Civil Engineering Reference
In-Depth Information
modal motion induced load contains three cross sectional terms
c
and
a
m
, pro-
portional to and in phase with structural displacement, velocity and acceleration, i.e. it is
assumed that
k
,
ae
ae
(
)
2
a
MC
i
K
a
η
=−
ω
+
ω
+
⋅
(4.21)
ae
ae
ae
Qae
i
i
i
i
i
where
⎡ ⎤
M
m
ae
i
⎡ ⎤
⎢ ⎥
ae
⎢ ⎥
2
⎢ ⎥
C
∫
c
dx
=
φ
(4.22)
⎢ ⎥
ae
i
ae
i
⎢ ⎥
L
⎢ ⎥
k
exp
⎢ ⎥
⎣ ⎦
K
ae
⎣ ⎦
ae
i
and where
k
,
c
and
m
are known constants. Introducing
a
from
ae
ae
ae
Qae
i
from Eq. 4.10) into Eq. 4.20, gath-
ering all motion dependent terms on the left hand side and dividing throughout the equa-
tion by
2
CM
ω ζ
2
K
M
Eq. 4.21 (as well as
=
and
=
ω
i
i
i
i
i
i
i
K
, the following is obtained
i
ˆ
i
()
H
ω
()
()
a
a
ω
=
⋅
ω
(4.23)
η
Q
i
K
i
i
where
−
1
⎡
2
⎤
⎛
⎞
⎛⎞
⎛
⎞
K
M
C
ω
ω
ae
ae
ae
ˆ
()
i
i
i
H
⎢
1
⎜
1
⎟
2
i
⎜
⎟
⎥
ω
=−
−−
⋅
+
ζ
−
⋅
(4.24)
⎜⎟
i
i
2
⎜
⎟
⎜
⎟
⎢
M
M
ω
2
M
ω
⎥
ω
ω
⎝⎠
⎝
⎠
i
⎝
⎠
i
⎣
i
i
i
i
i
⎦
is the non-dimensional modal frequency-response-function.
Introducing
(
)
,
and
2
M
M
K
M
C
2
M
μ
=
κ
=
ω
ζ
=
ω
, then
ae
ae
i
ae
ae
i
i
ae
ae
i
i
i
i
i
i
i
i
−
1
2
⎡
⎤
⎛⎞
ω
ω
(
)
(
)
ˆ
()
⎢
⎥
H
1
1
2
i
ω
=−
κ
−−
μ
⋅
+
ζ
−
ζ
⋅
(4.25)
⎜⎟
i
ae
ae
i
ae
i
i
i
⎢
ω
ω
⎥
⎝⎠
i
i
⎣
⎦
()
i
t
The single-sided spectrum of
η
is given by
2
ˆ
()
H
ω
1
1
(
)
(
)
i
()
*
*
S
lim
a
a
lim
a
a
ω
=
⋅
⋅
=
⋅
⋅
⋅
(4.26)
η
η
η
2
QQ
i
T
i
i
T
π
K
π
i
i
T
→∞
T
→∞
i
2
ˆ
i
()
H
ω
()
()
S
S
⇒
ω
=
⋅
ω
(4.27)
η
2
Q
i
K
i
i
