Learning Gears: How They Work, Terms and Manufacturing, through to Ordering from a Tool Maker (Basic to Advanced)

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Calculators are on the gear pages: Spur gears / Helical gears / Span measurement / Over-pin, Module table, Machining methods, Gear-cutting tools.

BasicHistory of gears: from wooden teeth to ground teeth

1. 4th century BCThe earliest mention of gears in writing is often said to be Aristotle's “Mechanical Problems”. Who invented them, and when, is not known^[68]
2. c. 2nd–1st century BCThe Antikythera mechanism: a bronze gear device found in a shipwreck. It is thought to have used gears to calculate the positions of the Sun and Moon and the cycles of solar and lunar eclipses (estimated dates differ somewhat between sources)^[69][70]
3. 16th centuryGears were still made by hand, cut one tooth at a time with a cutter shaped like the tooth space. Examples of hand-cranked gear-cutting machines also survive^[71]
4. 1551–1612Mechanical clocks reach Japan. To match Japan's seasonal-hour system, in which the length of an hour changes with the season, Japanese clocks (wadokei) are developed^[78]
5. 1754–1767Euler discusses gear tooth shapes mathematically (said to be the first to present the involute tooth profile)^[72][73]
6. 1796“Karakuri Zui” is published, a book with drawings explaining how to make Japanese clocks and tea-serving karakuri dolls. A later survey of tea-serving dolls reported that the shape of the wooden gear teeth was almost the same as today's involute^[79][80]
7. 1825Airy publishes a general theory of tooth profiles. He shows that with the involute, meshing is maintained even when the center distance changes.^[73][74]
8. 1835In Whitworth's British patent, the mechanism for cutting gears with a hob is depicted^[71][75]
9. 1838, 1841Willis publishes an instrument for drawing tooth profiles and a design method, and recommends the involute tooth profile in his book^[74]
10. 1851Tanaka Hisashige's Myriad Year Clock (Mannen Jimeisho). Designated an Important Cultural Property^[81]
11. 1889Grant in the US patents a hobbing machine for spur gears (the earliest known hobbing machine patent)^[71][75]
12. 1896–1897Fellows in the US puts the gear shaper into practical use. It was the only machine that could cut internal gears at a practical speed^[76][75]
13. 1897Pfauter in Germany patents a hobbing machine that can cut both spur and helical gears^[71][77]
14. 1913Maag is founded in Switzerland. It is known for its gear-cutting machines that cut teeth with a rack-type cutter and its gear grinding machines with dish-shaped wheels^[59][58]
15. 1945Reishauer in Switzerland launches the continuous generating gear grinding machine, which grinds continuously with a threaded grinding wheel^[62]
16. 1990The last Maag gear machine is delivered, and production ends^[58][60]
17. 1998–1999The gear JIS standards are rewritten to align with ISO (accuracy: JIS B 1702-1; standard basic rack: JIS B 1701-1; and so on. “Reference pitch circle diameter” becomes “reference circle diameter”)^[82][83]
18. 2000s onwardFor automotive transmission gears, carburizing and quenching followed by finishing with continuous generating grinding becomes the mainstream^[21][24]

BasicA gear's job is to change the speed, force and direction of rotation

A gear is a part that transmits rotation through meshing teeth. Everything is determined by the ratio of the numbers of teeth of the two gears.

Speed and force are traded offSlowing down increases force, and speeding up reduces force. A gear is not a device that creates energy; it changes how “speed” and “force” are shared out (in practice a little is lost to friction on the tooth surfaces and so on).

BasicWhy is the tooth shape the “involute”?

If the tooth shape is arbitrary, the rotation speeds up and slows down during meshing, causing vibration and noise. To transmit rotation at a constant speed at all times, the tooth shape must meet certain conditions, and the typical curve that meets them is the involute^[10][11].

