# 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
#### 3.2.2 Lateral Static Equilibrium and Stiffness Modeling  
At the static equilibrium position, all four spring components of the vibration isolator remain in a compressed state. When the bearing mass \(m\) undergoes lateral displacement \(x\), the spring components on the +x side (in the same direction as the deformation) experience further compression, whereas the spring components on the \(-\)x side (opposite to the deformation) begin to unload and elongate. When the spring components on the \(-\)x side return to their initial free length, the lateral displacement \(x\) reaches a critical value \(x_{1}\), where \(x_{1}\) satisfies Equation (7).  
\[(L\cos\theta_{0}\cos 45^{\circ}+x_{1})^{2}+(L\cos\theta_{0}\cos 45^{\circ})^{2}+ (L\sin\theta_{0}-a)^{2}={L_{0}}^{2} \tag{7}\]  
From Equation (7), the value of \(x_{1}\) can be determined as:  
\[x_{1}=L\bigg{(}\sqrt{1-\cos^{2}\theta_{0}\big{(}\cos^{2}45^{\circ}+\tan^{2} \theta_{1}\big{)}}-\cos\theta_{0}\cos 45^{\circ}\bigg{)} \tag{8}\]  
where \(\theta_{1}\) is the spatial angle between the central axis of the hinge component and the horizontal plane in the static load balance state.  
Evidently, when \(x<x_{1}\), all four spring components remain compressed, meaning that they collectively provide restoring elastic forces to the bearing mass \(m\). Based on geometric analysis, when the isolator undergoes lateral deformation \(x\) from the static equilibrium position, the length of the spring components on the +x and -x sides (denoted as \(L_{1}\) and \(L_{2}\) respectively) can be expressed as follows:  
\[L_{1}=\sqrt{\left(L\cos\theta_{0}\cos 45^{\circ}-x\right)^{2}+\left(L\cos \theta_{0}\cos 45^{\circ}\right)^{2}+\left(L\sin\theta_{0}-a\right)^{2}} \tag{9}\]  
\[L_{2}=\sqrt{\left(L\cos\theta_{0}\cos 45^{\circ}+x\right)^{2}+\left(L\cos \theta_{0}\cos 45^{\circ}\right)^{2}+\left(L\sin\theta_{0}-a\right)^{2}} \tag{10}\]  
Consequently, the lateral elastic restoring force \(F_{x}\) in the x-direction is as follows:  
\[F_{x}=2K_{n}\bigg{(}(L-L_{1})\frac{L_{0}\cos\theta_{0}\cos 45^{\circ}-x}{L_{1}} -(L-L_{2})\frac{L_{0}\cos\theta_{0}\cos 45^{\circ}+x}{L_{2}}\bigg{)}\ (0\leq x\leq x_{1}) \tag{11}\]  
Similarly, by applying nondimensionalization to Equation (11), we obtain the following:  
\[\widetilde{F}_{x}=2\delta\left(\begin{array}{l}\Bigg{(}\frac{1}{\sqrt{ \left(\cos\theta_{0}\cos 45^{\circ}-\widetilde{x}\right)^{2}+\left(\cos \theta_{0}\cos 45^{\circ}\right)^{2}+\left(\sin\theta_{0}-\lambda\right)^{2}}}-1 \Bigg{)}(\cos\theta_{0}\cos 45^{\circ}-\widetilde{x})\\ -\Bigg{(}\frac{1}{\sqrt{\left(\cos\theta_{0}\cos 45^{\circ}+\widetilde{x} \right)^{2}+\left(\cos\theta_{0}\cos 45^{\circ}\right)^{2}+\left(\sin\theta_{0}- \lambda\right)^{2}}}-1\Bigg{)}(\cos\theta_{0}\cos 45^{\circ}+\widetilde{x})\end{array}\right) \tag{12}\]  
where \(\widetilde{x}=x/L\). By differentiating Equation (12) with respect to \(\lambda\), the transverse dimensionless stiffness of the three-directional vibration isolator is obtained as:  
\[\widetilde{K}_{x}=2\delta\left(\begin{array}{l}2+\frac{\left(\cos\theta_{0} \cos 45^{\circ}-\widetilde{x}\right)^{2}}{\left(\left(\cos\theta_{0}\cos 45^{ \circ}-\widetilde{x}\right)^{2}+\left(\cos\theta_{0}\cos 45^{\circ}\right)^{2}+ \left(\sin\theta_{0}-\lambda\right)^{2}\right)^{\frac{3}{2}}}\\ +\frac{\left(\cos\theta_{0}\cos 45^{\circ}+\widetilde{x}\right)