## 3 Effect of Cyclic Loading
### Amplitude of Cyclic Loading  
To clearly demonstrate the impact of cyclic loading amplitude on the natural frequency, the dimensionless coefficient \(\xi_{\text{L}}\) was defined using Equation (6) to represent various cyclic loading amplitudes. As referenced in the work of [PERSON] et al. (2016) and [PERSON] et al. (2017), this equation defines the ultimate bearing capacity Su as the lateral load corresponding to a lateral displacement of 0.1D (where D denotes the pile diameter) at the pile's mud surface. The accumulated deformation of the monopile is included in fatigue limit state or service limit state, so the maximum dimensionless coefficient \(\xi_{\text{L}}\) is about 0.47 ([PERSON] et al., 2017; [PERSON] et al., 2017).  
\[\xi_{L}=\frac{S_{cyc}}{S_{u}} \tag{6}\]  
where \(\xi_{\text{L}}\) is the dimensionless coefficient of the cyclic loading, \(S_{\text{cyc}}\) represents the cyclic horizontal displacement amplitude at the pile's mudline (m), and \(S_{\text{u}}\) denotes the ultimate horizontal displacement capacity at the mudline (0.1\(D\), \(D\) is the pile diameter).  
In both medium sand and dense sand environments, how cyclic loading amplitude affects the natural frequencies of offshore wind turbine structures was examined. When the soil site was the medium-dense sand, the lateral displacement loadings of 1 m, 1.5 m, 2 m, 3 m, and 3.5 m were applied on the top of the wind turbine structures, respectively. Similarly, in the dense sand site, the lateral displacement loadings of 1 m, 2 m, 3 m, 3.5 m, 4 m were, respectively, applied to the wind turbine structures' top. Then the structural displacement response results are shown in Figure 9. It can be seen that the structural displacement response period increases as the dimensionless coefficient of the cyclic loading increases. This means that as the dimensionless coefficient of cyclic loading increases, the natural frequency of offshore wind turbine structures decreases. The reasons lie in the fact that an increasing amplitude intensifies the pronounced nonlinearity of the surrounding soil and heightens cyclic stress characteristics. This indicates that the surrounding soil's secant modulus degrades, ultimately shortening the structure's response period. This trend is found in the soil cyclic tests conducted by [PERSON] et al. (2017).  
Figure 9: Response displacement of tower-top under the different amplitude of cyclic loading.  
Using the tower-top response displacement described earlier, the offshore wind turbine's natural frequency was derived. Figure 9 illustrates how different cyclic loading amplitudes affect natural frequency. It can be seen that in medium-dense sand and dense sand, the structure's natural frequency decreases linearly as the cyclic loading's dimensionless coefficient increases. In other words, natural frequency declines as cyclic loading amplitude grows. The fitting function is obtained from Figure 10, as shown in Equation (7). When it is the service limit state, the dimensionless coefficient of cyclic loading is about 0.47 [40]. At this time the natural frequency is reduced by about 18%. \(\xi_{\rm L}\) = 0.3 is the fatigue limit state [40]; at this point the natural frequency is reduced by about 13%.  
\[f_{\xi_{L}}=f_{0}-\alpha\xi_{L} \tag{7}\]  
where \(f_{0}\) is the natural frequency when the soil around pile is in an elastic state, \(f_{\xi{\rm L}}\) denotes the natural frequency when the dimensionless cyclic loading coefficient is \(\xi_{\rm L}\), and \(\alpha\) quantifies how cyclic loading amplitude influences natural frequency.  
To determine the coefficient \(\alpha\) in Equation (7), we explored how cyclic loading's dimensionless coefficient affects \(\alpha\), with the findings shown in Figure 11. It was indicated that coefficient \(\alpha\) does not change with changing the dimensionless coefficient of cyclic loading. In other word the coefficient \(\alpha\) is a constant value, which is 0.136 in this study.