Belt, rope and chain drives

In short

Belt, rope and chain drives transmit power between shafts that are too far apart for gears, using a flexible member wrapped over pulleys, sheaves or sprockets. Power is carried by the difference in tension between the tight and slack sides; friction (belts, ropes) or positive engagement (chains, timing belts) creates that difference. The topic is heavily examined in Theory of Machines and in GATE/ESE numericals on ratio of tensions, belt length and condition for maximum power, and in design vivas on belt selection. In Indian industry it is everywhere — Fenner India and Pix Transmissions V-belts, Rolon (LG Balakrishnan) and TI Diamond roller chains, two-wheeler final-drive chains, agricultural threshers, lifts and mine haulage ropes.

Explained

The family of flexible drives

• Flat belt: rectangular section on plain (crowned) pulleys; good for long centre distances (up to about 8 m), tolerant of misalignment, slips under overload.
• V-belt: trapezoidal section running in a grooved pulley; the wedge action multiplies friction, so it carries far more power in a compact space (centre distance roughly up to 4 m). Classified A, B, C, D, E by section (IS 2494); also wedge (SPZ/SPA/SPB) and ribbed/poly-V belts.
• Timing (toothed) belt: teeth mesh with a toothed pulley — positive drive, no slip, used for camshaft drives and printers.
• Round belt: circular section in a V-groove, light loads.
• Rope drive: fibre ropes (cotton, manila) in multiple V-grooves for long distances; wire ropes (IS 2266, typical 6×19 construction) for lifts, cranes, mine haulage and winders.
• Chain drive: steel links on toothed sprockets — positive, compact, high efficiency (96-98%), for short centre distances (under about 4 m).
• Types of chain: *roller chain* (bush-roller, the common power chain; simplex/duplex/triplex), *silent or inverted-tooth chain* (smoother, quieter), *leaf chain* (lifting), *hoisting/conveyor chains*.

Arrangements and velocity ratio

• Open belt: shafts parallel, pulleys rotate in the same sense. Crossed belt: opposite sense, but the belt rubs where it crosses, so it wears fast and runs slower.
• Quarter-turn (right-angle) drive, stepped/cone pulley (speed change in steps, as on lathes and drilling machines), compound drive (several pulleys in series), idler or jockey pulley (takes up slack, increases arc of contact, allows short centres).
• Velocity ratio from equal surface speeds: π d₁N₁ = π d₂N₂ → N₂/N₁ = d₁/d₂. Including belt thickness t, use (d₁+t)/(d₂+t).
• Slip is relative motion between belt and pulley surface due to insufficient friction; it reduces output speed and wastes power. Total slip s = s₁ + s₂.
• Creep is the alternate stretching and relaxing of the belt as it passes from the tight side (higher stress) to the slack side; it also lowers the velocity ratio slightly and is a function of belt material modulus E.
• Crowning the pulley face (slight convexity) keeps a flat belt tracking centrally; belts climb to the highest point.

Geometry: belt length and angle of contact

• Open belt, pulley diameters d₁ (small) and d₂, centre distance x: sin α = (d₂ − d₁)/2x; arc of contact on the smaller pulley θ = π − 2α radians — always use the smaller pulley, since slip starts there.
• Crossed belt: sin α = (d₁ + d₂)/2x and θ = π + 2α, the same for both pulleys — one reason a crossed belt can transmit more power.
• Length formulas follow from adding the two arcs and the two straight tangents (see formula boxes).
• Standard practice: keep x ≥ about 1.5–2 times (d₁+d₂)/2 for V-belts, and the arc of contact above ~120° to avoid heavy slip.

Tensions, centrifugal tension and power

• Ratio of tensions (derivation outline): take a short element of belt subtending dθ at the centre, about to slip. Resolve forces normal to the element → R_N = T dθ; along the element → dT = μR_N. Hence dT/T = μ dθ; integrating from the slack to the tight end over the arc θ gives ln(T₁/T₂) = μθ, i.e. T₁/T₂ = e^μθ.
• V-belt and rope: the belt touches two inclined faces, so each normal reaction R_N gives a radial component R_N sin β (β = half the groove angle). The same analysis gives T₁/T₂ = e^{μθ cosec β} — a 2β = 38° groove effectively triples μ.
• Centrifugal tension T_c = m v² (m = mass per metre, v = belt speed) acts all round the belt, adds equally to both sides, and does not contribute to power — but it does consume part of the allowable tension.
• Maximum tension is set by the belt material: T_max = σ·b·t (σ = permissible tensile stress). So T₁ = T_max − T_c.
• Power P = (T₁ − T₂)·v. Substituting T₁ = T_max − mv² and maximising with respect to v gives T_c = T_max/3, i.e. v = √(T_max/3m) — the speed for maximum power. At that condition T₁ = (2/3)T_max.
• Initial tension T₀ = (T₁ + T₂ + 2T_c)/2 (for a belt obeying a linear stress-strain law); too little gives slip, too much gives bearing load and short belt life.
• Practical belt speeds: flat belt 10–22 m/s, V-belt up to about 25–30 m/s; efficiency ~94-97% (V-belt), lower with slip.

