Civil Engineering Reference
In-Depth Information
where
k
and
c
are the element stiffness and damping matrices, defined by
L
⎫
T
∫
kNkN
=
dx
⎪
⎪
⎬
⎪
0
0
(9.19)
L
T
∫
dx
cNcN
=
⎪
⎭
0
0
By introducing
d
(
)
( )
( )
el
x t
,
x
t
δ
rNd
and
=⋅
δ
r
=
N
⋅
then the external work (see
Eq. 9.9) is given by:
L
L
⎛
⎞
(
)
T
T
(
)
T
T
W
∫
dx
∫
dx
=
δ
dF
⋅
−
NdmNd
⋅
δ
⋅
=
δ
d F
⋅
⎜
−
NmN d
⋅
⎟
ext
tot
0
tot
0
⎜
⎟
⎝
⎠
0
0
(9.20)
Thus, introducing the element mass matrix
L
T
=
mNmN
dx
(9.21)
0
0
then the external work is given by
(
)
dF md
T
W
=
δ
⋅
−
⋅
(9.22)
ext
tot
By setting
WW
=
int
ext
(
)
(
)
(9.23)
T
T
δ
dkd+cd dF md
⋅
⋅
⋅
=
δ
⋅
−⋅
tot
tot
cancelling out
δ
d
and rearranging, then the following element equilibrium condition is
obtained:
md
++
cd
kd
=
F
(9.24)
tot
tot
Since
()
()
t
t
FFF
this equation may be split into a mean (static)
part and a fluctuating (dynamic) part:
ddd
and
=+
=+
tot
tot
⎫
kd
=
F
⎪
⎬
++=
(9.25)
md
cd
kd
F
⎪
⎭
