Mechanical Engineering Principles

122 MECHANICAL ENGINEERING PRINCIPLES

However, from Table 7.1, page 91, the polar second
moment of area of a circle,

J=

πR^4

2

,

hence, T=

Gθ

L

J

or

T

J

=

Gθ

L

( 10. 5 )

Combining equations (10.3) and (10.5) gives:

τ

r

=

T

J

=

Gθ

L

( 10. 6 )

For a solid section of radiusRor diameterD,

J=

πR^4

2

or

πD^4

32

( 10. 7 )

For a hollow tube of circular section and of internal
radiusR 1 and external radiusR 2

J=

π

2

(

R 24 −R^41

)

( 10. 8 )

or, in terms of external and internal diameters ofD 2
andD 1 respectively, (see Problem 12, Chapter 7,
pages 93),

J=

π

32

(

D^42 −D^41

)

( 10. 9 )

The torsional stiffness of the shaft,k, is defined by:

k=

GJ

L

( 10. 10 )

The next section of worked problems will demon-
strate the use of equation (10.6).

10.3 Worked problems on the twisting

of shafts

Problem 1. An internal combustion engine

of 60 horsepower (hp) transmits power to the

car wheels of an automobile at 300 rev/min

(rpm). Neglecting any transmission losses,

determine the minimum permissible diameter

of the solid circular section steel shaft, if the

maximum shear stress in the shaft is limited

to 50 MPa. What will be the resulting angle

of twist of the shaft, due to the applied

torque, over a length of 2 m, given that the

rigidity modulus,G=70 GPa? (Note that

1hp= 745 .7 W).

Power=60 hp=60 hp× 745. 7

W

hp

=44742 W

From equations (9.1) and (9.2), page 110,

power=Tω= 2 πnTwatts, wherenis the speed

in rev/s

or 44742 = 2 π

rad

rev

×

300

60

rev

s

×T

= 31. 42 Trad/s

from which, T=

44742 W

31 .42 rad/s

i.e. torqueT=1424 N m

(since 1 W s=1Nm)

From equation (10.6),

τ

r

=

T

J

i.e.

50 × 106

r

N

m^2

=

1424 N m

πr^4

2

and

r^4

r

=

1424 × 2

π× 50 × 106

m^3

from which, r^3 =

1424 × 2

π× 50 × 106

or shaft radius,r=^3

√(

1424 × 2

π× 50 × 106

)

= 0 .0263 m

Hence, shaft diameter,d= 2 ×r= 2 × 0. 0263

= 0 .0526 m

From equation (10.6),

τ

r

=

Gθ

L

from which,θ=

τL

Gr

=

50 × 106

N

m^2

×2m

70 × 109

N

m^2

× 0 .0263 m