2.6 Energy Method and Free Torsional Vibration
39
thickness of the disc, then
ρ =
4 W
π D 2 tg
(2.46)
I P = ρ J D · t =
4 W
π D 2 tg
·
π D
4
32
t =
W D
2
8g
(2.47)
where J D is the polar moment of inertia of the disc. In free torsional vibration of the
shaft, the spring stiffness is provided by the shaft and it is given by
k =
π d
4 G
32 L
(2.48)
where G is shear modulus of elasticity of the material of the shaft.
If φ be the angular displacement at any instant of time, the kinetic energy of the
system is
T E =
1
2
I P ˙
φ
2
(2.49)
The corresponding potential energy is nothing but the strain energy in the spring
and is given by
U =
φ
0
kφ dφ =
1
2
kφ
2
(2.50)
Substituting the values of T E and U from Eqs. (2.49) and (2.50), respectively, into
Eq. (2.41), we get
d
dt
1
2
I P ˙
φ
2
+
1
2
k ˙
φ
2
= 0
(2.51)
or
I P ¨
φ + k ˙
φ = 0
Equation (2.51) is the free torsional vibration equation of the shaft.
Equation (2.51) is similar to Eq. (2.2), and its solution is given by
φ = φ 0 cos pt +
˙
φ 0
p
sin pt
(2.52)
where φ 0 and ˙
φ 0 are the initial angular displacement and the initial angular velocity,
respectively.