Pulley shaft load from belt span tensions

What does this quantity describe?

Both belt spans pull along their own directions. The law of cosines gives FW = √(F₁²+F₂²−2F₁F₂ cos β), where β is the angle between span-force vectors directed away from the pulley center.

Two ropes leaving the same ring in nearly opposite directions load the ring by almost the sum of their magnitudes. If they point nearly the same way, the resultant approaches their difference. β captures this directional effect.

Formula and variables

FW = √(F₁² + F₂² − 2F₁F₂ cos β)

• FW = √(F₁²+F₂²−2F₁F₂ cos β)
• β = arccos[(F₁²+F₂²−FW²)/(2F₁F₂)]
• at β = 180°: FW = F₁+F₂
• at β = 0°: FW = |F₁−F₂|

Choose the inputs correctly

F₁ and F₂ are operating span tensions. β is the included angle of outward force vectors; for a simple open two-pulley drive it equals the wrap angle at the pulley. Confirm the convention with a free-body diagram.

How to use the calculator

Select FW and enter both tensions and β. Use the result as an external radial load in subsequent shaft and bearing analysis, adding actual pulley overhang, load direction, other pulleys and service factors.

Worked example

For F₁ = 1,000 N, F₂ = 400 N and β = 165°, FW = 1,390.2 N. At β = 180° it would be exactly 1,400 N; at β = 0°, 600 N.

Understand the result and units

A 1.39 kN resultant acts at the pulley center. An overhung pulley also creates a bending moment at the bearing. The load direction follows from vector geometry and is not supplied by this magnitude-only result.

Forces use newtons and angles are evaluated in radians internally; degrees can be selected. For a later bending moment, multiply FW by pulley-to-bearing offset in metres.

From pulley force to bearing force

FW is an external force at the pulley center. With a pulley between two bearings, reactions split according to distances. With an overhung pulley outside the bearings, reactions and shaft bending rise. Bearing-life input therefore comes only after resolving all external loads into each bearing reaction.

Typical applications

Shaft load supports bearing-life, shaft-deflection and gearbox overhung-load checks, and shows the effect of changed pretension. Gates explicitly recommends comparing calculated overhung load with the gearbox manufacturer's limit.

Assumptions, limits and common mistakes

This is a planar static resultant. It excludes misalignment axial force, weight, shock/service factors and spatial superposition of other elements. Actual bearing reactions also require bearing locations and static equilibrium.

Common mistake: Do not use F₁−F₂ as bearing load. Do not assume 180° without checking geometry, and do not confuse wrap angle with the smaller angle between tangent lines. Treat forces for multiple belts consistently.