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
.
 
Search WWH ::




Custom Search