Civil Engineering Reference
In-Depth Information
Substituting the discretized approximation for
u
(
x
,
t
),
the integral on the left
becomes
L
L
2
u
N
i
(
x
)
∂
A
d
x
=
A
N
i
(
N
1
¨
u
1
+
N
2
¨
u
2
)d
x
i
=
1, 2
(10.57)
t
2
∂
0
0
where the double-dot notation indicates differentiation with respect to time. The
two equations represented by Equation 10.57 are written in matrix form as
N
1
d
x
¨
21
12
¨
L
N
1
N
2
=
AL
6
u
1
¨
u
1
¨
A
=
[
m
]
{¨
u
}
(10.58)
N
2
u
2
u
2
N
1
N
2
0
and the reader is urged to confirm the result by performing the indicated integra-
tions. Also note that the mass matrix is symmetric but not singular. Equa-
tion 10.58 defines the
consistent
mass matrix for the bar element. The term
con-
sistent
is used because the interpolation functions used in formulating the mass
matrix are the same as (consistent with) those used to describe the spatial varia-
tion of displacement. Combining Equations 10.56 and 10.58 per Equation 10.55,
we obtain the dynamic finite element equations for a bar element as
21
12
1
u
1
u
2
f
1
f
2
AL
6
AE
L
u
1
¨
¨
−
1
+
=
(10.59)
u
2
−
11
or
(10.60)
and we note that
AL
=
m
is the total mass of the element. (Why is the sign of
the second term positive?)
Given the governing equations, let us now determine the natural frequen-
cies of a bar element in axial vibration. Per the foregoing discussion of free
vibration, we set the nodal force vector to zero and write the frequency equa-
tion as
[
m
]
{¨
u
}+
[
k
]
{
u
}={
f
}
2
[
m
]
|
[
k
]
−
|=
0
(10.61)
to obtain
k
2
m
3
2
m
6
k
−
−
+
=
0
(10.62)
k
2
m
6
2
m
3
−
+
k
−
2
Expanding Equation 10.62 results in a quadratic equation in
k
2
k
2
2
m
3
2
m
6
−
−
+
=
0
(10.63)
