Belt Drive Design: Velocity Ratio, Belt Length and Power Transmitted

Belt drives connect a motor to a machine when the shafts are some distance apart and an exact speed ratio is not essential. They are cheap, quiet, absorb shock and protect the motor by slipping under overload. You will find them on flour mills, lathes, compressors, washing machines and agricultural pump sets. This article covers the geometry of open and crossed belts, the friction-based tension ratio, and a complete calculation of the power a flat belt can transmit, followed by the V-belt advantage.

Types of Belts

Belt	Section	Features	Typical use
Flat belt	Rectangular	Long centre distances, high speed, some slip	Older line shafts, conveyors
V-belt	Trapezoidal, runs in grooved pulley	Wedging action gives high grip; several belts can run side by side	Motors, compressors, pump sets
Ribbed (poly-V)	Many small V ribs on a flat back	Flexible, compact, high speed	Automobile accessory drives
Timing (synchronous) belt	Teeth mesh with toothed pulleys	No slip, exact ratio	Engine camshafts, 3D printers, CNC axes

Speed Ratio and Belt Length

Velocity ratio             N2 / N1 = D1 / D2
With belt thickness t      N2 / N1 = (D1 + t) / (D2 + t)
With total slip s (%)      N2 / N1 = (D1 / D2) * (1 - s/100)

Open belt length    L = pi (D1 + D2)/2 + 2C + (D2 - D1)^2 / (4C)
Cross belt length   L = pi (D1 + D2)/2 + 2C + (D1 + D2)^2 / (4C)
Angle of contact on smaller pulley (open belt)
                    theta = 180 deg - 2 * asin( (D2 - D1) / (2C) )
Belt speed          v = pi * D1 * N1 / 60

A crossed belt reverses the direction of rotation and gives a larger angle of contact, but the belt rubs on itself where it crosses, so it is used only at low speeds with flat belts.

Example 1: Speed and length

A motor at 1440 rpm carries a 150 mm pulley and drives a 450 mm pulley. The centre distance is 1000 mm.

Driven speed (no slip)    = 1440 * 150 / 450 = 480 rpm
Driven speed (2 % slip)   = 480 * 0.98 = 470 rpm

Open belt length  = pi * 600/2 + 2000 + 300^2 / 4000
                  = 942.5 + 2000 + 22.5 = 2965 mm
Cross belt length = 942.5 + 2000 + 600^2 / 4000 = 3032.5 mm

Angle of contact  = 180 - 2 * asin(300 / 2000)
                  = 180 - 2 * 8.63 = 162.7 deg = 2.840 rad

Belt Tensions and Power

The belt grips the pulley by friction. The tension on the tight side T1 is higher than on the slack side T2, and their ratio depends on the coefficient of friction and the angle of contact on the smaller pulley, which is where slip occurs first.

Tension ratio (flat belt)    T1 / T2 = e^(mu * theta)
Tension ratio (V-belt)       T1 / T2 = e^(mu * theta / sin(beta))
                             (2 * beta = groove angle)
Centrifugal tension          Tc = m * v^2      (m = mass per metre, kg/m)
Power transmitted            P = (T1 - T2) * v
Maximum tension              T_max = T1 + Tc
Speed for maximum power      v = sqrt( T_max / (3m) )

Example 2: Power a flat belt can transmit

For the drive above, take mu = 0.3, maximum allowable belt tension 1500 N and belt mass 0.3 kg/m.

v       = pi * 0.15 * 1440 / 60 = 11.31 m/s
T1/T2   = e^(0.3 * 2.840) = 2.345

Ignoring centrifugal tension:
  T1 = 1500 N, T2 = 1500 / 2.345 = 640 N
  P  = (1500 - 640) * 11.31 = 9.73 kW

Including centrifugal tension:
  Tc = 0.3 * 11.31^2 = 38.4 N
  T1 = 1500 - 38.4 = 1461.6 N, T2 = 1461.6 / 2.345 = 623.4 N
  P  = (1461.6 - 623.4) * 11.31 = 9.48 kW

Speed for maximum power = sqrt(1500 / (3 * 0.3)) = 40.8 m/s

At 11.3 m/s centrifugal tension costs only about 2.6 percent of the power. At higher belt speeds it grows with the square of speed and eventually leaves no tension for transmitting power, which is why there is an optimum belt speed.

Example 3: The V-belt advantage

For a V-belt with a 40 degree groove (beta = 20 degrees), the effective friction coefficient becomes 0.3 / sin 20 = 0.877. The tension ratio jumps to e^(0.877 x 2.840) = about 12.1, compared with 2.35 for the flat belt. For the same maximum tension the V-belt carries much more power per belt, which is why V-belts dominate industrial motor drives.

Common Mistakes

Using the angle of contact of the larger pulley; slip always starts on the smaller one.
Leaving theta in degrees inside the exponential; it must be in radians.
Ignoring centrifugal tension at high belt speed.
Over-tightening belts, which overloads motor and machine bearings.
Mixing old and new V-belts in a matched set; the new belts carry most of the load.

Summary

Speed ratio comes from pulley diameters, belt length from the open or cross belt formula, and grip from the exponential tension ratio. Power is the effective tension multiplied by belt speed, reduced by centrifugal tension at high speed. The wedge action of V-belts multiplies the effective friction, letting a compact drive transmit far more power than a flat belt.