Page 328 - Wind Energy Handbook

302 DESIGN LOADS FOR HORIZONTAL-AXIS WIND TURBINES

(2) Calculate the damping logarithmic decrement, ä, for the tower first mode. The
aerodynamic component is given by
þ
2
^ c c a (r)ì (r)dr
1
ä a ¼ 2ðî a ¼ 2ð c a1 ¼ (5:126)
2m T1 ø 1 2m T1 n 1
where m T1 is the generalized mass of the tower, nacelle and rotor (including the
contribution of rotor inertia) with respect to the first mode given by Equation
(5.110), and n 1 is the tower natural frequency in Hz. For a stall-regulated
machine facing the wind, the rotor contribution to aerodynamic damping is
simply rUC D A R =2m T1 n 1 where A R is the rotor area.

(3) Calculate the standard deviation of resonant nacelle displacement according to
ð p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ó x1 ó u
¼ 2 p ffiffiffiffiffiffi R u (n 1 ) K Sx (n 1 ) (5:6)
U 2ä
x 1
(4) Calculate the ratio º M1 , which relates the ratio of the standard deviation of
resonant tower base moment to the mean value to the corresponding ratio for
nacelle displacement as follows:

ó M1 ó x1
¼ º M1 (5:7)
M x 1
If ì 1 (r) is set to unity for the rotor, º M1 is given by:
( )
ð H ð H U(z) 2
m(z)ì 1 (z)z dzC d A R þ C f d(z)ì 1 (z)dz
0 0 U
º M1 ¼ ( ð ) (5:127)
H U(z) 2 z
m T1 HC d A R þ C f d(z) dz
0 U H

where z is the height up the tower measured from the base, and H is the hub
height. If the loading on the tower is relatively small, this approximates to
ð H
m(z)ì 1 (z)z dz
º M1 ¼ 0 (5:127a)
m T1 H
which is close to unity because the tower head mass dominates the integral.

(5) Calculate the size reduction factor for the root bending moment quasistatic or
background response, K SMB , which reflects the lack of correlation of the wind
fluctuations along the blades and tower. K SMB may be derived from a similar
expression to that for the resonant size reduction factor given in Equation
x
(5.125), but with the exponential function modified to exp[ s=0:3L ].
u