Environmental Engineering Reference
In-Depth Information
with
cs,
j
-TT
(
j
= 1, 2) being the
j
th coupled system (CS) mode shape component
a
t
the top of the tower,
Φ
K
is the
j
th modal stiffness of the coupled system and
D,
f
is the
j
th modal mean drag force. Because modal/generalized quantities are
used in eqn (10), it is assumed that the free vibration parameters obtained from the
tower-rotor system are from a classically damped one. The modal mean drag force
on a structure (i.e. the tower or the blade) is obtained as
CS,
j
H
1
⎛
⎞
∫
2
f
=
r
C
() () ()
z B z v z
Φ
()d
z
z
(11 )
⎜
⎟
D,
j
D
CS,
j
⎝
⎠
2
0
where
H
is the length over which drag is to be calculated (i.e. the total height of the
tower or the length of the b
la
de),
C
D
(
z
) is the drag coeffi cient,
B
(
z
) is the width of
the tower (or blades), and
CS,j
(
z
) is the
j
th
mode shape component of the coupled system, all as a function of the spatial vari-
able
z
. The expected maximum displacement may be obtained as the product of a
peak factor,
vz
is the mean wind velocity and
()
Φ
(using fi rst passage analysis, as in [44]) and the root mean square
(RMS) of the displacement response at the top of the tower,
s
X
. This RMS dis-
placement response, which includes a second mode of vibration, may be obtained
by taking the square root of the area under the displacement response PSD func-
tion,
S
XX
(
f
) The PSD function
S
XX
(
f
) is found as the sum of the products of the
modal wind drag force PSD functions with their appropriate squared amplitude of
the modal mechanical admittance functions [43].
The modal drag force PSD function may be obtained from the expression:
Ψ
HH
2
∫∫
S
()
f
=
S
()
f
r
CzCzBzBzvzvz
() ( )()( )()( )
MF MF
j
j
VV
D
1
D
2
1
2
1
2
00
(12)
FF
()
z
( )(, ;)dd
z
R z
z
f
z
z
CS,
j
1
CS,
j
2
1
2
1
2
where
S
VV
(
f
) denotes the wind velocity PSD function at the top of the tower [37],
r
is the density of air, and
R
(
z
1
,
z
2
;
f
) is the spatial coherence function between
elevations
z
1
and
z
2
[19]. The mechanical admittance function at the top of the
tower due to a unit force at that point for the
j
th mode may be obtained as
F
F
CS,
j
−
TT
CS,
j
Hf
()
=
(13 )
D,
j
22
⎡
2
⎤
4
p
f
M
1 (/
−
f f
)
+
2
i
x
(/
f f
)
⎣
⎦
CS,
j
CS,
j
CS,
j
CS,
j
CS,
j
where
F
CS,
j
is the
j
th modal force due to a u
ni
t force placed at the top of
th
e
tower,
f
CS,
j
is the
j
th natural frequency,
M
is the
j
th modal mass
CS,
j
H
2
with
m
(
z
) as the mass distribution of the structure
and,
x
CS.
j
is the
j
th modal damping ratio.
Two procedures have been proposed by Murtagh [43] based on how the value of
s
X
. may be calculated. It may be computed by numerically evaluating an integral
or it may also be obtained in closed form based on some approximation. For the closed
form calculation, a method of decomposition can be employed, in which it is assumed
that the variance of the displacement response PSD function may be separated into
two components: a background component and a resonant component. Contrary to
(
M
=∫
m z
( )
Φ
( )d )
z
z
CS,
j
0
CS,
j
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