Civil Engineering Reference
In-Depth Information
Single mode single component response calculations
A single mode single component response calculation is in the following only considered
relevant for displacements in the
z
direction and in torsion. Thus, it is only mode shapes
that primarily contain either
z
or
θ
components that are relevant. I.e., it is taken for
granted that any of the following two conditions apply
T
⎫
()
[
]
φ
x
≈
0
φ
0
⎪
⎬
⎪
i
z
i
T
(6.85)
()
[
]
φ
x
≈
00
φ
i
⎭
θ
i
Off diagonal terms in Eq. 6.78 will then vanish, rendering all covariance quantities
obsolete, and
r
S
will simply contain the response variances of the excitation of each
mode on its diagonal. Thus, the response spectrum and the displacement variance
associated with the excitation of an arbitrary mode
i
are given by
2
⎫
ˆ
()
2
( )
()
()
S
x
H
S
ωφ
=
⋅
ω
⋅
ω
ˆ
⎪
⎪
⎬
⎪
r
n
r
η
Q
n
n
n
z
⎧
n
=
(6.86)
⎨
⎩
∞
θ
2
()
∫
σ
=
Sd
ω
ω
r
r
n
n
⎪
⎭
0
where
1
−
⎫
2
⎡
⎤
⎛
⎞
ω
ω
(
)
⎪
ˆ
()
⎢
⎥
H
ω
=−
1
+⋅
2
i
ζ
−
ζ
⋅
⎜ ⎟
nae
⎪
η
n
n
ω
ω
⎢
⎥
⎝ ⎠
n
n
⎣
⎦
⎪
⎪
⎬
(6.87)
()
2
()
∫
S
ωφ
⋅
x dx
q
n
⎪
⎪
n
L
exp
()
S
2
D
ωλ
=
⋅
ˆ
Q
n
(
)
2
⎪
⎪
⎭
2
M
ω
nn
and where aerodynamic damping properties may be extracted from Eq. 6.68, rendering
2
2
⎫
⎪
∫
dx
∫
dx
φ
φ
z
z
2
⎡
⎤
C
2*
2
BH
ρ
ρ
B
⎛ ⎞
σ
L
L
ae
zz
exp
exp
1
z
⎪
K
⎢
1
⎥
ζ
=
=
⋅
=
⋅
−
⋅
⎜ ⎟
ae
a
z
4
m
2
4
m
z
a D
2
2
ω
M
∫
dx
⎢
⎥
∫
dx
⎪
φ
φ
⎝ ⎠
zz
z
z
z
z
⎣
⎦
z
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎭
L
L
(6.88)
⎬
2
2
∫
dx
∫
dx
φ
φ
θ
θ
⎡
2
⎤
C
4*
4
ρ
BA
B
⎛ ⎞
σ
ρ
L
L
ae
exp
exp
2
θθ
K
⎢
1
θ
⎥
ζ
=
=
⋅
=
⋅
−
⋅
⎜ ⎟
ae
a
4
m
2
4
m
a
2
θ
2
M
∫
dx
θ
⎢
⎥
∫
dx
ω
φ
φ
⎝ ⎠
θ
θ
θ
θθ
⎣
⎦
θ
θ
L
L