Selection practice and Indian standards

• Design power = rated motor power × service factor (1.0-1.8 for shock, hours of running, idler use). Number of V-belts = design power ÷ (power per belt × arc-of-contact factor × belt-length factor), from the manufacturer's/BIS tables.
• Relevant Indian standards (confirm the current revision/part before design): IS 2494 endless V-belts for industrial drives, IS 3142 grooved (V) pulleys, IS 1891 textile conveyor/elevator belting, IS 2403 short-pitch precision transmission roller chains and chain wheels (aligned with ISO 606), IS 2266 steel wire ropes for general engineering. Indian catalogues also quote ISO/BS designations such as 06B, 08B, 12B, 16B chain.
• Safety and law: the guarding duties below were Sections 21, 22 and 24 of the Factories Act, 1948, repealed from 21 November 2025 by the OSH Code, 2020; they continue through the Code's rules and the State Factories Rules saved under it (s.143(3)). Section 21 required transmission machinery — belts, pulleys, chains and their nip points — to be securely fenced. Section 24 required suitable striking gear (belt shifters) to move belts to and from loose and fast pulleys and forbids idle belts riding on shafting in motion; Section 22 barred cleaning, lubricating or adjusting transmission machinery in motion except by a trained adult worker under the prescribed conditions.

Chain drives in detail

• Pitch p = centre distance between adjacent roller centres. Pitch-circle diameter D = p / sin(180°/T), T = number of teeth. Chain speed v = T·N·p/60 (p in m).
• Velocity ratio N₂/N₁ = T₁/T₂; keep it below about 6:1 (7:1 maximum) in one stage, pinion teeth ≥ 17 (use 21–25 for smooth, high-speed drives), and preferably an odd number of sprocket teeth with an even number of chain links for even wear and no offset link.
• Chordal action: the chain forms a polygon on the sprocket, so the linear speed fluctuates between π D N/60 (at pitch radius) and that value × cos(180°/T). Fewer teeth → larger fluctuation, more noise and vibration.
• Wear elongation ("chain stretch") is pin/bush wear, not metal stretching; it raises the chain onto a larger effective pitch circle and causes jumping. Replacement at about 1.5–2% elongation (2-3% for low-speed drives).
• Slack and lubrication: sag on the slack side about 2% of centre distance; drip, bath or forced-feed lubrication by speed. The driven (slack) strand should be at the bottom; use a tensioner for long spans.
• Failure modes: fatigue of link plates, pin/bush wear, roller galling, and sudden shock overload.

Comparison, advantages and limitations

• Belt: cheap, quiet, absorbs shock, acts as an overload fuse (slips), tolerates misalignment; but slip and creep mean no exact ratio, needs tension and more space, and generates bearing load.
• Rope: best for very long distances and heavy loads; wire rope is the only practical member for lifts and winders, but needs sheaves of large diameter.
• Chain: exact ratio, compact, high efficiency, works in oil/heat and at short centres; but noisy, needs lubrication and alignment, and chordal action limits very high speeds.
• Gear: exact ratio and largest torque in the shortest span, but only at short, fixed centre distances and higher cost.

