## 1 Introduction
Their work confirmed that nonlinear stiffness design can transcend the bandwidth limitations inherent to linear TMD configurations. To address time-varying loads and uncertainties in practical engineering, multi-objective robust optimization methods have gained prominence. [PERSON] and [PERSON] [10] introduced Pareto frontier solution theory in bridge-TMD design, simultaneously optimizing displacement suppression rates under high wind speeds and mass block travel constraints. Recent advancements focus on intelligent optimization algorithms (e.g., genetic algorithm, particle swarm optimization) for coordinated TMD-structure tuning in deep-water jacket platforms, addressing wave-structure nonlinear coupling effects. By decoupling multimodal resonance peaks from the multi-peak characteristics of nonlinear dynamic equations, these advancements have expanded the operational applicability boundaries of passive TMD systems in deep-water engineering [11; 12]. Notably, the placement position of TMDs significantly influences vibration suppression efficiency. [PERSON] et al. [13] demonstrated that installing TMDs on platform decks maximizes inertial force effectiveness, providing theoretical guidance for TMD deployment in deep-water jacket platforms. While alternative solid dampers, such as FD and SMA, exhibit potential application, their performance is constrained by nonlinear characteristics or environmental sensitivity. For instance, SMA dampers rely on temperature-phase transition coupling for energy dissipation, which can induce distortions in the hysteresis loop under extreme conditions [14]. Conversely, FD systems face damping ratio degradation due to frictional interface wear during prolonged service [15]. In contrast, TMDs, with their linear dynamic characteristics and operational robustness, prove better suited for complex time-varying load scenarios in deep-water electrical platforms. Fluid-based dampers, including Tuned Liquid Dampers (TLDs) and Tuned Liquid Column Dampers (TLCDs), dissipate energy through the resonant sloshing of liquids. However, their performance is constrained by fluid nonlinearities and spatial installation limitations. [PERSON] et al. [16] demonstrated that TLDs can mitigate seismic responses in jacket platforms, with larger mass ratios yielding more significant vibration suppression. Yet, this approach conflicts with structural lightweighting imperatives. [PERSON] et al. [17] identified optimal TLD placement at the top deck level, where increasing mass ratios enhance control effectiveness. [PERSON] et al. [18] proposed semi-active hydraulic dampers to reduce deck accelerations, shear forces, and foundation displacements. Nevertheless, the sealing integrity and corrosion resistance requirements of fluid dampers escalate maintenance costs in deep-water environments. Viscous Dampers (VDs), leveraging broadband vibration suppression and hermetic sealing, represent classical liquid-based solutions. However, their velocity-dependent performance causes response deficiencies under low-frequency wave loads. [PERSON] et al. [19] systematically compared viscoelastic, viscous, and friction dampers for wave-induced jacket platform responses. [PERSON] et al. [20] optimized VD deployment configurations using long-term metocean data to enhance structural longevity. [PERSON] et al. [21] emphasized that VD fatigue life optimization requires precise phase synchronization with platform motions. Critically, multimodal coupling vibrations in deep-water jackets may induce localized VD overloading. In contrast, TMDs maintain stable dissipation efficiency across low-frequency excitations due to their inertial forces' direct proportionality to structural accelerations [22]. This study investigates the vibration mitigation performance of Tuned Mass Dampers (TMDs) on deep-water jacket platforms under typhoon-induced wind and wave loads. The jacket platform was modeled in OpenSees using P-y, T-z, and Q-z nonlinear springs to capture soil-structure interaction (SSI).