Civil Engineering Reference
In-Depth Information
4.4 General multi-mode response calculations
In the final section of this chapter it is assumed that a full multi-mode approach is re-
quired. The basic assumptions from chapter 4.1 are that
(
)
( )
( )
x t
x
t
r
=
Φη
(4.50)
⋅
⎧
T
( )
()
r
xt
,
=
⎡
r
r
r
⎤
⎪
⎣
⎦
yz
θ
⎪
⎡
⎤
x
...
...
where
Φφ φ φ
=
(4.51)
⎨
⎣
1
i
N
⎦
mod
⎪
⎪
T
()
⎡
⎤
x
...
...
η
=
⎣
ηηη
1
i
N
⎦
⎩
mod
T
()
x
⎡
⎤
φ
and where
mo
N
is the number of modes chosen to be in-
cluded in the calculations. Still adopting the assumptions regarding motion induced load
effects as presented in chapter 4.2, the cross sectional load is
=
⎣
φφφ
⎦
i
y
z
θ
i
(
)
(
)
xt
,
xt
, , , ,
q q
=
+
q
rrr
(4.52)
tot
ae
⎧
T
(
)
xt
,
⎡
q
q
q
⎤
q
=
⎣
(4.53)
⎪
⎨
⎪
y
z
⎦
θ
where
T
(
)
xt
,,,,
⎡
q
q
q
⎤
q
rrr
=
⎣
⎦
ae
y
z
θ
⎩
ae
Thus, the time domain modal equilibrium equation is given by (see also Eq. 4.9)
()
()
()
()
(
)
M η C η K η QQ
⋅
t
+⋅
t
+⋅
t
= +
t
t
η ηη
,,,
(4.54)
0
0
0
M
C
and
K
are
where
,
N
by
N
diagonal matrices defined in
0
mod
mod
Eq. 4.10, and the modal
N
by one flow induced load vector is given by
mod
T
Q
()
⎡
⎤
t
=
⎣
Q
...
Q
...
Q
(4.55)
1
i
N
⎦
mod
(
)
T
∫
φ q
Where
Q
=
⋅
dx
(4.56)
i
i
L
exp
Taking the Fourier transform on either side of Eq. 4.54
(
)
()
()
(
)
2
i
,,,
−
MCKa
ω
+
ω
+
⋅
ω
=
a
ω
+
a
ω η η η
(4.57)
0
0
0
η
Q
Q
ae
