# Mechanical Modeling and Dynamic Characteristics of a Three-Directional Vibration Absorber
## 3 Results and Discussion
### Influence of Nonlinear Damping on Dynamic Performance
#### 3.4.1 Effect of Nonlinear Damping on Dynamic Response Behavior
Based on the above discussion, the total equivalent damping force in the vertical direction of the vibration isolator, resulting from the Coulomb friction forces generated by the various sliding pairs and rotational joints within the hinge assemblies, is given as follows:  
\[F_{c}=F_{c1}+F_{c2}=(4f_{c1}\sin\theta+8f_{c2}\sin\theta)\text{sgn}\left( \frac{d\widetilde{z}}{dt}\right) \tag{34}\]  
Let the external excitation be a harmonic force, defined as \(F_{n}=F_{0}\cos(\omega t)\). Considering the nonlinear damping force, the dynamic equation of the vibration isolator is expressed as follows:  
\[m\frac{d^{2}\widetilde{z}}{dt^{2}}+F_{c}+c\frac{d\widetilde{z}}{dt}+ \widetilde{F}_{z}\cdot kL-mg=F_{0}\cos(\omega t) \tag{35}\]  
Applying nondimensionalization to Equation (34) yields the following:  
\[\frac{d^{2}\widetilde{z}}{d\tau^{2}}+2\xi_{c}f_{c}\text{sgn}\left(\frac{d \widetilde{u}}{d\tau}\right)+2\xi\frac{d\widetilde{z}}{d\tau}+f_{n}=F_{0}\cos (\Omega\tau) \tag{36}\]  
where \(\xi_{c}=f_{c1}/2m\omega_{n}\) represents the equivalent Coulomb friction factor in the kinematic pairs, and \(f_{c}\) denotes the dimensionless equivalent vertical damping coefficient, which corresponds to the nonlinear damping force in the vibration isolator's kinematic pairs. By combining Equations (32) and (34), the specific expression can be derived as follows:  
\[f_{c}=\frac{\cos\theta_{0}\tan\theta_{1}-\widetilde{z}}{L\sqrt{\left(\cos\theta_ {0}\tan\theta_{1}-\widetilde{z}\right)^{2}+\left(\cos\theta_{0}\right)^{2}}} \left(4+8\frac{f_{c2}}{f_{c1}}\right) \tag{37}\]  
From Equation (37), it follows that \(f_{c}\) is continuous at \(\widetilde{z}=0\). By setting \(f_{c1}=0.5\) and \(f_{c2}=1\), the function \(f_{c}\) at \(\widetilde{z}=0\) can be approximated using a third-order Taylor expansion, which yields the following:  
\[f_{c}=\gamma_{1}+\gamma_{2}\widetilde{z}+\gamma_{3}\widetilde{z}^{2} \tag{38}\]  
where \(\gamma_{i}\) (\(i=1,2,3\)) represents the Taylor series expansion coefficients of \(f_{c}\) at the origin. By comparing the Taylor expansion curve of the dimensionless equivalent vertical damping coefficient \(f_{c}\) with the original curve, as shown in Figure 13, it can be observed that within a small displacement range, the Taylor series approximation closely follows the original function curve. This confirms that using the Taylor expansion to approximate the dimensionless equivalent vertical damping coefficient is a valid and effective approach. Thus, the expression for the kinetic energy can be represented as follows:  
\[\frac{d^{2}\widetilde{z}}{d\tau^{2}}+2\xi_{c}(\gamma_{1}+\gamma_{2}\widetilde {z}+\gamma_{3}\widetilde{z}^{2})\text{sgn}\left(\frac{d\widetilde{z}}{d\tau} \right)\frac{d\widetilde{z}}{d\tau}+2\xi\frac{d\widetilde{z}}{d\tau}+\alpha_{1 }\widetilde{z}+\alpha_{2}\widetilde{z}^{2}+\alpha_{3}\widetilde{z}^{3}+\alpha_ {4}\widetilde{z}^{4}+\alpha_{5}\widetilde{z}^{5}=F_{0}\cos(\Omega\tau) \tag{39}\]  
The solution procedure follows the methodology outlined in Section 4, which establishes the relationship between the vertical amplitude-frequency and phase-frequency responses while accounting for the nonlinear damping force:  
\[\left\{ \begin{array}{l}\left(\frac{2\xi\alpha\Omega-2\xi_{c}\gamma_{1} \Omega-\frac{1}{2}a^{2}\gamma_{3}\Omega}{f_{0}}\right)^{2}+\left(\frac{-a \Omega^{2}+\alpha_{1}a+\frac{3}{2}\alpha_{3}a^{2}+\frac{10}{2}\alpha_{5}a^{ 5}}{f_{0}}\right)^{2}=1\\ \tan(\phi)=\frac{2\xi\alpha\Omega-2\xi_{c}\gamma_{1}\Omega-\frac{1}{2}a^{3} \gamma_{3}\Omega}{-a\Omega^{2}+\varphi_{1}a+\frac{3}{2}\varphi_{3}a^{3}+\frac {10}{2}\varphi_{5}a^{5}}\end{array} \right. \tag{40}\]  
The parameters of the vibration isolator are consistent with those discussed in previous research, with the linear viscous damping coefficient set to \(\xi=0.05\) while varying the equivalent Coulomb friction factor \(\xi_{c}\).