# Mechanical Modeling and Dynamic Characteristics of a Three-Directional Vibration Absorber
## 3 Results and Discussion
### Dynamic Response Analysis of the Three-Directional Vibration Isolator
#### 3.3.2 Nonlinear Amplitude-Frequency Response and Resonance Behavior  
The vertical and transverse amplitude-frequency response curves of the three-directional vibration isolator for different excitation force amplitudes \(f_{0}\) are illustrated in Figure 8.  
Figure 8: Vertical and lateral amplitude–frequency characteristic curves of the three-directional vibration isolator (\(\theta_{0}=37.6^{\circ}\), \(\xi=0.05\)): (**a**) Vertical amplitude–frequency characteristic curve; (**b**) Lateral amplitude–frequency characteristic curve. Figure 7: Exact vs. Approximate Expression Curves for \(f_{z}\) and \(f_{x}\). For small excitation force amplitudes, the amplitude-frequency response of the system closely resembles that of a linear system under simple harmonic excitation, exhibiting linear characteristics, where each excitation frequency \(\omega\) corresponds to a unique dynamic displacement. As the excitation force amplitude increases, the response curve bends towards the lower frequency region, displaying a softening characteristic, similar to that observed in a Softening Duffing system. This results in a single jump phenomenon sweeping the excitation frequency from low to high frequencies or vice versa. With a further increase in the excitation force amplitude, the system's amplitude-frequency response curve initially bends towards the lower frequency region (indicating a softening effect) and subsequently bends towards the higher frequency region (indicating a hardening effect). This transition leads to the appearance of two jump phenomena during frequency sweeps. At large excitation force amplitudes, the system's amplitude-frequency response curve bends towards the higher frequency region, exhibiting a hardening behavior similar to that observed in a Hardening Duffing system. Consequently, a single jump phenomenon occurs during frequency sweeps. The transverse amplitude-frequency characteristics of the three-directional vibration isolator exhibit similar trends to those observed in the vertical direction, demonstrating comparable softening and hardening behaviors. It is important to note that the tri-directional amplitude-frequency response curves of the three-directional vibration isolator, as shown in Figure 8, are derived from Equation (18). These curves represent an approximate vibrational system, which is obtained by applying a fifth-order Taylor series expansion to Equation (16). Consequently, they differ from the original vibrational system described by Equation (16). As shown in Figure 8, when the excitation force amplitude is large, the system's dynamic displacement also increases significantly. However, as the dynamic displacement grows, the discrepancy between the exact and approximate expressions for \(f_{z}\) and \(f_{x}\) also increases. In this regime, the approximate expressions for \(f_{z}\) and \(f_{x}\) no longer accurately represent their exact counterparts, implying that the amplitude-frequency response curves of the approximate vibrational system shown in Figure 9 may not precisely reflect those of the original vibrational system when subjected to large excitation force amplitudes. Therefore, it is essential to validate the effectiveness of the approximate vibrational system represented by Equation (18) and assess the accuracy of the analytical solutions derived using the harmonic balance method. A comparison between the analytical solutions for the system's dynamic response, obtained via the harmonic balance method, and the numerical solutions derived from the fourth-order Runge-Kutta [23,24] method, is presented in Figure 8.  
As shown in Figure 9a-c, when the excitation force amplitude is relatively small, the system's dynamic displacement remains moderate. Within this displacement range, the discrepancy between the exact and approximate expressions for \(f_{z}\) and \(f_{x}\) is negligible. The analytical solutions obtained via the harmonic balance method and the numerical solutions from the fourth-order Runge-Kutta method exhibit good agreement across the entire frequency band.