# 2.1.2 Engagement with Sprockets

Although chains are sometimes pushed and pulled at either end by cylinders, chains are usually driven by wrapping them on sprockets. In the following section, we explain the relation between sprockets and chains when power is transmitted by sprockets.

- Back tension

First, let us explain the relationship between flat belts and pulleys.
Figure 2.5 shows a rendition of a flat belt drive. The circle at the top is a pulley, and the belt hangs down from each side. When the pulley is fixed and the left side of the belt is loaded with tension (*T _{0}*), the force needed to pull the belt down to the right side will be:

*T _{1}* =

*T*× e

_{0}^{µθ}

For example, *T _{0}* = 100 N: the coefficient of friction between the belt and pulley, µ = 0.3; the wrap angle θ = π (180°).

*T _{1}* =

*T*× 2.566 = 256.6 N

_{0}In brief, when you use a flat belt in this situation, you can get 256.6 N of drive power only when there is 100 N of back tension. For elements without teeth such as flat belts or ropes, the way to get more drive power is to increase the coefficient of friction or wrapping angle. If a substance, like grease or oil, which decreases the coefficient of friction, gets onto the contact surface, the belt cannot deliver the required tension.

In the chain's case, sprocket teeth hold the chain roller. If the sprocket tooth configuration is square, as in Figure 2.6, the direction of the tooth's reactive force is opposite the chain's tension, and only one tooth will receive all the chain's tension. Therefore, the chain will work without back tension.

**Figure 2.5** Flat Belt Drive

**Figure 2.6** Simplified Roller/Tooth Forces

**Figure 2.7** The Balance of Forces Around the Roller

But actually, sprocket teeth need some inclination so that the teeth can engage and slip off of the roller. The balance of forces that exist around the roller are shown in Figure 2.7, and it is easy to calculate the required back tension.

For example, assume a coefficient of friction µ = 0, and you can calculate the back tension (T_{k}) that is needed at sprocket tooth number k with this formula:

*T _{k}* =

*T*× {sin ø ÷ sin (ø + 2β)}

_{0}^{k-1}

Where:

*T*= back tension at tooth k_{k}*T*= chain tension_{0}*ø*= sprocket minimum pressure angle 17 - 64/N(°)*N*= number of teeth*2β*= sprocket tooth angle (360/N)*k*= the number of engaged teeth (angle of wrap × N/360); round down to the nearest whole number to be safe

By this formula, if the chain is wrapped halfway around the sprocket, the back tension at sprocket tooth number six is only 0.96 N. This is 1 percent of the amount of a flat belt. Using chains and sprockets, the required back tension is much lower than a flat belt.

Now let's compare chains and sprockets with a toothed-belt back tension.

Although in toothed belts the allowable tension can differ with the number of pulley teeth and the revolutions per minute (rpm), the general recommendation is to use 1/3.5 of the allowable tension as the back tension (F). This is shown in Figure 2.8. Therefore, our 257 N force will require 257/3.5 = 73 N of back tension.

Both toothed belts and chains engage by means of teeth, but chain's back tension is only 1/75 that of toothed belts.

**Figure 2.8** Back Tension on a Toothed Belt

- Chain wear and jumping sprocket teeth

The key factor causing chain to jump sprocket teeth is chain wear elongation (see Basics Section 2.2.4). Because of wear elongation, the chain creeps up on the sprocket teeth until it starts jumping sprocket teeth and can no longer engage with the sprocket. Figure 2.9 shows sprocket tooth shape and positions of engagement. Figure 2.10 shows the engagement of a sprocket with an elongated chain.

In Figure 2.9 there are three sections on the sprocket tooth face:

- Bottom curve of tooth, where the roller falls into place;
- Working curve, where the roller and the sprocket are working together;
- Where the tooth can guide the roller but can't transmit tension. If the roller, which should transmit tension, only engages with C, it causes jumped sprocket teeth.

The chain's wear elongation limit varies according to the number of sprocket teeth and their shape, as shown in Figure 2.11. Upon calculation, we see that sprockets with large numbers of teeth are very limited in stretch percentage. Smaller sprockets are limited by other harmful effects, such as high vibration and decreasing strength; therefore, in the case of less than 60 teeth, the stretch limit ratio is limited to 1.5 percent (in transmission chain).

**Figure 2.9** Sprocket Tooth Shape and Positions of Engagement

**Figure 2.10** The Engagement Between a Sprocket and an Elongated Chain

**Figure 2.11** Elongation Versus the Number of Sprocket Teeth

In conveyor chains, in which the number of working teeth in sprockets is less than transmission chains, the stretch ratio is limited to 2 percent. Large pitch conveyor chains use a straight line in place of curve B in the sprocket tooth face.