THEORY OF MACHINES

FLAT BELT DRIVES

1. VELOCITY RATIO IN FLAT BELT DRIVE

Velocity ratio can be defined as the ratio of velocities of the driver and the follower/ driven.
Let, d1, d2 = Diameter of the driver and the follower respectively,
and N1, N2 = Speed of the driver and the follower resp. in r.p.m.
∴ Length of the belt that passes over the driver, in one minute = π d1. N1
Similarly, length of the belt that passes over the follower, in one minute = π d2 .N2
Since the length of belt that passes over the driver in one minute is equal to the length of belt
that passes over the follower in one minute, therefore
π d1. N1 = π d2 .N2

When the thickness of the belt (t) is considered, then velocity ratio becomes

Sometimes the power is transmitted from one shaft to another, through a Compound Belt
Drive i.e. via a number of pulleys, as shown in figure

Let d1 d2, d3, d4 = Diameter of the pulley 1, 2, 3 and 4 respectively.

and N1, N2, N3, and N4 = Speed of the 1,2,3 and 4 respectively. in r.p.m.,
Then, the velocity ratio of pulleys 1 and 2 will be

and the velocity ratio of pulleys 3 and 4 will be

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 Therefore

A little consideration will show, that if there are six pulleys, then

Slip in the Belt Drive

The motion of belts over the pulleys is due to a firm frictional grip between the belts and the
shafts. But sometimes, the frictional grip becomes insufficient. This may cause some forward
motion of the driver without carrying the belt with it, called slip of the belt. The result of the
belt slipping is the reduced velocity ratio of the system. As the slipping of the belt is a
common, the belt drive should never be used where a constant velocity ratio is required (as in
the case of hour, minute and second arms in a watch).

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Creep of the Belt
When the belt passes from the slack side to the tight side, a certain portion of the belt extends
and it contracts again when the belt passes from the tight side to slack side. Due to these
changes of length, there is a relative motion between the belt and the pulley surfaces. This
relative motion is termed as creep. The total effect of creep is to reduce slightly the speed of
the driven pulley or follower. Considering creep, the velocity ratio is given by

2. OPEN BELT DRIVE CALCULATIONS

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Power Transmitted by a Belt
In the open belt drive shown in Fig. A is the driving pulley (or driver) and B is the driven
pulley (or follower) B. The driving pulley pulls the belt from one side called the tight side
while the other side remains loose is the slack side of the belt.

Let,
T1 and T2 be the tensions on the tight and slack side of the belt respectively.
r1 and r2 be the radii of the driver and follower respectively,
v = Velocity of the belt in m/s.
The effective turning (driving) force at the circumference of the follower is the difference
between the two tensions (i.e. T1 – T2).
∴ Work done per second W = (T1 – T2) v. N-m/s
Power transmitted, P = (T1 – T2) v. W
Torque exerted on the TD = (T1 – T2) r1.
driver pulley is
Torque exerted on the TF = (T1 – T2) r2.
follower pulley

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Ratio Of Driving Tensions
Consider a flat belt drive in which the driven pulley is rotating in the clockwise direction.
Let, T1 = Tension in the belt on the tight side,
T2 = Tension in the belt on the slack side, and
θ = Angle of contact in radians (i.e. angle subtended by the arc AB, along
which the belt touches the pulley at the centre)

Now consider a small portion of the belt PQ, subtending an angle δθ at the centre of the
pulley. The belt PQ is in equilibrium under the following forces :
1. Tension T in the belt at P,
2. Tension (T + δ T) in the belt at Q,
3. Normal reaction RN, and
4. Frictional force, F = μ × RN ,
where μ is the coefficient of friction between the belt and pulley.

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The above expression gives the relation between the tight side and slack side tensions, in
terms of coefficient of friction and the angle of contact.

Angle of Contact
When the two pulleys of different diameters are connected by means of an open belt, then the
angle of contact or lap (θ) at the smaller pulley must be taken into consideration.
Let, r1 = Radius of larger pulley,
r2 = Radius of smaller pulley, and
x = Distance between centres of two pulleys (i.e. O1 O2).

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3. CENTRIFUGAL TENSION
Since the belt continuously runs over the pulleys, centrifugal force is caused, whose effect is
to increase the tension on both, tight as well as the slack sides. The tension caused by
centrifugal force is called centrifugal tension. At lower belt speeds (less than 10 m/s), the
centrifugal tension is very small, but at higher belt speeds (more than 10 m/s), its effect is
considerable and thus should be taken into account. Consider a small portion PQ of the belt
subtending an angle dθ the centre of the pulley.

Let m = Mass of the belt per unit length in kg,

v = Linear velocity of the belt in m/s,
r = Radius of the pulley over which the belt runs in metres, and
TC = Centrifugal tension acting tangentially at P and Q in newtons.
We know that the length of the belt PQ = r. dθ
and mass of the belt PQ = m. r. dθ
∴ Centrifugal force acting on the belt PQ,

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Maximum Tension in the Belt
The maximum tension in the belt is equal to the total tension in the tight side of the belt (Tt1).
Let σ = Maximum safe stress in N/mm2,
b = Width of the belt in mm, and
t = Thickness of the belt in mm.
We know that maximum tension in the belt,
T = Maximum stress × cross-sectional area of belt = σ. b. t
When centrifugal tension is neglected, then
T (or Tt1) = T1, i.e. Tension in the tight side of the belt
and when centrifugal tension is considered, then
T (or Tt1) = T1 + TC

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4. CONDITION FOR MAXIMUM POWER TRANSMISSION
We know that power transmitted by a belt,
P = (T1 – T2) v ...(i)
where T1 = Tension in the tight side of the belt in newtons,
T2 = Tension in the slack side of the belt in newtons, and
v = Velocity of the belt in m/s.
The ratio of driving tensions is

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