Environmental Engineering Reference
In-Depth Information
Equilibrium of out-of-plane moments about the teeter axis requires that
R
R
R
m
b
r
2
dr
- gW
2
m
b
r
2
dr
0 =
p
a
,
x
r dr
- g
(5-41a)
-
R
-
R
-
R
where
p
a
,
x
= axial air pressure loading per unit of blade span (N/m)
m
b
= blade mass per unit of span (kg/m)
g,g² = teeter deflection (rad) and acceleration (rad/s
2
), respectively
The limits -
R
to
R
indicate that the integration is over both blades. The first term is the net
(unbalanced) aerodynamic moment on the rotor, the second is the inertial moment, and the
third is the moment of centrifugal forces which oppose the teetering motion. We can analyze
the
rigid-body motion
of the rotor by rewriting Equation (5-41a) as an
equation of motion
and
setting the aerodynamic forces to zero (
i.e.
teetering in a vacuum), which gives
I
tt
g +
I
tt
W
2
g =
M
a
,
net
= 0
(5-41b)
where
I
tt
= mass moment of inertia of the rotor about the teeter axis (kg-m
2
)
M
a
,
net
= unbalanced aerodynamic out-of-plane moment on the rotor (N-m)
Equation (5-41b) describes simple harmonic motion, with a frequency equal to W (
1P
), so
g
=
g
1
sinW
t
+ g
2
cosW
t
= g
1
siny + g
2
cos y
(5-41c)
where
g
1
, g
2
= teeter amplitudes determined by the wind speed spatial distribution (rad)
y = blade azimuth (rad)
Returning to a consideration of air loads, the teeter motion will cause a wind velocity com-
ponent
opposing the motion
, which can reduce or eliminate variations in wind speed across
the rotor. To illustrate this, assume a vertical, linear wind shear gradient. The axial wind
velocity,
V
x
, seen by the teetering blade is
V
x
=
U
0
+
D
U
(5-42a)
2
R
r
siny (1
-
a
)
-
r
W (g
1
cosy
-
g
2
siny )
where
U
0
= free-stream wind velocity at hub elevation (m/s)
D
U
= vertical change in wind velocity across the rotor diameter (m/s)
Variations in wind speed from
U
0
are eliminated at the rotor by a small teetering motion that
is 90 degrees out-of-phase with the wind shear gradient, with the component amplitudes
2
= -
(1 -
a
)
D
U
U
g
1
= 0
g
(5-42b)
2 l
In practice, teeter deflections are restrained by
teeter stops
that prevent the blade from
hitting the tower during extreme gusts or during starting and stopping. Teetered rotors may
also be
coned
, and sometimes the teeter axis is not perpendicular to the blade axis (the angle
of inclination is called the d
3
angle). While these modifications complicate the aerodynamic
analysis, the benefits of teetering remain the same: reductions in cyclic loads on the blades
and, most importantly, on the nacelle and tower. Analysis of rotor loads must include teeter
motion, but mean power output can be determined without including teetering.
Search WWH ::
Custom Search