Environmental Engineering Reference
In-Depth Information
-- tension stiffening
-- centrifugal forces
--
Coriolis accelerations
The tension stiffening that affects primarily the blades is induced by steady centrifugal and
gravity forces. The centrifugal and Coriolis terms are a direct result of using a rotating
coordinate system.
Equations of Motion
The total acceleration at a point in the rotating structure, with respect to fixed
coordinates, can be represented in terms of the acceleration in the rotating coordinate system
by an equation similar to Equation (11-18), as follows:
a
t
=
u
+ 2W
x
u
+ W
x
[W
x
(
r
+
u
)]
(11-42)
where
a
t
=
total acceleration vector, excluding gravity (m/s
2
)
r
= original position vector of an element; origin at lower bearing (m)
u
= displacement vector; observed in the rotating coordinate system (m)
W
=
constant angular velocity vector of the rotating coordinate system (rad/s)
Using this expression for the acceleration in the
equation of motion
for a structure, the usual
damping and stiffness matrices are altered from those of a static structure. The resulting
differential equations are represented by
M
u
+
C
u
-
S
u
+
K
u
=
F
c
+
F
g
(11-43)
where
M =
normal (unaltered) mass matrix (kg)
u =
displacement observed in the rotating coordinate system; dot signifies
derivative with respect to time (m)
C
= additional Coriolis matrix (N-s/m)
S
= additional centrifugal softening matrix (N/m)
K =
normal (unaltered) stiffness matrix (N/m)
F
c
=
additional static load vector representing steady centrifugal force (N)
F
g
= gravitational forces; steady in a VAWT (N)
The Coriolis matrix,
C
, is
skew-symmetric
and results from the second term on the right
side of Equation (11-42). Here we have ignored any structural (internal) damping because
most VAWTs are lightly damped. Therefore, the
C
matrix is totally the result of Coriolis
effects. The centrifugal softening matrix,
S
, comes from the variable portion (
i.e.
,
dependent on
u
)
of the last term on the right in Equation (11-42).
F
c
also comes from the
last term in Equation (11-41). To obtain the mode shapes and frequencies of the turbine
as observed in the rotating system, Equation (11-41) is reduced to the following form:
M
u
+
C
u
+ (
K
+
K
G
-
S
)
u
= 0
(11-44)
where
K
G
= geometric stiffness matrix resulting from steady centrifugal and
gravitational loadings (N/m)
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