208
6 Free Vibration of Multiple Degrees of Freedom System
Fig. 6.12 Example 6.6
Indicating the joints as 0, 1 and 2, the following transfer matrices can be written
x
N
L
1
=
1 1/k 1
0 1
x
N
R
0
x
N
R
1
=
1 0
0 1
x
N
L
1
x
N
L
2
=
1 1/k 2
0 1
x
N
R
1
and
x
N
R
2
=
1 0
− mp
2 1
x
N
L
2
Therefore, from the above relations, the following equation can be written as
x
N
R
2
=
1 0
− mp
2 1
1 1/k 2
1 1
1 0
0 1
1 1/k 1
0 1
x
N
R
0
or
x
N
R
2
=
1
1
k 1
+
1
k 2
− mp
2
− mp
2
(1/k 1 + 1/k 2 ) + 1
x
N
R
0
(a)
The boundary conditions of the problem indicate
x 0 = 0 and N
R
2 = 0
( b )
Therefore, the above equation yields