## 1 Introduction  
A jacket platform is a fixed-type marine engineering structure composed of a steel truss (jacket), pile foundations, and an upper module (deck). It is widely used in offshore oil and gas exploitation, offshore wind turbine support, and oceanographic observation. Notably, there are approximately 7000 offshore platforms worldwide for marine energy production, with 93% being jacket-type platforms constructed by welding tubular members [1]. The deployment of marine structures in deep-water environments intensifies the coupled effects between complex hydrodynamic loads (including wind-wave-current interactions) and structural dynamic responses. Under extreme marine environmental excitations, structural vibrational energy may excite higher-order modal responses, leading to distortion of local stress fields and fatigue crack initiation/propagation, thereby significantly compromising structural integrity. Crucially, such vibrations persist until the energy is fully dissipated through damping mechanisms. However, conventional passive damping systems relying on frictional effects at joint interfaces and microcrack friction energy dissipation in reinforced concrete exhibit notably insufficient energy dissipation efficiency during elastic working stages [2]. Traditional vibration mitigation design paradigms primarily rely on structural stiffness enhancement principles, achieving natural frequency regulation through large-scale material augmentation. Although this mass-driven solution can attain predefined frequency response control objectives, it induces exponentially escalating construction costs in deep-water engineering practices, significantly undermining economic feasibility [3]. Consequently, dynamic control methodologies such as vibration suppression technologies and energy dissipation systems have demonstrated substantial mitigation efficacy through experimental validation [4]. Vibration control strategies are categorized into active and passive modes. Active control systems depend on real-time sensor-actuator networks and external energy supply [5]. However, for offshore infrastructure exposed to harsh marine environments, their complexity, energy dependency, and maintenance sensitivity render them prohibitively challenging. Passive control systems have gained widespread adoption due to their energy autonomy and operational robustness, classified into two categories based on energy dissipation media: solid dampers (e.g., Tuned Mass Damper (TMD), Friction Damper (FD), Shape Memory Alloy Damper (SMA)) and fluid-based dampers (e.g., Tuned Liquid Damper (TLD), Tuned Liquid Column Damper (TLCD), Hydrodynamic Buoyant Mass Damper (HBMD)) [6]. Among these, TMD has emerged as a primary research focus in solid damper applications owing to its parameter tunability, engineering maturity, and adaptability to multimodal vibrations in deep-water jacket platforms. As a classic passive control device, the Tuned Mass Damper (TMD) was initially proposed by [PERSON] (1909) [7] to mitigate ship vibrations induced by ocean waves. The TMD neutralizes primary structural vibrations through the counter-phase inertial forces generated by its mass-spring-damper system. Typically, a TMD with a mass equivalent to 1-2% of the structure's first modal mass is installed at optimal elevations and tuned to the structure's fundamental frequency. When excited at this frequency, the damper mass absorbs and dissipates vibrational energy through out-of-phase motion relative to the host structure. [PERSON] et al. [8] implemented TMDs on offshore platforms under seismic loads. The vibration suppression performance of passive TMDs critically depends on precise matching of core parameters: mass ratio, frequency ratio, and damping ratio. [PERSON] et al. [9] developed a frequency-domain iterative optimization framework for nonlinear TMDs, quantitatively revealing the influence of primary structural damping ratios and excitation intensities on parametric sensitivity.