Abstract
Due to severe environmental conditions in the marine area,the FWTs suffer from excessive three-dimensional vibrations which will cause potential instability issues and severe fatigue damage of the structure and mooring cables.To reduce the vibrations caused by wind and wave loading,structural vibration control method is being studied to mitigate the vibrations of offshore wind turbines[1-3].While most of the current research focuses on reducing the fore-aft vibrations,OWTs suffer from significant side-side vibrations due to vortex induced vibrations or misalignment between wind and wave loading.To address this issue,Lackner and Rotea[4]used dual linear TMDs to control bi-directional vibrations of both fixed and floating wind turbines.The controller was shown to be effective,however the TMD installed in the FWT required a large stroke.To overcome the limitations of dual linear TMDs,Sun and Jahangiri[5]developed a 3D-PTMD to control bi-directional vibrations of a fixedbottom OWT.
In this paper,a three-dimensional pendulum tuned mass damper(3DPTMD)and dual linear pounding tuned mass dampers(2PTMDs)are used to mitigate the three-dimensional vibrations of a spar type FWT.An analytical model of a spar-type FWT coupled with the 3DPTMD and 2PTMDs is established using Euler-Lagrangian equation.The NREL 5MW OC3-Hywind spar buoy wind turbine is used to examine the performance of the 3D-PTMD and 2PTMDs.Results show that the 3D-PTMD with a mass ratio of 0.1%can reduce the FWT tower vibrations by around 32%.It is also found that the 2PTMDs can reduce the spar RMS response by around 18%in roll direction and 50%in pitch direction with an allowable stroke in the spar.
Figure 1 illustrates the schematic model of the FWT with a 3D-PTMD in the nacelle and dual linear pounding TMDs(2PTMDs)installed in the floater.Totally,the coupled system contains 17 degrees of freedom(DOF).
In terms of Euler-Lagrangian equation,the system's EOMs can be established and written in a matrix form as:
Figure 1 Schematic model of the spar FWT coupled with a 3D-PTMD and 2PTMDs
where
are the system mass,damping and stiffness matrices.
is the restoring stiffness matrix,
is the generalized force caused by nonlinearity of pendulum,and
is the pounding force.Parameters
are the wind,wave,buoyancy and mooring loadings respectively.The responses of the FWT can be determined by solving Eq.(1).Figure 2 portrays the flowchart for solving the response of the FWT system.
Figure 2 Flow chart of solving the spar FWT system
The established model is validated by comparing the natural frequencies of the established model with that reported in reference[6].Table 1 shows that the established model is valid.
Table 1 Model verification
According to reference[5],a suggested mass ration of 0.1%is chosen with a frequency ratio of 0.97 and a damping ratio of 8%.Figure 3 illustrates the relative displacement of the nacelle with and without the 3D-PTMD under misalignment angles of 30°and 60°.It can be found that the 3D-PTMD can reduce the RMS response by around 32%in foreaft direction.Also,it can be seen that the 3D-PTMD is ineffective in reducing the structural response of the nacelle in sideside direction.This is due to the reason that the frequency of the 3D-PTMD is tuned to the natural frequency of the tower,while the dominant frequency in side-side direction is around 0.1 Hz caused by wave loading.
Figure 3 Tower response comparison with and without 3D-PTMD under 30°and 60°of misalignment angle
The 2PTMDs are adopted and located at sea water level to mitigate the response of the spar in roll and pitch directions.A suggested mass ratio of 2%is chosen for the 2PTMDs with a frequency ratio of 0.94 and a damping ratio of 11%.Also,the 2PTMDs properties are optimized and the pounding stif fness is chosen as 104(N/m 1.5)and coefficient of restitution is 0.3,the gap distance is considered to be 3 m.Figure 4 shows the roll and pitch response of the spar with and without 2PTMDs under 30° and 60° of misalignment angle.It can be found that the 2PTMDs reduce the RMS response by around 18%in roll direction,and 50%in pitch direction.Figure 5 illustrates the TMD stroke in X direction.Although,the TMD stroke is relatively large in X direction,it is much smaller than that of a regular linear TMDs studied in reference[4]where the TMD stroke is up to 20 m.
Figure 4 Spar response comparison with and without 2PTMDs under 30°and 60°of misalignment angle
Figure 5 2PTMDs stroke under 30°misalignment angle in xr direction