Key points

• Power transmitted - depends on the tension *difference* (T₁ − T₂) and belt speed, not on absolute tension.
• T₁/T₂ = e^μθ - limiting friction condition; θ in radians, taken on the smaller pulley.
• V-belt ratio - T₁/T₂ = e^{μθ cosec β}; the wedge effect is why a V-belt beats a flat belt of the same size.
• Centrifugal tension T_c = mv² - adds to both sides, transmits no power, but eats into T_max.
• Condition for maximum power - T_c = T_max/3 and v = √(T_max/3m); T₁ = 2T_max/3.
• Initial tension - T₀ = (T₁ + T₂ + 2T_c)/2.
• Slip - belt surface sliding on the pulley; creep - belt stretching and relaxing between the two sides. Both reduce output speed.
• Crossed belt - larger angle of contact (π + 2α) and reversed rotation, but rapid wear at the crossing.
• Crowning - a convex pulley face that keeps a flat belt centred.
• Idler/jockey pulley - maintains tension and increases the arc of contact.
• Timing belt and chain - positive drives with no slip; flat/V-belts are friction drives.
• Chordal action - speed fluctuation in a chain drive, equal to 1 − cos(180°/T); reduced by more sprocket teeth.
• Sprocket PCD - D = p / sin(180°/T).
• Design power - rated power × service factor; number of belts follows from corrected power per belt.
• Chain stretch - wear at pins and bushes; replace at about 1.5–2% elongation.
• Factories Act 1948 (repealed by the OSH Code, 2020 from 21 November 2025) - former Sec. 21 transmission machinery securely fenced; Sec. 24 striking gear/belt shifters; the duties continue under the Code's rules.
• Efficiency - chain 96-98%, V-belt 94-97%, flat belt 90-95% (lower with slip).

Formulas and examples

Example 1 - flat belt drive (complete numerical). Driver d₁ = 450 mm at 200 rpm, driven d₂ = 900 mm, x = 2 m, belt 100 mm × 8 mm, density 1000 kg/m³, μ = 0.3, σ = 2 MPa, slip 2%.

• N₂ = 200 × (450/900) × 0.98 = 98 rpm.
• L = 1.5708(1.35) + 4 + (0.45)²/8 = 2.121 + 4 + 0.0253 = 6.146 m.
• sin α = 0.45/4 = 0.1125 → α = 6.46° = 0.1127 rad; θ = π − 0.2254 = 2.916 rad.
• T₁/T₂ = e^0.3×2.916 = e^0.875 = 2.40.
• v = π × 0.45 × 200/60 = 4.71 m/s; m = 0.1 × 0.008 × 1000 = 0.8 kg/m; T_c = 0.8 × 4.71² = 17.8 N.
• T_max = 2×10⁶ × 0.1 × 0.008 = 1600 N → T₁ = 1582 N, T₂ = 1582/2.40 = 660 N.
• P = (1582 − 660) × 4.71 = 4.35 kW. Initial tension T₀ = (1582 + 660 + 35.6)/2 = 1139 N.

Example 2 - maximum power for the same belt. v = √(1600/(3×0.8)) = √666.7 = 25.8 m/s; T_c = 533 N, T₁ = 1067 N, T₂ = 1067/2.40 = 445 N → P = (1067 − 445) × 25.8 = 16.1 kW. Almost four times the Example 1 output, from the same belt run at its optimum speed.

Example 3 - V-belt selection. Groove angle 2β = 38° (β = 19°), μ = 0.25, θ = 165° = 2.880 rad, belt area 230 mm², density 1000 kg/m³, allowable tension 850 N, v = 15 m/s. Motor 20 kW, service factor 1.3.

• μθ cosec β = 0.25 × 2.880 / 0.3256 = 2.211 → T₁/T₂ = e^2.211 = 9.13 (a flat belt would give only e^0.72 = 2.05).
• m = 0.23 kg/m; T_c = 0.23 × 225 = 51.8 N → T₁ = 798 N, T₂ = 798/9.13 = 87 N.
• Power per belt = (798 − 87) × 15 = 10.7 kW. Design power = 20 × 1.3 = 26 kW → belts = 26/10.7 = 2.4 → use 3 belts (catalogue arc and length correction factors would be applied in real design).

Example 4 - roller chain drive. 12B chain, p = 19.05 mm; pinion T₁ = 21 teeth at 1000 rpm, wheel T₂ = 42 teeth, x ≈ 600 mm.

• N₂ = 1000 × 21/42 = 500 rpm; v = 21 × 1000 × 0.01905/60 = 6.67 m/s.
• D₁ = 19.05/sin 8.571° = 127.8 mm; D₂ = 19.05/sin 4.286° = 254.9 mm.
• Chordal fluctuation: pinion 1 − cos 8.571° = 1.1%; wheel 1 − cos 4.286° = 0.28%.
• Links K = 1200/19.05 + 31.5 + (3.342)² × (19.05/600) = 63.0 + 31.5 + 0.35 = 94.85 → 96 pitches, L = 96 × 19.05 = 1829 mm.
• If allowable working tension is 2400 N (breaking load ÷ factor of safety), P = 2400 × 6.67 ≈ 16 kW — final rating must come from the maker's power-rating chart, which also covers fatigue and lubrication.

