## 2 Principles of Local and Dual-Connected SSDI Techniques
### Dual-Connected SSDI
A direct advantage of the dual-connected SSDI over the local SSDI is that at least half of the numbers of the semi-active circuits can be saved. Here we consider harmonic excitation and compute the steady-state response in Equation (7). To solve this nonlinear structural dynamic problem, we employ the multi-harmonic balance method with a few numerical treatments to enhance the convergence. We have developed an in-house code which passes several test cases and has already been applied to analyze local SSDI [28], SSDNC [29] and dry friction dampers [37]. This method is much faster than the numerical integral method such as the Runge-Kutta method or the Newmark method to calculate the steady-state response. For the sake of brevity, we will not describe the method in this paper, and the readers can find detailed information in the aforementioned literature. To illustrate the general trend of the results, we define some nondimensional parameters. They are: nondimensional time \(\omega t\); nondimensional frequency \(\frac{\omega}{\sqrt{\frac{k_{1}}{m_{1}}}}\); nondimensional force \(\frac{f}{\max(f_{1})}\); nondimensional displacement \(\frac{\lambda k_{1}}{f_{1}}\). These parameters will be used in Figures 4-7. Figure 4: Steady-state response of the nonlinear force of dual-connected SSDI and the relative displacement: (**a**) signal plotted in time domain; (**b**) the force-displacement hysteresis loop. Figure 5: Nonlinear force of dual-connected SSDI in different voltage inversion factor (\(\gamma\)). Figure 6: Vibration reduction performance of the dual-connected SSDI (PP) with respect to the phase difference: (**a**) amplitude of x1 and x2 DOFs with respect to the phase difference; (**b**) steady-state response of x1 and x2 when the phase difference of excitation is 180 degree. Figure 7: Frequency response function of the system with and without the dual-connected SSDI (PP). The steady-state response of the nonlinear force and the relative displacement are plotted in Figure 4b. It can be seen that every extremum of displacement response corresponds to a voltage-mutation (or inversion) from positive value to negative value or inversion. It is the voltage mutation that induces a force opposite to the relative velocity. The loop of the force with respect to the relative displacement is shown in Figure 4b. The area represents the energy dissipated in a vibrating period, and demonstrates the damping nature of the force. When the voltage inversion factor \(\gamma\) varies, the nonlinear force generated by the piezo-patch voltage will vary accordingly, which is illustrated in Figure 5. The higher the voltage inversion factor is, the closer the waveform of the nonlinear force is to the rectangle and the larger the damping effect is. In the following, the damping effect of the shared SSDI shunt is studied through adjusting the phase-difference of two excitations (with the same amplitude). The evolution of response amplitudes of two masses under excitation with the phase difference are plotted in Figure 6a. When the phase difference of two excitations is 0 degrees, the two masses vibrate identically at any moment, and there is no voltage difference across the switch. The switch is out of action in this case and the vibration is not reduced at all. When the phase-difference of two excitations is 180 degrees, the two masses vibrate in opposite phase (as shown in Figure 6b), which means that the displacement-difference of the two masses keeps a great value at any moment, so the voltage difference across the switch reaches the maximum. As a result, the vibration amplitudes of the both two masses achieve the minimum, which means an optimum vibration damping effect. The results signify that the greater the displacement-difference of the masses, the better the damping effect of dual-connected SSDI. This conclusion makes special sense to study the suppression of local vibration of mistuned blisk.