A three-legged pendulum escapement is a gravity escapement that uses a 3-toothed escape wheel to alternately lift two pivoted gravity arms, which in turn deliver impulse to the pendulum from their own falling weight rather than from the wheel train. The Westminster clock at the Palace of Westminster — Big Ben — runs on Edmund Beckett Denison's double three-legged version of this design. It exists to isolate the pendulum from friction, weather load, and train irregularity in large turret clocks. The result is sub-second-per-day accuracy on a clock driving four 7-metre dials exposed to wind and snow.
Three-legged Pendulum Escapement Interactive Calculator
Vary gravity-arm mass, lift, beat time, and bob mass to see impulse energy, power, wheel stepping, and equivalent pendulum kick.
Equation Used
The calculator uses the gravity-arm energy equation. A raised arm of mass m_arm falling through lift h releases E = mgh. Dividing by beat time gives average impulse power, and equating that energy to bob kinetic energy gives an equivalent velocity kick.
- One gravity arm falls through the stated lift per beat.
- Standard gravity g = 9.80665 m/s^2.
- Losses at pivots, impulse pin, and locking faces are neglected.
- Equivalent bob velocity assumes all impulse energy reaches the pendulum bob.
Operating Principle of the Three-legged Pendulum Escapement
The three-legged gravity escapement separates two jobs that earlier escapements like the deadbeat tried to do at once. The wheel train no longer pushes the pendulum directly. Instead, the train lifts a small gravity arm a few millimetres, and that arm — falling under its own weight — gives the pendulum a precisely repeatable kick. Because the impulse comes from a falling weight inside the case, not from the dirty, wind-loaded wheel train outside, the pendulum stops caring what the wheels are doing. That is the whole point of the design.
Mechanically, you have a 3-legged escape wheel rotating slowly, two gravity arms (one each side of the pendulum), and two locking blocks. As the pendulum swings right, it picks up the right gravity arm and unlocks the escape wheel. The wheel rotates 60° until the next leg lands on the left locking block. During that rotation, a lifting pallet on the wheel raises the left gravity arm by a fixed amount — typically 3 to 4 mm of lift at the impulse pallet. The pendulum, swinging back left, now collects the impulse from the falling left arm. Right side does the same on the next swing. Net result: each swing receives identical impulse energy regardless of train torque variation.
Get the geometry wrong and the clock misbehaves in very specific ways. If the gravity arm lift is too high, the pendulum has to do more work on the upswing than the falling arm returns on the downswing — circular error grows and the rate slows on cold mornings when the oil thickens. Too little lift and the arm bounces off the locking face, the escape wheel trips through two legs at once, and you get a tripping or galloping fault. The fly fan on the escape arbor must absorb the kinetic energy released during the 60° wheel rotation; a fly that is too small lets the wheel slam into the locking block and chip the leg tip. Denison sized the Westminster fly to limit wheel speed to roughly 1 revolution per 6 seconds during release.
Key Components
- Three-Legged Escape Wheel: A wheel with 3 equally spaced legs at 120°, rotating one leg per pendulum-swing pair. The leg tips run on the locking blocks of the gravity arms. Tip radius and locking-face flatness must hold within roughly 0.05 mm or the wheel will skip a leg under low train torque.
- Gravity Arms (Pair): Two pivoted arms, one each side of the pendulum, which deliver impulse by falling under their own weight. Arm mass is typically 30 to 80 g for a 1-second pendulum. The pivot must be jewelled or run on a hardened pin — a worn brass pivot bushing changes the effective lever arm and alters rate by several seconds a day.
- Locking Blocks: Hardened pads on each gravity arm that catch the leg tips of the escape wheel. The locking face sits at the angle of repose so the leg tip cannot self-release. A polished, hardened steel face is mandatory — soft brass blocks will burr within a year and cause tripping.
- Lifting Pallets: Surfaces on the escape wheel that raise the gravity arm during the 60° wheel rotation. Pallet rise determines impulse energy. For a 1-second beat with a 6 kg bob, lift is typically 3 to 4 mm — a tolerance band of about ±0.2 mm before rate stability degrades visibly.
- Fly Fan: Air-resistance governor on the escape wheel arbor. It absorbs the kinetic energy released when a leg drops from one locking block to the next. Without a properly sized fly, the impact knocks the leg tip out of true within weeks.
- Pendulum and Impulse Pin: The pendulum carries an impulse pin or crutch fork that engages the gravity arms near the centre of swing. Engagement geometry must keep the pendulum free for at least 80% of its arc — any earlier contact means the train is influencing the pendulum, which defeats the whole escapement.
Real-World Applications of the Three-legged Pendulum Escapement
You find the three-legged gravity escapement and its double-three-legged sibling almost exclusively in large public turret clocks where the dials are exposed to weather and the hands carry serious wind load. That is the niche it was invented for, and it has not been displaced by anything mechanical since 1854. Smaller domestic regulators rarely use it — a deadbeat is cheaper and equally accurate at small scale. Where it shines is where train torque varies wildly: ice on the dials, a 4-metre minute hand catching gusts, or remontoire failures upstream.
