Engineering Mechanics

(^676) „„„„„ A Textbook of Engineering Mechanics
33.10.LENGTH OF THE BELT
It means the total length of the belt, required to connect a driver and a follower. Here we shall
discuss the lengths of the following two types of belt drives :

1. Open belt drive. 2. Cross belt drive.

33.11.LENGTH OF AN OPEN BELT DRIVE

Fig. 33.6. Length of an open belt drive.

In this case, both the pulleys rotate in the same direction as shown in Fig. 33·6.

Let O 1 and O 2 = Centres of the two pulleys,

r 1 and r 2 = Radii of the larger and smaller pulleys,

l= Distance between O 1 and O 2.

Let the belt leave the larger pulley at E and G and the smaller pulley at F and H as shown in

Fig. 33·6. Through O 2 draw O 2 M parallel to EF. From the geometry of the figure, we find that O 2 M

will be perpendicular to O 1 E.

∴ sin α=^12

rr–

l

... (where α is the angle MO 2 O 1 in radians.)

Since the angle α is very small, therefore

sin α=^12

rr–

l

α= ...(i)

∴ Length of the arc JE= 1

2

r

⎛⎞π

⎜⎟+α

⎝⎠

...(ii)

Similarly length of the arc FK= 2 –

2

r

⎛⎞π

⎜⎟α

⎝⎠

...(iii)

and EF=MO 2 = lrr^22 –( – ) 12 =

2

l 1–rr^12 –

l

⎛⎞

⎜⎟

⎝⎠

Expanding this equation by Binomial theorem :

EF=

2

1 –^112 – .....

2

rr

l

l

⎡⎤⎛⎞

⎢⎥⎜⎟+

⎢⎥⎣⎦⎝⎠

=

2

• (– )^12
  2

rr

l

l

...(iv)