Mechanical Engineering Principles

124 MECHANICAL ENGINEERING PRINCIPLES

Figure 10.5, wherea>b. Determine the

maximum resulting torque acting on the shaft

and then draw the torque diagram.

a

A

T 1 T 2

T b

c

B

x

Figure 10.5

From equilibrium considerations, (see Figure 10.5),

clockwise torque=the sum of the

anticlockwise torques

T=T 1 +T 2 ( 10. 11 )

LetθC= the angle of twist at the pointC.

From equation (10.6),

T

J

=

Gθ

L

from which, T=

GθJ

L

Therefore T 1 =

GθCJ

a

( 10. 12 )

and T 2 =

GθCJ

b

( 10. 13 )

Dividing equation (10.12) by equation (10.13) gives:

T 1

T 2

=

b

a

or T 1 =

bT 2

a

( 10. 14 )

Substituting equation (10.14) into equation (10.11)
gives:

T=

bT 2

a

+T 2 =

(

1 +

b

a

)

T 2

orT 2 =

T

(

1 +

b

a

)=

T

(

a+b

a

)=

Ta

(a+b)

( 10. 15 )

However,a+b= L

Therefore T 2 =

Ta

L

( 10. 16 )

Substituting equation (10.16) into equation (10.14)

gives:

T 1 =

b

(

Ta

L

)

a

=

bT

L

( 10. 17 )

Asa>b,T 2 >T 1 ; therefore, maximum

torque=T 2 =

Ta

L

The torque diagram is shown in Figure 10.6.

T

−Ta/L −Ta/L

Tb/L Tb/L

0 0

Figure 10.6 Torque diagram

Now try the following exercise

Exercise 50 Further problems on the

twisting of shafts

1. A shaft of uniform solid circular section
   is subjected to a torque of 1500 N m.
   Determine the maximum shear stress in
   the shaft and its resulting angle of twist, if
   the shaft’s diameter is 0.06 m, the shaft’s
   length is 1.2 m, and the rigidity modulus,
   G = 77 × 109 N/m^2. What power can
   this shaft transmit if it is rotated at 400
   rev/min? [35.4 MPa, 1.05°, 62.83 kW]


2. If the shaft in Problem 1 were replaced
   by a similar hollow one of wall thickness
   10 mm, but of the same outer diameter,
   what would be the maximum shearing
   stress in the shaft and the resulting angle
   of twist? What power can this shaft
   transmit if it rotated at 500 rev/min.
   [44.1 MPa, 1.31°, 78.5 kW]


3. A boat’s propeller shaft transmits 50 hp
   at 100 rev/min. Neglecting transmission
   losses, determine the minimum diame-
   ter of a solid circular section phosphor
   bronze shaft, when the maximum permis-
   sible shear stress in the shaft is limited to
   40 MPa. What will be the resulting angle