## 2 System Model
### Dynamic Modeling of the Coupled System
\begin{table}
\begin{tabular}{c c c c} \hline \hline \begin{tabular}{c} **Current** \\ **(A)** \\ \end{tabular} & \begin{tabular}{c} **Torsional Stiffness** \\ **(N \(\cdot\) m/rad)** \\ \end{tabular} & \
\begin{tabular}{c} **Natural Frequency** \\ **(Hz)** \\ \end{tabular} &
\begin{tabular}{c} **Amplitude** \\ **(\({}^{\circ}\))** \\ \end{tabular} \\ \hline
1 & 0.3843 & 0.210 & 54.27 \\ \hline
2 & 0.7686 & 0.292 & 38.89 \\ \hline
3 & 1.153 & 0.353 & 32.24 \\ \hline
4 & 1.537 & 0.402 & 28.28 \\ \hline
5 & 1.922 & 0.444 & 25.67 \\ \hline
6 & 2.306 & 0.480 & 23.74 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Variable stiffness control effect of the system. Figure 4: Frequency response of the system. Figure 5: Comparison of system frequency-shift effect under different conditions: single joint and dual joints. With the joint torsional stiffness being changed at the resonant peak, the vibration suppression effect is most obvious. Therefore, it can be supposed that the exciting frequency is the first natural frequency of the system when the joint receives DC-2 A power. Figure 6a shows the time domain response of the system when the external excitation frequency is 0.292 Hz, corresponding to the point A in Figure 4, and when the joint torsional stiffness is 0.7686 N - m/rad in Table 2. In Figure 6a, the exciting frequency is the first natural frequency of the system under DC-2 A, the resonance appears, and then the angle amplitude reduces from 38.89\({}^{\circ}\) (0 A) to 1.88\({}^{\circ}\) (4 A) and 0.97\({}^{\circ}\) (6 A), and the vibration isolation can be realized by the frequency-shifting phenomenon. Furthermore, as shown in Figure 4, we see the external excitation frequency of 0.266 Hz (point B) and 0.316 Hz (point C) near 0.292 Hz (point A). In Figure 6b, the external excitation frequency reaches 0.266 Hz, and the angle of magnitude lowers from 42.78\({}^{\circ}\) (0 A) to 9.904\({}^{\circ}\) (2 A), 1.61\({}^{\circ}\) (4 A) and 0.88\({}^{\circ}\) (6 A). In Figure 6c, the excitation frequency reaches 0.316 Hz, and the angular amplitude lowers from 35.93\({}^{\circ}\) (0 A) to 9.964\({}^{\circ}\) (2 A) to 2.33\({}^{\circ}\) (4 A) and 0.98\({}^{\circ}\) (6 A). It is important to note that, although the points B and C are not at the resonance peak, the increase in joint torsional stiffness causes the system resonance frequency to shift (or increase), causing the points B and C to produce a certain vibration isolation effect, as shown in Figure 6b,c. In Figure 7, when DC-2 A current is continuously applied to both joints, the system maintains a certain torsional stiffness constraint throughout. The external excitation frequency is located on the right side of the system resonance frequency when DC-2 A is applied. It is demonstrated that the higher the external excitation frequency is, the smaller the joint rotation angle is. Figure 6: Time domain response of the system excitation at (**a**) 0.292 Hz, (**b**) 0.266 Hz and (**c**) 0.316 Hz. In Figure 8, when a DC-6 A current power supply is applied to both joints, the first eight modes of the system are performed. In Figure 8, the first mode (0.48 Hz) is dominated by the rigid body motion, supplemented by the bending deformation of the plate, and the rigid body moves in the same direction as the bending deformation of the plate. The second mode is the bending mode (5.47 Hz), and the third mode is the coupled bending-twisting and vibration mode (8.51 Hz). After the third mode, the modal frequency is higher. However, in this study, we mainly focused on the low-frequency vibration below 1 Hz, not on the high-order modes above 1 Hz, thereby leaving only the rigid body mode and pure bending mode. Therefore, in this study, it is reasonable to simplify the two-dimensional plate into a one-dimensional beam. Figure 8: _Cont._  
Figure 7: Comparative results of rotation angle response excitation at higher frequency. (**a**) Time domain response; (**b**) Frequency domain response.