1. The force direction never changes: the contact point moves only along a single straight line called the “line of action”, and the force pushing on the teeth keeps the direction of that line. The load on the bearings stays steady
2. Tolerant of center distance errors: even if the center distance between two gears shifts slightly because of assembly error or heat, the speed ratio stays equal to the ratio of the numbers of teeth (what changes is the backlash and the operating pressure angle)^[11]
3. Can be made with a tool with straight cutting edges: if you increase the number of teeth of an involute gear without limit, the tooth flank becomes a straight-sided “rack”. That is why a single tool with straight cutting edges (a hob or rack cutter) can accurately make gears with any number of teeth. The biggest reason the involute is used all over the world is that the tools are easy to make and the gears are easy to measure

Cycloid tooth profiles, which allow thicker tooth roots, are also used in clocks and similar devices. But they stop meshing correctly if the center distance shifts, so almost all power-transmitting machines use the involute^[11].

BasicA map of gear types

BasicTerms to learn first

45 gear terms, including profile shift, contact ratio, TIF and grinding notch, are collected with English and Chinese equivalents in the "Gears" section of the glossary.

On overseas drawings, diametral pitch DP (teeth per inch of reference circle diameter) is sometimes used instead of the module. m = 25.4 ÷ DP (conversion table).

IntermediateProfile shift and undercut: a remedy for gears with few teeth

When a rack-type tool cuts a gear with few teeth, the tool tip removes extra material from the tooth root and the root is cut away. This is undercut, which makes the root thinner and easier to break, and also reduces the range over which the teeth can mesh. With a standard tooth with a 20° pressure angle, it occurs when the number of teeth is roughly fewer than 17 (theoretical value; in practice, down to about 14 is often allowed)^[10].

So the tool is moved slightly away from the center of the gear when cutting. This is profile shift, and the shift distance divided by the module is the profile shift coefficient x . A positive profile shift thickens the tooth root and reduces undercut, but if overdone the tooth tip becomes pointed.

Minimum number of teeth without undercut z_min = 2(1 − x) / sin²α

This figure calculates the tooth profile in your browser by moving the rack tool in small steps. It assumes a standard tool with a tip radius of 0.38m and a dedendum of 1.25m. If the actual tool shape (tip radius, tooth depth) differs, the way the root is cut away will also differ.

Profile shift is also used to match the center distance to a specified value, to equalize tooth strength between the two gears, and to reduce sliding. The center distance is determined from the sum of the profile shift coefficients of the two meshing gears (Spur gear calculator).

IntermediateHelical gears: don't mix up “normal” and “transverse”

Gears with slanted tooth traces — Helical gears — mesh gradually, so they are quiet and strong, and are the mainstay of automotive transmissions and steering pinions. In return, they produce axial force (thrust).

• Helix angle β: the inclination of the tooth trace on the reference circle. Hand of helix (right/left): viewed with the axis upright, it is a right-hand helix if the tooth trace rises to the right
• External gears meshing on parallel shafts have the same helix angle but opposite hands
• In helical gears, the transverse pressure angle is larger, so undercut is less likely than in spur gears even with few teeth

The most common mix-up in quotes and ordersEven if a drawing says “module 2”, the tool is a completely different one depending on whether it is normal or transverse. For the specifications of a helical gear, please always state whether the module is normal or transverse (the Helical gear calculator can give you both).

IntermediateHow to measure tooth thickness and accuracy

Tooth thickness cannot be measured accurately with calipers. On the shop floor it is controlled using the following two “indirect dimensions”.

Accuracy is expressed by the shape of a single tooth (profile deviation), the direction of the tooth trace (helix deviation), the spacing between teeth (pitch deviation / cumulative pitch deviation), the offset of the tooth spaces (runout), and the tolerance for each sets the accuracy grade (see accuracy grade / error items). A gear testing machine records the tooth profile and helix as curve charts. When ordering tools too, decide at the outset how pass/fail is judged on these charts, and you will avoid disputes at acceptance.

IntermediateHow gears are made: cut → heat-treat → finish

For gears that need strength, the teeth are cut while the material is still soft (soft machining), the surface is hardened by carburizing and quenching, and then the distortion caused by heat treatment is removed by grinding or honing (hard machining). The key is to leave the grinding stock (the amount to be ground off) on the teeth when gear cutting, and that is where the “protuberance” (the “bump”) in the next section comes in.

AdvancedWhat Is a Protuberance (the “Bump”)?

In briefA tool with a protuberance is a pre-machining hob or pinion cutter with a small “bump” on the corner of the tooth tip, shaped to cut a deliberately shallow relief into the root of the gear tooth. When the gear is later ground, shaved, or honed, the tip of the finishing tool does not touch the tooth root, so no step is left at the end of the finished surface^[7][15][19].

