Tower design and analysis - Wind Power Generation and Wind Turbine Design - page 534

with S vkvk ( f ) and S vkvk ( f ) being the velocity PSD functions at nodes k and l respec-
tively and coh( k , l ; f ) is the spatial coherence function between nodes k and l . The
terms S vkvk ( f ) and S vkvk ( f ) are functions of frequency f and may be calculated
using the Kaimal spectra [37]. A coherence function suggested by Davenport
[ 19 ], coh( k , l ; f ), which relates the frequency dependent spatial correlation between
nodes k and l , is represented as

⎛ kl ⎞

−

coh( , ; kl f ) = exp −

(6)

⎜ ⎟

⎝ L ⎠

s

where | k - l | is the spatial separation and L S is a length scale given by

v

L = (7 )

S

fD

with

ˆ

v = 0.5( v + v )

(8 )

k l

and D is a decay constant. The fl uctuating component of the modal force acting
on the tower may be obtained by employing the DFT technique. The mean nodal
drag force component is obtained by transforming the nodal mean drag force time-
histories into modal force time-histories using the modal matrix. The mean modal
drag force is added to the modal fl uctuating component to obtain the total modal
drag force time-history.

2.6 Response of tower including blade-tower interaction

In order to couple the tower and rotating blades, equations of motion for the
tower that includes the blade shear forces is necessary to be considered. This is
represented by

V

{} {}

(9 )

[ Mxt ] () +[ Cxt ] () +[ Kxt ]{ ()}={ Ft ()}+{ Vt ()}

T T T T B

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

are the displacement, velocity and
acceleration vectors respectively, { F T ( t )} is the total wind drag loading vector act-
ing on the tower and

xt xt xt

V ′ is the effective blade base shear vector transmitted
from the root of the rotating blades and acting at the top of the tower. The set of
equations cannot be solved directly in time domain as the base shear is dependent
on the motion of the tower (due to coupling) and hence is not known explicitly. An
alternative way to solve the equations is to convert the set into a set of algebraic
equations by FFT and subsequently solve by inverse FFT [30].
A numerical example [30] is presented for a steel wind turbine tower of height
60 m with three blades of rotor radius 30 m. The total mass of the nacelle and rotor
system is 19,876 kg. The average wind speed at the top of the tower is 20 m/s.
Figure 3 shows the displacement response time-history at the top of tower when

{( }