# Mechanical Modeling and Dynamic Characteristics of a Three-Directional Vibration Absorber
## 4 Conclusions  
To achieve effective three-directional vibration isolation, a novel vibration isolator was designed, utilizing inclined spring assemblies as the core elastic and damping elements. The inclined configuration of the spring assemblies under static equilibrium ensures balanced support and stiffness along all three axes. This study systematically investigated the isolator's static and dynamic behavior through theoretical modeling, numerical simulation, and experimental validation. The main conclusions are summarized as follows:  
1. Structural design and stiffness characteristics: The inclination angle of the spring assemblies at static equilibrium is \(35.26^{\circ}\), ensuring three-directional isolation. The system exhibits weak nonlinear stiffness characteristics, where vertical stiffness increases and lateral stiffness decreases as displacement grows. The stroke of the isolator is determined by the initial inclination angle of the spring assemblies, while the load-bearing capacity is influenced by both spring stiffness and the initial inclination angle. 2. Approximate dynamic modeling and validation: The nonlinear restoring forces in vertical and horizontal directions were approximated using Taylor series expansions, and the harmonic balance method was used to obtain an analytical amplitude-frequency response. Comparisons with numerical solutions via the fourth-order Runge-Kutta method and experimental results confirmed the validity of the approximation under small excitation amplitudes. 3. Influence of Excitation Force Amplitude on Isolation Performance: As the excitation force amplitude increases, the peak dynamic displacement of the isolator increases, while the force transmissibility fluctuates but generally follows an increasing trend. Compared to a linear isolator, the three-directional isolator exhibits a larger peak dynamic displacement but a lower force transmissibility, indicating enhanced vibration isolation capability. Both the peak dynamic displacement and force transmissibility increase with the initial inclination angle of the spring assemblies. 4. Impact of Nonlinear Damping due to Coulomb friction: The incorporation of Coulomb friction damping captures the nonlinear dissipation behavior within the hinge pairs. Figure 19: The comparison between the prototype frequency sweep test and the theoretical force transmission ratio curve. Under small-amplitude harmonic excitation, an increase in the equivalent Coulomb friction factor leads to a decrease in response amplitude and force transmissibility before resonance, but an increase beyond the resonance frequency. Additionally, an increase in Coulomb friction factor shifts the dynamic response and force transmissibility curves downward and to the right, raising the system's resonance frequency and reducing the effective isolation range. These results highlight the dual effect of nonlinear damping, where optimizing damping strength is critical for achieving effective isolation performance. * Experimental verification of static and dynamic models: Quasi-static and harmonic excitation tests were conducted on a prototype isolator. The static model accurately captured the loading-unloading force-displacement characteristics, while the dynamic model effectively reproduced the frequency response curves. Hysteresis loops due to Coulomb friction were clearly observed, validating the nonlinear damping assumptions. This research offers a comprehensive framework for analyzing nonlinear stiffness and damping in multi-axis isolation systems, supported by both theoretical derivation and experimental validation. The findings provide theoretical foundations for isolator design and performance tuning in aerospace, precision equipment, and intelligent suspension applications. Future studies will focus on multi-parameter optimization, adaptive damping control, and robustness under varying operational conditions. Conceptualization, Z.W. ; methodology, Z.W. ; software, M.C., Q.L. and C.L. ; validation, Z.W., M.C. and Z.Y. ; formal analysis, G.X.