1] Velocity Ratio

Velocity Ratio is the ratio of speeds of the driver and driven pulley. Let

N1, d1 = speed (rpm) and diameter of the driving pulley

N2, d2 = speed (rpm) and diameter of the driven pulley

t = thickness of the belt

Velocity Ratio, $\frac{N_{2}}{N_{1}}=\frac{d_{1}}{d_{2}}$

If the thickness of the belt is considered,

Velocity Ratio, $\frac{N_{2}}{N_{1}}=\frac{d_{1}+t}{d_{2}+t}$

Slip of belt over the pulleys reduces the velocity ratio. Let

s1 = percentage slip between driver and belt

s2 = percentage slip between belt & driven pulley

s = total percentage slip

Velocity Ratio, $\frac{N_{2}}{N_{1}}=\frac{d_1+t}{d_{2}+t}\left[1-\frac{S}{100}\right]$

2] Angle of Wrap

Angle of wrap on smaller (as) and larger pulleys (al) are given by,

$a_{s}=180^{\circ}-2 \sin ^{-1}\left(\frac{D-d}{2 C}\right)(\text { for open belt drive })$

$a_{1}=180^{\circ}+2 \sin ^{-1}\left(\frac{D-d}{2 C}\right)(\text { for open belt drive })$

$a_{s}=a_{1}=180^{\circ}+2 \sin ^{-1}\left(\frac{D+d}{2 C}\right)(\text { for Cross belt drive })$

3] Belt Tension

To increase friction and avoid slip, some initial tension is provided in the belt. Even when the belt is not transmitting any power, there is some initial tension (Ti ) in it, provided to avoid slipping of the belt.

Consider a flat belt drive, with the smaller pulley as a driver and the larger pulley as driven. When the driver starts rotating (say clockwise), it applies a tangential force on the belt and tends to rotate it. This leads to an increase in tension on one side and a decrease in tension on the other side of the belt by the same amount, $dT$, which equals the tangential force applied by the pulley. Side of the belt in which the tension increases is called tight side (lower side in this case) and the side in which tension decreases is called slack side (as shown in the figure below).

Tension in tight side, T1 = Ti + dT

Tension in slack side, T2 = Ti – dT

Therefore, $T_{i}=\frac{T_{1}+T_{2}}{2}$

The ratio of tension in tight side and slack side is given by,

$\frac{T_{1}}{T_{2}}=e^{\mu \theta}$ (for Flat belt drive), $\frac{T_{1}}{T_{2}}=e^{\mu \theta} \csc \beta$ (for Cross belt drive)

$\begin{array}{ll}{\text { Where, }} & {\mu=\text { coefficient of friction between belt } \& \text { pulley }} \\ {} & {\theta=\text { angle of contact }} \\ {} & {\beta=\text { half of the groove angle of v-belt }}\end{array}$

When the belt operates at higher speeds, centrifugal force acts on it, which increases the tension in the belt. This additional tension in the belt due to the centrifugal force is called centrifugal tension and it can be proved that, centrifugal tension,

$\begin{array}{ll}{\mathrm{Tc}=\mathrm{mv}^{2}} & {} \\ {\text { where, }} & {\mathrm{m}=\mathrm{mass} \text { of the belt per unit length }} \\ {} & {\mathrm{v}=\text { velocity of the belt in } \mathrm{m} \mathrm{m} / \mathrm{s}}\end{array}$

Therefore the tensions in the tight and slack side increase by an amount equal to $T_c$. The maximum tension in the belt then becomes,

$$T = T_1 + T_C$$

If b and t are width & thickness of a flat belt and $[\sigma] $ is the maximum allowable stress in the belt, maximum permissible tension for it can be given by,

$$ \text{[T] = maximum stress} \times \text{Cross - Sectionare} = [\sigma] b t $$

For the belt to run safely, maximum tension in the belt should not cross the permissible limit.

4] Belt Material:

• Leather: Oak tanned or chrome tanned.
• Rubber: Canvas or cotton duck impregnated with rubber. For greater tensile strength, the rubber belts are reinforced with steel cords or nylon cords.
• Plastics: Thin plastic sheets with rubber layers
• Fabric: Canvas or woven cotton ducks

The belt thickness can be built up with a number of layers. The number of layers is known as ply.

The belt material is chosen depending on the use and application. Leather oak tanned belts and rubber belts are the most commonly used but the plastic belts have a very good strength almost twice the strength of the leather belt. Fabric belts are used for temporary or short period operations.