Three ways to carry rotation from one shaft to another

A drive transfers motion and turning effort from a driver to a driven shaft. Belts and chains bridge a gap. Gears engage directly and fit into a compact housing. Start with the job the drive must do, then choose the parts.

Gears, chains and timing belts are positive drives: engagement maintains a tooth-count ratio without ordinary frictional slip. Flat and V-belts are friction drives: their grip depends on friction and tension, so slip can change the speed ratio.

Positive engagement does not mean perfect positioning. Tooth clearance, elastic deflection and chain speed ripple still matter. For ratios, compound trains and losses, use Gear Ratios. For turning effort and power, use Torque and Power and Efficiency.

Trace the whole torque path, including where each pulley, sprocket or gear meets its shaft. Keys and keyways are one way to connect a hub to a shaft; they are not a feature of every drive. Bearings have a different job: supporting and locating the shaft relative to its housing.

Compare timing, space and upkeep side by side

Belt contact running dry does not mean the shaft bearings need no lubrication. A sealed gearbox can keep lubricant away from a product; an exposed chain generally cannot. Special materials and lubrication-free products need their own ratings.

Read the parts in detail: Belt Drives, Chain Drives and Gear Drives and Tooth Geometry.

A key joins the shaft slot to the hub slot to transmit torque

A key is a separate piece fitted between a shaft and the bore of a hub, the part of a pulley, sprocket or gear that fits around the shaft. Contact between the key and slot faces transfers turning force. Here, keyseat means the recess in the shaft and keyway means the matching slot in the hub. Other references sometimes use keyway for both; always identify which part is being cut.

Trace the turning load through the connection: shaft slot face → key → hub slot face, or the reverse when the hub drives the shaft. At the rim, the pulley, sprocket or gear transfers the load through its belt, chain or mating gear teeth. Not every hub is keyed; check how the actual assembly connects the hub to the shaft.

Identify the key by its cross-section and seating shape

Supporting detail: Addendum 2 — Public reference for key forms.

Practice identifying keys

Check key fit and axial retention as separate requirements

For a parallel key, the side fit controls how the turning load is carried. Excessive side clearance permits movement; an overly tight fit can prevent assembly. Verify the specified top clearance too. A tapered key has a different seating action, so do not apply a parallel-key fit rule to it.

Use the drawing's key designation, dimensions and tolerances. Do not choose a key size from shaft diameter alone or assume that half the key height belongs in each part.

The key does not by itself define how the hub is held against movement along the shaft. Check the assembly drawing for axial retention as well as torque transmission. A hub that turns with the shaft is not necessarily secured against sliding along it.

Bearings support the shaft relative to the housing; the keyed hub transfers torque between the shaft and the attached rotating part. For the bearing arrangement and the shaft's axial location, see Shaft Support and Axial Adjustment.

To make and inspect the recesses, continue with Machining Keyseats and Keyways. That lesson connects shaft milling to broaching a through hub keyway.

Flat and V-belts grip; timing belts engage teeth

Flat belt

A broad, flat band transmits force through friction against a pulley. Useful for smooth, quiet motion and long spans when the drive is designed for them. Tracking and tension matter.

V-belt

A trapezoidal section runs in a matching V-groove. Contact at the sides gives a wedging action that increases grip. The belt should not rely on rubbing the bottom of the groove.

Timing / synchronous belt

Teeth on the belt engage matching grooves in the pulleys. Load is carried by this engagement, so the drive keeps its tooth-count ratio without frictional slip in normal operation.

Timing does not come from extra tension. It comes from matching teeth. An undertensioned or overloaded timing belt can jump teeth or strip them; that is a fault, not its normal way of transmitting power. A notched V-belt remains a friction belt: relief notches for bending do not make it a timing belt.

Belt contact transfers load at the pulley rim; the pulley still needs a connection to its shaft. For a keyed pulley, follow the torque path through the key and hub. Other pulley mountings use different connections.

Read length, tension and wrap before ordering

Pulley / sheave

The wheel that carries a belt. “Sheave” is commonly used for a grooved belt pulley. A timing pulley has matching tooth grooves.

Pitch and pitch length

Timing-belt pitch is the spacing between adjacent tooth centers along the pitch line. Pitch length is the loop length along that reference line, near the tensile cords. It is not the outside circumference. Match pitch and tooth profile to the pulley.

Tension

The pulling force along the belt. The difference between the loaded and return spans transmits torque. Installation tension keeps the belt seated; too little encourages slip or tooth jumping, and too much loads belts and bearings.

