A Double Tri-toothed Pendulum Escapement is a clock escapement that uses two cooperating escape wheels, each carrying three teeth, to deliver impulse to a single pendulum on alternating swings. The pendulum receives one push per half-cycle from whichever wheel is unlocked, while the other wheel rests dead. The split-wheel layout halves the per-tooth load and reduces recoil compared to a single 30-tooth escape wheel. Builders use it in precision regulators where amplitude stability and low circular error matter — it can hold rate variation under 0.5 seconds per day on a temperature-stable 1-second pendulum.
Double Tri-toothed Pendulum Escapement Interactive Calculator
Vary tooth count and pallet geometry to see wheel pitch, clocking offset, drop margin, and setup error on an animated twin-wheel escapement.
Equation Used
This calculator checks the core escapement geometry: tooth pitch comes from the number of teeth per wheel, and the second wheel is clocked by half a pitch so its teeth bisect the first wheel gaps. It also compares drop, locking face, and impulse face against the article's recommended setup values.
- Two identical escape wheels are equally toothed and mounted on a common arbor.
- Wheel B is clocked to bisect Wheel A tooth gaps.
- Recommended drop range is 1.5 deg to 2.0 deg.
- Recommended locking face is 12 deg and impulse face is 2 deg to 3 deg.
How the Double Tri-toothed Pendulum Escapement Actually Works
The mechanism splits the job of a conventional escape wheel across two thin wheels mounted on a common arbor, each cut with only three teeth spaced 120° apart and clocked so the second wheel's teeth bisect the first wheel's gaps. A pair of pallets — one per wheel — sits on the pendulum's crutch. As the pendulum swings right, pallet A unlocks wheel A, takes a brief impulse on the impulse face, then locks again on the next tooth. On the return swing, pallet B does the same job on wheel B. You get one impulse per half-period, the same as a Graham dead-beat, but the per-tooth angular load is roughly halved because torque enters through two parallel paths.
Why build it this way? Three reasons. First, with only three teeth per wheel the tooth pitch is large (120°), so you can grind the locking and impulse faces to tighter geometry without the crowded interference you get on a 30-tooth wheel. Second, the dead wheel provides a stable mechanical reference — the pallet not in use rests against a locking face that does not move, which kills recoil. Third, the wider tooth pitch tolerates more pivot wear before drop becomes pathological.
Geometry has to be right or the clock fights itself. The locking face on each tooth must sit at 12° to the radial line, the impulse face at 2° to 3°, and the drop — the freewheel angle between unlocking and the next lock — must land between 1.5° and 2.0°. Push drop above 2.5° and you waste impulse energy as audible tick. Squeeze it below 1° and the pallet clips the heel of the next tooth on a hot day when the brass arbor expands. Common failure modes are pivot-hole wear in the escape arbor (shows up as inconsistent beat), pallet jewel chipping at the locking corner (shows up as galloping rate), and crutch-pin slop (shows up as low amplitude).
Key Components
- Twin Escape Wheels: Two thin steel wheels, typically 28 mm to 40 mm pitch diameter, each with 3 teeth at 120° spacing, mounted on a common arbor and clocked 60° to one another. Wheel concentricity must hold within 0.01 mm TIR or impulse becomes asymmetric between left and right swings.
- Paired Pallets: Two ruby or hardened-steel pallets carried on the pendulum crutch, one engaging each wheel. Locking face at 12° to radial, impulse face at 2°-3°. Pallet span is set so only one pallet engages at a time — the other sits clear by 0.3 mm minimum.
- Crutch and Suspension Spring: Transfers impulse from pallets to pendulum rod and back. The crutch fork must be square to the suspension spring within 0.1° or you get beat error that no amount of adjustment cleans up. Suspension spring thickness is typically 0.10 mm to 0.15 mm for a 1-second pendulum.
- Pendulum Rod and Bob: Invar or wood-rod with a brass bob, tuned to T = 1 s exactly for a seconds regulator. Bob mass 5 kg to 8 kg gives enough inertia that the small impulses do not visibly disturb amplitude. Rod length 994 mm at 20 °C for invar.
- Escape Arbor and Pivots: The arbor carrying both escape wheels runs in jewelled or hardened bushings. Pivot diameter 1.0 mm to 1.5 mm with radial play under 0.005 mm. Wear here is the single most common cause of rate drift in service.
Where the Double Tri-toothed Pendulum Escapement Is Used
You see this escapement almost exclusively in precision pendulum regulators where rate stability matters more than serviceability. It is rare in domestic clocks because the parts count is high and the setup is fussy, but in observatory regulators, time-distribution master clocks, and a handful of modern horological revivals it earns its place. The split-wheel design also crops up in some torsion-pendulum demonstrators and in custom builds where a horologist wants to show the recoil-free behaviour visually.
