Page 304 - Wind Energy Handbook

278 DESIGN LOADS FOR HORIZONTAL-AXIS WIND TURBINES

where the coupled terms have been transferred to the right hand side. Multiplying
through by ì(r) and integrating over the length of the blade gives:

ð ð
R R
€
€
_
2
m 1 f J (t) þ c 1 f J (t) þ m 1 ø f J (t) ¼ ì(r)q(r, t)dr m(r)ì(r)ì TJ (r)dr:f T (t)
f
f
f
0 0
ð
R
_
^ c c(r)ì(r)ì TJ (r)dr: f T (t) (5:108)
f
0
By analogy with Equation (5.70), the equation of motion of the tower is
ð
H
_
€
2
f
m T1 f T (t) þ c T1 f T (t) þ m T1 ø f T (t) ¼ ì T (z)q(z, t)dz (5:109)
f
T
0
Here ì T is the tower first mode shape and 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
ð
H
2
m T1 ¼ m T (z)ì (z)dz þ m N þ m R þ I R =L 2 (5:110)
T
0
Here m T (z) is the mass per unit height of the tower, m N and m R are the nacelle and
rotor masses, and I R is the inertia of the rotor about the horizontal axis in its plane,
which is constant over time for a three-bladed rotor. For a two-bladed, fixed-hub
rotor it varies with rotor azimuth, and for a teetering rotor it is omitted altogether.
The major component of the loading on the tower, q(z, t), is the load fed in at hub
height, H, from the blades. The inertia forces on the blades due to rigid body
motion associated with the tower first mode have been accounted for by including
rotor mass and inertia in m T1 , and the corresponding damping forces can be
accounted for in the calculation of the damping coefficient, c T1 . However, the
aerodynamic loads on the blades and the inertia and damping forces associated
with blade flexure – all of which are transmitted to the tower top – have to be
included in the right-hand side of Equation (5.109) as

dì T
ì T (H):F þ :M ¼ F þ M=L (5:111)
dz H
where

X X ð R X ð R
_
€
f
f
F ¼ q(r, t)dr m 1 (r)ì(r)dr:f J (t) ^ c c(r)ì(r)dr: f J (t) (5:112)
N N 0 N 0
and