## 5 Conclusions  
To conclude, a new electromagnetic joint actuator for vibration control with tunable magnetically controlled stiffness/damping is proposed. The variable stiffness stems from the fact that when the adjusting coils of the joint mechanism are supplied with DC power, the output torque of the joint is nearly linear with the rotation angle. The variable damping derives from the eddy current damping, which results from the relative motion between the permanent magnet rotor and the joint stator core. Moreover, the greater the joint loading current and the rotating speed, the greater the eddy current damping torque.  
Firstly, the tuning mechanism of the joint stiffness/damping parameters are explained for the coupling system of dual joints and a single flexible plate. The coupling dynamic equations are acquired by adopting the Lagrange equation and the assumed mode approach. Then, the semi-active vibration control simulations are conducted. The frequency-shift effect of the system is studied for the variable stiffness joint. Finally, the experimental platform of dual joints and a single flexible plate is built. The non-contact permanent magnet excitation is employed at the root of the flexible plate to simulate the single-frequency excitation. The non-contact airflow excitation is imposed on the surface of the flexible plate to simulate the multi-frequency excitation. The experimental results indicate that, for dual joints and a single flexible plate system, in the range of 0.06-0.343 Hz, magnetic-controlled damping dominates and magnetic-controlled stiffness is auxiliary. The primary reason is the fact that when dual joints are loaded with DC-2 A, the external excitation frequency is less when compared with the natural frequency of the system. There is no obvious effect of magnetic-controlled stiffness of the system. In addition, it is also experimentally verified that the dual joint actuation has good synchronization. In conclusion, the proposed control method is simple and effective. Only through the joint actuation, the low-frequency vibration of the flexible plate can be efficiently hindered, and the plate surface does not need any other intelligent equipment.  
Furthermore, this study focuses on the basic research on the common problems abstracted from solar panel structures in space microgravity environment, which is realized by a scaled flexible plate model. The theoretical simulations and experimental verifications are adopted, and the semi-active vibration control effect of magnetic joints is demonstrated by building an experimental platform. The above-stated research results are significant for guiding the vibration attenuation of flexible space solar panels.  
Conceptualization, methodology, writing--original draft, W.H.; data curation, software, writing review & editing, J.W.; validation, supervision, formal analysis, Q.Z.; visualization, project administration, J.S.; investigation, funding acquisition, X.Z. All authors have read and agreed to the published version of the manuscript.  
This research was funded by Shanghai Sailing Program, grant No. 20 YF1417100, and China Postdoctoral Science Foundation, grant No. 2021M692066. The APC was funded by 20 YF1417100.  
The study did not require ethical approval.  
Not applicable.  
The authors declare no conflict of interest.