- Observatory timekeeping: Riefler-pattern and Strasser & Rohde precision regulators of the late 19th century used twin-wheel layouts to reduce per-tooth wear in continuously-running observatory references.
- Horological education: Patek Philippe Museum demonstration regulators and the British Horological Institute teaching collection use double-wheel escapements to make the alternating impulse path visible to students.
- Master clock systems: Synchronome and Shortt-pattern free-pendulum slave timing distribution in railway and broadcast facilities through the 1950s, where the master ran a tri-toothed split wheel for low maintenance.
- Modern independent horology: Bespoke regulator builds by makers like Buchanan Clocks in Australia, where the visible split-wheel impulse is part of the aesthetic.
- Scientific instrumentation: Geophysical reference clocks used pre-quartz for timestamping seismograph drum recorders, where the dead-wheel reference reduced vibration coupling into the pendulum.
- Auction-grade restoration: Restoration of late-Victorian English regulator clocks signed by makers such as E. Dent & Co. where a worn single escape wheel is sometimes replaced with a tri-toothed split pair to extend service life between overhauls.
The Formula Behind the Double Tri-toothed Pendulum Escapement
The figure that matters most for this escapement is the impulse energy delivered to the pendulum per half-cycle, because that energy must exactly replace what the pendulum loses to air drag and suspension flexure. Too little and amplitude collapses below the isochronous sweet spot of 1.5°. Too much and amplitude climbs past 3° where circular error dominates and rate becomes amplitude-dependent. At the low end of typical drive torque the pendulum runs lean and the clock loses time as the mainspring unwinds; at the high end the clock gains. The sweet spot for a 1-second regulator with a 6 kg bob sits around an impulse energy of 80 to 120 microjoules per half-swing.
Variables
| Symbol | Meaning | Unit (SI) | Unit (Imperial) |
|---|---|---|---|
| Eimp | Impulse energy delivered per pendulum half-cycle | J (joules) | ft·lbf |
| Tw | Torque at the escape wheel arbor | N·m | oz·in |
| θimp | Angular sweep of the impulse face during unlocking | rad | rad |
| Nt | Number of teeth engaged per pendulum cycle (2 for a double tri-toothed pair) | dimensionless | dimensionless |
Worked Example: Double Tri-toothed Pendulum Escapement in a marine chronometry research bench at a polar science institute
A polar research institute in Tromsø is commissioning a tri-toothed pendulum regulator as a mechanical backup time reference for an ice-core drilling timestamp logger. The pendulum is a 1-second invar rod with a 6 kg bob. The escape wheel arbor sees 1.8 mN·m of net torque after train losses, the impulse face sweep is 2.5° (0.0436 rad), and you want to confirm the impulse energy lands in the isochronous band before signing off on the build.
Given
- Tw = 1.8 mN·m
- θimp = 2.5 ° (0.0436 rad)
- Nt = 2 teeth per cycle
Solution
Step 1 — at nominal torque of 1.8 mN·m, compute the impulse energy delivered per half-cycle:
That sits at the lower edge of the isochronous sweet spot for a 6 kg bob. The pendulum will run, but amplitude will hover near 1.5° and any extra friction in the suspension will push it below isochronism.
Step 2 — at the low end of typical mainspring-state torque, around 1.2 mN·m near the end of the going period:
52 µJ is too lean. Amplitude collapses toward 1.0° and the clock loses 3 to 5 seconds per day as the spring unwinds. You can see this on a strip-chart rate plot as a downward sloping line over the 7-day going period.
Step 3 — at the high end, freshly wound, around 2.4 mN·m:
105 µJ pushes amplitude up past 2.5°, where circular error starts to dominate. The clock now gains 2 to 3 seconds per day for the first 24 hours after winding. The remoise-style fix is a remontoire or a fusee — anything that clamps Tw to a tighter band than the raw mainspring delivers.
Result
Nominal impulse energy is 79 µJ per half-cycle, which sits at the lean end of the isochronous band for this pendulum. In practice you'll see a steady amplitude near 1.5° at mid-wind, drifting to 1.0° as the spring runs down and climbing past 2.5° just after winding — a daily rate spread of roughly 5 to 8 seconds across the going period unless you add torque regulation. If your measured rate spread is wider than predicted, the three usual culprits are: (1) escape arbor pivot wear above 0.01 mm radial play, which lets the wheel wobble and randomises drop, (2) a suspension spring that is too thick (over 0.15 mm for a 6 kg bob), which dumps energy into flexure rather than swing, or (3) crutch-fork misalignment greater than 0.2°, which biases impulse to one swing direction and shows up as audible beat error on a timing microphone.
