12.2 Torsional Vibration of Shafts
447
Fig. 12.2 Example 12.1
Eq. (6.5).
[M] { ¨
X } + [K ] {X } = {0}
(12.23)
Assuming a solution
{X } = ae
ipt
{φ}
(12.24)
and following the steps similar to that given in Chap. 6, the final equation becomes
[K ]
− 1
[M] {φ} = λ {I ] {φ}
(12.25)
where
λ =
1
p 2
Equation (12.25) is a typical eigenvalue problem and its solution procedures have
been discussed in Chap. 6.
Example 12.1 A uniform shaft having one end fixed and the other end free is divided
into two equal elements (Fig. 12.2). Determine the natural frequency in free torsional
vibration.
After assembling the two elements, the structure stiffness matrix is
[K ] =
2G J
L
1 − 1
− 1 2
and the total mass matrix is given by
[M] =
I L
12
2 1
1 4
[K ]
− 1
[M] =
L
2G J
2 1
1 1
I L
12
2 1
1 4
=
I L
2
24 G J
5 6
3 5