- Public Horology: The Great Clock of Westminster (Big Ben), 1859, uses Denison's double three-legged gravity escapement driving a 4-metre pendulum.
- Civic Tower Clocks: Royal Liver Building clock, Liverpool, built by Gent & Co., uses a three-legged gravity escapement on dials of 7.6 m diameter.
- Heritage Restoration: Smith of Derby retrofits double three-legged gravity escapements during turret-clock recommissioning where the original pinwheel escapement cannot maintain rate under modern wind-loaded hands.
- University Observatories: The Cambridge University clock at Trinity College carries a three-legged variant for a tower clock subject to bell-strike vibration.
- Cathedral Clocks: St Paul's Cathedral clock, London, Smith of Derby installation, uses a Grimthorpe-pattern gravity escapement to handle the heavy minute hands on the south clock face.
- Museum Demonstrations: The Science Museum London exhibits a working three-legged gravity escapement model from the original Dent workshop trials of 1853.
The Formula Behind the Three-legged Pendulum Escapement
The useful number to compute is the impulse energy delivered to the pendulum per beat — that determines amplitude stability. At the low end of typical lift (around 2.5 mm) the gravity arm barely overcomes pivot friction and the pendulum amplitude wanders with temperature. At the nominal design point (around 3.5 mm lift with a 50 g arm) you get a stable 1.5° to 2° amplitude that holds across seasons. Push lift past 5 mm and circular error starts dominating — the rate goes non-linear with amplitude and you lose the very isolation property that justified using this escapement in the first place.
Variables
| Symbol | Meaning | Unit (SI) | Unit (Imperial) |
|---|---|---|---|
| Eimp | Impulse energy delivered to the pendulum per beat | J | ft·lbf |
| marm | Mass of one gravity arm | kg | lb |
| g | Local gravitational acceleration | m/s² | ft/s² |
| hlift | Vertical lift of the impulse face during one wheel-leg release | m | in |
| Lcg | Distance from arm pivot to arm centre of gravity | m | in |
| Limp | Distance from arm pivot to impulse contact point on the pendulum | m | in |
Worked Example: Three-legged Pendulum Escapement in a recommissioned cathedral turret clock
A cathedral works department in Salisbury is recommissioning an 1878 Dent turret clock fitted with a double three-legged gravity escapement driving a 2-second pendulum. The gravity arms have been re-bushed and the escape wheel re-tipped. They need to verify the impulse energy per beat lands in the design window before regulating the rate. Each gravity arm weighs 65 g, the lifting pallet rise is 3.5 mm at nominal, the arm centre of gravity sits 110 mm from the pivot, and the impulse contact sits 85 mm from the pivot. Local g is 9.812 m/s².
Given
- marm = 0.065 kg
- g = 9.812 m/s²
- hlift = 0.0035 m
- Lcg = 0.110 m
- Limp = 0.085 m
Solution
Step 1 — at the nominal 3.5 mm lift, compute the raw potential energy stored in the lifted gravity arm:
Step 2 — apply the lever ratio to convert the arm's centre-of-gravity drop into actual impulse delivered at the pendulum contact point:
That is the design point — roughly 2.9 mJ per beat. On a 2-second pendulum carrying a 200 kg bob this maintains an amplitude of about 1.8°, which is exactly where the Grimthorpe geometry sits cleanly inside its linear range.
Step 3 — at the low end of the typical lift range, 2.5 mm (worn pallets, dirty oil, or sagging gravity arm bushings):
That is a 29% drop in impulse energy. In practice the pendulum amplitude falls toward 1.4°, the clock starts losing 1 to 3 seconds a day, and on cold mornings the escapement may trip a leg because the pendulum no longer reliably unlocks the gravity arm.
Step 4 — at the high end, 5.0 mm lift (over-zealous pallet adjustment during overhaul):
Amplitude climbs above 2.3°, circular error becomes significant, and the rate becomes amplitude-dependent — exactly the disease the gravity escapement was designed to cure.
Result
The nominal impulse energy is 2. 89 mJ per beat. That puts the pendulum amplitude at roughly 1.8°, which is the sweet spot for Denison-pattern double three-legged escapements — comfortably above the unlocking threshold and well below the onset of measurable circular error. Pull lift down to 2.5 mm and you drop to 2.06 mJ with the clock losing a few seconds a day in cold weather; push to 5.0 mm and you climb to 4.13 mJ with amplitude-dependent rate variation. If your measured rate drifts despite a clean impulse calculation, suspect: (1) a worn locking block face causing recoil during the release — visible as a polished arc rather than a sharp edge, (2) a fly fan with a bent blade or excess shake on its arbor letting the escape wheel overspeed and slam the locking block, or (3) the impulse pin on the pendulum crutch sitting too low so it engages the gravity arm before the pendulum reaches its release angle.
