6-
37
Problems and Solutions Section 6.5 (6.40 through 6.47)
6.40
Calculate the natural frequencies and mode shapes of a clamped-free beam.
Express your solution in terms of
E
,
I
,
ρ
, and
l
.
This is called the cantilevered
beam problem.
Solution:
Clamped-free boundary conditions are
w
t
w
t
w
l t
w
l t
x
x
xx
xxx
( , )
( , )
( , )
( , )
0
0
0
0
−
=
=
=
and
assume
E, I,
ρ
, l
constant.
The equation of motion is
∂
∂
+
∂
∂
=
2
2
4
4
0
w
t
EI
A
w
x
ρ
assume separation of variables
)
(
)
(
)
,
(
t
q
x
t
x
w
φ
=
to get
EI
A
q
q
ρ
φ
φ
ω
′′′′
= −
=
˙˙
2
The spatial equation becomes
′′′′ −
=
φ
ρ
ω φ
A
EI
2
0
define
0
that
so
4
2
4
=
−
′
′
′
′
=
φ
β
φ
ω
ρ
β
EI
A
which has the solution:
x
C
x
C
x
C
x
C
β
β
β
β
φ
cosh
sinh
cos
sin
4
3
2
1
+
+
+
=
Applying the boundary conditions:
φ
φ
φ
φ
( )
( )
( )
( )
0
0
0
0
=
′
=
′′
=
′′′
=
and
l
l
yields that
C
2
+
C
4
= 0
C
1
+
C
3
= 0

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