# Theoretical Design of a Novel Vibration Energy Absorbing Mechanism for Cables
## 4 Numerical Simulation
### Non-Linear Stiffness Isolator  
When the magnetic repulsive force is less than the minimum working force of the hydraulic cylinder, only the non-linear stiffness vibration isolator composed of the magnetic springs works. In this section, we will individually alter the key parameters of the coil springs and PMs used in the non-linear vibration isolator in order to observe the influence of the parameters \(\mathrm{k_{s}}\), \(\mathrm{k_{b}}\), \(\mathrm{b}\), and \(\zeta\) on the vibration isolation performance of the system and summarize the design experience. In the first case, the linear stiffness of the coil springs is set as 1000 Nm\({}^{-1}\), 2000 Nm\({}^{-1}\), and 3000 Nm\({}^{-1}\), while the other parameters are kept at the default values that are shown in Table 1, and the displacement transmissibility of the non-linear stiffness isolator module is observed. The simulation results are shown in Figure 9. It can be observed that the simulated transmissibility curve moves to the high frequency region with the increase in linear stiffness of the spring. This is accompanied by an increase in the peak value of transmissibility. When the stiffness is 1000 Nm\({}^{-1}\), the maximum displacement transmissibility of 3.9 occurs at a frequency of 16.5 Hz. When the stiffness is 3000 Nm\({}^{-1}\), the maximum displacement transmissibility of 4.5 occurs at a frequency of 21.7 Hz. This shows that increasing the linear stiffness of the spring in the VEAM can effectively reduce the displacement transmissibility in the low frequency region. This means that the vibration isolation performance of the VEAM at a tuning frequency can be optimized using springs with higher stiffness values. Figure 8: Simulation results for the acceleration and the corresponding Fourier spectrum of the VEAM. The results of the second case are presented in Figure 10, which shows the effect of different equivalent linear stiffness obtained using different magnetic springs on the displacement transmissibility of the VEAM. The solid, dotted, and dash-dot-dash curves represent the magnetic spring equivalent linear stiffness values of 2000 Nm\({}^{-1}\), 3500 Nm\({}^{-1}\), and 5000 Nm\({}^{-1}\), respectively. It can be seen that the shifting of the transmissibility curve is similar to that observed while varying the linear stiffness of the spring; i.e., as the equivalent linear stiffness increases the curve shifts to the high frequency region. At frequencies below 18.5 Hz, the VEAM sub-system with a k\({}_{\rm b}\) value of 5000 Nm\({}^{-1}\) has a lower transmissibility than that of the sub-system with a k\({}_{\rm b}\) value of 2000 Nm\({}^{-1}\). At 18.5 Hz, the transmissibility in both cases is equal with a value of 3.2. Beyond this frequency, the transmissibility with a k\({}_{\rm b}\) value of 5000 Nm\({}^{-1}\) becomes higher than that with a k\({}_{\rm b}\) value of 2000 Nm\({}^{-1}\). Therefore, according to these results the VEAM sub-system with larger k\({}_{\rm b}\) value will have better vibration isolation performance in the low frequency range, but it will not work satisfactorily in the high frequency range. b is the only non-linear stiffness parameter in the VEAM system, so it is necessary to consider its influence on the vibration isolation performance of the whole sub-system. The solid, dotted, and  
Figure 10: Transmissibility of the non-linear stiffness isolator module of VEAM with varying values of k\({}_{\rm b}\). Figure 9: Transmissibility of the non-linear stiffness isolator module of VEAM with varying values of k\({}_{\rm s}\). dash-dot-dash curves in Figure 11 indicate the simulated transmissibility of the VEAM sub-system with respect to b values of 0 (no non-linear stiffness component in the system), default value (\(8.1\times 10^{6}\)), and \(10\times 10^{9}\) (an ideal value chosen in order to clearly see the effect of increasing the value), respectively. It can be observed that, as the value of b increases, the peak value of the transmissibility decreases and shifts towards the high-frequency region.