Page 283 - Wind Energy Handbook

BLADE DYNAMIC RESPONSE 257
(
X ð R ð R
1
€
_
f
m(r)ì i (r)ì j (r) f j (t)dr þ ^ c c(r)ì i (r)ì j (r) f j (t)dr
f
j¼1 0 0
ð ð
R R
2
þ m(r)ø ì i (r)ì j (r) f j (t)dr ¼ ì i (r)q(r, t)dr (5:67)
j
0 0
The undamped mode shapes are orthogonal as a result of Betti’s law, so they satisfy
the orthogonality condition:
ð
R
m(r)ì i (r)ì j (r)dr ¼ 0 for i 6¼ j (5:68)
0
c
If we assume that the variation of the damping per unit length along the blade, ^ c(r),
c
is proportional to the variation in mass per unit length, m(r), i.e., ^ c(r) ¼ a:m(r), then
ð
R
^ c c(r)ì i (r)ì j (r)dr ¼ 0 for i 6¼ j (5:69)
0
As a result, all the cross terms on the left-hand side of Equation (5.67) drop out, and
it reduces to

ð
R
_
€
2
f
f
m i f i (t) þ c i f i (t) þ m i ø f i (t) ¼ ì i (r)q(r, t)dr (5:70)
i
0
Ð R
2
where m i ¼ m(r)ì (r)dr and is known as the generalized mass, c i ¼
Ð 0 Ð i
R 2 R
i
0 ^ c c(r)ì (r)dr and 0 ì i (r)q(r, t)dr ¼ Q i (t) is termed the generalized fluctuating
load with respect to the ith mode. Equation (5.70) is the fundamental equation
governing modal response to time varying loading.
Blade flexural vibrations occur in both the flapwise and edgewise directions (i.e.,
about the weak and strong principal axes respectively). Blades are typically twisted
some 158, so the weak principal axis does not, in general, lie in the plane of rotation
as assumed above. Consequently blade flexure about one principal axis inevitably
results in some blade movement perpendicular to the other. This is illustrated in
Figure 5.23, in which the maximum blade twist near the root has been exaggerated
for clarity. Point P represents the undeflected position of the blade tip, point Q
represents the deflected position as a result of flexure about the weak principal axis,
and the line between them is built up of the contributions to the tip deflection made
by flexure of each element along the blade, M(R r)˜r=EI.
The interaction between flexure about the two principal axes can be explored
with the help of some simplifying assumptions. If the moment M varies as (R r)
2
for the first mode and I varies as (R r) , then each of the tip deflection contribu-
tions referred to above are equal, so that, for a linear twist distribution, the line PQ
is the arc of a circle. If the twist varies between zero at the tip and a maximum value
of â towards the root, then the tip deflection, ä 12 , in the direction of the weak
principal axis, at the blade section with maximum twist, is â=2 times the tip
deflection, ä 11 , perpendicular to this axis. Hence in the case of â ¼ 158, the ratio