Why a Step Is a Problem

A grinding wheel only grinds the tooth flank that the mating gear actually touches (the active flank) and does not go down to the tooth bottom. If you grind a gear that was pre-machined without a protuberance, a step about as deep as the grinding stock is left where the wheel runs off the flank. This is called a grinding notch. The tooth root is where bending stress is highest, so the notch becomes a stress concentration and lowers the rated bending strength of the tooth. The lower on the root fillet the notch sits, the greater its effect, and increasing the grinding stock makes the notch deeper^[16][20].

The same goes for shaving: if the tip of the shaving cutter hits the fillet left by pre-machining, the cutter wears quickly and the tooth profile is damaged. With the relief from the bump, the shaved flank and the unshaved root connect smoothly^[7][18][19].

How to Decide the Size of the Bump

Protuberance amount pr = finishing stock q + relief u remaining after finishing

• Finishing stock q (per side, perpendicular to the flank): the bump must always be larger than this. The difference u becomes the relief that remains at the root even after finishing^[15][17][19]
• In one tool manufacturer's proven profile, with a 20° pressure angle, a 10° bump angle, a tool addendum of 1.4m, and a tip radius of 0.4m, the values are q ≈ 0.09 + 0.0125m, pr ≈ 0.129 + 0.029m [mm] (module 7 and below). For module 3, q ≈ 0.13 mm, pr ≈ 0.22 mm, and the remaining relief u ≈ 0.09 mm^[15]
• Another tool manufacturer gives guideline values for shaving stock, in terms of tooth thickness, of 0.05–0.08 mm for m2.25 and below, 0.06–0.10 mm for m2.25–3.5, and so on^[7]
• Keep the top of the relief (root form circle dFf) below the lowest circle that the mating tooth tip reaches (dNf = SAP, TIF). If the bump is made too large, dFf rises into the contact range and the tooth root also becomes thinner^[15][19]
• Allow a margin for heat-treatment distortion and for centering error on the grinding machine (equalizing the stock removed from the left and right flanks)^[15]
• For gears with very few teeth, the undercut that naturally forms during generation is enough, and a tool without a bump may be used^[17][19]

The symbols are taken from a tool manufacturer's technical documents (per JIS B 4350)^[7]. There is no single standard reference form for the protuberance; it is designed for each application^[15].

With a protuberance, which manufacturer's gear grinder should you use?

Answer: having a protuberance does not mean you need a particular manufacturer's grinder.In the materials we checked, we found no document in which a grinder manufacturer specifies the pre-machining tool. Tool manufacturers say the bump amount is set on the tool side, based onthe finishing method, grinding stock, heat-treatment distortion, and the mating gear, while the grinder side matches it by adjusting the height and radius of the wheel tip (dressing) and the split of the stock between the flanks (phase alignment) to fit^{[7][15][16][20]}. Conversely, once the grinder and wheel are decided, pass that information (how deep the wheel goes in) to the tool manufacturer: that is the right order.

In mass production of automotive transmission gears, a threaded wheel is rotated in mesh with the gear: continuous generating grinding is the mainstream, and when gears are ground, shaving is omitted^[24][28][21]. Production machines use a sensor to measure the position of the tooth spaces that shifted during hardening, and align the gear so that the left and right flanks are ground equally before grinding^[23][24][29].

Model names and ranges were checked against each company's official pages and technical documents (October 2026). Profile grinding (one tooth space at a time with a disc wheel) is slower, so it suits large-diameter, small-batch, large-module gears^[24][27]. Gears with a small diameter and closely spaced shaft steps, such as steering pinions are hard to grind after hardening because the wheel interferes, and in some cases they are finished after hardening with a carbide hob^[21]. Grinders that use a small-diameter wheel have also appeared^[26].

Comparing the Maag Method and the Reishauer Method

The history of gear grinding is easy to follow through the methods of two Swiss companies. Both are Zurich companies; Maag ground gears with dish-shaped wheels, and Reishauer with threaded wheels.

