Belt, rope and chain drives

Why it matters

Belts, ropes and chains transmit power between shafts that are too far apart for gears: from motors to compressors, fans, conveyors, machine tools and two-wheelers. They are cheap, tolerate misalignment and (belts) absorb shock, but their capacity is limited by friction and by the tension the belt or chain can carry. Designing one means finding the tensions, the power and the speed at which the drive works best.

Key ideas

Types.

• Flat belts: rectangular section, run on crowned pulleys; suit long centre distances and high speeds; efficiency about 95–98 %.
• V-belts: trapezoidal section wedging into a grooved pulley. The wedge action multiplies the normal force, so a V-belt transmits more power for the same tension and suits short centre distances and multiple-belt drives. Wedge friction and flexing make them slightly less efficient than flat belts.
• Rope drives: fibre or wire ropes in grooved sheaves, used to carry large power over long distances (mill drives, hoists, cranes); the analysis is the same as for a V-belt.
• Timing (toothed) belts: teeth engage the pulley, giving a positive drive with no slip.
• Chain drives: roller chains on sprockets give a positive, no-slip drive with a fixed average speed ratio, carry high loads at moderate speeds and do not need initial tension, but need lubrication and are noisier.

Velocity ratio, slip and creep. Without slip the belt speed is the same on both pulleys, so N₂/N₁ = d₁/d₂. Including belt thickness t, the effective diameters are d + t. Slip is the bulk sliding of the belt on the pulley when friction is insufficient; it is expressed as a percentage s and reduces the driven speed. Creep is the small relative motion caused by the belt stretching more on the tight side than the slack side; it is always present in a friction belt and also lowers the speed slightly.

Open and crossed belts. In an open drive both pulleys turn the same way and the smaller pulley has the smaller angle of contact, so it governs slipping. In a crossed drive the pulleys turn in opposite directions and both have the same, larger angle of contact.

Tension ratio. As the belt wraps the pulley, friction lets the tension rise from T₂ (slack side) to T₁ (tight side). Integrating over the arc gives the capstan equation T₁/T₂ = e^(μθ), with θ in radians on the pulley where slipping would start. For a V-belt or rope, the wedge raises the normal reaction by 1/sin α (α = half the groove angle), so μ is replaced by μ/sin α.

Centrifugal tension. At speed, each element of belt needs a centripetal force; this appears as an extra tension Tc = m·v² in both sides. It does not change the power (it cancels in T₁ − T₂) but uses up part of the allowable tension. With T_max fixed, T₁ = T_max − Tc, and the power is maximum when Tc = T_max/3, at v = √(T_max/(3m)).

Initial tension. A belt is fitted with an initial tension T₀; in use the tight side gains and the slack side loses roughly equally, so T₀ = (T₁ + T₂)/2, or (T₁ + T₂ + 2Tc)/2 when centrifugal tension is counted in T₁ and T₂.

Chains. A sprocket of z teeth and chain pitch p has pitch diameter d = p/sin(180°/z). The chain wraps as a polygon, so the chain speed fluctuates between v_max and v_max·cos(180°/z) every tooth (chordal or polygon action). More teeth on the small sprocket (17 or more is common) reduce this fluctuation, noise and wear.

Formulas

Velocity ratio: N₂/N₁ = (d₁ + t)/(d₂ + t) × (1 − s/100) — N in rev/min, d and t in m, s total slip in %.

Belt speed: v = π·d·N/60 (m/s, d in m, N in rev/min).

Open belt: sin α = (r₂ − r₁)/C; angle of contact on the small pulley θ = π − 2α; length L = π(r₁ + r₂) + 2C + (r₂ − r₁)²/C.

Crossed belt: sin α = (r₁ + r₂)/C; θ = π + 2α; length L = π(r₁ + r₂) + 2C + (r₁ + r₂)²/C.

Flat belt tension ratio: T₁/T₂ = e^(μ·θ) (θ in rad). V-belt or rope: T₁/T₂ = e^(μ·θ/sin α), α half groove angle.

Power: P = (T₁ − T₂)·v (W, T in N, v in m/s).

