# Theoretical Design of a Novel Vibration Energy Absorbing Mechanism for Cables
## 3 Theoretical Modeling of the VEAM
### Automatic Switching Between Non-Linear Stiffness and Viscous Damping Modules  
Figure 4(a) presents a simplified single degree of freedom theoretical model of a subsystem of the VEAM along the direction of its central axis. The thick line in the middle indicates the cable, with its two ends fixed between the deck and the pylon under a tensile force (T) acting at each end. The total length of the cable is L. The VEAM is fixed at a distance L1 from the deck. The cable fixing hex is connected to the cable and the damping system structure is fixed to the deck. In the figure, \(y_{c}\) represents the excitation of the stay cable and \(y_{d}\) represents the absolute displacement of the deck. As we are only considering the single degree of freedom case, the role of the ball joint and the rotary joint of the hydraulic cylinder is not considered. Hence, VEAM is simplified as a model in which the magnetic force that is generated by the PMs is connected in series with the hydraulic cylinder and then in parallel with the spring.  
The minimum working pressure of the hydraulic cylinder is mainly caused by the friction of the piston seal, which is mostly dependent on the manufacturing and assembly accuracies, etc. In recent years, in order to improve the efficiency of hydraulic cylinders, researchers have tried to minimize this minimum working pressure through various methods, but static friction that is caused by seals is inevitable. With the development of industrial technology, it is not difficult to maintain the minimum working pressure at a fixed value by controlling the precision of machining and seal manufacturing. This minimum working pressure can be easily determined through experiments. The VEAM realizes automatic switching between the non-linear vibration isolator and the hydraulic viscous damper based on the minimum working pressure of the hydraulic cylinder.  
Figure 5: \(Cont\).  
For each subsystem, it can be observed from Figures 2 and 4 that one PM is fixed on the piston rod and the other is connected to the vibration excitation side, such that like the poles of the two PMs are opposite to each other. When a large amplitude vibration occurs at the excitation side, the distance between the PMs becomes small and the repulsive force increases. This repulsive force is transmitted to the piston rod as a thrust force (\(\mathrm{F_{m}}\)). When this thrust force is less than the minimum working force (\(\mathrm{F_{f}}\)) of the hydraulic cylinder only the non-linear vibration isolator works. Conversely, only the hydraulic viscous damper works when the thrust force is greater than the minimum working force of the hydraulic cylinder. [PERSON] et al. [52] proposed that the repulsive force between two PMs is inversely proportional to the nth power of the distance between them. The two magnets that are discussed in this article are identical and, therefore, satisfy the conditions for the application of this relationship. Thus, the repulsive force between the two PMs can be taken as,  
\[\mathrm{F_{m}}=\mathrm{a(1-y)^{-n}} \tag{9}\]  
Figure 5: Simplified theoretical model of the VEAM.  
where, a (Unit: \(\rm{Nm^{-n}}\)) is the equivalence coefficient of the repulsive force, n denotes the characteristic index, 1 (Unit: m) represents the preset distance between the PMs, and y (Unit: m) denotes the relative displacement between the cable and ground. When only the magnetic non-linear stiffness module works, the inequality \(\rm{F_{m}}\leq\rm{F_{f}}\) exists. Substituting Equation (9) into this inequality, we get the relationship between the preset distance of the two PMs and the excitation amplitude:  
\[1\leq y+\sqrt[n]{\frac{a}{F_{f}}} \tag{10}\]