Civil Engineering Reference
In-Depth Information
where the zero index indicate that they contain structural properties in vacuum or in still
air, and where the modal mass, damping and stiffness matrices are given by
(
)
⎧
T
M
∫
φ M φ
dx
=
⋅
⋅
⎫
diag
⎡⎤
M
i
i
0
i
M
=
⎪
⎪
⎪
0
⎣
⎪
⎪
⎡⎤
i
L
diag C
CM
K
2
C
=
where
=
ωζ
(4.10)
⎬
⎨
⎪
0
⎣⎦
⎪
⎡⎤
⎪
i
i
i
i
i
2
=
ω
diag
K
K
=
⎪
⎪
⎩
i
i
i
⎣⎦
⎭
0
i
The modal load vector in Eq. 4.9 is given by
T
Q
⎡
⎤
=
⎣
QQQ
...
...
(4.11)
tot
1
i
N
⎦
mod
tot
where:
(
)
T
Q
∫
φ q
dx
=
⋅
(4.12)
i
i
tot
tot
L
exp
In Eq. 4.10
ω
are the eigen-frequencies and
ζ
are the damping ratios, each associated
with the corresponding eigen-mode. It is in the following assumed that the structural
damping ratios
ζ
are known quantities, chosen from experimental experience or an
acknowledged code of practice, and that a pertinent mode shape variation has been
adopted (e.g. a Rayleigh type of frequency dependency). The three by three mass matrix
i
()
()
()
M
=
diag m
⎡
x
m
x
m
x
⎤
(4.13)
⎣
⎦
0
y
z
θ
contains the cross sectional mass properties associated with the
y
,
z
and
degrees of
freedom, all taken with respect to the shear centre. (It may often be more convenient to
calculate modal mass matrix
M
in Eq. 4.9 directly from the nodal mass lumping used in
the preceding finite element eigen-value solution and the corresponding eigen-vectors,
instead of the formal calculation procedures indicated above, as these already contain all
the structural properties that are necessary for such a calculation.)
The cross sectional load vector
to
q
contains the total drag, lift and moment loads per
unit length (see Fig. 4.1) including flow induced as well as motion induces loads, i.e.
θ
T
q
=
⎣
⎡
qqq
⎤
(4.14)
⎦
tot
y
z
θ
tot
