Belt and chain drives

Why it matters

Belts and chains transmit power over centre distances too long for gears, cheaply and with some tolerance for misalignment. Flat and V-belts drive pumps, fans, compressors and machine-tool spindles from motors; chains drive conveyors, bicycles, motorcycles and agricultural machines where slip is unacceptable. Their capacity is limited by friction and belt tension (belts) or by link fatigue and wear (chains), so the tension analysis is the heart of the design.

Key ideas

Belt drives - geometry. For driver diameter d₁ at speed n₁ and driven diameter d₂:

• Speed ratio n₂/n₁ = d₁/d₂ without slip; with belt thickness t and total slip s, n₂/n₁ = ((d₁ + t)/(d₂ + t))·(1 − s). Creep (elastic stretch difference between tight and slack sides) causes a small additional speed loss.
• Angle of wrap on the smaller pulley of an open belt: θ = π − 2·sin⁻¹((D − d)/(2C)). It is the smaller pulley that limits capacity. A longer centre distance increases θ on the small pulley; idlers on the slack side also increase it.
• Belt length: open L = π(d₁ + d₂)/2 + 2C + (d₂ − d₁)²/(4C); crossed L = π(d₁ + d₂)/2 + 2C + (d₁ + d₂)²/(4C).

Tension ratio. Friction on the wrapped arc lets the tight-side tension T₁ exceed the slack-side T₂: T₁/T₂ = e^(μθ) (flat belt, about to slip). For a V-belt with groove angle 2β, the wedge raises the normal force, so the effective friction coefficient is μ/sin β and T₁/T₂ = e^(μθ/sin β). That is why V-belts transmit more power at lower tension.

Centrifugal tension. At belt speed v, each element needs a centripetal force supplied by an extra tension Tc = m·v² (m = mass per metre) on both sides. It does not help transmit power: the friction relation applies to (T₁ − Tc)/(T₂ − Tc) = e^(μθ). With maximum allowable tension Tmax (from belt strength), T₁ = Tmax.

Power and the optimum speed. P = (T₁ − T₂)·v. With Tmax fixed, power is a maximum when Tc = Tmax/3, i.e. at v = √(Tmax/(3m)). At higher speeds centrifugal tension eats the capacity.

Initial tension. Belts are installed with T₀ ≈ (T₁ + T₂)/2 (plus Tc if centrifugal effect is included); too little gives slip and heat, too much overloads shafts and bearings.

Belt types. Flat belts (high speed, long centres), V-belts (compact, high power, selected from manufacturer's tables with service and correction factors), ribbed (poly-V) belts, and toothed (timing) belts for positive, slip-free drive in positioning systems (next topic).

Chain drives (roller chain) give a positive drive with no slip, high efficiency (about 96-98%) and compactness, but need lubrication and are noisier.

• Pitch circle diameter D = p/sin(180°/z); average chain speed v = z·p·n/60.
• Chordal (polygonal) action: the chain wraps the sprocket as a polygon, so the chain speed fluctuates, causing vibration and impact; it falls as the number of teeth rises. Use at least about 17-19 teeth on the small sprocket (odd numbers help even wear).
• Length in pitches: Ln = 2C/p + (z₁ + z₂)/2 + ((z₂ − z₁)/(2π))²·p/C; round to an even whole number of links (to avoid an offset link). Recommended centre distance is about 30-50 pitches.
• Selection uses power ratings from the chain standard or catalogue, corrected by service and tooth factors.

Formulas

n₂ / n₁ = d₁ / d₂ (no slip) · n₂ / n₁ = ((d₁ + t)/(d₂ + t))·(1 − s)

• n: speed (rpm); d: pulley diameter (m); t: belt thickness (m); s: slip fraction.

θ = π − 2·sin⁻¹((D − d)/(2C)) (small pulley, open belt)

• θ: angle of wrap (rad); D, d: large and small pulley diameters (m); C: centre distance (m).

