Page 295 - Wind Energy Handbook

BLADE DYNAMIC RESPONSE 269

Hence
ð R
1
Q i ¼ rÙ dC l ì i (r)u(r, t)c(r)dr (5:87)
2 dÆ 0
, can be derived by a method
An expression for the standard deviation of Q i , ó Q i
analagous to that given in Section A5.4 of the Appendix for a non-rotating blade,
yielding
2ð ð ð 1
R R
o
ó 2 ¼ 1 rÙ dC l S (r 1 , r 2 , n)dn ì i (r 1 )ì i (r 2 )c(r 1 )c(r 2 )r 1 r 2 dr 1 dr 2 (5:88)
Q i 2 dÆ 0 0 0 u
0
Here S (r 1 , r 2 , n) is the rotationally sampled cross spectrum for a pair of points on
u
the rotating blade at radii r 1 and r 2 . Equation (5.88) is parallel to Equation (A5.16) in
2
the Appendix with r and r9 replaced by r 1 and r 2 and (rUC F ) replaced by
1
2
( rÙ(dC l )=(dÆ)) . From this it can be deduced that the power spectrum of the
2
generalized load with respect to the ith mode is
ð ð
2 R R
(n) ¼ 1 rÙ dC l o (5:89)
u
S Q i S (r 1 , r 2 , n)ì i (r 1 )ì i (r 2 )c(r 1 )c(r 2 )r 1 r 2 dr 1 dr 2
2 dÆ 0 0
In practice, this expression is evaluated using summations to approximate to the
integrals.
Power spectrum of tip deflection

The expression for the amplitude of the ith mode blade tip response in response to
excitation by a harmonically varying generalized load is given by Equation (A5.4)
in the Appendix. Hence the power spectrum of the tip displacement is related to
the power spectrum of the generalized load by

(n) 1
S xi (n) ¼ 2 (5:90)
S Q i
2
2
2 2
2 2
k [(1 n =n ) þ 4î n =n ]
i i i i
2 2
(n)=k )[DMR] where DMR stands for the dynamic
i
This can be written S xi (n) ¼ (S Q i
magnification ratio. n i is the ith mode natural frequency in Hz.
Figure 5.28 shows the power spectrum of first mode tip deflection, S x1 (n), for
Blade TR operating at 30 r.p.m. in a mean wind of 8 m=s. A lower damping ratio of
0.1 has been selected so that the effect of dynamic magnification is emphasized and
the turbulence intensity has been arbitrarily set at 12.5% so that ó u ¼ 1m=s. Also
shown is the first mode tip deflection spectrum ignoring dynamic magnification,
2
S Q1 (n)=k , which, when multiplied by the square of the dynamic magnification ratio
i
(also plotted), yields the S x1 (n) curve. The standard deviation of first mode tip
Ð 1
deflection, ó x1 ¼ S x1 (n)dn, comes to 54 mm, a 24 percent increase compared
0
with the value without dynamic magnification. The former is at a minimum,