Centrifugal tension: Tc = m·v² (N, m mass per metre of belt in kg/m). T₁ = T_max − Tc; T_max = σ_allow·b·t for a flat belt.

Maximum power: Tc = T_max/3, v_opt = √(T_max/(3m)).

Initial tension: T₀ = (T₁ + T₂ + 2Tc)/2 (with Tc excluded from T₁ and T₂).

Chain: d = p/sin(180°/z); average speed v = z·p·N/60 (m/s, p in m); speed variation v_min/v_max = cos(180°/z).

Worked examples

Example 1 (standard). An open flat belt connects a 400 mm driving pulley at 720 rev/min to a 1000 mm driven pulley; centre distance 2 m. μ = 0.3, belt mass 0.8 kg/m, maximum permissible tension 2000 N. Neglecting slip and thickness, find the power transmitted.

1. v = π × 0.4 × 720/60 = 15.08 m/s; driven speed N₂ = 720 × 0.4/1.0 = 288 rev/min.
2. Tc = m·v² = 0.8 × 15.08² = 181.9 N; T₁ = 2000 − 181.9 = 1818.1 N.
3. sin α = (0.5 − 0.2)/2 = 0.15, α = 8.63°; θ = 180 − 2 × 8.63 = 162.75° = 2.840 rad (small pulley).
4. T₁/T₂ = e^(0.3 × 2.840) = 2.345; T₂ = 1818.1/2.345 = 775.4 N.
5. P = (T₁ − T₂)·v = (1818.1 − 775.4) × 15.08 = 15 720 W.
6. P ≈ 15.7 kW. (Belt length L = π × 0.7 + 4 + 0.3²/2 = 6.244 m.)

Example 2 (GATE level). A V-belt has a groove angle of 40°, μ = 0.25, angle of contact 165° on the smaller pulley, mass 0.35 kg/m and maximum allowable tension 900 N. Find the belt speed for maximum power and that power.

1. v_opt = √(T_max/(3m)) = √(900/(3 × 0.35)) = 29.28 m/s.
2. Tc = T_max/3 = 300 N; T₁ = 900 − 300 = 600 N.
3. θ = 165° = 2.880 rad; μθ/sin α = 0.25 × 2.880/sin 20° = 2.105; T₁/T₂ = e^2.105 = 8.207.
4. T₂ = 600/8.207 = 73.1 N.
5. P = (600 − 73.1) × 29.28 = 15 430 W.
6. v ≈ 29.3 m/s, P_max ≈ 15.4 kW. A flat belt with the same μ and θ would have T₁/T₂ = e^(0.72) = 2.05 only, which shows the wedge advantage.

Common mistakes

• Using θ in degrees in e^(μθ).
• Taking the angle of contact of the larger pulley in an open drive; slipping starts on the smaller pulley.
• Using the full groove angle instead of half of it in the V-belt formula.
• Forgetting that T_max includes centrifugal tension, so T₁ = T_max − Tc.
• Including Tc in the effective pull: Tc cancels in T₁ − T₂, but it lowers the T₁ available from a fixed T_max, so it limits power at high speed.
• Mixing pulley diameter and radius in the length and angle formulas.
• Treating chain drives as slip-free and therefore constant-velocity; the average ratio is exact, but the instantaneous speed fluctuates.

For GATE PI

Expect the tension ratio for flat and V-belts, power from T₁ and T₂, centrifugal tension and the speed for maximum power, open and crossed belt angle of contact and length, speed ratio with slip, and sprocket pitch diameter or chain speed. Practise the e^(μθ) calculation with θ in radians and checking which pulley governs.

Quick check

1. A flat belt has μ = 0.3 and θ = π rad. If T₂ = 100 N, what is T₁?
2. At what fraction of the maximum tension is centrifugal tension when power is maximum?
3. Why does a V-belt transmit more power than a flat belt for the same tensions?
4. In which drive do both pulleys have the same angle of contact?
5. What is the pitch diameter of a 20-tooth sprocket for a 12.7 mm pitch chain?

Answers: 1. about 256.6 N; 2. one third; 3. the wedge raises the normal force, so the effective friction coefficient becomes μ/sin α; 4. the crossed belt drive; 5. about 81.2 mm.