L = π(d₁ + d₂)/2 + 2C + (d₂ − d₁)²/(4C) (open) · L = π(d₁ + d₂)/2 + 2C + (d₁ + d₂)²/(4C) (crossed)

(T₁ − Tc)/(T₂ − Tc) = e^(μθ) (flat) · e^(μθ / sin β) (V-belt, β = half groove angle) · Tc = m·v²

• T₁, T₂: tight and slack side tensions (N); Tc: centrifugal tension (N); μ: coefficient of friction; m: mass per unit length (kg/m); v: belt speed (m/s).

P = (T₁ − T₂)·v · v_opt = √(Tmax / (3m))

• P: power (W); Tmax: maximum allowable tension (N).

D = p / sin(180°/z) · v = z·p·n / 60 · Ln = 2C/p + (z₁ + z₂)/2 + ((z₂ − z₁)/(2π))²·p/C

• p: chain pitch (m); z: sprocket teeth; Ln: chain length in links.

Worked examples

Example 1 (standard). An open flat belt connects a 200 mm motor pulley (1440 rpm) to a 500 mm pulley at C = 1.2 m. μ = 0.3, maximum belt tension 1200 N, belt mass 0.5 kg/m. Find the power transmitted.

1. Wrap on small pulley: θ = π − 2·sin⁻¹((500 − 200)/(2 × 1200)) = π − 2 × 0.1253 = 2.891 rad (165.6°).
2. Belt speed: v = π·d·n/60 = π × 0.2 × 1440/60 = 15.08 m/s.
3. Centrifugal tension: Tc = m·v² = 0.5 × 15.08² = 113.7 N.
4. e^(μθ) = e^(0.3 × 2.891) = 2.380. T₁ − Tc = 1200 − 113.7 = 1086.3 N → T₂ − Tc = 1086.3/2.380 = 456.3 N (T₂ = 570.0 N).
5. P = (T₁ − T₂)·v = (1086.3 − 456.3) × 15.08 = 9.50 kW.

Example 2 (GATE level). A V-belt (groove angle 40°, mass 0.2 kg/m, maximum tension 600 N) runs with μ = 0.25 and 160° wrap. Find the speed for maximum power and that power per belt.

1. Effective friction: μ/sin 20° = 0.25/0.3420 = 0.731; θ = 160° = 2.793 rad; e^(0.731 × 2.793) = 7.70.
2. Optimum speed: v = √(Tmax/(3m)) = √(600/0.6) = 31.6 m/s; Tc = Tmax/3 = 200 N.
3. T₁ − Tc = 400 N; T₂ − Tc = 400/7.70 = 51.9 N.
4. P = (T₁ − T₂)·v = (400 − 51.9) × 31.62 = 11.0 kW per belt.

Example 3 (chain, quick). Sprockets of 19 and 57 teeth, pitch 15.875 mm, C = 40 pitches: Ln = 80 + 38 + (38/(2π))² × (1/40) = 118.9 → 120 links; small sprocket D = 15.875/sin(180°/19) = 96.4 mm.

Common mistakes

• Using the large pulley's wrap angle; the smaller pulley usually governs.
• Taking θ in degrees inside e^(μθ).
• Applying e^(μθ) to T₁/T₂ when centrifugal tension is significant - subtract Tc first.
• Forgetting the sin β factor for V-belts, or using the full groove angle instead of half.
• Claiming a longer centre distance reduces the wrap angle (it increases it).
• Leaving an odd number of chain links, or using too few sprocket teeth (rough, noisy running).

For GATE ME

Expect the tension ratio e^(μθ) for flat and V-belts, power transmitted, centrifugal tension and the condition for maximum power (Tc = Tmax/3), wrap angle and belt length, velocity ratio with slip, and chain pitch diameter and length. Practise the maximum-power derivation; it is a frequent conceptual question.

Quick check

1. Driver 150 mm at 1000 rpm, driven 300 mm, 2% slip. Driven speed?
2. Condition for maximum power in a belt drive?
3. Why does a V-belt transmit more power than a flat belt at the same tension?
4. Pitch 12.7 mm, 20 teeth. Sprocket pitch diameter? Answers: 1. 490 rpm. 2. Tc = Tmax/3. 3. The wedge raises the normal force; effective μ becomes μ/sin β. 4. 81.2 mm.