## 4 Simulation Analysis
### Case Study  
Using harmonic wave \(A_{0}cos(2\pi\omega t)\) to simulate the working conditions, the damping coefficient of the horizontal damper is set as 400 N/m, the load force is set as 269.5 N, the excitation amplitude is set to 0.01 m, and the excitation angular frequency \(\omega\) is set as 5\(\pi\) rad/s (\(T=0.4\) s), 4\(\pi\) rad/s (\(T=0.5\) s), and 2\(\pi\) rad/s (\(T=1\) s), respectively. The simulation results of the response displacement are plotted in the Figure 9a-c. As \(\omega\) decreases, the isolation efficiency gradually decreases. When \(\omega\) reaches 2\(\pi\) rad/s, the isolation of QVIP is still effectively.  
Figure 7: Comparison of theoretical and simulation results of static stiffness force: (**a**) simulation model in Adams, (**b**) setting of motion pairs and constraints, (**c**) theoretical and simulation results with different \(\mu\), and (**d**) theoretical and simulation results with different \(\
u\) (parameters are set as \(L=0.1\), \(\varepsilon=1\), \(\lambda=1\), \(\xi=1\), \(\theta=\pi/6\), \(k=2000\)).  
To realistically simulate ocean wave conditions in Adams simulation environment, we developed a composite vibration excitation profile (\(\Delta h\)) comprising (1) a step function representing transient wave impacts, and (2) two superimposed harmonic functions with identical amplitudes but distinct frequencies to mimic characteristic wave frequency variations. This multi-component excitation approach effectively reproduces the complex spectral characteristics of actual offshore wave dynamics while maintaining controlled experimental conditions for precise isolator performance evaluation. The definition of \(\Delta h\) is  
\[\Delta h=\left\{\begin{aligned} & A_{0}\cdot\cos(2\pi\cdot 0.366t)& 0<t\leq 3\\ &\text{step}& 3<t\leq 21\\ & A_{0}\cdot\cos(2\pi\cdot 0.8t)& 21<t\end{aligned}\right. \tag{22}\]  
Figure 8: The types of main floating platforms and the application of the proposed QVIP in the vibration control of the floating platform: (**a**) four classic types of the offshore floating platform (barge, tension leg, semi-submersible, and spar), and (**b**) installation schematic of the QVIP on the spar type floating platform.  
The excitation amplitudes were systematically varied at 0.009 m, 0.012 m, and 0.015 m to evaluate the QVIP's performance under different loading conditions, with the corresponding displacement-time relationships presented in Figure 9d-f. Simulation results demonstrate a positive correlation between excitation amplitude and isolation efficiency within the system's recoverable deformation range, indicating enhanced vibration isolation performance at larger amplitudes. However, simulation studies reveal an operational threshold beyond which excessive excitation amplitudes (typically \(>\)0.015 m in our configuration) prevent the QVIP from achieving steady-state operation, suggesting the existence of a critical amplitude for stable performance.  
As vibration frequency decreases, the difficulty of isolation gradually increases. The vibrations in Figure 9a-c are regular cosine waves, which can roughly simulate ocean waves. However, real ocean waves are far more complex than simple cosine waves. Figure 9d-f employs a more realistic wave model that better approximates actual sea conditions, and the wave frequencies used are also lower. Consequently, the vibration attenuation amplitude in Figure 9d-f is smaller than that in Figure 9a-c.  
To investigate the influence of the different damping coefficient \(c\) on the QVIP, the damping coefficient of the horizontal damper was modified. Based on the dynamic simulation results of the QVIP, the displacement-time curves before and after vibration isolation were plotted, as shown in Figure 10. Under the same excitation conditions, a smaller damping coefficient results in a lower amplitude of the platform after vibration isolation. Simulation results indicate that when the damping coefficient is too small, the platform fails to reach a stable state quickly.