# Mechanical Modeling and Dynamic Characteristics of a Three-Directional Vibration Absorber
## 3 Results and Discussion  
### Structural Design of the Three-Directional Vibration Isolator  
To achieve effective three-directional vibration isolation while maintaining static load balance, the proposed vibration isolator incorporates four sets of spring components as its core elements. As illustrated in Figure 1, these spring components are arranged in an inclined configuration and are connected via spherical hinges and interface installations, allowing multiple degrees of freedom necessary for achieving simultaneous isolation along all three orthogonal axes. The simplified structural model of the proposed isolator and its hinge assembly configuration is illustrated in Figure 2.  
The mechanical properties of the three-directional vibration isolator are predominantly governed by its stiffness distribution along the three principal axes. Based on force equilibrium analysis, the vertical stiffness in the z-direction can be approximated as follows:  
\[K_{z}=4K_{n}\sin^{2}\theta_{0} \tag{1}\]  
where \(K_{n}\) is the axial stiffness of an individual spring, and \(\theta_{0}\) is the initial inclination angle of the spring assembly relative to the horizontal plane.  
Figure 1: Three-dimensional modeling of the three-directional vibration isolator.  
Similarly, the stiffness in the horizontal x- and y-directions is expressed as follows:  
\[K_{x}=K_{y}=2K_{u}\cos^{2}\theta_{0} \tag{2}\]  
Thus, the ratio of horizontal (X or Y direction) to vertical stiffness can be expressed as follows:  
\[r_{k}=\frac{K_{x}}{K_{z}}=\frac{K_{y}}{K_{z}}=\frac{1}{2}\cot^{2}\theta_{0} \tag{3}\]  
where \(r_{k}\) represents the stiffness ratio between the horizontal and vertical directions. Figure 3 illustrates the relationship between the stiffness ratio and the inclination angle of the spring assembly. As shown, a smaller spatial angle (i.e., a larger inclination) results in greater transverse stiffness. When the inclination angle \(\theta_{0}\) = 35.26\({}^{\circ}\), the ratio \(r_{k}\) = 1, meaning that the lateral stiffness equals the vertical stiffness. Therefore, the optimal inclination angle for the vibration isolator is determined as 35.26\({}^{\circ}\).  
Owing to the Z-axis symmetry of the simplified model, the Y-direction forces remain inherently balanced. Consequently, the system can be reduced to a two-degree-of-freedom model in the X-Z plane for analyzing both vertical (Z-axis) and lateral (X-axis) static characteristics. Notably, owing to symmetry equivalence, this simplified model maintains equal validity in the Y-Z plane while preserving X-direction force equilibrium. This simplification process is clearly illustrated in Figure 4, which depicts the isolator's operational schematic under both vertical and lateral loading conditions. The dotted lines in the diagram represent the free length of the spring assembly \(L\), the initial tilt angle \(\theta_{0}\), and the spring assembly  
Figure 3: Approximate variation curve of stiffness ratio vs. tilt angle.  
Figure 2: Simplified model of the three-directional vibration isolator spring assembly.  
stiffness \(K_{n}\). The solid lines indicate the spring assembly at static equilibrium, where the bearing mass \(m\) is balanced with a static deformation \(a\). The inclination angle between the spring assembly axis and the horizontal plane is set to \(35.26^{\circ}\), ensuring an optimized stiffness distribution, and the length of the spring assembly is \(L_{0}\). The displacement response of the bearing mass \(m\) from the static equilibrium position is represented by x, y, and z corresponding to the three orthogonal directions of motion.  
In addition to elastic stiffness, the vibration isolator incorporates viscous damping to dissipate vibrational energy. The system damping is modeled as linear viscous damping, characterized by a damping coefficient \(c\). The damping effect primarily influences the system's dynamic response and will be further analyzed in subsequent sections.