Tower design and analysis - Wind Power Generation and Wind Turbine Design - page 544

Table 3: Properties of the TMD.

Rotational frequency (rev/min)

15 30

Mass ratio (%) 1 1

Tuning ratio 0.99 0.99

Natural frequency (rad/s) 4.45 4.55

Mass (kg) 997 997

Stiffness constant (kN/m) 20.64 19.74

Damping constant (kNs/m) 0.45 0.44

Damping ratio (%) 5 5

optimal tuning ratio of the TMD. Ghosh and Basu [55] extended the theory based
on “fi xed-points” to obtain closed form expression for optimal tuning ratio in case
of a damped structure. This was used by Murtagh et al. [49] designing an optimal
TMD for a wind turbine tower. The expression for the optimal tuning parameter
n opt for a wind turbine tower with damping ratio x n in the fundamental mode of
vibration is [ 49 , 55 ]:

2 2

14 −− xmx (2 − )

n n

n =

opt (23 )

3

(1 + m )

The optimal tuning ratio together with an optimal damping ratio in the TMD will
minimize the maxima of the displacement transfer function of a wind turbine tower.
Murtagh et al. [49] considered a tower of hub height 60 m and blades with
radius 30 m for a three-bladed wind turbine and designed a TMD for suppression
of the tip displacement. The mean wind speed at the top of the tower was assumed
to be 20 m/s. The fi rst three modal damping ratios of the tower were assumed to be
1% of the critical. A mass ratio of 1% was assumed for the TMD, giving the
damper a damping ratio of 5% of critical. Thus, when used in conjunction with eqn
(23), an optimal tuning ratio of 0.99 is obtained. The forced vibration responses of
the coupled tower-blades model including and excluding the TMD were calcu-
lated and compared. Two rotational frequencies of the rotor system were consid-
ered, and the blades are perturbed under the action of rotationally sampled wind
turbulence [30]. The design parameters of the dampers designed for the two cases
are presented in Table 3.
Figure 8 presents the tip displacement transfer function amplitudes obtained for
the coupled tower and rotating blades model ( = 15 rev/min) with and without

the damper. When contrasting the two transfer functions obtained, it is evident that
the presence of the damper causes the peak to split and decrease substantially in
magnitude. Figure 9 presents the simulated wind-induced response of the coupled
blade-tower model, at the top of the tower, including and excluding the damper.
From this fi gure, it is evident that the damper has been effective in suppressing the
vibrations, particularly in the earliest portion of the time-history, where the

Ω