We investigate to apply tuned mass dampers (TMDs) (one in the fore-aft direction, one in the side-side direction) to suppress the vibration of a monopile wind turbine tower. Using the spectral element method, we derive a finite-dimensional state-space model Sigma(d) from an infinite-dimensional model Sigma of a monopile wind turbine tower stabilised by a TMD located in the nacelle. Sigma and Sigma(d) can be used to represent the dynamics of the tower and TMD in either the fore-aft direction or the side-side direction. The wind turbine tower subsystem of Sigma is modelled as a non-uniform SCOLE (NASA Spacecraft Control Laboratory Experiment) system consisting of an Euler-Bernoulli beam equation describing the dynamics of the flexible tower and the Newton-Euler rigid body equations describing the dynamics of the heavy rotor-nacelle assembly (RNA) by neglecting any coupling with blade motions. Sigma(d) can be used for fast and accurate simulation for the dynamics of the wind turbine tower as well as for optimal TMD designs. We show that Sigma(d) agrees very well with the FAST (fatigue, aerodynamics, structures and turbulence) simulation of the NREL5-MW wind turbine model. We optimise the parameters of the TMD by minimising the frequency-limitedH(2)-norm of the transfer function matrix of Sigma(d) which has input of force and torque acting on the RNA, and output of tower-top displacement. The performances of the optimal TMDs in the fore-aft and side-side directions are tested through FAST simulations, which achieve substantial fatigue load reductions. This research also demonstrates how to optimally tune TMDs to reduce vibrations of flexible structures described by partial differential equations.