## 4 Simplified Calculation Method
### Evaluation of the Long-Term Natural Frequency  
A case study was undertaken to evaluate the long-term evolution of natural frequency under sustained cyclic loading. A monopile supported offshore 3.6 MW wind turbine structures in the Walney 1 wind farm was used to conduct the case study, which can provide some reference for engineering design. These required parameters in the calculation are shown as follows in Table 2.  
Based on the parameters mentioned above, the natural frequency of the fixed-base structural system is 0.422 Hz. Equations (13)-(16) are used to obtain the foundation stiffness coefficients \(\mathrm{C_{R}}\) and \(\mathrm{C_{L}}\), which are 0.837 and 0.997, respectively. Therefore, when pile-soil interaction is taken into account, the natural frequency of the wind turbine structures is 0.351 Hz.  
The seabed where this 3.6 MW wind turbine support structure system is located is in the dense sand [46], so using Equation (10) to evaluate the long-term natural frequency of the 3.6 MW wind turbine structures, the influence coefficient \(\mathrm{c}\) is \(-0.016\) in Equation (10). Meanwhile, an assessment is also made of how cyclic loading amplitudes of varying magnitudes affect natural frequency, so the influence coefficient \(\alpha\) is \(-0.136\), and the dimensionless ratio \(\xi_{\mathrm{L}}\) is 0.15-0.3. The calculated processes are as shown in Equations (18) and (19). The calculated results are shown in Figure 16.  
\[f_{N}=f_{\eta}N^{-0.016} \tag{18}\]  
\[f_{N}=\left(f_{\eta}-0.136\xi_{L}\right)N^{-0.016} \tag{19}\]  
Figure 16 shows that the natural frequency of wind turbine structures decreases as the number of cycles increases. The natural frequency exhibits a similar trend across different cyclic loading amplitudes. In the early stage of cyclic loading (\(\mathrm{N<10^{5}}\)), the natural frequency drops significantly by approximately 20%. Thereafter, it remains nearly unchanged with further increases in cycle count. In this case study, the natural frequency stabilized at a constant value after about half a month. To explain this trend, soil stiffness variations under different cycle numbers were analyzed (Figure 17). Results show that soil stiffness variation ranges increase with cyclic loading but cease to expand after \(10^{5}\) cycles, which signifies that foundation stiffness has stabilized, and consequently, the natural frequency has also stabilized.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**OWT Dimension and Foundation Stiffness** & **Symbol/Unit** & **Value** \\ \hline Top mass & \(m_{\mathrm{t}}\)/ton & 234.5 \\ Tower mass & \(m_{\mathrm{T}}\)/ton & 260 \\ Tower height & \(L_{\mathrm{T}}\)/m & 67.3 \\ Tower bottom diameter & \(D_{\mathrm{b}}\)/m & 5 \\ Tower top diameter & \(D_{\mathrm{t}}\)/m & 3 \\ Tower wall thickness & \(r_{\mathrm{T}}\)/mm & 41 \\ Young’s modulus of tower material & \(E\)/GPa & 210 \\ Monopile diameter & \(D_{\mathrm{p}}\)/m & 6 \\ Monopile wall thickness & \(t_{\mathrm{p}}\)/mm & 80 \\ Lateral stiffness & \(K_{\mathrm{L}}\)/GN\(\cdot\)m\({}^{-1}\) & 1.53 \\ Rock stiffness & \(K_{\mathrm{R}}\)/ GN m\(\cdot\)rad\({}^{-1}\) & 205.72 \\ Coupling stiffness & \(K_{\mathrm{LR}}\)/GN & \(-\)13.88 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Parameters for soil layer [46].  
Because of the soft-stiff region being very small, it is observed that the natural frequency of the wind turbine structures will tend toward 1P and even draw close to it. Therefore, in order to ensure the long-term normal operation of the wind turbine, the natural vibration frequency of the wind turbine should be biased towards 3P in the design soft-stiff region. That is, the design natural frequency should be in a slightly higher frequency area to avoid the resonance risk during the long-term operation of the wind turbine. As the number of cyclic loading increases, the natural frequency would be moving towards the medium point in the design soft-stiff region.