## 2 Principles of Local and Dual-Connected SSDI Techniques
### Dual-Connected SSDI  
The proposed dual-connected configuration is illustrated in Figure 1b,c. In this configuration, the control rules of the switching circuit are different from the local SSDI described in Figure 1a. The extreme of the difference between two displacement-signals of two piezo-patches should be detected, other than one displacement-signal from one piezo-patch. When the relative displacement reaches the maximum or the minimum, the switch is closed and then it is open instantaneously. There are two ways of interconnection due to different alignment of the polling direction, termed the PP (positive-to-positive, Figure 1b) and PN (positive-to-negative, Figure 1c) connections respectively. In the PP connection, the positive poles of both PZT (Lead-Zirconate-Titanate) patches are connected together by the inductor and the switching. In the PN connection, the polling direction of structure 2 is reversed, so the positive pole of pzt1 and negative pole of pzt2 are connected in the SSDI circuit. In both the PP and PN connections, the switching is controlled by the same way, i.e., by the relative displacement of two structures. In the dual-connected configuration, the intrinsic capacitance of the two piezo-patches are connected in series, and the equivalent capacitance in the electric field decreases by 50%. The electrical quality factor and the voltage inversion factor are deduced, according to Equations (1)-(3). The closed time of the switch becomes shorter. Thus, a larger quality factor and a higher voltage inversion factor are obtained in the dual-connected SSDI. We first conduct some numerical analysis on a rather simplified model to show the characteristics of the dual-connected SSDI (with PP connection). The model is shown in Figure 3 where \(m_{1}\) and \(m_{2}\) represent the two modal mass, \(k_{1}\) and \(k_{2}\) represent the two modal stiffness, and \(c_{1}\), \(c_{2}\) are the structural damping coefficients of the corresponding mechanical system, respectively. The mass \(m_{1}\) and the mass \(m_{2}\) are subjected to the external periodic excitation \(f_{1}(t)\) and \(f_{2}(t)\) respectively. Figure 3: A 2-DOF (degree-of-freedom) mechanical system with dual-connected SSDI. The governing equations of the system are:  
\[\left\{ \begin{array}{l}m_{1}\ddot{x}_{1}+c_{1}\dot{x}_{1}+(k_{1}+k_{\rm e})x_{1 }+f_{\rm nl}\Big{(}x_{1}-x_{2},\dot{x}_{1}-\dot{x}_{2},t\Big{)}=f_{1}(t)\\ m_{2}\ddot{x}_{2}+c_{2}\dot{x}_{2}+(k_{2}+k_{\rm e})x_{2}-f_{\rm nl}\Big{(}x_{1 }-x_{2},\dot{x}_{1}-\dot{x}_{2},t\Big{)}=f_{2}(t)\end{array} \right. \tag{7}\]  
where \(k_{\rm e}\) represents the stiffness of the piezo-patch, and \(f_{\rm nl}\) is the nonlinear force generated by the dual-connected SSDI. The expression of \(f_{\rm nl}\) can be derived in the similar manner as \(f_{shunt}\) in Section 2.1, and the only difference is to replace terms \(x\) and \(\dot{x}\) in Equation (6) into \(x_{1}-x_{2}\) and \(\dot{x}_{1}-\dot{x}_{2}\) respectively:  
\[f_{\rm nl}\Big{(}x_{1}-x_{2},\dot{x}_{1}-\dot{x}_{2},t\Big{)}=-\frac{(1+\gamma _{net})\alpha_{p}^{\ 2}}{(1-\gamma_{net})C_{p}}\cdot(x_{1}-x_{2})_{M}\cdot\text{sign} \Big{(}\dot{x}_{1}-\dot{x}_{2}\Big{)}-\frac{\alpha_{p}^{\ 2}}{C_{p}}(x_{1}-x_{2}) \tag{8}\]  
Note that the dual-connected SSDI generates a pair of self-equilibrium forces to the connected structures. The force applied to each structure is in opposite direction to the relative velocity thus plays the role of damping. In analogy with the dry friction dampers, a local SSDI is similar to an under-platform friction damper which can be modelled as a contact to the ground. In this case the damping force is only determined by the vibration of the local blade itself. However, a dual-connected SSDI is similar to a blade shroud, whose damping force is determined by the relative displacement. Besides the performance advantage of SSDI over dry friction [31], one additional advantage of the dual-connected SSDI is that one can choose which two blades to be connected, and this is more flexible than the shroud.