Wrap angle

The angle through which the belt contacts a pulley. Less wrap gives a friction belt less contact and a timing belt fewer teeth in engagement. Check the smaller pulley in particular.

Catalogs may specify inside, outside, datum, effective or pitch lengths. Use the length convention and measuring method for that belt family; do not substitute a tape measurement of the outside for the catalog value.

One-stage reminder: n_out = n_in × D_driver / D_driven using matching effective/pitch diameters and assuming no slip. For timing belts, use tooth counts N in place of diameters. Write the reduction as i:1, where i = input speed / output speed. Follow Gear Ratios for the ratio method and Rotation and Linear Speed for belt linear speed.

Here N means teeth and n means rotational speed. Some class slides use n for teeth and ω for angular speed; check the quantity, not just the letter.

Choose a belt for the load and environment

Belts can span separated shafts, soften vibration and avoid oil at the transmission contact. Flat and V-belts suit jobs where a small speed variation is acceptable. Choose a timing belt when the shafts must stay synchronized. Compare the alternatives in Belt, Chain and Gear Drives.

Check belt section or tooth profile, width, minimum pulley size, available length, wrap, power rating, temperature and chemical compatibility. There is no single maximum speed or temperature range for every belt. Reinforcement cords limit stretch, but the belt still needs correct alignment and installation tension.

Some specially designed flat-belt layouts connect nonparallel shafts. That does not make misalignment acceptable in an ordinary V-belt or timing-belt installation. Frictional slip may soften an overload, but it can also generate heat and destroy the belt; it is not a guaranteed overload-protection device.

Use the wear pattern to find the cause

Inspect the pulleys as well as the belt. Replacing the worn part without fixing its cause often repeats the failure. Follow the product's tension and inspection instructions; use Equipment Operating Procedure before touching the drive.

Practice a timing-pulley speed

Supporting detail: Addendum 3 — Manufacturer references for belt geometry and faults.

Sprocket teeth engage a chain of moving joints

A chain drive is a positive drive: sprocket teeth engage the chain instead of relying on frictional grip. It can bridge separated shafts and carry substantial load, but its joints and sprockets need attention.

Pitch, p

The center-to-center distance between adjacent chain pins. A chain's pitch must match its sprocket.

Roller

A sleeve that turns around a bush as it contacts a sprocket tooth. Some small chains are bush chains and have no separate roller.

Link

A connecting unit of the chain. Ordinary roller chain alternates inner links, carrying bushes and rollers, with outer links, carrying pins and side plates.

Sprocket

The toothed wheel that engages the chain. Tooth count sets the speed relationship; tooth shape and width must suit the chain series.

Master / connecting link

The link used to join the ends into a loop. Use the specified type and installation method; retention and strength depend on the product.

The chain engages the sprocket teeth, but torque must also pass between each driven or driving sprocket and its shaft. A keyed sprocket uses a key between the shaft and hub; tooth engagement alone does not explain that connection.

Read #25, #35 and #40 as pitch codes

For the common ANSI short-pitch chain numbers here, remove the final digit of the base number and divide the remaining number by 8 to get pitch in inches. Thus #40 uses 4/8 in, not 40/8 in. #100 uses 10/8 in.

A suffix such as -1 or -2 counts parallel strands: simplex or duplex. #40-2 still has a 0.500 in pitch. Heavy-series suffixes and manufacturer prefixes carry other information; do not apply the pitch shortcut to every chain product code.

Supporting detail: Addendum 5 — Checked public wording behind the ANSI numbers.

“Metric chain” still needs a series and a pitch

ISO chain data are commonly tabulated in millimeters. That does not make every pitch a whole number of millimeters. ISO 606 covers both A-series chains derived from American practice and B-series chains derived from European practice.

The common 06B, 08B and 10B sizes correspond to 6/16, 8/16 and 10/16 in pitch. Use the table for the exact designation: small sizes such as 05B do not follow that shortcut. B identifies the series; -1 identifies one strand.

Same pitch does not mean interchangeable. #40 and 08B both have 12.7 mm pitch, but roller diameter and inner width differ. Check series, pitch, width, roller diameter and strand count when matching a sprocket or replacement link.

Supporting detail: Addendum 6 — ISO scope and the source of the millimeter dimensions.

Positive engagement still permits wear and speed ripple

Chain is worth considering when separated shafts need a fixed average ratio and lubricant and noise are acceptable. Select it for the actual power, speed, starts, shock loads and environment. Tensile breaking strength is not an allowable continuous working load.

