Subject: Kinematics of Machinery

Topic: Belts, Chains and Brakes

Difficulty: High

An open chain drive system connecting the two sprockets it shown in fig. we have already discussed in art. That the length of belt for an open belt drive connecting the two pulleys of radii $r_{1}$ and $r_{2}$ and a centre distance x, is

$L=\pi(r_{1}+r_{2})+2x+\frac{( r_{1}-r_{2})^2}{x}$…...........(i)

If this expression is for determining the length of chain, the result will be slightly greater than the required length. This is due to the fact that the pitch lines A B C D E F G and P Q R S of the sprockets are the parts of a polygon and not that of a circle. The exact length of the chain may be determined as discussed below;

Let $T_{1}$= Number of teeth on the larger sprocket,

$T_{2}$= Number of teeth on the smaller sprocket, and

P = Pitch of the chain

We have discussed in art. That diameter of the pitch circle,

$d=pcosec(\frac{180^{\circ}}{T})$ ot $r=\frac{p}{2}pcosec(\frac{180^{\circ}}{T})$

For larger sprocket,

$r_{1}=\frac{p}{2}pcosec(\frac{180^{\circ}}{T_{1}})$

For larger sprocket

$r_{1}=\frac{p}{2}pcosec(\frac{180^{\circ}}{T_{1}})$

Since the term $\pi(r_{1}+r_{2})$ is equal to half the sum of the circumferences of the pitch circles, therefore the length of chain corresponding to

$\pi(r_{1}+r_{2})=\frac{p}{2}(T_{1}+T_{2})$

Substituting the values of $(r_{1},r_{2})$and $\pi(r_{1}+r_{2})$ in equation (i) the length of chain is given by

$L=\frac{p}{2}(T_{1}+T_{2})+2x+(\frac{p}{2}cosec\frac{180^{\circ}}{T_{1}})-(\frac{p}{2}cosec\frac{180^{\circ}}{T_{2}})^{2}/x$

$L=\frac{p}{2}(T_{1}+T_{2})+2m+(\frac{p}{2}cosec\frac{180^{\circ}}{T_{1}})-(\frac{p}{2}cosec\frac{180^{\circ}}{T_{2}})^{2}/4m$= p.k

Where k = Multiplying factor,

$k=\frac{(T_{1}+T_{2})}{2}+2m+(\frac{p}{2}cosec\frac{180^{\circ}}{T_{1}})-(\frac{p}{2}cosec\frac{180^{\circ}}{T_{2}})^{2}/4m$

The value of multiplying factor (k) may not be a complete integer. But the length of the chain must be equal to an integer number of times the pitch of the chain. Thus the value of k should be rounded off to next higher integral number.