Solution of Homework #2, 1998 Fall
MEAM 501
Analytical Methods in Mechanics and Mechanical Engineering
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
r
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
W
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 M1 degree polynomial:
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
(
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
W
e
= (x
1
, x
M
)
Noting that the weighted residual formulation of the equation of the motion and the
boundary condition may be represented by the integral form
(
29
(
29
(
29
(
29
w
fwdx
L
t
w
L
t
u
k
t
w
t
u
k
dx
x
w
x
u
EA
w
t
u
A
L
R
L
L
2200
=
+
+
∂
∂
∂
∂
+
∂
∂
∫
∫
,
,
,
0
,
0
,
0
0
2
2
r
that is
(
29
(
29
(
29
(
29
w
fwdx
L
t
w
L
t
u
k
t
w
t
u
k
dx
x
w
x
u
EA
w
t
u
A
e
e
E
e
N
e
R
L
N
e
2200
=
+
+
∂
∂
∂
∂
+
∂
∂
∑
∫
∑
∫
=
Ω
=
Ω
,
,
,
0
,
0
,
1
1
2
2
r