## 3 Establishment of a Single-Degree-of-Freedom Eddy Current Damping Vibration Suppression Model
### Analysis of Vibration Characteristics of Eddy Current Damping Vibration Suppression Model  
To analyze the vibration suppression characteristics of the plate-type electromagnetic damper system, forced vibration analysis on the damping system is performed. According to the 'Specifications for Acceptance of Quality in Highway Engineering Construction', the maximum allowable deviation for the elevation difference of a second-level cement road is \(\pm 5\) mm/m. Therefore, in this simulation experiment, a 5 mm displacement amplitude sine wave was use as the excitation source. As the excitation frequency coincides with the natural frequency of the damped structure, it will cause resonance phenomenon, which will significantly amplify the vibration response. In order to investigate the vibration suppression characteristics of the damped structure, it is first necessary to determine the natural frequency of the damped structure. Since the structure is lightweight and has sufficient stiffness, the local deformations can be neglected, and each component can be treated as a rigid motion. Extract the natural frequency of the system using Equations (14) and (15). The extracted natural frequency of the damping structure is shown in Table 1. The selection of parameters is shown in Table 2 and Equation (28). Considering that the human body is sensitive to vertical vibrations in the frequency range of 4-8 Hz, and the natural frequency of the damping structure intersects with this frequency band, this frequency band as the damping range of interest should be prioritized. By applying excitation within this frequency band, it can be used to analyze the characteristics of the electromagnetic damping obtained by varying the current for different weight groups. The operating conditions applied to the damping system are shown in Table 2. \begin{table}
\begin{tabular}{c c} \hline
**Lead/kg** & **Frequency/Hz** \\ \hline
60 & 4.59 \\
70 & 4.25 \\
80 & 3.98 \\
90 & 3.75 \\
100 & 3.56 \\ \hline \end{tabular}
\end{table} 
Table 1: Natural frequency at different loads. \begin{table}
\begin{tabular}{c c c} \hline
**Frequency/Hz** & **Magnetic Potential/AT** & **Load/kg** \\ \hline
4–10 & 0, 2000, 4000, 8000, 10,000 & 60, 70, 80, 90, 100 \\ \hline \end{tabular} Where zero AT is the control group without eddy current damping, and the load is the mass of the human body plus the upper part of the seat.  
\end{table} 
Table 2: Simulation condition parameters. Consider the supporting effect of the spring and apply the spring force to the system:  
\[F=k\times(x_{1}-x_{0}) \tag{28}\]  
where \(\mathbf{k}=50\) N/mm, \(\mathbf{x_{1}}-\mathbf{x_{0}}\) is the vertical displacement difference between the copper plate and the electromagnetic array. Considering the air and spring damping, set the damping coefficient \(\mathbf{c_{0}}\) to 0.2 N\(\cdot\)s/m. From the above model, the relationship between the amplitude response and the magnetic potential for different loads is calculated as shown in Figures 5-9. Figure 5: The amplitude response at 60 kg in theoretical model. Figure 6: The amplitude response at 70 kg in theoretical model. Figure 7: The amplitude response at 80 kg in theoretical model. From Figures 5-9, it can be seen that the resonance band gradually shifts to the left and the vibration isolation band gradually becomes wider as the mass increases; the increase of the magnetic potential significantly improves the suppression of the resonance peak, but at the same time increases the amplitude in the non-resonance band. The general trend of the amplitude response with excitation frequency and magnetic dynamic potential is shown in Figure 10.  
In this section, the motion model for a single-degree-of-freedom eddy current damping vibration structure is established, the motion differential equations are derived and solved, and the parameters that influence vibration control are analyzed.