Civil Engineering Reference
In-Depth Information
Introducing Eq. 9.128 the following is obtained
2
⎡
⎤
σ
"
Cov
"
Cov
F
F F
F F
1
1
i
1 12
⎢
⎥
⎢
#
%#
$#
⎥
⎢
⎥
⎡
T
⎤
2
E
Cov
Cov
Cov
=
F
⋅
F
=
"
σ
"
⎢
⎥
FF
⎣
n
n
⎦
⎢
FF
F
FF
nn
i
1
i
i
12
⎥
#
$#
%#
⎢
⎥
(9.134)
⎢
⎥
2
12
Cov
Cov
"
"
σ
⎣
FF
FF
i
⎦
12 1
12
T
T
T
=⋅
mCov
m cCov
+⋅
c kCov k
+⋅
n
n
n
n
n
d d
n
dd
dd
nn
nn
nn
T
T
+⋅
mCov k kCov
+⋅
m
n
n
n
n
dd
dd
nn
nn
where
Cov
are defined in Eq.
9.130. It should be noted that if damping has been defined at a global level (e.g. in the
form of Rayleigh damping
Cov
,
Cov
,
Cov
,
Cov
and
d
nn
d
nn
d
nn
d
nn
d
nn
CMK
), then the damping properties at element level
should comply with the same choices of damping properties(i.e.
=+
α
β
c
=
α
mk
+
β
). It is
n
n
n
L
is sufficiently small then the
also worth noting that if the chosen element length
T
Cov
k
Cov
k
mass and damping terms above will be small, i.e.
≈⋅
.
F F
n
d d
n
nn
nn
9.8
Frequency domain response calculations in modal coordinates
9.8 Freq uency do mai n respo nse calculat ions
The dynamic equilibrium condition in original discretised coordinates (see Eq. 9.80)
() (
) () (
) ()
()
t
t
t
t
Mr
+−
C C
r
+−
K K
r
=
R
(9.135)
ae
ae
may readily be transformed into a modal format by choosing
()
()
t
t
r Φη
(9.136)
=⋅
where the vector
()
η
contains
N
modal coordinates
mod
T
()
⎡
⎤
η
t
=
⎣
ηη
""
(9.137)
η
η
12
i
N
⎦
mod
and
Φ
contains the mode shapes
