Hirth coupling indexing is a face-toothed rotary joint that locks two coaxial discs together using radial teeth that mesh across the full face. The teeth wedge into each other when axial clamping force is applied, self-centring the two halves and locking the angular position. We use it in machine tools, 5-axis tilt heads, and aerospace shaft joints to deliver indexing repeatability inside ±1 arc-second and load capacities exceeding 30,000 Nm without any backlash.
Hirth Coupling Indexing Interactive Calculator
Vary tooth count and desired rotation to see the nearest Hirth coupling index position, step size, tooth move count, and angular error.
Equation Used
The tooth count sets the discrete angular pitch of the Hirth coupling. The calculator rounds the requested angle to the nearest whole tooth position, then reports the indexed angle and remaining geometric error in arc-seconds.
- The coupling indexes only to whole-tooth positions.
- Tooth count is rounded to the nearest integer.
- Angular error shown is geometric command error, not manufacturing repeatability.
- Default values mirror the 72-tooth, 60 deg indexing example.
How the Hirth Coupling Indexing Works
A Hirth coupling is two flat discs with identical radial teeth cut across their faces — picture a pair of bevel gears flattened into pancakes, with teeth running from the bore outward like spokes. When you push the two halves together axially, every tooth on one disc slides into the gap between teeth on the other. Because the teeth are radial and symmetric, the two halves cannot do anything except self-centre. That is the whole trick — there is no key, no taper, no dowel pin doing the alignment work. The teeth do it themselves, and they do it every time you re-engage.
The self-centring behaviour comes from the tooth flank angle, typically 60° included, which converts axial clamp load into radial wedging force. The number of teeth determines your indexing resolution — a 72-tooth Hirth gives you 5° increments, a 360-tooth gives you 1° increments. You can only index to discrete tooth positions, which is the trade-off you accept for the precision. If the tooth pitch is wrong by even a few microns across the face, the teeth bind on one side and gap on the other, and you lose both the centring and the load-sharing across teeth. We grind Hirth couplings as a matched pair on the same machine in the same setup for this reason — the tolerance budget across the full face is typically 2-3 µm.
When tolerances drift, the failure modes are predictable. Worn flanks cause the indexing repeatability to wander by 5-10 arc-seconds. Insufficient axial clamp force lets the teeth chatter under cutting load and you'll see micro-fretting at the tooth tips within 100 hours. Contamination — a single chip trapped between the faces — lifts the coupling off its seat and throws the indexing position out by tens of arc-minutes. The rule we tell customers: clean the face with a lint-free wipe and a drop of light oil before every re-index, and verify clamp force with a calibrated hydraulic gauge, not feel.
Key Components
- Tooth-faced disc (driving half): Carries radial teeth ground to a 60° included flank angle, with tooth root and tip relief typically 0.05-0.10 mm to prevent tip-on-root contact. The tooth count sets indexing resolution — 72, 96, 180, or 360 are common.
- Tooth-faced disc (driven half): Identical mirror geometry to the driving half, ground in the same setup as a matched pair to hold pitch error below 3 µm across the face. Any pitch mismatch concentrates load on a few teeth and destroys the equal load-sharing that gives Hirth its capacity.
- Axial clamping system: Hydraulic piston, Belleville stack, or central tie-bolt providing the axial preload that wedges the teeth together. Typical clamp force runs 50-200 kN for a 200 mm coupling. Drop below the design preload and the joint loses stiffness; the teeth talk and fret.
- Lift mechanism: Hydraulic or pneumatic actuator that separates the two halves by 0.5-2 mm during indexing. Lift travel must clear the tooth tip relief plus a safety margin for thermal growth — a 1 mm lift is typical on a 250 mm coupling.
- Radial servo drive: Rotates the upper half to the next tooth position while the coupling is lifted. Position accuracy of the drive only needs to be within half a tooth pitch — the teeth themselves do the final alignment when the coupling re-engages.
Who Uses the Hirth Coupling Indexing
Hirth couplings show up wherever you need to lock and unlock a rotary axis at exact angular positions, with zero backlash, and you need it to repeat thousands of times. The classic case is the rotary indexing table on a machine tool, but the same principle drives turbine shaft joints, satellite antenna pointing mechanisms, and high-end optical instruments. You'll see Hirth serrations, face spline couplings, and Hirth joints used interchangeably as terms — they all refer to the same radial-tooth self-centring coupling.
