## 5 Vibration Reduction of Bladed Disks by Dual-Connected SSDI
### Dynamic Model  
Here a lumped parameter model of bladed disk is considered, and it has three DOFs per sector as shown in Figure 17. Let \(N\) be the number of sectors of the model. \(m_{\text{b1}}\), \(m_{\text{b2}}\) and \(m_{d}\) represent the mass of blade body, blade root and the disk-sector, respectively. \(k_{\text{b1}}\), \(k_{\text{b2}}\) and \(k_{d}\) represent the stiffness of them. \(c_{\text{b1}}\), \(c_{\text{b2}}\) and \(c_{d}\) are the mechanical damping factors of them. \(k_{c}\) is the coupling stiffness between adjacent disk sectors. A piezo-patch is supposed to be inserted at the root of blade, the position is between  
Figure 16: Response vs. phase difference (degree) of excitation. and \(m_{d}\) in the lumped parameter model in each sector. The piezo-patch is simulated to be parallel to \(k_{\rm b2}\) in the lumped parameter model. \(m_{e}\) and \(k_{e}\) are the mass and the stiffness of each piezo-patch. Applying the piezoelectric SSDI shunt technique in blisk means that every piezo-patch will be connected with a SSDI shunt. But with the shared SSDI shunt, every two blades are connected in series with one SSDI circuit as shown in Figure 17. Thus, at least half of the needed switches can be saved. The governing equations of the bladed disk can be expressed on a single sector (the Jth) based on [PERSON]'s law:  
\[\left\{ \begin{array}{l}m_{\rm b1}\ddot{x}_{\rm b1,j}+k_{\rm b1}x_{\rm b1,j}- k_{\rm b1}x_{\rm b2,j}=f_{\rm b1,j}(t)\\ (m_{\rm b2}+m_{e})\ddot{x}_{\rm b2,j}-k_{\rm b1}x_{\rm b1}+(k_{\rm b1}+k_{\rm b2 }+k_{\rm e})x_{\rm b2,j}-(k_{\rm b2}+k_{\rm e})x_{\rm d,j}+f_{\rm nl,j}=f_{\rm b 2,j}(t)\\ m_{\rm d}\ddot{x}_{\rm d,j}-(k_{\rm b2}+k_{\rm e})x_{\rm b2,j}+(k_{\rm b2}+k_{ \rm e}+k_{\rm d}+2k_{\rm c})x_{\rm d,j}-k_{\rm c}x_{\rm d,j-1}-k_{\rm c}x_{\rm d,j+1}=f_{\rm d,j}(t)\end{array} \right. \tag{9}\]  
where \(f_{\rm nl,j}\) is the nonlinear force produced by dual-connected SSDI and can be expressed as:  
\[f_{\rm nl,j}=-\frac{(1+\chi)}{(1-\chi)}\cdot\frac{\alpha^{2}x_{\rm a,j}^{\rm M }}{C_{\rm p}}\cdot\mathrm{sign}\Big{[}\dot{x}_{\rm b2,j+1}-\dot{x}_{\rm b2,j} \Big{]}-\frac{\alpha^{2}}{C_{\rm p}}\cdot\Big{[}x_{\rm b2,j+1}-x_{\rm b2,j} \Big{]},\ j=1,3,5,\cdots N-1, \tag{10}\]  
\[f_{\rm nl,j}=-f_{\rm nl,j-1},\ j=2,4,6,\cdots N \tag{11}\]  
and \(x_{\rm b1,j}\) and \(x_{\rm b2,j}\) are the displacements of the blade body and blade root in sector \(j\) respective; \(x_{d,j}\) represents the displacement of disk sector \(j\); terms \(f_{\rm b1,j}\), \(f_{\rm b2,j}\) and \(f_{d,j}\) are the exciting forces from the airflow on the blade body, blade root and disk sector \(j\), respectively. To make the results more general, the parameters in Equation (9) are normalized by:  
\[\left\{\begin{array}{l}\tau=\omega t\\ \omega_{\rm b1}=\sqrt{k_{\rm b1}/m_{\rm b1}}\\ \lambda=\omega/\omega_{\rm b1}\\ \gamma_{\rm b}=k_{\rm b2}/k_{\rm b1}\\ \gamma_{\rm d}=k_{\rm d}/k_{\rm b1}\\ \gamma_{\rm c}=k_{\rm c}/k_{\rm b1}\\ \gamma_{\rm e}=k_{\rm e}/k_{Therefore, we obtain the following non-dimensional dynamic equations:  
\[\left\{ \begin{array}{l}\lambda^{2}y_{\text{b1},j}^{\prime\prime}+y_{\text{b1},j}- y_{\text{b2},j}=g_{j}(\tau)\\ \lambda^{2}(\delta_{\text{mb}}+\delta_{\text{me}})y_{\text{b2},j}^{\prime\prime }-y_{\text{b1},j}+(1+\gamma_{\text{b}}+\gamma_{\text{e}})y_{\text{b2},j}-(y_{ \text{b}}+\gamma_{\text{e}})y_{\text{d},j}+g_{\text{nl},j}=\delta_{\text{fb}}g _{j}(\tau)\\ \lambda^{2}\delta_{\text{md}}y_{\text{d},j}^{\prime\prime}-(\gamma_{\text{b}}+ \gamma_{\text{e}})y_{\text{b2},j}+(\gamma_{\text{b}}+\gamma_{\text{e}}+\gamma _{\text{d}}+2\gamma_{\text{c}})y_{\text{d},j}-\gamma_{\text{c}}y_{\text{d},j-1} -\gamma_{\text{c}}y_{\text{d},j+1}=\delta_{\text{fd}}g_{j}(\tau)\end{array} \right.