine
## 3 Results and Discussion
### Effects of the Operating Conditions  
In this section, the effects of the operating conditions on the stiffness coefficients are presented. Figure 9 shows the variation of stiffness coefficients with frequency at different rotational speeds. Both the direct and cross-coupling stiffness coefficients decreased with the rotational speed, which indicates that the increase of rotational speed caused a decrease of the tip clearance excitation effect. This was because the torque on the rotor decreased with increasing rotational speed and hence caused a decrease in the response forces in accordance with Equation (1). Moreover, the direct stiffness coefficients at lower frequencies were more easily affected compared to  
Figure 11 presents the variation of the stiffness coefficients with frequency with different rotor eccentricities in the y-direction. The eccentricity could be introduced using a mesh deformation. For an unshrouded turbine with a specific rotor eccentric ratio (\(ER\)) in the y-direction, the locations of nodes in part A were:  
\[\left\{ \begin{array}{c}X_{t}=X_{0}+x\\ Y_{t}=Y_{0}+ER\cdot t_{m}+y\end{array} \right.. \tag{13}\]  
The locations of the nodes in part B were:  
\[\left\{ \begin{array}{c}X_{t}=X_{0}+\frac{R_{t}-R_{0}}{R_{t}-R_{t}}x\\ Y_{t}=Y_{0}+ER\cdot t_{m}+\frac{R_{t}-R_{0}}{R_{t}-R_{t}}y\end{array} \right.. \tag{14}\]  
According to the results, the eccentricity had little impact on the direct stiffness coefficient, especially for the result at low frequencies. When the rotor eccentric ratio (\(ER\)) increased, the direct stiffness coefficient showed an accelerated decreasing trend from \(ER=0\)-\(0.3\) and then a decelerated decreasing trend from \(ER=0.3\)-\(0.7\). For the cross-coupling stiffness coefficients, the results at low frequencies increased with the eccentricity from \(ER=0\)-\(0.3\) and decreased rapidly from \(ER=0.3\)-\(0.7\). However, the results at high frequencies decreased continually with the eccentricity from \(ER=0\)-\(0.7\). These results indicate that the tip clearance excitation effect reduced when the unshrouded turbine rotor was at a high eccentricity. Figure 10: Variation of the rotordynamic coefficient with the inlet flow angle: (**a**) direct stiffness coefficient and (**b**) cross-coupling stiffness coefficient. Figure 9: Variation of stiffness coefficient with rotational speed: (**a**) direct stiffness coefficient; (**b**) cross coupling stiffness coefficient. Figure 12 shows the rotordynamic coefficient with different mean tip clearances. It was shown that the direct stiffness at selected frequencies decreased dramatically with the mean tip clearance, especially for the results with \(t_{m}/h=2\%\)-\(3\%\). In addition, the results at low frequencies were more easily affected by the mean tip clearance than at high frequencies. Moreover, it should be noted that the direct stiffness coefficients became negative at frequencies over 220 Hz for \(t_{m}/h=3\%\)-\(4\%\). The variation of the cross-coupling stiffness showed a different pattern compared to that of the direct stiffness. When \(t_{m}/h\) increased from \(1\%\) to \(2\%\), the cross-coupling stiffness coefficients increased for all frequencies, which indicates increased instability forces on the rotor. This was because too small of a tip clearance leads to a change of tip leakage and hence causes a change of torque on the blades. When \(t_{m}/h\) continued to increase from \(2\%\)-\(3\%\), the cross-coupling stiffness at low frequency (\(\Omega<160\) Hz) increased and that at high frequency (\(\Omega>160\) Hz) decreased. Meanwhile, the dependency of the cross-coupling stiffness with the whirling frequency reduced a lot. As \(t_{m}/h\) increased further from \(3\%\) to \(4\%\), the cross-coupling stiffness decreased for all frequencies. Therefore, when the mean tip clearance was too small (\(t_{m}/h=1\%\)), reducing the tip clearance excitation forces by increasing the mean tip clearance may not be feasible.