A braceless semi-submersible floating platform is proposed for a Technical University of Denmark (DTU) 10-MW wind turbine at moderate water depths with reference to an existing National Renewable Energy Laboratory (NREL) 5-MW braceless semi-submersible floating platform, and a servo control system for a 10-MW semi-submersible floating offshore wind turbine (FOWT) is introduced. To control the ultimate and fatigue loads of the FOWT, a fore-aft tuned mass damper (TMD) installed in the nacelle of the 10-MW semi-submersible FOWT was investigated for vibration alleviation and load reduction. Considering the hydrodynamic and mooring effect, a four degrees-of-freedom (DOFs) (platform surge and pitch motions, tower fore-aft bending, and TMD translation) simplified dynamic model for the 10-MW semi-submersible FOWT is established based on D’Alembert’s principle. Then, the parameter estimation is conducted based on the Levenberg–Marquardt (LM) algorithm, and the simplified dynamic model was further verified by comparing the output responses with FAST and the proposed model. Furthermore, the exhaustive search (ES) and genetic algorithm (GA) are embedded into the simplified dynamic model to optimize the TMD parameters. Finally, a fully coupled time-domain simulation for all the selected environmental conditions is conducted in FAST, and the vibration suppression performance of the optimized TMD design for the 10-W semi-submersible FOWT was further examined and analyzed.