Environmental Engineering Reference
In-Depth Information
Development of Natural Wind Excitation Testing
The structural dynamics of rotating VAWT machines are critical design conditions that
require experimental techniques to match the fidelity of the advanced analyses described
previously. The previous example using the 2-m VAWT showed the power of performing
step-relaxation testing during rotation. However, this approach is technically intensive and
would not be generally applicable for many operational scenarios. As mentioned previously,
the use of natural wind excitation is generally a more applicable approach and allows testing
in the actual operating environment. This section details the development of the natural exci-
tation technology in the form of the
natural excitation technique
or NExT [James
et al.
1995].
A more complete background on the historical development of the natural wind excitation
testing for wind turbines is also available [Carne and James 2008].
Theoretical Development of NExT
A critical step in the development of NExT was to find a relationship between modal pa-
rameters of structural interest (such as frequency and damping) and a function that could be
produced from measured response data. For NExT, the
cross-correlation function
between
two response-only measurements was used. The derivation of the relationship of modal
parameters and the cross-correlation function begins by assuming the standard matrix equa-
tions of motion:
[
M
]{
x
(
t
)} + [
C
]{
x
(
t
)} + [
K
]{
x
(
t
)} = {
f
(
t
)}
(11-50)
where
[
M
] = mass matrix (g)
[
C
] = damping matrix(N-s/m)
[
K
] = stiffness matrix (N/m]
{
f
} = vector of random forcing functions (N)
{
x
} = vector of random displacements (m)
t
= elapsed time (s)
˙,
¨ = first and second derivatives with respect to time (m/s, m/s
2
)
Equation (11-50) can be expressed in modal coordinates using a standard modal transfor-
mation after performing a standard matrix diagonalization (assuming proportional damping).
A solution to the resulting scalar modal equations can be obtained by means of a Duhamel
integral, assuming a general forcing function, {
f
}, with zero initial conditions (
i.e.
zero initial
displacement and zero initial velocity). The solution equations can be converted back into
physical coordinates and specialized for a single input force and a single output displacement
using appropriate mode shape matrix entries. The following equations result:
x
(
t
) =
n
r
=1
y
ir
y
kr
×
-¥
f
(t)
g
(
t
- t)
d
t
ò
r
(11-51)
k
ik
g
r
(
t
) = 0,
f or t
< 0
1
m
r
w
d
g
r
(
t
) =
- z
r
w
n
t
sin w
n
t
exp
,
f or t
³ 0;
where
w
n
(1 - z
r
2
)
1/2
is the damped modal frequency (rad/s)
w
d
=
w
n
=
rth
modal frequency (rad/s)
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