## 5 Conclusions  
This study addresses the critical challenge of low-frequency vibration control in floating platforms by proposing a novel quadrilateral-shaped vibration isolation platform (QVIP) based on a foldable quadrilateral mechanism. Through integrated theoretical modeling, parametric optimization, and dynamic simulations, the research systematically investigates the QVIP's vibration suppression mechanism and demonstrates its engineering feasibility for offshore applications. The key contributions include the following. The theoretical modeling and analysis demonstrate that the QVIP can generate an ideal quasi-zero stiffness (QZS) curve when the link length ratio is 1 (forming an equilat \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Platform Load (N)** & **Metric** & **Without Isolation** & **With QVIP** & **Reduction Percentage** \\ \hline \multirow{3}{*}{269.5} & Maximum Amplitude (m) & 0.00917 & 0.00553 & 39.68\% \\ \cline{2-5}  & Variance & 25.48 & 7.84 & 69.23\% \\ \cline{2-5}  & Vibration Isolation Efficiency & & 44.53\% & \\ \hline \multirow{3}{*}{325.3} & Maximum Amplitude (m) & 0.00914 & 0.0056 & 38.73\% \\ \cline{2-5}  & Variance & 25.87 & 7.79 & 69.89\% \\ \cline{2-5}  & Vibration Isolation Efficiency & & 45.13\% & \\ \hline \multirow{3}{*}{389.5} & Maximum Amplitude (m) & 0.00915 & 0.00554 & 39.48\% \\ \cline{2-5}  & Variance & 25.47 & 6.84 & 73.14\% \\ \cline{2-5}  & Vibration Isolation Efficiency & & 48.17\% & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The comparison of vibration data without vibration control and with QVIP. Figure 11: Vibration control effect of an optimally designed QVIP: (**a**) theoretical and simulation results of equivalent nonlinear stiffness forces, (**b**) response when load is 325.3 N, (**c**) response when the load is 389.5 N, and (**d**) vibration isolation performance comparison. eral rhombus structure). Parameter analysis further reveals the mechanism behind QZS formation in the QVIP and how to adjust parameters to achieve an extended QZS range. The results indicate that increasing the link length enhances load-bearing capacity while maintaining the QZS range. To validate the theoretical model, a rigid-body dynamics simulation was conducted using Adams software. The numerical simulation results align with the theoretical predictions, confirming the accuracy of the model and verifying that the QVIP can achieve tunable and controllable QZS characteristics. Numerical simulations were performed to analyze the vibration of a floating platform under wave action. Time-history analysis shows that the QVIP achieves vibration isolation efficiency exceeding 44% under different loads. By adjusting key parameters such as spring stiffness ratio, initial angle, and link length, the QVIP can adapt to varying loads while retaining its QZS vibration suppression properties. For instance, under a 325.3 N load, the isolation efficiency reaches 44.53%, while under a 389.5 N load, it improves to 48.17%, demonstrating excellent vibration isolation performance. Additionally, a systematic parametric study elucidates the influence of key QVIP parameters on the QZS bandwidth, providing practical design guidelines for engineering applications. In conclusion, this study introduces a QVIP based on a quadrilateral-shaped structure design, combining theoretical analysis and simulation to successfully address the challenge of ultra-low-frequency wave-induced vibrations in offshore floating platforms. Conceptualization, Z.G. and J.B.; methodology, Z.G., J.B. and L.L. ; software, N.S. ; validation, Z.G., J.B. and L.L. ; formal analysis, N.S. ; investigation, J.B.; resources, [PERSON] and J.B.; data curation, L.L. ; writing--original draft preparation, Z.G. ; writing--review and editing, J.B. and L.L. ; visualization, L.L. ; supervision, [PERSON] and J.B.; project administration, J.B.; funding acquisition, Z.G. and J.B. All authors have read and agreed to the published version of the manuscript.