The relationship to the protuberance is the same for bothEven if the grinder type differs, the idea is the same: the cutting edge of the wheel should run out within the relief of the bump on the pre-machined gear. What you tell the tool manufacturer is the grinder type, the diameter down to which the wheel reaches (the depth grinding reaches), and the grinding stock.^[15][16].

If You Grind All the Way Down the Root, Is the Bump Unnecessary?There are methods that grind even the root fillet, but grinding leaves tensile residual stress, so shot peening and an inspection for grinding burn are required; it is also expensive and used only for limited gears such as aircraft gears. When made correctly, an unground root is said to have almost the same bending fatigue strength^[22].

AdvancedFlank Modifications: Deliberately Altering the Shape Slightly

Under load, the teeth deflect, and so do the shafts. If the involute stays exactly as in theory, the tooth tips collide at the start of mesh, or only the ends of the teeth are loaded heavily. So the finished shape is deliberately altered, in units of microns.

Modifications are verified on a gear tester chart as “deviation from the target shape.” For gears finished by grinding, the modification is often applied on the grinder; for hob-finished gears, it is built into the hob profile. In either case, it is important to write the amount and range of the modification (the diameter where it starts) on the product drawing, and pass them on to the tool manufacturer and the grinding department together with TIF and SAP.

AdvancedGear Chamfering: Workpiece Side and Tool Side

A gear straight from gear cutting has sharp corners between the end face and tooth flank, and burrs remain. Sharp corners and burrs become brittle after carburizing and quenching, chip under load, and damage the transmission. Hardened burrs also shorten the tool life of later processes such as honing. If the end face is used as a datum or for clamping, burrs are not allowed, and chamfering also helps prevent injury when handling by hand^[63][64].

Workpiece side: where to chamfer, and in what shape

• The end-face chamfer has tapered, parallel (flank only), and parallel (including root) as its three basic shapes^[7][67]. The angle of a chamfer made by pressing is roughly 20–30°^[63]
• A gear with an inclined end face can also be chamfered, but beyond 15° a check by calculation is needed^[7][67]
• The product drawing should specify the chamfer's shape, amount (width and angle), and location, and the allowable amount of the secondary burr that appears on the flank side

Tool side: how to chamfer

The "secondary burr" of press chamferingSome of the crushed material also flows onto the tooth flank and forms a bulge (secondary burr). If the next process is shaving or continuous generating grinding, this is rarely a problem, but with honing it affects tool life. The countermeasure is either to push the secondary burr in with a tool whose half circumference is a burnishing (smoothing) section, or to split hobbing into two passes, chamfer in between, and remove the burr with the second hobbing pass^[63][64]. Chamfering units can also be built into hobbing machines, shaving machines and skiving machines^[66][64].

When ordering a chamfering tool, tell the maker the gear data (as-cut tooth thickness, TIF), the chamfer shape and location, whether there is a shoulder and the tilt of the end face, the mating gear, and the tool's number of teeth, material and mounting type (for details, see the chamfering tool guide).

AdvancedOrdering from the tool maker: what to tell them and what to hand over

A tool maker can design the tool once it knows the gear data, how the gear will be finished, and which machine will cut it. Conversely, if even one item is missing or mixed up, the tool you receive cannot cut the correct gear. A tool maker's inquiry or order form is broadly divided into three sections: "Workpiece (gear)", "Tool" and "Machine", as the table below shows^[35][36][7].

1. Prepare the product drawing and the gear data table: How to write the data table is defined in JIS B 0003 (gear drawings). It lists the tooth profile, module, pressure angle, number of teeth, profile shift coefficient, reference circle diameter, tooth thickness (span measurement or over-pin measurement), accuracy, mating gear, and so on^[39][37][38]
2. Add what the data table does not cover: TIF (the diameter from which the effective tooth profile starts), finishing stock, tip chamfer, order of heat treatment, and machine data. These are the items most easily left out^[35][7]
3. Request a quotation: Put the items in the table below into a request form and hand over the full set of materials
4. Check the tool drawing (approval drawing): Check the tool shape and dimensions calculated by the tool maker against the gear data and the machine
5. Manufacture, delivery and trial cutting: Measure the first gear on a gear tester and accept it against the pass/fail criteria decided beforehand

