Belt Drive and Band Brake Friction Calculator

Screen belt-drive tensions and shaft load, or estimate simple and differential band-brake forces from friction and wrap geometry.

Engineering screening tool—verify inputs and results independently. Use at your own risk.

Model inputs

Choose an application, enter values in the displayed units, then calculate.

Screening result

Values use the two-decimal display convention.

Shaft load—N

Resultant pulley load from belt tensions.

Driven speed · rpm—

Belt speed · m/s—

Belt length · m—

Small-pulley wrap · deg—

Tension ratio T₁/T₂—

Tight tension T₁ · N—

Slack tension T₂ · N—

Effective friction μ′—

Drum bearing load · N—

Nominal pressure · MPa—

Equations used

SI base units are used internally. Power and torque are positive magnitudes; angles are converted from degrees to radians.

V / push belt: μ′ = μ / sin(β/2); other belt modes: μ′ = μ

i = D₁/D₂; n₂ = in₁; v = πD₁n₁

L_b = 2a + (π/2)(D₁+D₂) + (D₂−D₁)²/(4a)

α = 2 acos[(D₂ − D₁)/(2a)]

F_t = P/v = T₁ − T₂; T₁/T₂ = exp(μ′α)

F_t = M/r; T₁/T₂ = exp(μα)

T₁ = F_t/[1−exp(−μα)]; T₂ = T₁exp(−μα)

F_a = √(T₁²+T₂²−2T₁T₂ cos α)

Simple: F_i = T₂(b/a); p_max = T₁/(Lr)

Differential: F_i = (b₂/a)[T₂ − (b₁/b₂)T₁]

Symbols: P is transmitted power; n₁/n₂ are driver/driven speeds; D₁/D₂ are pitch diameters; a is centre distance; β is included groove angle; μ/μ′ are actual/effective friction; α is wrap; M is brake torque; r is drum radius; L is band width; T₁/T₂ are tight/slack tensions; F_t is their difference; F_a is shaft or bearing load; and F_i is actuating force. A negative actuating force in differential mode indicates self-energizing reversal under this sign convention.

How to use

Select V / push belt, Flat / round / conveyor, Band brake, or Differential brake.
Enter positive dimensional inputs in the units shown and dimensionless friction/lever ratios.
For V/push belts, enter included groove angle β. A 180° setting removes wedge amplification.
For brake modes, enter positive torque magnitude and wrap; interpret differential F_i using its displayed sign note.
Select Calculate to update the canonical result and secondary metrics.
Correct any red finite-value, geometry, range, or friction–wrap exponent warning.
Select Reset example to restore the supplied defaults and recalculate.
Verify friction, strength, centrifugal effects, temperature, dynamics, and manufacturer limits independently. Do not use this model for safety-critical sizing or unsuitable cases listed below.

Worked example

1. With V/push belt inputs P=1.5 kW, n₁=3000 rpm, D₁=100 mm, D₂=200 mm, a=300 mm, β=40°, and μ=0.3, the shaft load is 112.84 N, belt speed is 15.71 m/s, and T₁/T₂ is 11.73.

2. With a band brake at M=100 N·m, r=150 mm, α=200°, μ=0.3, L=40 mm, and b/a=0.4, F_i is 144.17 N and p_max is 0.17 MPa.

Theory and method

The Euler–Eytelwein relation describes the limiting tension ratio of a flexible belt or band over a cylindrical surface. A V-groove increases the normal reaction and is represented by μ′. Pulley geometry supplies open-belt length and small-pulley wrap. Static force and moment balance then recover individual tensions, resultant bearing load, nominal pressure, and lever force.

Assumptions and limits

The model assumes steady impending slip, constant Coulomb friction, a flexible massless belt/band, rigid pulleys or drum, open-belt geometry, and static lever equilibrium. All required dimensions and power/torque magnitudes are finite and strictly greater than zero; lever ratios may be zero only where stated. Open-belt geometry requires a > |D₂−D₁|/2, and the friction–wrap exponent must remain below 700. It omits centrifugal tension, bending stiffness, creep, slip loss, pulley inertia, wear, heat, compliance, shock, transient stopping, fatigue, and material limits. Do not use it for crossed/toothed belts, safety brakes, life prediction, thermal rating, or variable-friction systems.

References

Whitney (1966), “Capstan Equation for Nofilament Friction,” doi:10.1177/004051756603600316.
Shigley’s Mechanical Engineering Design, belt-drive and band-brake statics.