One-stage reminder: n_out = n_in × N_driver / N_driven; reduction i = N_driven / N_driver, written i:1. N counts teeth and n is rotational speed. For the reasoning and further stages, use Gear Ratios.

The chain wraps a sprocket as a series of straight links, forming a polygon rather than a perfectly smooth circle. This chordal action produces speed ripple, especially with small sprockets. Backlash and elastic stretch also limit precise positioning, even without ordinary slip.

• Wear elongation: pin/bush wear increases joint clearance and measured chain length. The worn chain rides poorly on sprocket teeth and can jump. Take-up removes slack; it does not restore worn joints.
• Stiff or seized joints: check lubrication reaching the joints, contamination and corrosion.
• Fractured plates or pins: investigate fatigue, overload, misalignment and connecting-link installation.
• Hooked or uneven sprocket teeth: inspect the chain and sprockets together and follow replacement limits for that drive.

Use the specified lubricant, alignment and slack. A chain is not installed taut like a belt. Set inspection intervals and wear limits from the maker's guide; isolate the drive before measuring or servicing it. See Equipment Operating Procedure.

Practice reading chain pitch

Supporting detail: Addendum 7 — Chain selection and maintenance reference.

Spur teeth engage straight across; helical teeth engage progressively

Gears transmit rotation by direct tooth contact. They make a compact positive drive, but accurate mounting, tooth compatibility and lubrication matter. This page compares external spur and helical gears on parallel shafts.

Spur gear

Teeth run parallel to the shaft. It is comparatively simple to make and produces no axial thrust from the ideal tooth mesh. Engagement can be noisy at higher speed.

Helical gear

Teeth run at a helix angle around the gear. Contact enters progressively across the face and overlapping teeth share load. This generally gives smoother, quieter operation, with added manufacturing cost.

The inclined helical tooth force also pushes along the shaft. That axial thrust needs a suitable bearing arrangement. A spur gear still loads its bearings radially; “no axial thrust” does not mean “no bearing force.”

The driver supplies power. The pinion is the smaller gear in a pair; it can be the driver or the driven member. An idler passes motion between gears and still needs a shaft or other bearing support. Follow Gear Ratios for direction, tooth-count ratios and compound trains.

Tooth contact and shaft attachment are separate connections. Where a gear hub is keyed to its shaft, the key transfers torque at the hub. Check axial retention separately, especially when the tooth mesh produces thrust.

Measure teeth from the pitch circle

Pitch diameter, d

The diameter of the reference pitch circle. In a standard unshifted pair at its standard center distance, the pitch circles roll together without slipping.

Module, m

Metric tooth size: m = d/N, with d in millimeters and N teeth. Larger module means larger teeth.

Diametral pitch, P_d

Inch-based tooth size: P_d = N/d, with d in inches. More teeth per inch of pitch diameter means smaller teeth. Do not confuse it with circular pitch, the distance along the pitch circle between successive teeth.

Pressure angle, α

The angle between the line of action (the contact-force direction, neglecting friction) and the common tangent to the pitch circles at the pitch point. It describes tooth geometry, not the helix angle.

Addendum / dedendum

Radial tooth height from pitch circle to tip / radial depth from pitch circle to root. Neither is a diameter difference by itself.

Backlash

Clearance between mating tooth flanks. It allows clearance in the mesh but creates lost motion when the load reverses.

Supporting detail: Addendum 9 — Check the size definitions with one gear.

Matching pitch is necessary, but not sufficient

Ordinary spur gears intended to mesh must share module or diametral pitch and pressure angle. Matching outside diameter or tooth count does not establish compatibility. Check the tooth system, center distance, clearance and manufacturer's mating information too.

For external helical gears on parallel shafts, match tooth size, pressure angle and helix angle. Use one right-hand and one left-hand gear (teeth slanting in opposite directions), and confirm the manufacturer's mating specification. Equal tooth counts are not required.

Supporting detail: Addendum 10 — Normal and transverse measurements use different planes.

A larger tooth size can support more load when other conditions are comparable, but pitch alone cannot determine capacity, backlash or achievable reduction. Material, face width, heat treatment, alignment and speed remain part of the design.

Supporting detail: Addendum 11 — Primary references for the geometry and helical mesh.

Gear damage tells you where the load went

Choose gears for compact packaging, a positive ratio and a supported shaft arrangement. Choose helical teeth when smoothness and noise justify their cost and thrust bearings. Check tooth bending strength, surface durability, speed, lubricant, seals and bearing loads before approving a pair.

A symptom is a clue, not a unique diagnosis. Correct the cause before replacing parts. Most loaded metal gear sets need a controlled lubricant supply; some specially selected materials can run dry. Follow the product data and Equipment Operating Procedure for access.

