# 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  
In Section 4, the dynamic characteristics of the damper were analyzed by simplifying the damping force as a linear viscous damping force. However, in practical applications, dampers inherently exhibit significant geometric nonlinearities due to their structural design. Additionally, the presence of multiple sliding and rotating pairs within the internal moving components introduces friction damping, which further contributes to the nonlinear behavior of the system. Although various engineering strategies can be implemented to mitigate friction damping, it remains challenging to completely eliminate the influence of Coulomb friction forces within these kinematic pairs. As a result, it is crucial to conduct a more detailed analysis to account for the realistic nonlinear damping effects and their impact on the damper's dynamic response. A schematic diagram of the kinematic components within the vibration isolator is presented in Figure 12.  
Figure 11: Influence of initial angle on peak dynamic displacements and peak force transmissibility of the isolators (\(f_{0}=0.0025\), \(\xi=0.05\)). Figure 12: Schematic diagrams of the kinematic pairs in the vibration isolator. Based on the working principle of the vibration isolator, we established a model for the Coulomb friction force presented and equivalently applied it to the vertical direction of the isolator. Considering the influence of geometric nonlinearity on the structural response, the expression for the angle \(\theta\) between the central axis of the hinge assembly and the horizontal plane, as the relative displacement \(z\) varies from the static equilibrium position, is given as follows:  
Based on the working principle of the isolator, a Coulomb friction force model is established and equivalently applied to the vertical direction of the system. To account for the influence of geometric nonlinearity on the structural response, the expression for the inclination angle \(\theta\) between the central axis of the hinge assembly and the horizontal plane, as the relative displacement \(z\) varies from the static equilibrium position, is given as follows:  
\[\sin\theta=\frac{L\cos\theta_{0}\tan\theta_{1}-z}{\sqrt{\left(L\cos\theta_{0} \tan\theta_{1}-z\right)^{2}+\left(L\cos\theta_{0}\right)^{2}}} \tag{31}\]  
Then, the dimensionless relative displacement \(\widetilde{z}\) can be expressed as follows:  
\[\sin\theta=\frac{\cos\theta_{0}\tan\theta_{1}-\widetilde{z}}{\sqrt{\left(\cos \theta_{0}\tan\theta_{1}-\widetilde{z}\right)^{2}+\left(\cos\theta_{0}\right) ^{2}}} \tag{32}\]  
The Coulomb friction forces generated by the sliding pairs and spherical joints in each of the four inclined hinge assemblies are equivalently modeled as damping forces in the vertical direction of the vibration isolator, given as follows:  
\[ \begin{split} F_{c1}&=4\mu N_{1}\sin\theta\text{ sgn}\left(\frac{d\widetilde{z}}{dt}\right)=4f_{c1}\sin\theta\text{sgn}\left(\frac{d \widetilde{z}}{dt}\right)\\ F_{c2}&=8\mu N_{2}\text{sgn}[(d\alpha/dz)z^{\prime }]\sin\theta=8f_{c2}\sin\theta\text{sgn}\left(\frac{d\widetilde{z}}{dt}\right) \end{split}  \tag{33}\]  
where \(\mu\) represents the Coulomb friction damping constant, \(N_{1}\) denotes the normal contact pressure between the sliding components of the guide rod sliding pair, and \(f_{c1}\) is the Coulomb friction coefficient for each guide rod sliding pair. Additionally, \(\alpha\) refers to the inclination angle of the hinge assembly relative to the vertical direction, \(N_{2}\) indicates the normal contact pressure within each spherical joint, and \(f_{c2}\) is the Coulomb friction coefficient for each spherical joint.