Tower design and analysis - Wind Power Generation and Wind Turbine Design - page 543

where [ M T ], [ K T ] and [ C T ] are the mass, stiffness and damping matrices of the
tower/nacelle, respectively, {()},{()},{()} are the time-dependent displace-

ment, velocity and acceleration vectors respectively, { F T ( t )} is the total wind drag
loading acting on the tower,

xt xt xt

V ′ is the effective blade base shear acting at
the top of the tower and { F DAMP ( t )} is the damping force brought about by the
action of the TMD. Details on how to calculate the effective blade base shear
time-histories and total wind drag loadings may be found in Murtagh et al. [ 30 ].
The response time-histories of the tower can be obtained following a modal
decomposition of the tower response, transforming the set of equations in eqn (20)
in a Fourier domain and subsequently applying an inverse FFT [49].

{( }

4.2 Design of TMD

For designing a TMD two important parameters need to be considered, the damp-
ing ratio and the tuning ratio. For an effi cient performance of a TMD these two
ratios need to be optimized.
A number of approximate and empirical expressions are available for the evalu-
ation of the optimum damping ratio of the TMD. Given below is the simple
expression by Luft [51] for the optimum damping ratio of the TMD:

m

(21 )

x =

D ,opt

2

where m is the mass ratio of the damper (i.e. mass of the damper to the entire
mass of the assembly). In order to tune the damper, its natural frequency is
obtained as the product of a tuning ratio n , times the natural frequency of the
coupled tower-blades system, i.e.:

w

D

n =

w

CS,1

(22 )
where w CS,1 is the fundamental frequency of the coupled tower-rotating blades
assembly. It is possible to derive a closed form expression for the optimum tun-
ing ratio of the TMD attached to a damped structure based on the “fi xed- point”
theory of Den Hartog [53] which had been proposed for the case of undamped
structural systems subjected to sinusoidal excitation. In the optimal design of a
TMD attached to an undamped structural system subjected to sinusoidal excitation
[ 53 , 54 ], two “fi xed-point” frequencies were obtained at which the transmissibil-
ity of vibration is independent of the damping in the TMD. It was also observed
that the amplitude of the response transfer functions at the two fi xed points was
unequal and had a contrasting effect with the change in the tuning ratio. For a
structure subjected to an external force which has wide banded energy content or
which has dominant energy at the natural period of the structure, the maximum
response reduction is achieved when the area under the transfer function curve
is at a minimum. This implies that the values of the transfer function at the fi xed
points should be equal and the value of the tuning ratio for which this occurs is the