Set of materials to hand over

• Product drawing (with data table): The tool is designed from the workpiece's drawing number and data^[7][35]
• Pre-machining dimensions and finishing stock: tooth thickness after gear cutting, stock to be removed per side
• Mating gear data and center distance (if the drawing does not give the TIF, have it calculated from these)
• Process plan: the order and method of gear cutting → heat treatment → finishing (grinding, shaving, honing, hob finishing after hardening)
• Machine specifications: model, maximum tool diameter and length, shift amount, swivel angle, arbor (tool bore diameter)
• Inspection criteria: how to measure tooth thickness (ball diameter, number of teeth spanned), accuracy grade, purpose and tolerance of modifications
• Quantity and tool life targets: annual quantity, cycle time

Some gear design software can output a "tool quotation request" report that collects the gear and tool data^[56]. Drawings are often confidential, so check your company's procedures (such as a non-disclosure agreement) before handing them over.

AdvancedExample 1: Quotation request for a hob that cuts a steering pinion

Steering in passenger cars is mostly of the rack-and-pinion type^[2]. The pinion is a helical gear with few teeth and a large helix angle that meshes with the rack (a round bar with teeth cut in it). Because of how it is mounted in the vehicle, the pinion and rack axes are not at right angles but have a crossing angle, and the rack is pressed against the pinion by a spring, so under light load it meshes on both tooth flanks^[1].

How the pinion is made (as far as public sources tell)

• Gear cutting is mostly hobbing: A hob cuts intermittently with a row of teeth, so small ridges matching the feed (polygonal error) are always left on the tooth flank and cause the "sliding noise" of the steering. Lowering the feed reduces them but lowers productivity, so one study reduced them from 7.5 µm to 4.8 µm by revising the hob path and other factors^[42]
• Two examples of heat treatment: examples of cutting non-heat-treated steel and then induction hardening^[44], and of carburizing,^[45] appear in patents
• Finishing: The diameter is small and the shaft shoulders are close, so a grinding wheel easily hits them. Finishing with a carbide hob after heat treatment is used in some cases^[21][42]. Grinding with a small-diameter wheel is also possible^[26]

Here, taking the "specifications modeled on steering" given in a technical report^[1] as an example, let's draft a quotation request for a pre-machining hob (with protuberance) used in a hobbing → heat treatment → grinding process. The grinding stock, material, machine and so on are values assumed as examples in this article.

A small puzzle: The report's data table says only "module 2". But the reference circle diameter of 19.861 mm matches 9 × 2 ÷ cos 25°, so it can be seen that it is a normal module (for a transverse module it would be 9 × 2 = 18 mm). On a request form, always write "normal" so that the recipient does not have to work this out^[1].

The tooth thickness and over-ball dimension in the table are "theoretical values with no backlash", worked out with the same formulas as the Span measurement / Over-pin calculations. In practice, apply the tooth thickness tolerance on the drawing (the amount by which tooth thickness is reduced). The span measurement is taken perpendicular to the tooth flank, so adding the grinding stock q (perpendicular to the flank) on both sides gives the as-cut value.

Watch out for undercut on a 9-tooth pinionIn a gear with few teeth like this example, undercut is avoided through profile shift and tooth depth. Fixing the tool's tip radius and protuberance changes the root shape, so it is safer to ask the tool maker for a drawing that confirms the root form (trochoid), undercut and TIF. See the profile shift and undercut diagram, where you can move the number of teeth and the profile shift coefficient to see the shape.

AdvancedExample 2: Worm shaft and worm wheel of electric power steering

In column-type and pinion-type electric power steering (EPS), the motor's rotation is reduced by a steel worm and resin worm wheel set. Resin is used to reduce rattle noise on rough roads and improve the steering feel^[4][5]. There is a design in which the worm is pressed against the wheel by a spring so that backlash is always zero^[4][3].

First, decide the "type of worm tooth profile"

A worm looks like a "screw", but there are five types, depending on which cross-section the tooth profile is straight in, and the tools used to make them differ as well. The types are covered by ISO/TR 10828, DIN 3975 and AGMA 6022, and JIS B 1723 (dimensions of cylindrical worm gears) also has a clause on tooth profiles^[50][51][52].

For ZK and ZC, the "tool shape" is also part of the worm's definitionBecause the tool diameter and edge form determine the tooth profile, also write the outside diameter and edge form of the tool (grinding wheel) on the drawing and the request form. In the standard's worked examples, too, the wheel's outside diameter and the radius of the edge form are listed as data.^[50][51].

