## 6 Conclusions  
We have proposed the dual-connected SSDI to control the vibration of bladed disks in aero-engines. The dynamic model of the new circuit configuration is derived, and the damping performance is numerically demonstrated. The dual-connected SSDI is implemented by a low-power circuit board and the performance is experimentally validated by a double-beam test rig. The main finds are:  
* The dual-connected configuration can increase the voltage inversion factor compared with the local configuration. 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. The damping performance increases monotonously with the amplitude of the relative displacement. In this regard, the in-phase vibration cannot be damped by the proposed approach, and this can be a limitation in the general application. However, in the application of bladed disks, due to the complexity of the excitation and existing of many (20-30) blades, we can always avoid the worst working condition. This is one of the reasons why we think the dual-connected SSDI can be a good candidate for vibration reduction of bladed disks. Additionally, the dual-connected SSDI share the limitations with traditional local SSDI, such as the need of external power and the difficulty of reliably implanting piezoelectric materials into the blades.
* The dual-connected configuration can reduce at least half the number of switching shunts while maintain nearly the same performance as the conventional (local) SSDI, while applying to bladed disks. It not only reduces the vibration amplitude, but also mitigate the vibration localization. All the vibration peaks excited by multi-engine-order force can be effectively suppressed. Note that the proposed configuration does not extend the applicable frequency band of the SSDI technique. It basically shares the same frequency band of the local SSDI. As proved in the literature [24; 25; 26; 27; 28; 29; 30; 31; 32] that the SSDI can work in a relatively wide frequency band thus provides multi-modal damping. It is important to keep the voltage reverse happened in a very short time compared to the vibration period. To do this the inductance is chosen such that the oscillating frequency of the inductance with intrinsic capacitance is more than 20 times higher than the vibration frequency. The damping performance is also closely related to the modal electromechanical coupling factor (MEMCF) of the piezoelectric patch [39]. Once the geometric parameters of the piezoelectric patch are fixed, it can provide good MEMCF in a certain frequency band. For example, when elastic wave-length associated the vibration is too small or too long compared to the length of the patch, generally the  
Figure 22: Forced response of the most dangerous blades under multi-engine-order excitation (E = 0, 1, 2 at the same time).  
MEMCF is very low [40]. One should be aware these general limitations of SSDI and piezo-based techniques during practical implementation.
* Future work can focus on the optional connection in blades in order to further reduce the number of switching elements.  
Conceptualization, F.Z. and L.L.; methodology, F.Z. and J.L.; validation, F.Z. and J.L.; writing--original draft preparation, L.L. and F.Z.; writing--review and editing, Y.F.; visualization, F.Z. and Y.F.; supervision, L.L.; project administration, Y.F.; funding acquisition, L.L. and Y.F. All authors have read and agreed to the published version of the manuscript.  
This research was funded by National Natural Science Foundation of China, grant numbers 51675022 and 11702011.  
The authors declare no conflicts of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.