This academic article presents a comprehensive study on the numerical solving of the Duffing equation using the Euler method in the time domain and analytical investigation in the frequency domain, with a specific focus on its application to a Tension Leg Platform (TLP) offshore wind turbine system. The Duffing equation, a classical nonlinear mathematical model, serves as a valuable tool for characterizing the dynamic response of the TLP platform subjected to varying wind loads and wave excitations. In the numerical approach, the Euler method is employed to solve the Duffing equation in the time domain, offering an efficient and straightforward numerical solution. By discretizing the equation, the system's transient behavior is captured, enabling a detailed analysis of the TLP wind turbine's response to different operational conditions. Conversely, an analytical investigation of the Duffing equation in the frequency domain is conducted, providing valuable insights into the system's steady-state response under harmonic excitations. By applying mathematical techniques, the system's frequency response is obtained, enabling the identification of resonance and frequency-domain characteristics critical to the TLP wind turbine's performance.

In conclusion, the research outcomes contribute to the deeper understanding of the complex interactions between wind, waves, and the TLP platform, providing valuable knowledge for the further development and optimization of offshore wind energy systems. Additionally, the successful application of the Euler method and the analytical approach to the Duffing equation can be extended to other nonlinear systems in engineering and scientific disciplines.