Belt Drives

Belt drives transmit motion between pulleys over a useful center distance. Flat and V-belts rely on friction; synchronous belts use positive tooth engagement. Belts are quiet, inexpensive, and able to absorb shock, but preload adds radial load to shafts and bearings.

Kinematics and traction

Ignoring slip,

$$i=\frac{n_1}{n_2}=\frac{d_2}{d_1}.$$

For a friction belt the Euler relation is

$$\frac{F_1}{F_2}=e^{\mu\alpha},$$

or for a V-belt the effective friction is increased by the groove wedge action. Transmitted power is

$$P=(F_1-F_2)v.$$

Loads and sizing

The approximate shaft load is \(F_r\approx F_1+F_2\). Centrifugal tension becomes important at high belt speed. Selection uses design power, small-pulley speed, ratio, service factor, belt section, pulley diameters, center distance, wrap angle, and number or width of belts.

Serviceability

Check traction reserve, belt stress, temperature, vibration, alignment, and pulley balance. Too little preload causes slip and heat; too much overloads bearings and shortens belt life. Guarding, retensioning method, contamination, and matched belt sets are part of the design.

Belt types

Flat belts suit high speed and long center distances. V-belts gain traction from wedging in pulley grooves and are commonly used in industrial drives. Ribbed belts combine flexibility with groove guidance. Synchronous belts use teeth and are selected when phase accuracy and absence of slip are required.

Length and wrap angle

For an open drive with center distance \(a\), an approximate belt length is

$$L\approx2a+\frac{\pi}{2}(d_1+d_2)+\frac{(d_2-d_1)^2}{4a}.$$

The small pulley usually has the smaller wrap angle and therefore governs traction. An idler can increase wrap but also adds reverse bending or extra bearing cycles depending on its position.

Preload and operating tensions

Before power transmission, initial tension \(F_0\) exists in both spans. During operation, tight-side tension rises to \(F_1\) and slack-side tension falls to \(F_2\). A simple elastic model gives approximately \(F_1+F_2\approx2F_0\), while transmitted force is \(F_t=F_1-F_2\).

Bending and centrifugal effects

Each passage around a pulley bends the belt, so a smaller pulley increases cyclic bending strain and reduces life. At high speed, centrifugal tension

$$F_c=\rho_Av^2$$

reduces the portion of allowable tension available for useful power transmission.

Toothed belts

Synchronous belts transmit force through tooth engagement. Check allowable tooth load, minimum pulley tooth count, number of teeth in mesh, belt width, tension-member stress, and tooth-jump reserve. Installation tension still matters for accurate meshing and control of span vibration.

Failures and maintenance

Typical failures are glazing, cracking, cord fatigue, tooth shear, edge wear, delamination, overheating, and chemical degradation. Pulley alignment, groove wear, contamination, tension, ventilation, and guard design should be inspected. Multiple V-belts should be supplied as a matched set and replaced together.

Design sequence

1. Determine design power, speeds, ratio, duty, and center-distance limits.
2. Select belt type and standard section.
3. Choose pulley diameters and verify belt speed and minimum diameter.
4. Calculate length, wrap, required belt number or width, and tensions.
5. Find shaft and bearing loads and check pulley strength.
6. Specify tensioning, alignment, guarding, inspection, and replacement criteria.