## 5 Conclusions  
Using FLAC3D finite-difference software, the numerical model incorporates the soil degradation stiffness model (DSM) to account for long-term cyclic loading effects. Subsequently, a parametric analysis is performed to assess how cyclic loading amplitude and cycle number impact the natural frequency of offshore wind turbine structures. A simplified approach that considers the influence of cyclic loading is then introduced. Finally, a case study is conducted to evaluate the evolution of natural frequency over the course of long-term cyclic loading. This study's results yield the following conclusions.  
Figure 16: Long-term development of the natural frequency for offshore 3.6 MW wind turbine structures.  
Figure 17: Variation of soil stiffness on the different number of cyclic loading (\(\xi_{\rm L}=0.25\)).  
1. The natural frequency of monopile-supported offshore wind turbines decreases linearly as cyclic loading amplitude rises. In early cyclic loading stages, this frequency drops sharply with more cycles. Later, it barely changes with additional cycles, stabilizing at a constant value.
2. The long-term cyclic loading would make the natural frequency of the wind turbine structures reduction effect. In an effort to ensure the sustained and secure operation of the offshore wind turbine over the long run, the natural frequency specified in the design of the structural system should lean towards 3P.
3. This simplified method is developed by the numerical results. Furthermore, this method allows for the rapid acquisition of the offshore wind turbine's natural frequency, but soil layer is a homogeneous sandy soil, and pore-pressure is simplified, so the universality application of the simplified method needs to be further verified for complex soil layer. However, in the absence of monitoring data of the natural frequency, the proposed simplified method can provide some initial assessment in terms of frequency calculation.  
Conceptualization, Y.S. and R.C.; methodology, Y.S. and R.C.; software, H.Y., J.Z. and B.S.; validation, Y.S. and J.X.; formal analysis, H.Y.; investigation, B.S. and R.C.; resources, J.Z.; data curation, Y.S.; writing--original draft preparation, Y.S.; writing--review and editing, J.X.; visualization, Y.S.; supervision, Y.S.; writing--review and editing, Y.S. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the Zhejiang Provincial Natural Science Foundation of China (Grant No. LQ24E080013) and the National Natural Science Foundation Youth Fund of China (Grant No.52408387)and the Zhejiang Province Leading Earth Goose Program (Grant No. 2023C03122) and the Wenzhou Basic Public Welfare Research Program (Grant No. S2023001).  
Not applicable.  
The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding author.  
This article's findings owe to the supervisor's meticulous guidance, precious assistance of Yongqing LAI, and care from the research group's teachers. Their joint efforts enabled the authors to obtain lake test data.  
The author has no conflicts of interest.  
The following abbreviations are used in this manuscript:  
\begin{tabular}{l l} \(E_{0}\) & Initial elastic modulus of the soil \\ \(k\) & Coefficients determined by the function of relative density of sand \\ \(\sigma_{at}\) & Atmosphere Pressure \\ \(\sigma_{3}\) & Minimum principal stress of the soil \\ \(\lambda\) & Coefficients defined via the sand's relative density function \\ \(K_{0}\) & Coefficient of soil pressure at rest \\ \(z\) & Soil depth \\ \(\gamma\) & Natural bulk density of soil \\ \(E_{sN}\) & Soil secant modulus after \(N\)th cycle \\ \(E_{s1}\) & Soil secant modulus under after first cyclic \\ \(N\) & Number of cyclic \\ \(a\) & Proportional parameters of exponential degradation function \\ \end{tabular}