Page 285 - Wind Energy Handbook

BLADE DYNAMIC RESPONSE 259
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Tip deflection input to last iteration
ø 1 ¼
Tip deflection output from last iteration

If the mode shapes are orthogonal, advantage can be taken of this property to
simplify the derivation of the mode shapes and frequencies of the higher modes,
provided this is carried out in ascending order. A trial mode shape is assumed as
before, but before using it to calculate the inertia loadings, it is ‘purified’ so that it
does not contain any lower mode content. For example, ‘purification’ of a second
mode trial mode shape, ì 2T (r), of first mode content is achieved by subtracting
ð ð
R R
ì 1 (r)ì 2T (r)m(r)dr ì 1 (r)ì 2T (r)m(r)dr
ì 2C (r) ¼ ì 1 (r) 0 ð R ¼ ì 1 (r) 0 (5:71)
2
ì (r)m(r)dr m 1
1
0
from it. The modified second mode trial mode shape, ì 2M (r) ¼ ì 2T (r) ì 2C (r), then
satifies the orthogonality condition
ð
R
m(r)ì 1 (r)ì 2M (r)dr ¼ 0
0
After ‘purification’ of the trial mode shape, the Stodola method can be applied
exactly as before. Further ‘purification’ before succeeding iterations should not be
necessary if the lower mode shapes used for the initial ‘purification’ are accurate
enough. See Clough and Penzien (1993) for a rigorous treatment of the method.

5.8.3 Centrifugal stiffening

When a rotating blade deflects either in its plane of rotation or perpendicular to it,
the centrifugal force on each blade element exerts a restoring force which has the
effect of stiffening the blade and thereby increasing the natural frequency compared
with the stationary value. The centrifugal forces act radially outwards perpendicu-
lar to the axis of rotation, so in the case of an out-of-plane blade deflection, they are
parallel to the undeflected blade axis and act at greater lever arms to the inboard
part of the blade than they do in the case of in-plane blade deflection. This is
illustrated in Figure 5.24.
In order to take account of the effects of centrifugal loads, the equation of motion
for a blade element loaded in the out-of-plane direction is modified by the addition
of an additional term to become

2
@ @x @ 2 @ x
c
x
x
m(r)€ x þ ^ c(r) _ x N(r) þ EI ¼ q(r, t) (5:72)
@r @r @r 2 @r 2
where the centrifugal force at radius r, N(r), is the summation of the forces acting
P
2
on each blade element outboard of radius r, that is N(r) ¼ r¼R m(r)Ù r˜r.
r¼r