Homework #2
MEAM 501
Fall 1998
Due Date October 20
Onedimensional Finite Element Method and Related Eigenvalue Problems
Let us
consider axial vibration of an elastic bar, whose length is
L
while the axial rigidity is
EA
,
shown in Fig. 1:
u(t,x)
x
f(t,x)
k
L
k
R
EA(x)
Figure 1
Vibration of an Elastic Bar in the Axial Direction
Suppose that the left and right end points are supported by two discrete springs whose
spring constant is given by
k
L
and
k
R
. The equation of motion of this elastic bar is written as
(
29
L
in
f
x
u
EA
x
t
u
A
,
0
2
2
+
∂
∂
∂
∂
=
∂
∂
r
where
ρ
is the mass density, and the boundary condition is written by
.
0
L
x
at
u
k
x
u
EA
and
x
at
u
k
x
u
EA
R
L
=

=
∂
∂
=

=
∂
∂

We shall apply the weighted residual method that is constructed by the finite element
method to derive a discrete system of the axial vibration problem. To this end, let the
domain (0,
L
) be decomposed into
N
E
number of finite elements
Ω
e
,
e
= 1,
....
,
N
E
, and let each
finite element consist of
M
number nodes in which the axial displacement is assumed to be
a
M
1 degree polynomial:
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(
29
( 29
( 29
( 29
( 29
{
}
( 29
( 29
e
M
e
e
M
M
j
j
j
e
t
u
t
u
s
N
s
N
s
N
t
u
x
t
u
Nu
=
=
=
∑
=
:
.....
,
1
1
1
( 29
( 29
( 29
{
}
e
e
e
M
j
j
j
e
x
x
s
N
s
N
s
N
x
x
Nx
=
=
=
∑
=
3
1
3
1
1
:
.....
( 29
C
M
j
k
k
k
j
k
j
x
x
x
x
s
N
≠
=


=
1
1
2
3
4
M
s=1
s=+1
x
1
x
2
x
3
x
4
x
M
element
Ω
e
= (x
1
,x
M
)
Figure X
A Finite Element
Ω
e
= (x
1
, x
M
)
Noting that the weighted residual formulation of the equation of the motion and the
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 Fall '09
 Finite Element Method, Mathematica, stiffness matrix, KU

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