The stability of offshore wind turbines during operation is crucial, and as an emerging technology, the TetraSpar floating offshore wind turbine demonstrates commendable stability. Moreover, it can effectively reduce the levelized cost of energy (LCOE) in production and installation. However, There is relatively scarce research on its dynamic response and reliability under complex ocean conditions. To reduce the vibration response of floating offshore wind turbines in complex marine environments and improve the stability of the turbines, it is necessary to establish a fully coupled dynamic model to study the multi-field coupled dynamic response and vibration control of the turbines. In light of this, this paper establishes a multi-physical field coupled dynamic model for TetraSpar floating offshore wind turbine through an F2A coupling framework, integrating aerodynamics, servo-elasticity, hydrodynamics, and mooring dynamics. The study analyzes the vibration suppression effects of different Tuned Mass Dampers (TMDs) on the wind turbine system. Using the two main response frequencies, 0.0367Hz and 0.57Hz, in the front and rear directions of the wind turbine as the tuning frequencies for TMD. The best vibration reduction effect was achieved by comparing multiple sets of TMD with different parameters. Analysis shows that TMD can significantly reduce the standard deviation of bending moment and tension of steel bars before and after tower foundation, with maximum values of 38.76% and 73.02%, respectively. At the same time, TMDs also impose a burden on wind turbines, such as a maximum increase of 2.23% in the average values of tower base bending moment, platform pitch, and tower top longitudinal displacement, which also slightly increase under extreme sea conditions. However, compared to the significant vibration suppression effect of TMD, it can be ignored.