## 5 Vibration Reduction of Bladed Disks by Dual-Connected SSDI
### Dynamic Model
\tag{13}\]  
where the non-dimensional nonlinear force of shared SSDI shunt is expressed as follows:  
\[g_{\text{nl},j}=-\frac{(1+\chi)}{(1-\chi)}\cdot\gamma_{\text{p}}y_{\text{A},j} ^{\text{M}}\cdot\text{sign}(y_{\text{b2},j+1}^{\prime}-y_{\text{b2},j}^{\prime })-\gamma_{\text{p}}\cdot(y_{\text{b2},j+1}-y_{\text{b2},j}),\ j=1,3,5,\cdots N-1 \tag{14}\]  
\[g_{\text{nl},j}=-g_{\text{nl},j-1}\ j=2,4,6,\cdots N \tag{15}\]  
where \(y_{\text{A},j}^{\text{M}}\) represents the amplitude maximum of the relative displacement \(\left(y_{b2,j+1}-y_{b2,j}\right)\). It is better to express the governing equations in the matrix form:  
\[\lambda^{2}\mathbf{M}\mathbf{y}^{\prime\prime}(\tau)+\lambda\mathbf{C}\mathbf{y}^{{}^{\prime} }(\tau)+\mathbf{K}\mathbf{y}(\tau)+\mathbf{g}_{\text{nl}}(\mathbf{y}(\tau),\mathbf{y}^{{}^{\prime} }(\tau),\tau)=\mathbf{g}(\tau) \tag{16}\]  
where vector of generalized displacement \(\mathbf{y}=[\mathbf{y}_{1},\mathbf{y}_{2},\cdots,\mathbf{y}_{N}]^{T}\) and \(\mathbf{y}_{j}=[y_{b1,j},y_{b2,j},y_{d,j}]^{T}\). The mass matrix writes:  
\[\mathbf{M}=\text{Bdiag}(\mathbf{M}_{1},\mathbf{M}_{2},\mathbf{M}_{3},\cdots,\mathbf{M}_{N}) \tag{17}\]  
where Bdiag() represents a block diagonal matrix, and the sub-matrix has the form:  
\[\mathbf{M}_{j}=\left( \begin{array}{ccc}1&0&0\\ 0&\delta_{\text{mb}}+\delta_{\text{me}}&0\\ 0&0&\delta_{\text{md}}\end{array} \right) \tag{18}\]  
The stiffness matrix writes:  
\[\mathbf{K}=\text{Bcirc}(\mathbf{K}_{1},\mathbf{K}_{\text{c}},\mathbf{0},\cdots,\mathbf{0},\mathbf{K}_ {\text{c}}) \tag{19}\]  
where and Bcirc() represents a block circulant matrix, and the sub-matrix has the form:  
\[\mathbf{K}_{j}=\left( \begin{array}{ccc}1&-1&0\\ -1&1+\gamma_{\text{b}}+\gamma_{\text{e}}&-(\gamma_{\text{b}}+\gamma_{\text{e}}) \\ 0&-(\gamma_{\text{b}}+\gamma_{\text{e}})&\gamma_{\text{b}}+\gamma_{\text{e}}+ \gamma_{\text{d}}+2\gamma_{\text{c}}\end{array} \right) \tag{20}\]  
\[\mathbf{K}_{\text{c}}=\left( \begin{array}{ccc}0&0&0\\ 0&0&0\\ 0&0&-\gamma_{\text{c}}\end{array} \right) \tag{21}\]  
In this work, the Rayleigh damping model are used to describe the mechanical damping of the blisk:  
\[\mathbf{C}=\alpha_{\text{c}}\mathbf{M}+\beta_{\text{c}}\mathbf{K} \tag{22}\]  
where \(\alpha_{\text{c}}\) and \(\beta_{\text{c}}\) are the coefficients of Rayleigh damping. Random mistuning is supposed to occur in the stiffness of the blade. Hence, the non-dimensional equations of the writes:  
\[\mathbf{M}\mathbf{y}^{\prime\prime}(\tau)+\mathbf{C}\mathbf{y}^{{}^{\prime}}(\tau)+(\mathbf{K}+ \Delta\mathbf{K})\mathbf{y}(\tau)+\mathbf{g}_{\text{nl}}(\mathbf{y}(\tau),\mathbf{y}^{{}^{\prime} }(\tau),\tau)=\mathbf{g}(\tau) \tag{23}\]  
where he mistuning matrix \(\Delta\mathbf{K}\) of blade stiffness is given as follows:  
\[\Delta\mathbf{K}=\text{Bdiag}(\Delta\mathbf{K}_{1},\Delta\mathbf{K}_{2},\cdots,\Delta\mathbf{K} _{N}) \tag{24}\]and  
\[\Delta K_{j}=\left[ \begin{array}{ccc}1+\delta\gamma_{\text{b},j}&-(1+\delta \gamma_{\text{b},j})&0\\ -(1+\delta\gamma_{\text{b},j})&1+\gamma_{\text{b}}+2\delta\gamma_{\text{b},j}+ \gamma_{\text{e}}&-(\gamma_{\text{b}}+\delta\gamma_{\text{b},j}+\gamma_{\text{ e}})\\ 0&-(\gamma_{\text{b}}+\delta\gamma_{\text{b},j}+\gamma_{\text{e}})&\gamma_{ \text{b}}+\delta\gamma_{\text{b},j}+\gamma_{\text{e}}+\gamma_{\text{d}}+2 \gamma_{\text{c}}\end{array} \right] \tag{25}\]  
The bladed disks are subjected to the traveling wave excitation produced by up-stream vanes and the rotor rotation. If the traveling wave has only one harmonic, the excitation has the following expression:  
\[g_{j}(\tau)=\left[g_{\text{b}1,1}(\tau),g_{\text{b}2,1}(\tau),g_{\text{d},1}( \tau)\right]^{\text{T}}\otimes\dot{e}^{i(j-1)\cdot E\cdot\theta} \tag{26}\]  
where \(\left[g_{\text{b}1,1}(\tau),g_{\text{b}2,1}(\tau),g_{\text{d},1}(\tau)\right]^ {T}\) is the excitation vector applied to the reference sector. The symbol \(\otimes\) defines as the Kronecker tensor product, \(i\) is the imaginary unit and \(\theta=2\pi/N\) is the phase difference between the adjacent blades.