Compare family trade-offs in Belt, Chain and Gear Drives, and follow Power and Efficiency for power losses. A positive drive is not automatically lossless.

Count the teeth to find the speed

The driver supplies motion; the driven gear receives it. For one external gear mesh, each tooth passing on the driver advances the driven gear by one tooth space. A gear with twice as many teeth needs twice as many passing teeth to complete a turn.

This lesson uses reduction ratio i = input speed ÷ output speed, with speed magnitudes. A 2:1 reduction means two input turns for one output turn. Ratios above 1 reduce speed; ratios below 1 increase it. At zero speed, use tooth counts to find the ratio; do not divide 0 by 0.

Notation map: this page's original equations use N for teeth and n for speed. The course uses n for teeth, ω for angular velocity and τ for torque. Thus page N = course n; page speed n is written ω in the additions below. GR and i name the same reduction ratio.

In course notation, using magnitudes: GR = nout/nin = dout/din = ωin/ωout = τout/τin for an ideal single pair. Here d is pitch diameter; the torque equality assumes no losses. For a compound train, multiply the stage ratios; the first and last tooth counts alone do not give the overall ratio.

Supporting detail: Addendum 12 — Why tooth counts work, and what the dashed circles mean.

An idler and a compound shaft do different jobs

Same shaft means the same angular speed. A large B and a small C fixed to one shaft make the same number of turns in the same direction. Their rim speeds differ because their radii differ. They do not mesh with each other.

In the compound example, 20 → 40 teeth halves the speed. A second 20 → 40 stage halves it again: 120 → 60 → 30 rpm. The two external meshes make the final output turn in the same direction as the input.

An idler is not another reduction stage. Drop its tooth count when finding the end-to-end ratio, but count every external mesh for direction: odd means opposite; even means the same, viewed from the same end of the parallel shafts. Do not drop a gear that shares a shaft with a different gear in a compound train.

More torque costs speed; it does not create power

Torque is turning effort. In this ideal, lossless train at steady load, output torque magnitude = input torque magnitude × reduction ratio. A 2:1 reduction changes 120 rpm and 2 N·m into 60 rpm and 4 N·m. The 4:1 compound example gives 30 rpm and 8 N·m.

Rotational power is torque × angular speed: P = T × 2πn/60 when T is in N·m and n is in rpm. Both sides of either example carry 25.13 W ideally. Real friction and other losses leave less output power. Tooth strength, bearings, shafts, lubrication and the motor all limit usable torque.

At zero speed, mechanical power is zero. A loaded, stationary gear train can still carry torque. The lab shows ideal static torque balance in that case; it does not model motor stall behavior or promise a safe load.

Supporting detail: Addendum 13 — Primary references and model boundaries.

Connect the gears to Angular Motion, Rotation and Linear Speed and Torque.

Stage efficiencies multiply; losses accumulate

Each stage receives only the power that survived the stage before it. Use decimal efficiencies and multiply: η_total = η₁ × η₂ × η₃. Never add or average them. See Power and Efficiency for the general meaning of efficiency.

At 98% per stage, three stages give 0.98³ = 0.941192: about 5.88% total loss, not 6%. Fewer, higher-quality stages often beat many cheap stages for efficiency, provided they can meet the ratio and load requirements.

Supporting detail: Addendum 14 — Why adding losses is only an approximation.

Positive drives keep the speed ratio; losses reduce torque

A positive drive keeps input and output linked in a fixed ratio without slipping. Gears, chains and timing belts are positive drives. A slipping flat belt or V-belt is not. Follow Rotation and Linear Speed for the connection from rotation to belt travel.

The teeth still set the speed exactly in this model. Efficiency reduces output torque from its ideal value. It does not reduce output rpm as well. Use matching torque units on both sides: lbf·ft or N·m. The course conversion is 1 rpm ≈ 0.1047 rad/s when angular speed in rad/s is needed.

Supporting detail: Addendum 15 — Follow both stages of one loaded train.

Count stages for ratio, meshes for direction and stated losses

For a simple train A → I → B, the idler cancels from (NI/NA) × (NB/NI) = NB/NA. The two external meshes still reverse direction twice.

Follow the loss assumption in the problem. Real meshes lose some power, but a problem may explicitly ignore it. If every actual mesh is 96% efficient, A → I → B has ηtotal = 0.96². If a whole stage including its idler is assigned 96%, use that stage value once. If η = 1.00 is stated, ignore losses. Do not infer the loss count from the reduction-stage count when an idler is present.