Choosing the Three-legged Pendulum Escapement: Pros and Cons
The three-legged gravity escapement competes with two main alternatives in turret-clock work: the Graham deadbeat and the pinwheel escapement. Choose between them on the basis of dial exposure, hand size, and tolerance for setup complexity.
| Property | Three-Legged Gravity Escapement | Graham Deadbeat | Pinwheel Escapement |
|---|---|---|---|
| Train-torque isolation | Excellent — pendulum sees only gravity-arm impulse | Poor — pendulum receives train torque directly | Moderate — some isolation through pin geometry |
| Typical accuracy in turret use | ±0.5 to 1 s/day exposed dials | ±2 to 5 s/day exposed dials | ±1 to 3 s/day exposed dials |
| Setup complexity | High — lift, locking, fly all interdependent | Low — pallet depth and drop only | Medium — pin geometry must be true |
| Tolerance to wind-loaded hands | Very high — designed for it | Low — rate varies with hand load | Medium — tolerates moderate variation |
| Service interval | 5 to 10 years between adjustments | 2 to 5 years | 3 to 7 years |
| Cost to retrofit | High — requires fly, gravity arms, new wheel | Low — drop-in pallet replacement | Medium — wheel replacement only |
| Suitability for sub-2 m pendulum | Poor — overkill below 1-second beat | Excellent | Good |
Frequently Asked Questions About Three-legged Pendulum Escapement
Tripping with correct impulse numbers almost always points to the fly fan, not the gravity arms. When the escape wheel releases, the fly is what limits how fast the leg slams into the next locking block. A fly that is too small or has lost a blade lets the wheel arrive at the locking face with too much kinetic energy, the leg bounces, and you get a double-leg release — a trip.
Diagnostic check: time the wheel release with a stopwatch. A correctly sized Denison fly should take about 0.3 to 0.5 seconds to rotate 60° between legs. If you see 0.1 seconds, the fly is undersized or its blades are bent flat against the airflow. Re-pitching the blades to 30° from the arbor axis usually fixes it.
Yes for any turret clock with a pendulum bob under about 100 kg and dials under 3 m diameter. The double three-legged version exists specifically because Westminster's 4-metre pendulum and 7-metre dials needed more impulse energy per beat than a single 3-legged wheel can deliver at sensible lift heights.
Rule of thumb: if your impulse energy requirement exceeds about 4 mJ per beat at 3.5 mm lift with reasonable arm mass, go double. The double version uses two parallel 3-legged wheels offset by 60° on the same arbor, doubling impulse without doubling lift height.
The whole point of this escapement is constant impulse, so amplitude variation that big means something else in the case is breathing. Most common cause: temperature-driven changes in pivot oil viscosity at the gravity arm pivots. Over a 24-hour temperature swing of 15°C, clock oil viscosity can change by a factor of 2, and that changes how much of the falling arm's energy is wasted in the pivot versus delivered to the pendulum.
Second most common cause: the impulse pin on the pendulum crutch is engaging the gravity arm at slightly different positions on each swing because the crutch fork has shake. Take up the fork shake to under 0.05 mm and amplitude drift typically settles within a week.
Always preserve lift height in the 3 to 4 mm range and scale the arm mass instead. Lift height controls the linearity of the impulse — go below 2.5 mm and pivot friction starts dominating, go above 5 mm and circular error wrecks rate stability. Both extremes destroy the property you bought the escapement for.
Practical sizing: pick your target impulse energy from pendulum amplitude requirements, fix lift at 3.5 mm, then solve for arm mass. For most provincial turret clocks you land between 30 g and 70 g per arm. Use brass arms with steel locking blocks — full steel arms have too much rotational inertia and slow the unlock event.
Because the gravity arm is a low-energy device by design. The whole impulse is around 2 to 3 mJ per beat, and any extra drag at the arm pivot eats directly into amplitude. Winter oil at low temperature can carry 3 to 5 times the viscous drag of summer oil at the same temperature, and the arm response time slows enough that it engages the pendulum slightly later in the swing — late impulse always reads as a rate loss.
Fix: use a single synthetic clock oil rated across your local temperature band rather than seasonal swaps. Moebius 9020 or equivalent is what most heritage workshops have moved to for exactly this reason.
Leg tip wear that fast means the leg is hitting the locking block with measurable velocity rather than arriving gently. Two causes dominate. First, the fly fan is undersized or fouled, so the wheel overspeeds during the release event. Second — and more common in restorations — the locking block face is no longer at the correct angle of repose, so the leg slides on impact instead of stopping cleanly.
Check the locking block face angle with a sine bar. Denison specified between 12° and 15° from vertical depending on the wheel diameter. If yours sits at 5° to 8° from a previous repair, the leg is effectively skidding to a halt every release and grinding itself away.
References & Further Reading
- Wikipedia contributors. Gravity escapement. Wikipedia
Building or designing a mechanism like this?
Explore the precision-engineered motion control hardware used by mechanical engineers, makers, and product designers.