- Machine tools: Tilt-trunnion table on the Mazak Integrex i-400 5-axis turning centre — the trunnion B-axis uses a Hirth coupling to lock at 1° increments with ±2 arc-second repeatability
- Aerospace turbomachinery: Rolls-Royce Trent engine compressor and turbine disc stacks — Hirth-style curvic serrations on the disc drive faces transmit shaft torque while allowing disassembly for blade inspection
- Astronomy instruments: Filter wheel and grating turret indexing on the ESO VLT spectrographs — the Hirth coupling holds optical alignment to sub-arc-second accuracy across thermal cycles
- Satellite mechanisms: High-gain antenna pointing platforms on geostationary comsats — Hirth indexers provide deterministic pointing positions that survive launch loads without recalibration
- Heavy industry rotary indexers: Pegard horizontal boring mill rotary tables — 360-tooth Hirth couplings deliver 1° indexing under cuts up to 30,000 Nm
- Medical imaging: Rotating gantry locks on Elekta Versa HD radiotherapy systems — Hirth detents hold beam delivery angles to clinical sub-millimetre accuracy at isocentre
The Formula Behind the Hirth Coupling Indexing
The torque capacity of a Hirth coupling depends on how many teeth share the load, how high those teeth are, the mean radius where load is applied, and the allowable shear stress on the tooth flank. At the low end of the operating range — say a small 80 mm coupling with 72 teeth — you'll see capacities around 800 Nm, fine for a tooling turret but undersized for any real cutting. At the nominal middle of the range — a 200 mm, 180-tooth coupling — capacity climbs to roughly 12,000 Nm, which covers most CNC trunnion duty. At the high end — a 400 mm, 360-tooth coupling on a heavy boring mill — you cross 80,000 Nm. The sweet spot sits at the diameter and tooth count that just exceeds your peak cutting torque with about 1.5× safety, because oversizing wastes clamp-force budget and undersizing fails by tooth-tip fretting before you see catastrophic failure.
Variables
| Symbol | Meaning | Unit (SI) | Unit (Imperial) |
|---|---|---|---|
| Tmax | Maximum transmittable torque | N·m | lb·ft |
| z | Number of teeth (only half carry load in pure torque) | — | — |
| h | Effective tooth height (load-bearing flank height) | mm | in |
| b | Tooth radial length (face width) | mm | in |
| rm | Mean radius of tooth contact | mm | in |
| Ï„allow | Allowable shear stress on tooth flank material | MPa | psi |
Worked Example: Hirth Coupling Indexing in a satellite antenna pointing platform
You are sizing a Hirth coupling for the elevation-axis indexer on a 2.4 m geostationary comsat antenna built around a Honeywell pointing actuator. The coupling has 144 teeth, mean tooth contact radius of 60 mm, tooth height 1.2 mm, and a 15 mm radial tooth length. The teeth are case-hardened 18CrNiMo7-6 with allowable shear stress of 400 MPa. Peak holding torque under thermal-snap loads is specified at 950 Nm.
Given
- z = 144 teeth
- rm = 60 mm
- h = 1.2 mm
- b = 15 mm
- Ï„allow = 400 MPa
- Trequired = 950 Nm
Solution
Step 1 — count the load-sharing teeth. In pure torque transmission, only half the teeth carry shear load on each flank because the other half is engaged on the opposing flank:
Step 2 — compute the nominal torque capacity at full design parameters:
That is far above the 950 Nm requirement — the coupling has a safety factor of roughly 33×, which sounds wasteful until you remember that satellite mechanisms must survive launch shock loads of 50-100 g and 20+ years of thermal cycling without re-indexing.
Step 3 — at the low end of practical operating conditions, assume only 50% of teeth share load due to manufacturing pitch error of 5 µm across the face (typical when the pair is not co-ground):
Still well above the requirement, but you have just halved your fatigue margin. At the high end — a perfectly co-ground pair with sub-2 µm pitch error and full 72-tooth contact — you keep the full 31,100 Nm and the load distributes evenly enough that no single tooth sees stress concentration. That is the sweet spot for a flight-grade coupling.
Step 4 — required axial clamp force to maintain tooth engagement. With a 60° flank angle:
Result
The coupling delivers a nominal torque capacity of 31,100 Nm against a 950 Nm requirement, with 27. 4 kN of axial clamp force needed to keep the teeth seated. In practice, that means the antenna can absorb a worst-case thermal-snap event without any tooth slip, and the pointing position holds to better than 1 arc-second across the orbital day-night cycle. At the low end of manufacturing quality (5 µm pitch error, half the teeth sharing load) you still hold 15,550 Nm; at the high end with co-ground precision pairs you keep the full 31,100 Nm and the load spreads uniformly. If your measured holding torque comes in below prediction, the most likely causes are: (1) axial preload bolts under-torqued — verify with an ultrasonic bolt-stretch gauge, not a torque wrench, because thread friction scatter is ±25%; (2) tooth flank surface finish above Ra 0.8 µm, which lowers effective τallow by 15-20% through micro-pitting; or (3) the two halves were not lapped as a matched pair, so only the high-spot teeth carry load and the rest float.
