# Mechanical Modeling and Dynamic Characteristics of a Three-Directional Vibration Absorber
## 3 Results and Discussion
### Static Modeling and Analysis of the Three-Directional Vibration Isolator
#### 3.2.3 Comprehensive Analysis of Three-Directional Static Characteristics  
To achieve effective three-dimensional vibration isolation at the static equilibrium position, the spatial inclination angle of the spring assemblies in the three-directional vibration isolator should be 35.26\({}^{\circ}\), as determined by Equation (3). Under this condition, the isolator maintains equal stiffness in both the vertical and lateral directions, ensuring optimal isolation performance. Consequently, the three-directional isolator satisfies the following expressions:  
\[\left\{\begin{array}{l}K_{n}=\frac{mg}{4L\sin 35.26^{\circ}(1-\cos\theta_{0} \sqrt{1+\tan 35.26^{\circ}})}\\ \lambda=\frac{t}{\lambda}=\sin\theta_{0}-\cos\theta_{0}\tan 35.26^{\circ}\\ \delta=\frac{K_{n}}{K_{2}}=\frac{\sin\theta_{0}-\cos\theta_{0}\tan 35.26^{ \circ}}{4\sin 35.26^{\circ}\left(1-\cos\theta_{0}\sqrt{1+(\tan 35.26^{\circ})^{2}} \right)}\end{array}\right. \tag{14}\]  
(1) Influence of the stiffness ratio on load distribution \(\delta\)  
As shown in Figure 5, when \(\lambda=0.05\) and \(\theta_{0}=37.6^{\circ}\), it can be observed that an increase in the stiffness ratio results in a greater axial force generated by the spring elements for a given deformation. Consequently, both the vertical and horizontal elastic forces acting on the three-directional vibration isolator increase.  
Figure 5: Vertical and lateral static characteristic curves of the three-directional vibration isolator (\(\lambda=0.05\), \(\theta_{0}=37.6^{\circ}\)): (**a**) vertical force–displacement curve; (**b**) lateral force–displacement curve; (**c**) vertical stiffness–displacement curve; (**d**) lateral stiffness–displacement curve.  
However, as the spatial inclination angle gradually decreases with increasing displacement, the vertical force provided to the isolator diminishes. Moreover, the rate of force reduction surpasses the rate of displacement increase, leading to a gradual decrease in vertical stiffness as displacement increases. Conversely, under these conditions, the spatial inclination angle has a negligible effect on the horizontal stiffness in the transverse direction. These findings indicate that the load-carrying capacity of the three-directional vibration isolator is strongly influenced by the stiffness of the spring elements, emphasizing the critical role of stiffness ratio selection in optimizing the isolator's mechanical performance. (2) Impact of initial inclination angle on deformation and load capacity \(\theta_{0}\) As depicted in Figure 6, when \(\lambda=0.05\) and \(\delta=0.73\), Equation (14) indicates that the static deformation a of the three-directional vibration isolator increases with a larger initial inclination angle. This increase in static deformation results in greater compression or elongation of the spring elements, leading to a higher axial elastic force generated within the system. Consequently, the vertical elastic force provided to the isolator at static equilibrium also increases.  
Figure 6: Vertical and lateral static characteristic curves of the three-directional vibration isolator (\(\lambda=0.05\), \(\delta=0.73\)): (**a**) vertical force–displacement curve; (**b**) lateral force–displacement curve; (**c**) vertical stiffness–displacement curve; (**d**) lateral stiffness–displacement curve.  
These findings suggest that the load-carrying capacity and working stroke of the three-directional vibration isolator are directly influenced by the initial inclination angle. However, as the spatial inclination angle decreases with increasing displacement, the vertical force acting on the isolator diminishes rapidly. Additionally, the rate of force reduction exceeds the rate of displacement increase, leading to a progressive decrease in vertical stiffness as displacement increases.  
In contrast, variations in the spatial inclination angle exhibit a negligible impact on the horizontal stiffness under these conditions, suggesting that lateral stiffness remains relatively unaffected by changes in the initial inclination angle.