Environmental Engineering Reference
In-Depth Information
Fig. 1.3
Transverse vibration of a string due to the initial transverse force
F
at point
x
=
a
For a specific problem, the cause may come from one, two or all three of the
above-mentioned mechanisms. For example, the vibration of a string fixed at the
two ends and with no initial displacement and no initial speed comes only from the
internal sources.
=
Example.
Consider a string of length
l
and line density
ρ
fixed at the two ends
x
0
=
=
and
x
l
. A transverse force
F
is initially acting at point
x
a
of the string to pull
the string to a small height
h
at
x
a
. The force is then removed, and the transverse
vibration of the string begins. Find the PDS for describing the vibration of the string.
=
Solution.
The free vibration of the string comes exclusively from its initial displace-
ment at the instant when the force acts at
x
=
a
.Let
T
be the tension of the string.
Since
h
is small, we have (Fig. 1.3)
sin
α
1
≈
tan
α
1
,
sin
α
2
≈
tan
α
2
.
A transverse force balance thus yields
T
h
h
F
(
l
−
a
)
a
a
+
T
a
=
F
or
h
=
.
l
−
Tl
Thus the PDS is a mixed problem
⎧
⎨
a
2
u
xx
,
a
2
u
tt
=
0
<
x
<
l
,
0
<
t
,
=
T
/
ρ
,
⎧
⎨
F
(
l
−
a
)
x
,
0
≤
x
≤
a
,
Tl
u
(
x
,
0
)=
u
t
(
x
,
0
)=
0
,
⎩
Fa
(
l
−
x
)
⎩
,
a
<
x
≤
l
,
Tl
u
(
0
,
t
)=
u
(
l
,
t
)=
0
,
t
≥
0
.
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