Environmental Engineering Reference
In-Depth Information
N N
l l
NNN
+
−
⎡
⎤
1
1
…0
⎢
⎥
⎢
1
1
⎥
⎢
⎥
−
1
1
2
…0
⎢
⎥
l
l
l
⎢
⎥
1
1
2
(2 )
[
K=
]
BG
⎢
⎥
N
l
NNN
+
−
n
−
1
⎢
⎥
⎢
⎥
n
−
1
⎢
⎥
−
n
−
1
n
−
1
n
⎢
0
0
⎥
l
l
l
⎢
⎥
⎣
⎦
n
−
1
n
−
1
n
where
N
i
is the axial force at node '
i
' and
l
i
is the length of beam segment between
the nodes '
i
' and '
i
+ 1'. The magnitude of the tensile centrifugal axial force,
CT(
x
), along the axis of a continuous blade, may be found from the expression
given by Naguleswaran [10] as
(3)
2
2
CT( )
x
=
0.5
mL
Ω+
(
2
L R
−
2
R x
−
x
)
B
B
B
H
H
where
is the rotational
frequency of the blade, and
x
is the distance along the blade from the hub. This
continuous force distribution is discretized into nodal values (CT
i
) and used to
form the geometric stiffness matrix. The component of nodal blade gravity force
(self weight),
G
i
, acting axially may be obtained from geometry and depends on
the angle
q
that the longitudinal axis of the blade makes with the horizontal global
axis, in the plane of rotation. Values of
N
i
are obtained from the expression:
m
represents the mass per unit length of the blade,
Ω
=±
(4 )
N
CT
G
i
i
i
with the sign convention that tensile forces are positive and compressive forces
are negative.
2.3 Forced vibration analysis
Forced vibration analyses of structures may either be carried out in the time or
frequency domain, with each having its own distinct merits. Analysis through the
time domain allows for the inclusion of behavioural non-linearity and response
coupling. Due to limited availability of actual input time-histories as measured in
the fi eld, the designer has to generate relevant artifi cial time-histories using widely
published spectral density functions. The method for generating the artifi cial time-
histories can be divided into three categories, the fi rst based on a fast Fourier trans-
form (FFT) algorithm, the second based on wavelets and other time-frequency
algorithms and the third based on time-series techniques such as Auto-Regressive
Moving Average (ARMA) method. Suresh Kumar and Stathopoulos [15] simu-
lated both Gaussian and non-Gaussian wind pressure time-histories based on the
FFT algorithm. Kitagawa and Nomura [16] recently used wavelet theory to gen-
erate wind velocity time-histories by assuming that eddies of varying scale and
strength may be represented on the time axis by wavelets of corresponding scales.
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