42
2 Free Vibration of Single Degree of Freedom System
Fig. 2.23 Example 2.14
Example 2.13 A shaft 10 mm in diameter and 0.2 mm long connects a generator to
the main engine. If the mass moment of inertia of the generator rotor is 0.55 Nmms
2 , determine the natural frequency in torsion [G = 8 × 10
6 N/cm
2 ].
The mass moment of inertia of the engine is very large compared to that of the
generator rotor. As such the engine end is assumed to be fixed.
The stiffness of the system is given by
k =
π d
4 G
32L
=
π (1)
4
× 8 × 10
6
32 × 20
= 3927 Ncm/rad = 39.27 Nm/rad
The natural frequency is given by
f =
1
2π
39.27
0.55 × 10 − 3 = 42.53 cycle/s
Example 2.14 A shaft with two circular discs of uniform thickness at the ends is
shown in Fig. 2.23. Determine the frequency of torsional vibration of the shaft.
The discs at the two ends are given two opposite torques, which are immediately
withdrawn to start the torsional vibration. There is a particular section cd which
remains immovable during vibration. The left-hand and right-hand portions of the
shaft about cd will have the same time period. Therefore,
I P 1
k 1
=
I P 2
k 2
or
k 1
k 2
=
I P 1
I P 2
where k 1 and k 2 are the spring stiffnesses for the left and for the right-hand portions
of the shaft. We know that stiffnesses are inversely proportional to their lengths
Therefore,
k 1
k 2
=
b
a
=
I P 1
I P 2
(a)