Where it is used

• Two-wheelers and tractors: roller-chain final drives (Rolon, TI Diamond); belt CVTs in scooters; fan and alternator poly-V belts.
• Machine tools and workshops: cone/stepped pulleys on lathes and pillar drills, V-belt drives on compressors, pumps and lathe headstocks.
• Agriculture and food processing: flat and V-belt drives on threshers, chaff cutters, flour mills, sugar-mill and oil-mill drives.
• Material handling and construction: conveyor belting (IS 1891), bucket elevators, chain drives in crushers and batching plants.
• Lifts, cranes and mines: steel wire ropes on sheaves and drums (IS 2266) for elevators, EOT cranes and mine winders.
• Job roles: design engineer (drive selection, service factors), maintenance and reliability engineer (belt tensioning, chain wear surveys, alignment), plant safety officer (guarding under the OSH Code, 2020, formerly the Factories Act).

Interview questions

• Q: Why does the power depend on the difference of tensions and not on the tensions themselves? A: The driving force on the pulley rim is the net tangential force T₁ − T₂; equal tension on both sides produces no torque, only bearing load.
• Q: Why is the angle of contact taken on the smaller pulley? A: It has the smaller arc of contact, so slip begins there and it governs the limiting ratio of tensions.
• Q: What is the difference between slip and creep? A: Slip is sliding between belt and pulley surface due to insufficient friction; creep is the alternate elastic stretching and contraction of the belt as it moves from the tight side to the slack side.
• Q: Why does a V-belt transmit more power than a flat belt of the same width? A: The belt wedges into the groove, so the normal reactions on the two flanks are much larger than the radial pull, effectively multiplying friction by cosec β.
• Q: Does centrifugal tension transmit power? A: No — it adds equally to both sides, so it cancels in (T₁ − T₂), but it uses up part of the allowable maximum tension and so reduces transmissible power.
• Q: At what speed is maximum power transmitted? A: When centrifugal tension equals one-third of the maximum permissible tension, i.e. v = √(T_max/3m).
• Q: What is chordal action? A: The chain wraps the sprocket as a polygon, so the chain's linear speed fluctuates between πDN/60 and that value times cos(180°/T); it is reduced by using more teeth.
• Q: Why are pulleys crowned? A: A belt always runs to the highest point of the face, so a slight convexity keeps a flat belt tracking centrally.
• Q: Why does a chain "stretch" in service? A: Pins and bushes wear, increasing the effective pitch; the links themselves do not stretch. Replace at about 1.5–2% elongation.
• Q: Where would you prefer a belt drive to a chain drive, and why? A: Where shafts are far apart, quiet running matters, lubrication is impractical, or slip is wanted as overload protection — for example a workshop compressor drive.
• Q: Why is an even number of chain links preferred? A: An odd number needs an offset (cranked) link, which is weaker and a common fatigue failure point.
• Q: What legal requirement applies to belt and chain drives in an Indian factory? A: The OSH Code, 2020 (in force 21 November 2025) and the rules under it, which carry forward Section 21 of the repealed Factories Act, 1948 — transmission machinery securely fenced — and Section 24 — striking gear (belt shifters) so belts are not moved by hand or left riding on shafting in motion; and former Section 22's restriction on work on machinery in motion.

Common mistakes

• Taking the angle of contact on the larger pulley, or leaving θ in degrees inside e^μθ instead of radians.
• Using T₁ = T_max in the power equation and forgetting to subtract centrifugal tension.
• Adding T_c to the net driving force, or thinking it increases power.
• Swapping the open-belt and crossed-belt length formulas — (d₂−d₁)² for open, (d₁+d₂)² for crossed.
• Using the full groove angle 2β instead of the half angle β in cosec β for V-belts.
• Mixing units: belt thickness in mm with stress in MPa, or pitch in mm with speed in m/s in chain formulas.
• Treating slip as a design allowance rather than a loss — and quoting a chain drive as slipping, when a chain is a positive drive that jumps or breaks instead.

Related topics

Theory of machines: mechanisms and gears · Gear trains and epicyclic gearing · Flywheels and turning moment diagrams · Design of machine elements (shafts, keys, couplings) · Friction, clutches and brakes · Material handling equipment and wire ropes