How to cut the worm shaft, and tools

For worms for electric power steering, a whirling machine maker claims that it cuts "module 0.8–3, swivel angles up to 42°, dry, and reaches gear accuracy with no subsequent process" (a machine maker's claim)^[48][13]. When ordering whirling tools, tell the maker your machine and spindle so that the cutter body can be chosen, and send the workpiece drawing. The tool maker then calculates the lead angle and blade shape from the drawing and makes dedicated blades. If you cut a multi-start worm in one pass, consult the maker beforehand because of machine constraints^[46].

The hob that cuts the mating worm wheel

• As a rule, the hob has the same form as the worm: same hand of helix, same pressure angle, same tooth depth. Outside diameter = "worm tip circle + bottom clearance"^[49][17]
• A hob of the same diameter sometimes cannot be made: A patent describes this problem. The resin wheels in EPS have a small module, so a hob with the same outside diameter as the worm is hard to make. Cutting with a slightly larger hob makes the gap at the tooth contact (crowning) too large and raises the contact pressure. The patent avoids this by forming the wheel teeth by injection molding^[6]
• Making the hob slightly larger on purpose is also an idea: One source describes enlarging the hob slightly to move the contact toward the center of the face width, while another study finds that when regrinding shrinks the outside diameter the contact pattern changes, so making it larger has only a limited effect^[49][53][54]
• Setting angle: The hob setting angle β is basically "the wheel's lead angle γ_c − the hob's lead angle γ_a" and it changes when the hob diameter changes. In an EPS reducer, there is an example where optimizing this angle reduced the fluctuation of output torque by 30%^[3]

Common mistakes

1. Not stating the profile shift coefficient or tooth thickness: The tool's tooth depth and tooth thickness cannot be determined. Tool makers' request forms always ask for either the profile shift coefficient or the tooth thickness (span measurement, over-pin)^[7][35][37]
2. Mixing up normal and transverse (axial for worms): It is not unusual for a specification table to say just "module". By JIS, the worm uses the axial module, while the hob is usually expressed in the normal section^[1][52][49]
3. Forgetting to state the hand of helix and number of starts: A worm wheel hob must always match the worm's hand of helix^[49][35]
4. Stating only the finished tooth thickness: A tool for the preliminary cut is made to the dimensions "after gear cutting"^[35][7]
5. Not telling the grinding stock and TIF: Pre-cutting with a tool that has no protuberance (the 'bump') leaves a grinding notch, which can lower the bending strength^[16][7]
6. Not telling the machine's limits: Without the maximum tool diameter, length, shift range, and clearance to a shoulder, the tool may not fit or may hit the shoulder^[35][40][41]
7. Not telling the order of heat treatment: The expected heat-treatment distortion changes, and so does the tool grade if you cut after heat treatment^[7][42]
8. Not deciding the worm wheel hob diameter and regrinding: A larger hob makes the gap bigger, and regrinding changes the contact pattern^[17][53][54]
9. Not stating the tool diameter for worms such as ZK: The tool diameter changes the tooth form^[50][51]

Three short notes prevent most failures: ①Which surface the value is for (normal, transverse, axial), ②Which stage the value is for (after gear cutting, finished), ③Where the guarantee starts and ends (TIF, effective tip circle). Just adding these three notes greatly cuts the back-and-forth of inquiries.

FAQ

If the module is the same, will it mesh with any gear?

The module and the pressure angle must both be the same, and for helical gears the helix angle must also be the same (with opposite hand). In addition, the center distance must be right.

Do I have to decide the tool form (hob tooth profile) myself?

No. If you correctly tell the tool maker the gear specifications, the finishing method, and the machine, the tool maker calculates the tool form. What matters is avoiding missing or mixed-up information, and checking the tool drawing (approval drawing) from the tool maker.

Can a gear with a protuberance (the 'bump') only be finished on a particular maker's grinding machine?

No. The size of the protuberance is set on the tool side, from the finishing method, the grinding stock, and the mating gear. Once the grinding machine and wheel are decided, give that information to the tool maker (Grinding machine makers / The Maag method and the Reishauer method).