## 5 Vibration Reduction of Bladed Disks by Dual-Connected SSDI
### Forced Response Results  
A random mistuning pattern of the blade stiffness is shown in Figure 17(a). Its mean value is 0 and standard deviation is 2%. First, we do not apply any piezoelectric damping, and the response amplitudes of 24 blades are shown in Figure 17(b). It can be seen that some blades (for example, Blade 19) have larger response amplitude than others. Figure 17(c) shows the phase response associated with Figure 17(b). Since the excitation has engine-order three, we can anticipate an approximate 45 degree (\(360\times 3/24\)) phase difference of the consecutive blades. This can be verified in Figure 17(c). The response amplitudes of 24 blades in three cases (tuned, mistuned with and without dual-connected SSDI) are compared in Figure 19. To solve the nonlinear equations, multi-harmonic balance method is employed with a few additional techniques to improve the convergence [28; 31]. The vibration of blades is significantly reduced by the dual-connected SSDI (around 50% reduction). The blade originally with largest amplitude receives the largest amount of vibration reduction and becomes one of the blade with smallest vibration in the controlled case. On other hand, the vibration localization is also mitigated, as can be seen by the standard deviation of blade amplitudes shown in Figure 20. According to Figure 17(c), the interconnected blades has vibration phase around 45 degrees, and according to Figure 16, it is not the best situation but can still generate some damping. We would like to note that the results in Figure 16 are obtained where the two structures do not have any mechanical coupling. However, in Figure 19 there are mechanical coupling among the interconnected blades (by the disk). Moreover, in the real working situation of bladed disks, the phase difference can be different when the load and rotation speed changes. Therefore, it is difficult to tune the interconnection always to the best situation. The results in Figure 19 show that even though the phase difference may not be the best case, substantial damping can still be generated. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline
**Variables** & \(N\) & \(\delta_{\text{mb}}\) & \(\delta_{\text{md}}\) & \(\gamma_{\text{b}}\) & \(\gamma_{\text{d}}\) & \(\gamma_{\text{c}}\) & \(\gamma_{\text{e}}\) & \(\gamma_{\text{p}}\) & \(\alpha_{\text{c}}\) & \(\beta_{\text{c}}\) & \(\delta_{\text{fb}}\) & \(\delta_{\text{fd}}\) \\ \hline Value & 24 & 1 & 12 & 1 & 1.5 & 30 & 0.05 & 0.01 & \(10^{-6}\) & 0.05 & 1 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Non-dimensional parameters of the blisk. Figure 19: Response amplitude of 24 blades in three configurations (tuned, mistuned with and without dual-connected SSDI). Figure 18: The targeted vibration problem of a bladed disk: (**a**) the mistuned pattern of the blade stiffness; (**b**) response amplitude of 24 blades without SSDI circuit (excited by engine-order E = 3); (**c**) the response phase of the 24 blades (excited by engine-order E = 3 at the same frequency as (**b**)). We select the frequency-response curves among 24 blades with largest peak value under different engine-order excitations (E = 0, 1, 2 respectively) and plot them in Figure 21. Different vibration modes are excited by different engine order excitation. This means that not only the amplitudes but also the phases of the blades are different. The damping effect of the dual-connected SSDI is significant, despite a slightly vary with the different modes. Figure 22 shows the greatest frequency-response in 24 blades with dual-connected SSDI under multi-harmonic excitation (including engine-orders 0, 1 and 2 at the same time). Results of local SSDI are also plotted for the sake of comparison. Note that all vibration peaks excited by multi-harmonic excitation can be suppressed by the dual-connected SSDI shunt, the effect is comparable with that of the local SDI shunt. However, the number of semi-active circuits in the dual-connected SSDI shunt is only half those of the local SSDI.