Environmental Engineering Reference
In-Depth Information
where
m
= blade mass per unit length (kg/m)
e
h
=
h-coordinate of the section center of mass (m)
I
zz
m
= minimum mass moment of inertia of section, per unit length (kg-m)
I
hh
m
=
maximum mass moment of inertia of section, per unit length (kg-m)
Governing Equation of Motion
For this single-degree-of-freedom system, with flap displacement
v
as the only dynamic
motion, the standard equation of motion for an incremental length of the blade has the form
[
M
]
v
+ [
J
]
v
+ [
K
]
v
= [
L
]
(11-26)
where
[M]
= mass function (kg/m)
[J] =
damping function (N-s/m
2
)
[K] =
stiffening function (N/m
2
)
[L] =
loading function (N/m)
To develop this equation governing flap motion, it is necessary to begin with Equations (11-
6) defining the equilibrium-moment curvature relationships for a blade element. Equation
(11-6a) for flapwise bending will be converted into an equation of motion.
Spanwise Tension
The spanwise tension,
T
,
in Equation (11-6a) can be obtained by directly integrating
Equation (11-6d) to give
R
ò
m
(
r
W
2
+ 2
v
W
s
q
p
-
gc
y)
d
x
T
(x) =
(11-27)
x
For the integration limit, we neglect any difference between the tip radius R and the length
of the blade.
Modal Flap Displacement Model
In order to reduce the flap motion equation to an ordinary differential equation required
for a computerized solution, a
modal model
is used, in which the flap displacement is
assumed to be of the form
v
(x,
t
) =
k
s
k
(
t
)g
k
(x)
k
= 1, 2, ...
(11-28)
where
k
=
number of
mode shapes
in the displacement model
s
k
=
time-dependent displacement scaling factor for the
k
th
mode (m)
g
k
=
spanwise shape function for the
k
th
mode
The spanwise shape functions, which must satisfy the blade kinematic and natural (force)
boundary conditions, are usually prescribed as the mode shapes for free vibration of the
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