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Example 3

free response of a fixed/fixed string
Find the free vibration solution of a fixed/fixed homogeneous string having a constant
tension of T and mass/length of
!
if the string is released from rest with an initial
deformation shown to the right.
x = 0
x=L
A
u
0
(x)
x=L/2
EOM:
T
"
2
u
"
x
2
=
#
"
2
u
"
t
2
(1)
Solution form
The above is a homogeneous, partial differential equation in terms of independent
variable of x and t. Will write solution in a “separable form” of a product of functions of
x and t as
u x
,
t
(
) =
"
x
( )
T t
( )
(2)
If we substitute (2) into (1) we find:
TT t
( )
d
2
x
( )
dx
2
=
#"
x
( )
d
2
T t
( )
dt
2
$
T
d
2
x
( )
/
dx
2
x
( )
=
d
2
T t
( )
/
dt
2
T t
( )
(3)
The left hand side of (3) is strictly a function of x, and the right hand side of (3) is strictly
a function of time, t. This equation of left side with right side must be true for ALL x and
for ALL t. In order for this equality to the true, then both sides of the equation must be
CONSTANT in both x and t; that is,
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