Page 303 - Wind Energy Handbook

BLADE DYNAMIC RESPONSE 277

are calculated as if the blades were cantilevered from a rigidly mounted shaft, i.e.,
in the same way as before. The blade modes are not orthogonal to the tower modes,
so the equations of motion for the different modes are no longer independent of
each other, but contain coupled terms. Furthermore, the blade deflections arising
from excitation of the tower modes vary with blade azimuth, so a step-by-step
solution is required. The treatment which follows is limited to the fundamental
blade and tower modes, but could be extended to encompass higher modes.
The equation of motion of the blade is given by Equation (5.62). The blade
deflection for blade J may be written
x(r, t) ¼ ì(r): f J (t) þ ì TJ (r): f T (t) (5:105)

where ì(r) is the first blade mode shape for a rigid tower and ì TJ (r) is the
normalized rigid body deflection of blade J resulting from excitation of the tower
first mode. Assuming the normalization is carried out with respect to hub deflec-
tion,
r
ì TJ (r) ¼ 1 þ cos ł J (5:106)
L

where L is the depth below the hub of the intercept between the tangent to the top
of the deflected tower and the undeflected tower axis, as illustrated in Figure 5.31.
Substitution of Equation (5.105) into Equation (5.62) yields, with the aid of Equation
(5.65):
_
€
2
f
c
f
m(r)ì(r) f J (t) þ ^ c(r)ì(r)f J (t) þ m(r)ø ì(r) f J (t)
_
€
f
c
¼ q(r:t) m(r)ì TJ (r) f T (t) ^ c(r)ì TJ (r) f T (t) (5:107)
f
µ(r) µ TJ (r)
ψ J
r cos ψ J
µ T (z)

L

z

Figure 5.31 Fundamental Mode Shapes of Blade and Tower