# Mechanical Modeling and Dynamic Characteristics of a Three-Directional Vibration Absorber
## 3 Results and Discussion
### Dynamic Response Analysis of the Three-Directional Vibration Isolator
#### 3.3.3 Vibration Isolation Performance and Transmissibility Analysis  
The vibration isolation performance of the three-directional vibration isolator is evaluated using two key metrics: the peak dynamic displacement (\(a_{\max}\)) and the peak transmissibility (\(T_{\max}\)). To facilitate this analysis, the amplitude-frequency response equation (Equation (18)) is reformulated as a function of frequency as follows:  
\[P_{1}\Omega^{4}+P_{2}\Omega^{2}+P_{3}=0 \tag{19}\]  
Figure 9: Comparison of analytical and numerical Solutions for the dynamic response of the three-directional vibration isolator (\(\theta=37.6^{\circ}\), \(\xi=0.05\)): (**a**) \(f_{0}=0.0025\); (**b**) \(f_{0}=0.005\); (**c**) \(f_{0}=0.01\); (**d**) \(f_{0}=0.02\); (**e**) \(f_{0}=0.03\). \[\left\{ \begin{array}{l}P_{1}=a^{2}\\ P_{2}=4\varsigma^{2}a^{2}-2\varphi_{1}a^{2}-\frac{3}{2}\varphi_{3}a^{4}-\frac{5}{4 }\varphi_{5}a^{6}\\ P_{3}=a^{2}\varphi_{1}{}^{2}+\frac{3}{2}\varphi_{1}\varphi_{3}a^{4}+\frac{5}{4 }\varphi_{1}\varphi_{5}a^{6}+\frac{9}{16}\varphi_{3}{}^{2}a^{6}+\frac{30}{32} \varphi_{3}\varphi_{5}a^{8}+\frac{100}{256}\varphi_{5}{}^{2}a^{10}-(f_{0})^{2} \end{array} \right. \tag{20}\]  
Hence, the solutions can be derived as follows:  
\[\Omega_{1,2}=\sqrt{\frac{-P_{2}\pm\sqrt{P_{2}{}^{2}-4P_{1}P_{3}}}{2P_{1}}} \tag{21}\]  
When \(\Omega_{1}=\Omega_{2}\), the peak dynamic displacement (\(a_{\max}\)) of the system can be determined. Specifically, \(P_{2}{}^{2}-4P_{1}P_{3}=0\). Substituting Equation (20) into this expression, we obtain the following:  
\[\left(4\varsigma^{2}a^{2}-2\varphi_{1}a^{2}-\frac{3}{2}\varphi_{3}a^{4}-\frac {5}{4}\varphi_{5}a^{6}\right)^{2}-4a^{2}\left(a^{2}\varphi_{1}{}^{2}+\frac{3} {2}\varphi_{1}\varphi_{3}a^{4}+\frac{5}{4}\varphi_{1}\varphi_{5}a^{6}+\frac{9 }{16}\varphi_{3}{}^{2}a^{6}+\frac{30}{32}\varphi_{3}\varphi_{5}a^{8}+\frac{10 0}{256}\varphi_{5}{}^{2}a^{10}-(f_{0})^{2}\right)=0 \tag{22}\]  
The peak dynamic displacement (\(a_{\max}\)) is obtained through numerical solutions. By substituting this peak dynamic displacement (\(a_{\max}\)) into Equation (21), the corresponding resonance frequency (\(\omega_{n}\)) can be determined. Simultaneously, the dimensionless force \(f_{t}\) transmitted to the foundation by the three-directional vibration isolator can be derived from Equation (17) as follows:  
\[f_{t}=2\varsigma\widetilde{u}^{\prime}+\varphi_{1}\widetilde{u}+\varphi_{2} \widetilde{u}^{2}+\varphi_{3}\widetilde{u}^{3}+\varphi_{4}\widetilde{u}^{4}+ \varphi_{5}\widetilde{u}^{5}=f_{0}\cos(\Omega\tau)-\widetilde{u}^{\prime} \tag{23}\]  
where \(u\) corresponds to \(\widetilde{z}\) and \(\widetilde{x}\) in the vertical and horizontal dynamic equations, respectively. Following the computational procedure in Section 3.3.1, under the assumption of harmonic excitation, the periodic dynamic By differentiating Equation (27) with respect to frequency, the peak dynamic displacement of the linear isolator and its corresponding resonance frequency can be determined as follows:  
\[\left\{ \begin{array}{l}\Omega_{1}=\sqrt{1-2\xi^{2}}\\ a_{1\max}=\frac{f_{0}}{2\xi\sqrt{1-\xi^{2}}}\end{array} \right. \tag{28}\]  
Following the same computational method as Equation (25), The force transmissibility of the linear isolator under harmonic excitation can be obtained from Equation (26):  
\[T_{1}=\frac{\sqrt{1+\left(2\xi\Omega\right)^{2}}}{\sqrt{\left(1-\Omega^{2} \right)^{2}+\left(2\xi\Omega\right)^{2}}} \tag{29}\]  
By differentiating Equation (29) with respect to frequency, the peak force transmissibility of the linear isolator and its corresponding resonance frequency can be determined. The resulting expression is given as follows:  
\[\left\{ \begin{array}{l}\Omega_{2}=\frac{\sqrt{-1+\sqrt{1+8\xi^{2}}}}{2\xi^{ 2}}\\ T_{1\max}=\sqrt{\frac{8\xi^{4}}{8\xi^{4}-4\xi^{4}-1+\sqrt{1+8\xi^{2}}}}\end{array} \right.