V-Belt Calculator

Belt Speed · Power · Tension · Gear Ratio · Belt Length

V-Belt Calculator

Effective (pitch) diameter of the pulley
Typical motor speeds: 960, 1450, 2900 rpm

Formulas & Symbols

Core Formulas
Belt speed:
v = π × d × n / 60 000
v [m/s], d [mm], n [rpm]
Transmitted power:
P = F_u × v × η
P [W], F_u [N], v [m/s], η = efficiency
Belt tension (Euler-Eytelwein):
F₁ / F₂ = e^(μ × β)
F_u = F₁ − F₂
β in radians; μ = friction coefficient
Gear ratio:
i = d₂ / d₁ = n₁ / n₂
i > 1 = speed reduction; i < 1 = speed increase
Belt length (open drive):
L ≈ 2e + π/2 × (d₁+d₂) + (d₂−d₁)² / (4e)
e = center distance [mm]
Symbol Reference
vBelt speed [m/s]
dPulley diameter [mm]
nRotational speed [rpm]
PPower [W / kW]
F_uPeripheral (tangential) force [N]
F₁Tight-side tension [N]
F₂Slack-side tension [N]
μFriction coefficient [0…1]
βWrap angle [rad]
iGear ratio [–]
eCenter distance [mm]
ηEfficiency [0…1]


V-Belt – Fundamentals

What is a V-Belt?

A V-belt is a flexible, endless power transmission element with a trapezoidal cross-section that sits in matching grooves of V-belt pulleys. The wedging action causes the belt to press deeper into the groove under load, producing a friction force up to three times higher than a flat belt at the same initial tension.

V-belts are used in an enormous variety of applications: agricultural machinery, machine tools, compressors, fan drives, and automotive auxiliaries. Their key advantages are quiet operation, slip tolerance (overload protection), easy installation, and the ability to dampen shock loads and vibrations.

Advantages
  • High friction force from wedge effect
  • Quiet, vibration-damping operation
  • Low maintenance, cost-effective
  • Slip provides natural overload protection
  • Bridges large center distances
  • No lubrication required
Disadvantages
  • Slip: no exact synchronous ratio
  • Limited belt speed (~35 m/s)
  • Re-tensioning required after run-in
  • Sensitive to oil and UV exposure
  • Lower efficiency than toothed belt

V-Belt Profiles per DIN 2215 / ISO 4184

Z (10×6 mm) | A (13×8 mm) | B (17×11 mm) | C (22×14 mm) | D (32×19 mm) | E (40×25 mm)
Format: width × height. Narrow profiles (SPZ, SPA, SPB, SPC) are more compact with higher power density.

Detailed Formula Derivation

1. Belt Speed v
v = π × d × n / 60 000
d [mm], n [rpm] → v [m/s]
Example: d = 200 mm, n = 1450 rpm → v = π × 200 × 1450 / 60 000 ≈ 15.2 m/s
2. Transmitted Power P
P = F_u × v × η
Example: F_u = 500 N, v = 15 m/s, η = 0.95 → P = 500 × 15 × 0.95 ≈ 7 125 W ≈ 7.1 kW
3. Belt Tension – Euler-Eytelwein Equation
F₁ / F₂ = e^(μ × β)
F_u = F₁ − F₂
β must be in radians (° × π / 180)
Example: μ = 0.35, β = 170° = 2.967 rad → e^(0.35×2.967) ≈ 2.83
F_u = 500 N → F₂ = 500 / (2.83−1) ≈ 273 N, F₁ ≈ 773 N
4. Gear Ratio i
i = d₂ / d₁ = n₁ / n₂
Example: d₁ = 100 mm (motor), d₂ = 300 mm (machine), n₁ = 1450 rpm
i = 300/100 = 3.0 (speed reduction), n₂ = 1450/3 ≈ 483 rpm
5. Belt Length L (open drive)
L ≈ 2e + π/2 × (d₁+d₂) + (d₂−d₁)² / (4e)
Example: d₁=100, d₂=300, e=500 mm
L ≈ 2×500 + π/2×400 + 200²/2000 ≈ 1000 + 628 + 20 ≈ 1648 mm

Practical Example: Woodworking Band Saw

Task:

An electric motor (1450 rpm, 3 kW) drives a band saw via V-belt. Driver pulley d₁ = 120 mm, driven d₂ = 240 mm, center distance e = 600 mm, μ = 0.35, η = 0.94.

Solution:
  • v = π × 120 × 1450 / 60 000 = 9.11 m/s
  • i = 240/120 = 2.0 → n₂ = 1450/2 = 725 rpm
  • F_u = P / (v × η) = 3000 / (9.11 × 0.94) ≈ 350 N
  • β₁ = 180° − 2·arcsin((d₂−d₁)/(2e)) ≈ 168.5° = 2.94 rad
  • e^(μβ) = e^(0.35×2.94) ≈ 2.79
  • F₂ = 350 / (2.79−1) ≈ 196 N, F₁ ≈ 546 N
  • L ≈ 2×600 + π/2×360 + 120²/2400 ≈ 1200 + 565 + 6 ≈ 1771 mm

Frequently Asked Questions

V-belts transmit force by friction and allow a small amount of slip — great for shock absorption and overload protection. Timing (toothed) belts are form-fit with no slip, ideal for synchronous motion (camshaft drives, CNC axes), but require more precise installation and cannot tolerate oil contamination.

After initial installation the belt seats and stretches — check and re-tension after approximately 24 operating hours. After that, an annual inspection is sufficient under normal load conditions. Warning signs: squealing noises or visible speed difference between pulleys.

At least 120° on the smaller pulley. Below 120° the transmissible force drops significantly and premature slip occurs. Remedies: increase center distance, add a tensioning idler, or adjust the pulley ratio.

Profile selection depends on transmitted power and driver speed. Belt manufacturers (Optibelt, Gates, ContiTech) provide selection charts. General rule: Z/SPZ up to ~1 kW, A/SPA up to ~5 kW, B/SPB up to ~15 kW, C/SPC up to ~75 kW, D/E above.

Slip occurs when friction is insufficient to transmit the full load — the belt slides on the pulley. Elastic creep (0.5–2 %) is inherent; gross slip under overload causes rapid wear and heat buildup. Efficiency decreases with increasing slip. Proper pre-tension prevents excessive slip.

Summary

Belt Speed

v = π × d × n / 60 000
Max. ~25–35 m/s

Power

P = F_u × v × η
η ≈ 0.92–0.97

Gear Ratio

i = d₂ / d₁ = n₁ / n₂
Typical i = 1 … 7

Typical Applications
  • Machine tools – lathes, milling machines, grinders
  • Agricultural machinery – combines, tedders, pumps
  • Fans & compressors – HVAC, air compressors
  • Automotive – alternator, power steering pump (Poly-V belt)
  • Conveying – belt conveyors, bucket elevators

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