Page 340 - Wind Energy Handbook

314 DESIGN LOADS FOR HORIZONTAL-AXIS WIND TURBINES

A5.2.2 Frequency response function

If q(r, t) varies harmonically, with frequency ø and amplitude q 0 (r), then it can be
shown that:
ð
R
ì i (r)q 0 (r)dr
1 0
f i (t) ¼ q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos(øt þ ö )
i
(ø ø ) þ (c i =m i ) ø
m i 2 2 2 2 2
i
ð R
ì i (r)q 0 (r)dr
1 0
¼ q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos(øt þ ö ) (A5:2)
i
m i ø 2 2 2 2 2 2 2
i (1 ø =ø ) þ (c i =m i ø ) ø
i i
2
Defining k i ¼ m i ø , and noting that the damping ratio î i ¼ c i =2m i ø i , this becomes:
i
ð
R
ì i (r)q 0 (r)dr
1
0
f i (t) ¼ q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos(øt þ ö ) ¼ A i cos(øt þ ö ) (A5:3)
i
i
k i 2 2 2 2 2 2
(1 ø =ø ) þ 4î ø =ø
i i i
Ð R
The numerator ì i (r)q 0 (r)dr: is the amplitude of the equivalent loading at the tip
0
of the cantilever that would result in the same tip displacement as the loading
q(r, t), and is known as the generalized load with respect to the ith mode, Q i (t).
Thus the ratio between the tip displacement amplitude and the amplitude of the
generalized load is
A i 1
ð ¼ q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
R 2 2 2 2 2 2
ì i (r)q 0 (r)dr: k i (1 ø =ø ) þ 4î ø =ø i
i
i
0
1
¼ q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼jH i (n)j (A5:4)
2 2
2 2
2
(1 n =n ) þ 4î n =n 2
k i
i i i
The ratio jH i (n)j is the modulus of the complex frequency response function, H i (n),
and its square can be used to transform the power spectrum of the wind incident on
the blade into the power spectrum of the ith mode tip displacement. Thus, in the
case of the dominant first mode, the tip displacement in response to a harmonic
generalized loading, Q 1 (t), of frequency n is given by
x 1 (R, t) ¼ f 1 (t) ¼ Q 1 (t)jH 1 (n)j
2
and the power spectrum of the first mode tip deflection is S 1x (n) ¼ S Q1 (n)jH 1 (n)j .
In what follows, the simplifying assumption is made initially that the wind is
perfectly correlated along the blade.