# Theoretical Design of a Novel Vibration Energy Absorbing Mechanism for Cables
## Mechanism of the VEAM
In this figure, points A, B, C, D, E, and F represent the rotating centers of the bearings between the hydraulic cylinders and the system structure, while d and \(\
ot\) represent the magnitude and angle (acute angle with respect to the horizontal) of displacement of the center point of the cable fixing hex. Circles with black solid hatching represent the joint balls. G is a ball in the initial (default) position and G' is the final position of ball G after the displacement. Figure 3: \(Cont\). It can be observed that the displacement GG' is equal to d and the angle \(\angle\)G'GA is equal to \(\angle\)\(\theta\). The distances from the cable fixing hex center to the ball and to the rotating center of the bearing at the initial position are known. Thus, the distance between the ball at position G and the bearing rotating center at the initial position is GA = OA - OG. According to the law of cosines, G'A can be deduced as:  
\[\mathrm{G^{\prime}A}=GA^{2}+d^{2}-2\cdot GA\cdot d\cdot\cos\theta \tag{1}\]  
Here, we assume that the radial deformation of the spring is zero, which means that it can only perform telescopic movement along the axial direction. Through the law of sines, the rotation angle of the hydraulic cylinder can be expressed as:  
\[\angle\mathrm{A}=\arcsin\left(\frac{d\cdot\sin\theta}{G^{\prime}A}\right) \tag{2}\]  
From the geometric relationship, it can be seen that the rotation angle of the cable fixing hex side magnet around the ball center and the rotation angle of the hydraulic cylinder are alternate interior angles. Thus, combining Equations (1) and (2), the \(\angle\)A can be rewritten as:  
\[\angle\mathrm{A}=\arcsin\left(\frac{d\cdot\sin\theta}{GA^{2}+d^{2}-2\cdot GA \cdot d\cdot\cos\theta}\right) \tag{3}\]  
The rotation angles of the other hydraulic cylinders can also be calculated with Equations (4)-(8):  
\[\angle\mathrm{B}=\arcsin\left(\frac{d\cdot\sin\left(\theta+\frac{\pi}{3} \right)}{GA^{2}+d^{2}-2\cdot GA\cdot d\cdot\cos\left(\theta+\frac{\pi}{3} \right)}\right) \tag{4}\]  
\[\angle\mathrm{C}=\arcsin\left(\frac{d\cdot\sin\left(\theta+\frac{2\pi}{3} \right)}{GA^{2}+d^{2}-2\cdot GA\cdot d\cdot\cos\left(\theta+\frac{2\pi}{3} \right)}\right) \tag{5}\]  
Figure 3: Kinematic response of the VEAM to cable displacement. \[\angle\mathrm{D}=\arcsin\left(\frac{d\cdot\sin(\pi-\theta)}{GA^{2}+d^{2}-2\cdot GA \cdot d\cdot\cos(\pi-\theta)}\right) \tag{6}\]  
Figure 2:0  
Figure 2:1  
A conventional viscous damper forms a hydraulic circuit by connecting the two hydraulic chambers cut off by the piston by opening an orifice in the piston itself. In this case, mainly the friction between the hydraulic oil and the orifice provides the damping effect. However, because the orifice is inside the hydraulic cylinder, it is difficult to change the orifice size after assembly. In order to facilitate the adjustment of the damping ratio, this proposed design utilizes an external circuit design, as shown in Figure 2:2 The double triangle marks in the figure indicate the orifice regulating valves, while LL, LR, RL, and RR represent the hydraulic chambers divided by the pistons. When the cable fixing hex is in the default position, the left and right subsystems of the VEAM are in a balanced state and the pistons on both sides are positioned in the middle of their respective hydraulic cylinders with equal pressures on both sides of the pistons. The LL chamber is connected to the RR chamber and the LR chamber is connected to the RL chamber. The purpose of this inter-connection is to ensure the continuity of the damping process. Assuming that the repulsive force between the PMs is large enough, when the cable fixing hex moves leftwards the magnet pushes the left side piston, reducing the LL chamber volume and increasing its internal pressure. Therefore, the hydraulic oil in the LL chamber flows through the orifice and the piping to the RR chamber, which is at a relatively lower pressure. At the same time, the pressure in the LR chamber is reduced, which sucks in the hydraulic oil from the RL chamber.