8.4 Free Torsional Vibration of the Shaft
321
Drill pipe : Outside diameter = 110 mm.
Inside diameter = 80 mm.
The density of the material of the pipe and the bit is 7800 kg/m
3 . Determine the
fundamental natural frequency of the system. G = 84×10
10 N/m
2 .
The general solution for the torsional vibration problem is
θ(x, t) =
A cos
px
a
+ B sin
px
a
C 1 sin( pt − α)
(a)
where a =
√
G/ρ.
The boundary condition for the problem at x = 0 is θ = 0. Putting this condition
in Eq. (a) gives
A = 0
( b )
At x = L
J 0
∂
2
θ
∂ t 2 = −G J
∂θ
∂ x
(c)
Combining Eqs. (a), (b) and (c), we get
−B J 0 p
2 sin
pL
a
C 1 sin( pt − α) = −G J B
p
a
cos
pL
a
C 1 sin( pt − α)
or
tan
pL
a
=
G J
a J 0 p
=
G J
J 0 p
ρ
G
=
ρ J L
p J 0 L
G
ρ
=
I p a
J 0 pL
(d)
I p = mass moment of inertia of the pipe.
Therefore,
pL
a
tan
pL
a
=
I p
J 0
.
(e)
For the data given,
J =
π
32
(0.11
4
− 0.08
4
) = 1.03525 × 10
−5 m
4
I p = ρ J L = 7800 × 1.03525 × 10
−5
× 1000 = 80.75 kgms
2
J 0 = 7800 ×
π
32
(0.16
4
− 0.05
5
) × 10 = 4.971 kgms
2