When to Use a Double Tri-toothed Pendulum Escapement and When Not To
Picking this escapement over a single-wheel Graham dead-beat or a Riefler spring-detent design comes down to whether you need the load-sharing and visible symmetry badly enough to accept the parts count and setup time. Here is how it stacks up on the dimensions builders actually compare.
| Property | Double Tri-toothed Pendulum Escapement | Graham Dead-beat Escapement | Riefler Spring-detent Escapement |
|---|---|---|---|
| Daily rate stability (1s pendulum, regulated) | ±0.3 to ±0.5 s/day | ±0.5 to ±2 s/day | ±0.01 to ±0.05 s/day |
| Parts count in escapement | 7 to 9 critical parts | 4 to 5 critical parts | 9 to 12 critical parts |
| Typical service interval | 8 to 10 years | 5 to 7 years | 10 to 15 years |
| Tolerance on locking-face angle | ±0.25° | ±0.5° | ±0.1° |
| Sensitivity to drive-torque variation | Moderate (needs fusee or remontoire for best rate) | High (rate visibly drifts with mainspring state) | Very low (free pendulum is decoupled) |
| Build cost (skilled-hours equivalent) | 120 to 180 hours | 40 to 60 hours | 300+ hours |
| Best application fit | Mid-tier precision regulators, demonstration clocks | Domestic regulators, longcase clocks | Observatory and metrology references |
Frequently Asked Questions About Double Tri-toothed Pendulum Escapement
Almost always this is wheel clocking error. The two escape wheels must be indexed exactly 60° apart on the arbor. If the indexing pin or the squared seat lets one wheel rotate by even 0.5° relative to the other, the unlocking angle for one pallet shrinks and for the other grows. The result is asymmetric impulse and a beat error you can hear on a microphone trace.
Check it by removing the pallets and rotating the arbor slowly under a vernier protractor — tooth A1 should lead tooth B1 by exactly 60° ±5 arc-minutes. If it doesn't, the fix is a fresh keyway or a re-staked indexing pin, not pallet adjustment.
If your goal is best rate per dollar and you're not chasing observatory-grade performance, build the Graham 30-tooth. It is faster to cut, easier to set up, and a competent builder can hold ±1 second per day. Choose the tri-toothed split-wheel only if you want the visible symmetry for a demonstration piece, or if you specifically need the longer service interval that comes from halved per-tooth loading.
The decision tipping point is going-period and load. For 7-day clocks with 4 kg+ bobs the load-sharing matters and the tri-toothed pays back. For 30-hour or 8-day domestic clocks under 2 kg bob, you are paying for complexity you will not measure.
The two energy sinks the formula does not capture are suspension-spring hysteresis and air drag on the bob. A suspension spring thicker than 0.15 mm on a 6 kg bob can absorb 20 to 30 µJ per cycle just in internal damping — enough to drop amplitude by half a degree. Air drag on a flat-faced bob at 1.5° amplitude eats another 10 to 15 µJ per cycle.
Diagnostic: swap the suspension to a 0.10 mm spring and re-measure. If amplitude jumps by 0.3° or more, you found it. If not, the loss is in the pallet impulse face — likely a polish issue letting the tooth slide rather than push cleanly.
Galloping rate on this escapement almost always points to the locking corner of one pallet jewel. A microscopic chip or a rolled edge on the locking face causes the tooth to slip past intermittently before locking properly, dumping a half-impulse into the pendulum unpredictably.
Pull the pallets and inspect the locking corners under 40× magnification. You are looking for a bright-line reflection along the corner — a clean jewel shows a sharp line, a chipped one shows a broken or scattered reflection. Replacement is the only fix; you cannot polish a chip out without changing the locking angle.
Drop should sit between 1.5° and 2.0° measured at the escape wheel. Below 1° and you risk the pallet clipping the heel of the next tooth when the brass arbor expands on a hot day — typically above 25 °C in an unheated room. Above 2.5° and you waste 15 to 20% of the impulse energy as freewheel travel before the next lock, which shows up as a louder tick and a 1 to 2 second per day rate loss.
Set drop with the pendulum stationary and the train under light hand-load, using a feeler shim against the locking face. Aim for 1.7° as a target — that gives thermal headroom both ways.
Mechanically yes, but you will need to re-cut the pallet span and possibly re-bush the escape arbor for the wider wheel stack. The original arbor was sized for a single 1.5 mm thick wheel; doubling that plus a spacer brings stack height to 3.5 mm or more, and the existing pivot may not have the shoulder length. You also need to check that the going barrel still delivers torque in the band the new geometry expects — a Dent originally tuned for a 30-tooth dead-beat may run lean on a tri-toothed pair because the per-impulse energy demand is different.
Most restorers who try this end up replacing the escape arbor entirely and re-pitching the third wheel. Budget 40 to 60 skilled hours for the conversion, and only do it if the original wheel is unsalvageable.
References & Further Reading
- Wikipedia contributors. Escapement. Wikipedia
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