## 1 Introduction  
Driven by the continuous depletion of hydrocarbon resources, the international energy market is shifting toward eco-friendly and renewable energy solutions. Offshore wind energy is a clean and sustainable source. It has obtained extensive attention from all over the world because of the high availability. The global installation of offshore wind turbines is 32.5 GW in total by the end of 2020. However, by 2030 the global installation capacity will reach up to 234 GW. A marked sensitivity to dynamic loading is characteristic of offshore wind turbine structures. Consequently, to prevent resonance with dynamic loads, special heed ought to be paid to the natural frequency of the wind turbine generator structure during the design process. For this reason, conventional design guidelines stipulate that the natural frequency of the supported structure should fall within the \"soft-stiff\" range [1,2], namely between 1P and 3P. Here, 1P is induced by mass imbalance issues, while 3P corresponds to the frequency of the shadowing effect. Due to the narrow scope of the soft-stiff range, the design natural frequency of wind turbine structures must meet stringent requirements, as illustrated in Figure 1. In the offshore wind farms, waves and wind are the predominant loading. The cyclic loading will have a remarkable effect on the foundation stiffness. The natural frequency of OWT will be induced to change by the variation of foundation stiffness. If the natural frequency is changed to be closed to 1P or 3P, the structure will resonate with the wave or wind, and even collapse. The calculation of the natural frequency of offshore wind turbine structures has long been a focal point in research circles, with the aim of refining existing evaluation methodologies. [PERSON] [3] introduced a simplified analytical method to analyze the dynamic characteristics of offshore wind turbine monopiles and the influence of foundation stiffness on natural frequency. [PERSON] [4] developed a simplified calculation method considering soil-pile interaction, modeling it with a rotational soil spring and lumping the inertial effects of the top structure and blades into a rigid mass point. [PERSON] et al. [5] accurately simulated soil-pile interaction using horizontal and rocking stiffness springs, equivalenting the tower to an Euler-Bernoulli beam, and validated the method's effectiveness through model tests and numerical calculations. [PERSON] et al. [6] used three coupled soil springs to improve calculation accuracy. While these methods are simple and easy to operate, determining foundation stiffness springs remains complex. Therefore, [PERSON] et al. [7] and [PERSON] et al. [8] calculated natural frequencies based on p-y curve and Winkel spring models. Employing coupled springs to model the foundation, [PERSON] et al. [9] developed a Rayleigh-Ritz solution for deriving high-order natural frequencies and mode shapes of monopile offshore wind turbines. Nevertheless, these methods fail to consider the nonlinear changes in soil stiffness induced by cyclic loading. However, the offshore wind turbine structures are in the special loading conditions. The supported structure would be often pushed by cyclic loading such as wind and waves during its operation period [10]. In addition, it will encounter the threat of extreme loadings such as storms and earthquakes, that will lead to the variation of soil parameters and characteristics. [PERSON] et al. [11] examined how nonlinearity in soil-pile interaction affects the  
Figure 1: Typical wind–wave spectra and 1P/3P frequency bands of three-blade wind turbines. natural frequency of offshore wind turbine units and pointed out that the nonlinear strain of soil around pile will have a remarkable effect on the natural frequency. Additionally, [PERSON] et al. [12] found that natural frequency correlates closely with soil-pile interaction state. To account for the effect of long-term cyclic loading on soil, [PERSON] et al. [13] and [PERSON] et al.