# Theoretical Design of a Novel Vibration Energy Absorbing Mechanism for Cables
## 4 Numerical Simulation
### Preset Distance of Magnet  
In Section 3.1, we mentioned that the automatic switching of the non-linear stiffness vibration isolator and hydraulic viscous damper is mainly determined by the distance between the magnets (I) and the minimum working force (F\({}_{\text{f}}\)) of the hydraulic cylinder. However, different permanent magnets exhibit different magnetic repulsive force characteristics. We must understand the effect of the hydraulic cylinder working force on the distance between PMs under different PM parameters in order to design a VEAM that can switch modes when the input vibration amplitude is above the specified value. In order to evaluate this, we set the mode switching amplitude y as 0.002 m in this simulation. Figure 6 presents the relationship between the designed distance between the PMs and the minimum working force of the hydraulic cylinder according to Equation (10). We performed simulations that are based on three kinds of PMs with different magnetic repulsive force characteristics. The solid line represents the curve obtained using the PMs with a = 0.4 and n = 1.0, where a and n are consistent with Equation (9). The dotted line and the dash-dot-dash line represent the curves for when a = 0.6, n = 1.1, and a = 0.8, n = 1.2, respectively. Because the magnet calculation formula involves the power function calculation that may get an infinite value, the ordinate is only taken to an effective distance of 0.05 m. \begin{table}
\begin{tabular}{c c} \hline \hline
**Item** & **Value** \\ \hline Stiffness of the springs, \(\text{k}_{\text{s1}}=\text{k}_{\text{s2}}\left(\text{Nm}^{-1}\right)\) & 2000 \\ Stiffness of the linear item of magnetic spring, \(\text{k}_{\text{b}}\left(\text{Nm}^{-1}\right)\) & 3500 \\ Stiffness of the non-linear item of magnetic spring, \(\text{b}\left(\text{Nm}^{-3}\right)\) & \(8.1\times 10^{6}\) \\ Mass, m (kg) & 0.5 \\ Structural damping ratio of the springs, \(\zeta_{\text{s1}}=\zeta_{\text{s2}}\) & 0.02 \\ Damping ratio of the hydraulic cylinder module, \(\zeta_{\text{h}}\) & 0.2 \\ Amplitude of vibration excitation, \(\text{Y}_{\text{c}}\left(\text{m}\right)\) & 0.0005 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Default parameters for simulation. Figure 6: Graph showing the relationship between the preset distance between Permanent magnets (PMs) and the working force of the hydraulic cylinder. It can be observed that the three curves show an exponential downward trend as the minimum working force required by the hydraulic cylinder increases, which also reflects the characteristics of the magnet repulsive force. We can also observe that, for the curve with a larger magnetic repulsion factor, the allowable distance between PMs is also greater. Assuming that the minimum working force of the hydraulic cylinder is 200 N, the curve obtained using the magnet repulsive force factor of a = 0.4 and n = 1.0 shows a distance of 0.004 m, which is very close to the preset amplitude value. The reciprocating motion that is caused by vibration is likely to cause the PMs to collide with each other, which may cause damage to the magnet or other components of the VEM. During the design stage, the allowable distance between PMs should be greater than the preset amplitude to ensure the safety of the system. At the same minimum working force, the allowable distance on the dash-dot-dash curve with a strong magnetic repulsive force is 0.012 m, which is two times higher than the solid curve. Even so, when considering the limitations of geometry and safety factor in actual application, the allowable distance between PMs provided by the simple structural Neodymium magnets (Nd2 Fe14B) cannot fully guarantee the safety of the system. However, this work only provides a basis for the design optimization of the VEM. Based on these methods, we can devise a safer system using stronger magnets in the future by possibly utilizing some sort of motion limiting mechanism or collision prevention buffers between the magnets.