Gear manufacturing: hobbing, shaping and finishing

Why it matters

Every gearbox — automotive, machine tool, wind turbine, reducer — depends on gear teeth whose involute profile, pitch and helix are accurate to a few micrometres. The cutting method fixes how accurate, how fast and how cheaply a gear can be made, and the finishing method decides noise, load capacity and life after heat treatment.

Key ideas

Basic gear geometry (needed for setting up any process)

• Module m = d/Z (mm); pitch diameter d = m·Z; circular pitch p = π·m.
• Standard full-depth involute teeth: addendum = m, dedendum = 1.25·m, whole depth = 2.25·m, outside (blank) diameter da = m·(Z + 2).
• Gears of the same module and pressure angle mesh with each other, whatever their tooth numbers — which is why one hob cuts all tooth numbers of a module.

Forming vs generating

• Forming: the cutter has the exact shape of the tooth space and copies it, one space at a time, with indexing in between. Examples: form milling with a disc or end-mill involute cutter on a milling machine with a dividing head, broaching (internal gears in mass production), and form grinding. Because the involute shape changes with tooth number, a set of 8 cutters per module covers the range of tooth numbers, so a form-milled gear is only approximately correct; this is a jobbing / repair method.
• Generating: the cutter is itself a gear element (a rack or a pinion) and the cutter and blank are moved as if they were in mesh. The involute is generated as the envelope of successive cutter positions, so one cutter of a given module and pressure angle cuts any number of teeth correctly. Hobbing, gear shaping and rack shaping are generating methods; so are bevel-gear generators.

Gear hobbing

• A hob is a worm with gashes (flutes) and relieved teeth — essentially a rack wrapped into a helix. Hob and blank rotate continuously in a fixed ratio: for a hob with K starts and a gear of Z teeth, the work turns K/Z rev per hob revolution. The hob also feeds axially across the face width.
• The hob axis is tilted so that its thread is aligned with the tooth direction: for a spur gear by the hob lead angle λ; for a helical gear by ψ ∓ λ (minus when hob and gear are of the same hand, plus when opposite).
• Continuous indexing makes hobbing fast and accurate: the main process for external spur and helical gears, worm wheels and splines in batch and mass production. It cannot cut internal gears and needs runout room, so it cannot cut a gear close to a shoulder.

Gear shaping (Fellows type)

• A pinion-shaped cutter reciprocates parallel to the gear axis while cutter and blank rotate slowly together as if meshing; the cutter is withdrawn on the return stroke.
• Cuts internal gears, cluster gears and gears next to a shoulder, because it needs only a small overrun. Helical gears need a helical guide. Rack-type shapers (Sunderland, Maag) use a rack cutter instead of a pinion.

Other routes: bevel gears on Gleason-type generators; gear rolling, casting, powder metallurgy, stamping and injection moulding for lower-precision or high-volume parts.

Gear finishing

• Shaving (before hardening): a hardened, serrated shaving cutter meshes with crossed axes with the gear and scrapes fine slivers, correcting profile and lead errors. Fast and cheap; for soft gears.
• Grinding (after hardening): form or generating grinding; the most accurate route and the only one that removes heat-treatment distortion substantially.
• Gear honing (after hardening): an abrasive-impregnated plastic gear-shaped tool, mainly improves finish and removes nicks; lapping runs gears with a lapping compound to improve contact pattern; burnishing rolls the soft gear against hardened dies to smooth the surface.

Formulas

• d = m·Z, da = m·(Z + 2), h = 2.25·m — pitch diameter, outside diameter, whole depth (mm).
• Nh = 1000·V / (π·Dh) — hob speed, rev/min; V cutting speed in m/min, Dh hob outside diameter in mm.
• Nw = Nh·K / Z — work-table speed, rev/min; K = number of hob starts.
• t = (b + A) / (f·Nw) — hobbing time for one pass, min; b face width (mm), A approach + overtravel (mm), f axial feed per work revolution (mm/rev).
• tan λ = K·m / dh — hob lead angle; dh = hob pitch diameter (mm), m = module (normal module for helical gears).
• Hob swivel: spur gear = λ; helical gear = ψ − λ (same hand) or ψ + λ (opposite hand); ψ = gear helix angle.

Worked examples

Example 1 (standard) — hobbing time. A spur gear, m = 3 mm, Z = 40, face width b = 30 mm, is cut with a single-start HSS hob of 70 mm outside diameter at V = 30 m/min. Axial feed f = 1.5 mm per work revolution; approach + overtravel A = 15 mm. Find the blank diameter, tooth depth and cutting time.

1. da = m·(Z + 2) = 3 × 42 = 126 mm; h = 2.25·m = 6.75 mm.
2. Nh = 1000·V/(π·Dh) = 1000 × 30/(π × 70) = 136.4 rev/min.
3. Nw = Nh·K/Z = 136.4 × 1/40 = 3.410 rev/min.
4. t = (b + A)/(f·Nw) = (30 + 15)/(1.5 × 3.410) = 8.80 min.

Example 2 (GATE level) — hob setting and multi-start hob. A helical gear with normal module 3 mm, Z = 30, helix angle ψ = 20° (right hand) and face width 40 mm is hobbed with a hob of 64 mm pitch diameter. (a) Find the hob lead angle and swivel angle for a single-start right-hand hob and for a single-start left-hand hob. (b) If a two-start hob at 150 rev/min is used with f = 2 mm/rev of work and A = 15 mm, find the time per pass.

1. tan λ = K·m/dh = 1 × 3/64 = 0.046875 → λ = 2.68°.
2. Same hand (RH hob, RH gear): swivel = ψ − λ = 20 − 2.68 = 17.32°.
3. Opposite hand (LH hob): swivel = ψ + λ = 22.68°.
4. (b) Nw = Nh·K/Z = 150 × 2/30 = 10 rev/min.
5. t = (b + A)/(f·Nw) = (40 + 15)/(2 × 10) = 2.75 min. A two-start hob doubles the work speed and halves the time, at some cost in accuracy (two threads must match).

Common mistakes

• Using one form cutter for all tooth numbers — form cutters are numbered for ranges of Z; generating cutters are not.
• Writing Nw = Nh·Z instead of Nh·K/Z — the work turns much slower than the hob.
• Choosing hobbing for an internal gear or a gear next to a shoulder; use shaping.
• Getting the swivel sign wrong: same hand → ψ − λ; opposite hand → ψ + λ.
• Shaving a hardened gear — shaving is done before hardening; hardened gears are ground or honed.
• Using the pitch diameter of the gear for the cutting speed; the cutting speed is that of the hob (or the shaper stroke).

For GATE PI

• MCQs: forming vs generating, which process for internal gears / worm wheels / hardened gears, order of shaving and hardening.
• NAT on module, blank diameter, hob and work speeds, hobbing time with multi-start hobs, and hob swivel angle.
• Indexing for form-milling gears with a dividing head (links to the milling topic).

Quick check

1. Which process cuts an internal spur gear?
2. A gear has m = 4 mm and Z = 25. Blank diameter?
3. A 3-start hob runs at 120 rev/min on a 60-tooth gear. Work speed?
4. Is shaving done before or after hardening?
5. Why does one hob cut gears of any tooth number of a given module?

Answers: 1. Gear shaping (or broaching in mass production). 2. 4 × 27 = 108 mm. 3. 120 × 3/60 = 6 rev/min. 4. Before. 5. Hobbing generates the involute by simulating rack–gear meshing, and all gears of the same module and pressure angle mesh with the same rack.