## 2 System Model
### Principle of Tunable Magnetically Controlled Stiffness/Damping  
(a) Tunable magnetically controlled stiffness.  
The magnetic stiffness \(k\) comes from that, with the coils being loaded with DC power, the joint output torque is almost linearly proportional to the rotation angle. The equivalent magnetic stiffness is expressed as follows:  
\[k=\frac{dT_{drive}}{d\theta} \tag{1}\]  
where \(T_{drive}\) and \(\theta\) represent the driving torque and rotation angle of the joint, separately. For further details about the driving torque \(T_{drive}\), please see reference [36].  
(b) Tunable magnetically controlled damping.  
The magnetic damping \(c\) comes from the eddy current damping resulting from the relative motion between the permanent magnet rotor and the stator core of the joint. The equivalent magnetic damping of the joint can be controlled by actively adjusting the input current. The equivalent magnetic damping coefficient is obtained as  
\[c=\frac{dT_{damp}}{d\omega} \tag{2}\]  
where \(T_{damp}\) and \(\omega\) represent the eddy current damping torque and rotational speed of the joint, separately. The joint damping torque \(T_{damp}\) is related to the material and structure parameters of permanent magnets and coils as well as the input current. For more details about the damping torque, please see reference [36].  
However, whether it is the variable stiffness or variable damping, which contributes the most to the vibration isolation effect needs to be further explained.  
On one hand, at a low frequency, although the frequency is low, the rotation angle is large. This large angle leads to a higher speed of cutting the magnetic lines in unit time. Therefore, the eddy current damping is large. When the joint is loaded with DC-2 A, the current intensity is small, resulting in a small joint torsional stiffness. However, even so, the external excitation frequency is still less compared with the system natural frequency when dual joints are loaded with DC-2 A, resulting in no obvious effect of the variable stiffness control of the system. In such a case, variable damping is considered as the main element, while variable stiffness secondary.  
On the other, at high frequency, the rotation angle is small, which will lead to a lower speed of cutting magnetic lines in unit time. Therefore, the eddy current damping is small. Meanwhile, when the current increases, the joint torsional stiffness also increases. Consequently, variable stiffness is the leading element, while variable damping secondary.  
Moreover, compared with the coupling system of a flexible beam and single magnetic joint in reference [34], the plate in this paper has a longer length, wider width, and higher flexibility. Meanwhile, the torsional stiffness of dual joints is greater than that of single joint. Ideally, it can better reflect the effect of joint variable stiffness vibration suppression. Furthermore, there are usually two joints between the substrates of solar panels. Therefore, this study will focus on the tunable magnetically controlled stiffness/damping semi-active vibration control of the coupling system of dual joints and a single flexible plate.