Hirth Coupling Indexing vs Alternatives
Hirth coupling is one of three serious options when you need a precise, lockable rotary index. The other two are curvic couplings (a Hirth variant with curved teeth, ground rather than milled) and conventional worm-gear or roller-cam indexers with no positive lock. Each suits a different combination of load, accuracy, indexing speed, and budget.
| Property | Hirth Coupling | Curvic Coupling | Worm-Gear Indexer |
|---|---|---|---|
| Indexing accuracy / repeatability | ±1 to ±2 arc-sec | ±0.5 to ±1 arc-sec | ±15 to ±60 arc-sec |
| Torque capacity (200 mm size) | 10,000-30,000 Nm | 20,000-60,000 Nm | 2,000-8,000 Nm |
| Indexing increment resolution | Discrete (tooth pitch only) | Discrete (tooth pitch only) | Continuous (any angle) |
| Index-to-index cycle time | 1-3 sec (lift, rotate, clamp) | 1-3 sec (lift, rotate, clamp) | 0.3-1 sec (continuous) |
| Manufacturing cost | High — milled, matched pair | Very high — ground, matched pair | Moderate — standard worm + wheel |
| Backlash under reversing load | Zero | Zero | 5-30 arc-min unless preloaded |
| Service life | 106+ index cycles | 107+ index cycles | 105-106 cycles |
| Typical application fit | CNC trunnions, satellite mechanisms | Aerospace shaft joints, heavy mill tables | General automation indexers |
Frequently Asked Questions About Hirth Coupling Indexing
Almost always it is axial clamp force decay, not tooth wear. Belleville stacks relax 5-10% in the first 100 cycles and another 2-3% over the next 1,000 cycles. Once preload drops below about 70% of design, the teeth start micro-rocking under reversing torque and you see repeatability scatter widen from 1 arc-sec to 5-8 arc-sec.
Re-tension the clamp bolts to design stretch (use ultrasonic bolt gauging or hydraulic tensioners — torque wrenches are too imprecise for this), and switch to a hydraulic clamp circuit if the cycle count is high. If the issue persists after re-tensioning, lift the coupling and inspect the tooth tips with a 10× loupe for the dull burnished band that signals fretting wear.
Yes, but the tooth count climbs fast. 1° resolution needs 360 teeth; 0.5° needs 720; 0.25° needs 1,440. Above about 720 teeth on a coupling under 300 mm diameter, the tooth height drops below 0.5 mm and the load capacity collapses because tooth shear area scales with tooth height squared.
For sub-degree indexing, the practical answer is a hybrid — a 360-tooth Hirth giving 1° steps, with a fine servo on top that interpolates between Hirth positions and locks the Hirth at the nearest whole degree. Heidenhain and Fanuc both use this architecture on their high-end rotary tables.
Decide on three things: budget, accuracy spec, and tooth count. Hirth teeth are straight-flanked and milled, so a matched pair runs roughly half the cost of a curvic of the same size. Curvic teeth are curved and ground in a generating process, which gives you tighter pitch control (±1 µm vs ±3 µm) and roughly 2× the load capacity at the same diameter.
If you need ±2 arc-sec repeatability and your peak cut is below 15,000 Nm, Hirth is the right call. If you need ±1 arc-sec or below, or you are above 20,000 Nm, go curvic and absorb the cost.
That is a lift-and-rotate sequencing fault, not a coupling fault. The servo is rotating the upper half before the lift cylinder has cleared the tooth tips, so the teeth catch on a tip and skip over to the next tooth. You'll often hear a faint click when this happens.
Check the lift travel — it should be tooth height plus tip relief plus 0.3 mm safety margin, minimum. On a coupling with 1.2 mm tooth height and 0.1 mm relief, that is 1.6 mm of clear lift before rotation begins. Verify the lift-complete sensor is actually triggering at full lift, not at 80% of stroke. Pneumatic lift cylinders are the usual culprit because air compressibility lets the piston bounce on re-engagement.
The textbook Faxial = T / (rm × tan(α/2)) gives you the bare minimum to prevent tooth slip under static torque. In a real machine you need 1.8-2.5× that figure, because dynamic cutting forces include impact components the static formula ignores, and you need residual preload after thermal expansion of the clamp bolts at running temperature.
Rule of thumb: take the static-formula result, multiply by 2, then verify the bolts or the hydraulic piston do not exceed 60% of their yield stress at that load. If they do, increase coupling diameter rather than chase higher clamp force — diameter buys you torque capacity at the cube while clamp force buys it linearly.
Because they are not actually identical. When you grind a Hirth pair as a matched set, you mount both halves on the same fixture in the same setup and dress the grinding wheel against both. The tiny pitch errors that exist on every Hirth — typically 1-3 µm — end up being the same error on both halves, so they cancel when the teeth mesh.
Order two halves separately and the pitch errors add instead of cancel, so you end up with 5-6 µm of pitch mismatch. Only a few teeth carry load, the rest float, and your torque capacity drops to 30-50% of the matched-pair rating. The price premium for matched grinding is real but it is buying you the full rated capacity, not just paperwork.
References & Further Reading
- Wikipedia contributors. Hirth joint. Wikipedia
Building or designing a mechanism like this?
Explore the precision-engineered motion control hardware used by mechanical engineers, makers, and product designers.
