Structural dynamics of wind turbines - Wind Power Generation and Wind Turbine Design - page 122

Clearly assessment of local stresses and strains especially when dealing with
composites, requires more advanced modelling that account for the internal
material structure.
Solely the elastic DOF cannot describe the complete dynamics of a wind tur-
bine. There are also rigid DOF while the connections among the components fur-
ther complicate the dynamics of the system. The rotor blades are connected to the
hub (the one end of the drive train) allowing rotation with respect to the drive-train
axis while the nacelle which houses the drive train, is connected to the tower top
also allowing at least rotation in yaw. In addition to blade rotation and yaw there is
also the blade pitch. Pitch can vary either collectively or cyclically or even inde-
pendently for each blade. Finally in the case of two-bladed rotors, the rotor system
as a whole is allowed to teeter. At the connection points, specifi c compatibility
conditions are required which transfer the deformation state from one connected
component to the other. This calls for a multi-component analysis which we detail
in Section 4.
Most of the rigid body DOF are connected to the control system. At least in the
more recent designs there is clear preference for variable pitch and variable speed
control concepts. So when the wind turbine operates in closed loop, the variation
of the pitch and the rotor speed will satisfy the control equations which therefore
become part of the dynamic description of the system. In certain cases, there is
need to also include concentrated properties like point masses, springs or dampers.
For example, the yaw gear and the gear box in the drive train are modelled as com-
bined point masses and springs while the hub is modelled as concentrated mass.
Concentrated properties are associated to specifi c DOF for which we must formulate
appropriate dynamic equations as in the case of the control variables.
Although reference has been made to the formulation of dynamic equations we
did not specify the way this is done. The general framework is presented in Section 2.
One way to proceed is to base the derivation on Newton's second law which
requires the dynamic balance of forces and moments. It has the advantage of a
direct interpretation in relatively simple systems but becomes cumbersome when
dealing with complicated systems. Alternatively energy or variational principles,
like the principle of virtual work or Hamilton's principle offer a general and sys-
tematic way for deriving the dynamic equations of even very complicated mechan-
ical systems. It is true that depending on the complexity of the system, the
derivations can become quite lengthy. However, the current availability of sym-
bolic mathematics software has removed former reluctance in using variational
principles.
Once the dynamic equations of the complete wind turbine are formulated, the
next step is to include the aerodynamic loading on the blades. The main diffi culty
in this respect is that aerodynamic loads depend on the dynamics of the blades in
a non-linear manner. A further complication is due to the onset of stall which because
of the aeroelastic coupling becomes dynamic with clear lagging behaviour. A brief
account on aeroelastic coupling is given in Section 5.
The usual way to solve the coupled fi nal aeroelastic equations is to integrate
them in time. Due to their non-linear character in principle we need an implicit