Belt Drive Formulas: Pulley RPM, Speed Ratio, Belt Length & Torque

May 29, 2026

Formula 1: Driven Pulley RPM

The fundamental pulley speed relationship:

N2 = N1 × D1 / D2

Variable	Meaning	Units
N1	Driver (motor) RPM	RPM
N2	Driven pulley RPM	RPM
D1	Driver pulley diameter	inches or mm
D2	Driven pulley diameter	inches or mm

With belt slip:

N2 = N1 × (D1 / D2) × (1 − s/100)

Where s = slip percentage (typically 1–3% for V-belts; 0% for synchronous/timing belts).

Pitch diameter note: For V-belts, D1 and D2 should be pitch diameters (the diameter at which the belt’s tensile cord rides), not outside diameters. Pitch diameter ≈ outside diameter − belt cross-section factor (varies by belt type; typically 0.2–0.6 inches less than OD for classical V-belts).

Formula 2: Speed Ratio

Speed Ratio = D1 / D2 = N2 / N1

Speed Ratio	Meaning
> 1	Speed increase (driven shaft faster than motor)
= 1	1:1, no speed change
< 1	Speed reduction (driven shaft slower than motor)

Speed ratio ≠ torque ratio: Torque ratio = D2/D1 (the inverse). Speed-down = torque-up.

Formula 3: Belt Speed

Imperial:  V = π × D1 × N1 / 12       (ft/min, D1 in inches)
Metric:    V = π × D1 × N1 / 1000      (m/min, D1 in mm)

Maximum belt speeds by type:

Belt Type	Max Belt Speed
Classical V-belt (A–E)	4,500 ft/min (1,372 m/min)
Narrow V-belt (3V/5V/8V)	6,500 ft/min (1,981 m/min)
Flat belt (rubber/polyurethane)	8,000 ft/min (2,438 m/min)
Synchronous timing belt	6,000–10,000 ft/min (varies)

Exceeding rated belt speed causes excessive heat, vibration, centrifugal force losses, and rapid belt failure.

Formula 4: Belt Length (Open Belt Drive)

For two parallel-shaft pulleys with an open belt configuration:

L = 2C + (π/2)(D1 + D2) + (D1 − D2)² / (4C)

Variable	Meaning
L	Belt length
C	Center distance between shaft centers
D1	Driver pulley diameter
D2	Driven pulley diameter

All variables in the same unit (inches or mm).

Cross belt configuration (shafts parallel but belt crosses):

L = 2C + (π/2)(D1 + D2) + (D1 + D2)² / (4C)

Use the cross-belt formula only when the belt must reverse direction between shafts.

Formula 5: Arc of Contact

The arc of contact on each pulley determines the belt’s grip and how much torque can be transmitted before slip:

Smaller pulley arc (open belt): θ = 180° − 2 × arcsin[(D1 − D2) / (2C)]
Larger pulley arc (open belt):  φ = 180° + 2 × arcsin[(D1 − D2) / (2C)]

For identical pulleys (D1 = D2): arc of contact = 180° on both pulleys.

For well-designed drives, the smaller pulley arc should exceed 120°. Below 120°, belt slip and premature wear increase significantly.

Formula 6: Torque

Driver torque:   T1 = P × 63,025 / N1   (in·lb, P in HP)
                 T1 = P × 9,549 / N1     (N·m, P in kW)

Driven torque:   T2 = T1 × D2 / D1      (ideal, no losses)
                 T2 = T1 × D2 / D1 × η  (with drive efficiency η)

Variable	Meaning
T1	Driver shaft torque
T2	Driven shaft torque
P	Motor power (HP or kW)
N1	Driver RPM
η	Drive efficiency (typical: 0.95–0.98 for well-maintained V-belt drive)

Formula 7: Required Driven Diameter (Reverse Calculation)

Given a target output RPM, solve for the required driven pulley diameter:

D2 = D1 × N1 / N2

Example: 1750 RPM motor, 4-inch driver pulley, target 875 RPM output:

D2 = 4 × 1750 / 875 = 8 inches

Belt Drive Design Quick Reference

Parameter	Typical Range	Notes
Speed ratio	1:7 max per stage	Higher ratios require intermediate shafts
Belt slip	1–3% (V-belt)	0% for synchronous belts
Drive efficiency	95–98%	Higher with proper tension and alignment
Minimum pulley diameter	Varies by belt type	Check manufacturer catalog
Maximum belt speed	4,500–6,500 ft/min	Depends on belt type
Tension ratio	2:1 to 5:1	Tight side / slack side

Use the Pulley RPM Calculator to solve these formulas without manual computation.