6- 52
Problems and Solutions Section 6.7 (6.53 through 6.63)
6.53
Calculate the response of Example 6.7.1 for
l
= 1 m,
E
= 2.6
×
10
10
N/m
2
and
ρ
=
8.5
×
10
3
kg/m
3
.
Plot the response using the first three modes at
x
=
l
/2,
l
/4, and
3
l
/4.
How many modes are needed to represent accurately the response at the
point
x
=
l
/2?
Solution:
w x t
l
e
t
x
n
n
t
n
n
n
n
( , )
.
(
)
cos
sin
.
=
−
+
−
=
∞
∑
0 02
1
2
2
1
0 01
2
1
σ
ω
σ
ω
Where
σ
π
ω
σ
ρ
ω
ω
n
n
n
dn
n
n
l
E
=
−
=
=
(
)
.
2
1
2
0 9999
For
l
= 1 m
E
=
×
=
×
2 6
10
8 5
10
10
3
.
.
N/m
kg/m
2
3
ρ
Response using first three modes at
x
l
l
l
=
2 4
3
4
,
,
plotted below.
Three modes accurately represents the response at
x
l
=
2
.
The error between a
three and higher mode approximation is less than 0.2%.

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6- 53