## 2 Cantilever-Type ECTMD
### Dynamic Characteristics  
The dynamic characteristics of the ECTMD can be estimated theoretically in advance. The estimation results can guide the process of selecting appropriate materials to fabricate the ECTMD. The force resulting from ECTMD, and acting on the main structure takes a form of [29]  
\[f_{\text{ECTMD}}=c_{\text{d}}v+k_{\text{d}}x \tag{1}\]  
where \(c_{\text{d}}\) and \(k_{\text{d}}\) are the damping coefficient and stiffness of the ECTMD respectively, and \(v\) and \(x\) are the relative velocity and displacement of the mass to the main structure respectively. A simplified method to predict the damping ratio of the ECTMD is presented below. Firstly, the magnetic flux density at the copper plate is required. For the cylindrical magnet in Figure 3, the magnetic flux density, \(B\), at the point \((0,0,z)\) is [34]  
\[B_{u}=B(0,0,z)=B_{0}\frac{\frac{z}{\left[\frac{r_{0}^{2}+z^{2}}{2}\right]^{1/2} }-\frac{z-t}{\left[\frac{r_{0}^{2}+(z-t)^{2}}{2}\right]^{1/2}}}{\frac{t}{\left[ \frac{r_{0}^{2}+t^{2}}{2}\right]^{1/2}}},\ \ \ \ \ z\geq t \tag{2}\]  
where \(B_{0}\) is the magnetic flux density at the point \((0,0,t)\), \(r_{0}\) and \(t\) are the radius and thickness of the cylindrical magnet. Assume that the center of copper plate is located at the point \((0,0,z)\), and a uniform magnetic field is assumed to be across the copper plate. Then the damping coefficient of the eddy current can be estimated by [35]  
\[c_{\text{d}}=\frac{\alpha\pi D^{2}{B_{u}}^{2}t}{4\rho} \tag{3}\]  
where \(\alpha\) is a non-dimensional parameter, \(D\) is the diameter of the magnet, and \(\rho\) is the electrical resistivity of copper. With the assumption that the internal resistance of the copper plate at the magnetic field is equal to the external resistance, \(\alpha\) is set as 0.5. The corresponding damping ratio of the ground-mounted ECTMD can be obtained by  
\[\zeta_{a}=\frac{\pi B_{u}^{2}D^{2}t}{16\rho\omega m} \tag{4}\]  
where \(m\) is the mass of magnet, and \(\omega\) is the natural circular frequency of the ground-mounted ECTMD. The elastic restoring force of ECTMD is provided by the cantilever, and can be approximately computed by  
\[f_{k}=\frac{3 EI}{L^{3}}x \tag{5}\]  
where \(E\) is the Young's modulus of the cantilever, \(I\) is the second area moment of the cross-section, and \(L\) is the length of the cantilever. The natural frequency of a ground-mounted ECTMD, designated as the frequency of the ECTMD, can be predicted, which is based on an idealized model of a cantilever  
Figure 3: Magnetic flux density along centerline of cylindrical magnet. beam with a tip mass. Note that the mass of the cantilevered beam is relatively small and negligible. Thus, the corresponding natural circular frequency can be computed by  
\[\omega=\sqrt{\frac{3 EI}{mL^{3}}} \tag{6}\]  
The parameters of the ECTMD are listed in Table 1.  
Dynamic characteristics of the ground-mounted ECTMD were identified through free vibration tests. The experimental setup of the free vibration tests is shown in Figure 4. To achieve a free vibration of the magnet, the magnet is given an initial displacement and is then released. A computer vision-based approach with corresponding identifying method [36] for dynamic displacement response is employed in the free vibration test of the ground-mounted ECTMD. A Huawei Mate 20 Pro cellphone was employed to record the video of motion, and the frame rate is 235 frames per second. The vibration responses are firstly filtered with the frequency component near the fundamental frequency left, and then processed by Hilbert transform [37]  
\[y(t)=\frac{1}{\pi}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\  
the damping ratios of the ECTMD in the two directions are very close. The difference between the results in two directions is less than 2% for natural frequency and 8% for damping ratio, which verifies the direction independent characteristic of the ECTMD used herein.