# BELT FRICTION

A belt is passing over a pulley and hence the belt is in contact with the surface of the pulley. If the surface of the pulley is perfectly smooth, the tension in the belt on both sides* of the pulley will be same (i.e., the tension throughout the belt will be constant). Also for the perfectly smooth surface, there will be no frictional resistance and hence no driving torque** will be developed.

But if the surface of the pulley is rough, the tension in the belt will not be constant. The tension will vary throughout the length of the belt which is in contact with pulley. This variation in tension is due to frictional resistance. The frictional resistance depends on the co-efficient of friction (i.e., value of µ) between the belt and pulley surface. It will be shown in the next articles that

Where T1 = Tension in the belt on tight side,

T2 = Tension in the belt on slack side,

µ = Co-efficient of friction, and

θ = Angle of contact in radians.

3.6.1. Ratio of Belt Tensions. Fig. 3.38 shows a driver pulley A and driven pulley B rotating in the clockwise direction. Fig. 3.39 shows only the driven pulley B. Consider the driven pulley B.

Let T1 = Tension in the belt on the tight side

T2 = Tension in the belt on the slack side

θ = Angle of contact, i.e., the angle subtended by the arc EF at the centre of the driven pulley.

µ = Co-efficient of friction between the belt and pulley.

* This means that T1 = T2 = T = constant where T1 and T2 are tensions on both sides of the belt.

** Torque = (T1– T2)×r where r is radius. As in this case T1 = T2 and hence will be zero.

The ratio of the two tensions may be found by considering an elemental piece of the belt MN subtending an angle δθ at the centre of the pulley B as shown in Fig. 3.39. The various forces which keep the elemental piece MN in equilibrium are:

(i)                           Tension T in the belt at M acting tangentially,

(ii)                        Tension T+δT in the belt at N acting tangentially,

(iii)                      Normal reaction R acting radially outward at P, where P is the middle point of MN.

(iv)                      Frictional force F = µR acting at right angles to R and in the opposition direction of the motion of pulley.

In equation (3.9), θ should be taken in radians. Here θ is known as angle of contact. For an open belt or for a crossed belt the angle of contact is determined as given below.

3.6.2 Angle of Contact for Open Belt Drive. With an open belt drive, the belt will begin to slip on the smaller pulley, since the angle of lap is smaller on this pulley than on the large pulley. The angle θ should be taken as the minimum angle of contact. Hence in equation (3.9), the angle of contact of lap (θ) at the smaller pulley must be taken into consideration.

Angle of contact, θ = (180 – 2α)                                                  … (3.10)

But the value of α is given by,

where r1 = Radius of larger pulley,

r2 = Radius of smaller pulley, and

x = Distance between the centres of two pulleys.

3.6.3 Angle of Contact for Crossed Belt Drive. For a crossed belt drive, the angle of lap on both the pulleys is same.

where r1 = Radius of larger pulley,

r2 = Radius of smaller pulley, and

x = Distance between the centres of two pulleys.