Civil Engineering Reference
In-Depth Information
The Fourier transform in Eq. 4.20 as well as the assumption regarding
a
in Eq. 4.21
Qae
i
are also still valid, but again, modal motion induced mass, damping and stiffness are
now given by
⎡
T
i
⎤
⎡ ⎤
M
C
φ M φ
φ C φ
φ K φ
⋅
⋅
ae
ae
i
⎢
⎥
⎢ ⎥
T
∫
dx
=
⎢
⋅
⋅
⎥
(4.40)
⎢ ⎥
ae
i
ae
i
⎢
⎥
⎢ ⎥
L
T
K
exp
⋅
⋅
⎢
⎥
⎢
⎣ ⎦
ae
⎣
i
ae
i
⎦
i
where
a
M
,
a
C
and
a
K
are three by three coefficient matrices associated with the
motion induced part of the loading. To justify a mode by mode approach it is necessary
to avoid the introduction of any motion induced coupling between modes, and therefore
a
M
,
a
C
and
a
K
must in this particular case be diagonal, i.e.
⎫
M
diag m
m
m
=
⎡
⎤
⎣
⎦
ae
y
z
θ
ae
⎪
⎪
C
diag c
⎡
c
c
⎤
=
(4.41)
⎬
⎣
⎦
ae
y
z
θ
ae
⎪
K
=
diag k
⎡
k
k
⎤
⎪
⎣
⎦
⎭
ae
y
z
θ
ae
Thus, altogether nine frequency domain motion dependent coefficients are required. In
wind engineering
a
M
is most often negligible.
Modally we are still dealing with a single-degree-of-freedom system, and thus, Eqs.
4.24 - 4.27 are valid. Linearity implies that the Fourier amplitudes of the displacement
components at an arbitrary position
x
are given by
a
⎡⎤
⎡
()
()
()
x
φ
⎤
r
y
y
r
⎢⎥
⎢
⎥
(
)
()
( )
()
x
,
a
x
a
x
a
a
ω
=
⎢⎥
⎢
=
φ
⋅
ω
=
φ
⋅
ω
(4.42)
⎥
r
η
r
η
r
r
z
r
r
i
i
i
z
⎢⎥
⎢
⎥
x
φ
a
⎢⎥
⎣
θ
r
⎦
⎣⎦
r
i
θ
The cross spectral density matrix of the three components is then
1
(
)
(
)
*
T
S
x
,
ω
=
lim
a
⋅
a
r
r
i
r
i
i
π
T
T
→∞
1
1
*
T
⎧
(
) (
)
⎫
(
)
*
T
lim
φ
a
φ
a
φ
lim
a
a
φ
=
⋅
⋅
⋅
=
⋅
⋅
⋅
(4.43)
⎨
⎬
r
η
r
η
r
η
η
r
i
i
i
i
i
i
i
i
π
T
π
T
⎩
⎭
T
→∞
T
→∞
