Page 282 - Wind Energy Handbook

256 DESIGN LOADS FOR HORIZONTAL-AXIS WIND TURBINES

to focus on the blade dynamic behaviour itself. The treatment is further limited to
the response of blades in unstalled flow because of the inherent difficulty in
predicting stalled behaviour.
The equation of motion for a blade element at radius r subject to a time varying
load q(r, t) per unit length in the out-of-plane direction is

2
@ 2 @ x
x
c
x
m(r)€ x þ ^ c(r) _ x þ EI(r) ¼ q(r, t) (5:62)
@r 2 @r 2
where the terms on the left-hand side are the loads on the element due to inertia,
damping and flexural stiffness respectively. I(r) is the second moment of area of the
blade cross section about the weak principal axis (which for this purpose is
assumed to lie in the plane of rotation) and x is the out-of-plane displacement. The
expressions m(r) and ^ c(r) denote mass per unit length and damping per unit length
c
respectively.
The dynamic response of a cantilever blade to the fluctuating aerodynamic loads
upon it is most conveniently investigated by means of modal analysis, in which the
the excitations of the various different natural modes of vibration are computed
separately and the results superposed:

X
1
x(t, r) ¼ f j (t)ì j (r) (5:63)
j¼1

where ì j (r) is the jth mode shape, arbitrarily assumed to have a value of unity at
the tip, and f j (t) is the variation of tip displacement with time. Equation (5.62) then
becomes

1
X d 2 d ì j (r)
2
_
€
f
f
m(r)ì j (r)f j (t) þ ^ c(r)ì j (r)f j (t) þ EI(r) f j (t) ¼ q(r, t) (5:64)
c
dr 2 dr 2
j¼1
For low levels of damping the beam natural frequencies are given by

2
d 2 d ì j (r)
2
m(r)ø ì j (r) ¼ dr 2 EI(r) dr 2 (5:65)
j
so Equation (5.64) becomes
X
1
€
_
2
c
f
fm(r)ì j (r)f j (t) þ ^ c(r)ì j (r) f j (t) þ m(r)ø ì j (r) f j (t)g¼ q(r, t) (5:66)
f
j
j¼1
Multiplying both sides by ì i (r), and integrating over the length of the blade, R,
gives: