Page 149 - Analysis and Design of Machine Elements

Belt Drives
Therefore, in order to ensure belt to transmit power P at speed v, an effective tension 127
F or frictional force F must be guaranteed between the belt and pulley. In other words,
e f
a minimum initial tension must be maintained on the belt.
In summary, the effective tension F is the sum of distributed frictional forces at the
e
interface between the belt and pulley. It also equals the difference between the tensions
on the two strands of belt. The greater the difference in belt tension, the greater the
power transmitted. Since the frictional force has a critical or maximum value that limits
the power a belt drive can transmit, it is therefore necessary to determine the maximum
frictional force or the maximum effective tension.
6.2.2.3 Critical or Maximum Effective Tension, F
ec
Consider a differential belt element shown in Figure 6.7d: the forces acting on the belt
element are tensions F, F + dF and normal force dN. Assume frictional force on the belt
is proportional to the normal force, the maximum frictional force at the point of slip is
fdN along the contact arc. Neglecting centrifugal tension in the belt for the time being,
according to the force equilibrium in both radial and tangential directions we have
{ d d
dN = F sin +(F + dF) sin
2 2
fdN + F cos d =(F + dF) cos d
2 2
For small angles,
{ d d
sin ≈
2 d 2
cos 2 ≈ 1
We then have
dF
= fd
F
Integrating this equation over the contact angle ,gives

dF
F 1
= fd
∫ F ∫
F 2 0
Which leads to
F
1 = e f (6.10)
F
2
where
f — coefficient of friction between the belt and pulley;
— contact angle, rad;
e — 2.718, the basis of the natural logarithm.
Equation (6.10) describes the relation between the tight tension F and slack tension
1
F on the verge of slipping. From Eqs. (6.5), (6.6) and (6.10) the maximum effective ten-
2
sion canbeexpressed as
1 − 1∕e f
F = 2F (6.11)
ec 0
1 + 1∕e f
Equation (6.11) gives a fundamental insight into belt drive forces. Factors affecting the
maximum effective tension F include initial tension F ,contact angle and coefficient
0
ec
of friction f .