## 3 Effect of Cyclic Loading
### Number of Cyclic Loading  
In this section, we studied how the number of cyclic loadings influences natural frequency. In medium-dense sand, cyclic loadings with dimensionless coefficients of 0.14 and 0.21 were applied to the tops of offshore wind turbine structures, respectively. Additionally, in dense sand, cyclic loadings with dimensionless coefficients of 0.17 and 0.28  
Figure 11: The relation between coefficient \(\alpha\) and dimensionless coefficient \(\xi_{\rm L}\).  
Figure 10: The influence of the cyclic loading’s dimensionless coefficient on natural frequency.  
were used to explore how the number of cyclic loadings affects these structures' natural frequency. The wind turbine's response results under varying numbers of cyclic loadings are presented in Figures 12 and 13. As observed in these figures, the response period of offshore wind turbine structures lengthens as the number of cyclic loadings increases. That is to say, as the number of cyclic loadings increases, the natural frequency of the structural system becomes lower than before. This is because the soil's secant stiffness decreases as the number of cyclic loadings rises. This result is found in the soil cyclic tests of [PERSON] et al. (2019), so when increasing the number of cyclic loading, the natural frequency decreases.  
From Figure 14, the natural frequencies of offshore wind turbines under different cyclic loading cycles are obtained. As depicted in the figure, in both medium-dense and dense sand conditions, the natural frequencies of the wind turbine structures exhibit a gradual decline as the number of cyclic loadings increases. Significantly, in the early stages of cyclic loading (N < 100), offshore wind turbine structures in medium-dense sand undergo a notable 4% reduction in natural frequency. Figure 14 analysis further shows that natural frequency changes linearly with the logarithm of cyclic loading cycles. Leveraging this insight, Equation (8) was developed to model how natural frequency evolves with the number of cyclic loadings.  
\[f_{N}=f_{1}N^{c} \tag{8}\]  
\[\log_{10}\biggl{(}\frac{f_{N}}{f_{1}}\biggr{)}=c\log_{10}(N) \tag{9}\]  
Figure 12: Response displacement of tower-top under different number of cyclic in dense sand.  
Figure 13: Response displacement of tower-top under different number of cyclic in medium dense sand.  
where \(f_{1}\) denotes the natural frequency of the monopile-supported offshore wind turbine in the first cycle, \(f_{\text{N}}\) represents its natural frequency in the \(N\)th cyclic, and \(c\) stands for the influence coefficient of cyclic loading cycles on natural frequency.  
[PERSON] [41] and [PERSON] et al. [42] observed that foundation stiffness primarily dictates the natural frequency of offshore wind turbine structures. That is, there is a certain positive correlation between the natural frequency of offshore wind turbine and the foundation. Equation (3) shows that there is an exponential relation between the secant modulus of soil and the number of cyclic loadings, that is similar to Equation (8). It is illustrated that Equation (8) is rational to express the relation between the natural frequency and number of cyclic loadings.  
According to Equation (9), the above numerical calculation results are processed to obtain the coefficient \(c\). The fitting curve of the influence coefficient \(c\) in medium dense sand and dense sand is shown in Figure 13. The slope of the fitting curve is the influence coefficient \(c\), so it is obtained in this study from Figure 15 that coefficient \(c\) is \(-0.012\) in medium dense sand and coefficient \(c\) is \(-0.016\) in dense sand.  
By combining Equations (7) and (8), how long-term cyclic loading affects natural frequency can be expressed by Equation (10), as follows.  
\[f_{N}=(f_{0}-\alpha\xi_{L})N^{c} \tag{10}\]  
Figure 14: Effect of number of cyclic loadings on natural frequency.  
Figure 15: Effect of number of cyclic loadings on coefficient c.