A pendulum escapement lets an escape wheel advance in steps as its pallets alternately hold and release the teeth. Explore the anchor and crutch arrangement, calculate ideal pendulum timing, and compare the wheel speed for a selected tooth count.
Pendulum Escapement Timing Explorer
Choose ideal effective length, local gravity, half-amplitude and escape-wheel tooth count. The worksheet calculates exact ideal pendulum timing and a declared one-tooth-per-full-period wheel index.
Equation Used
- The pendulum is an ideal point mass on a massless inextensible suspension.
- The selected angle is peak amplitude on each side of vertical.
- The escape wheel advances half a tooth per beat and one tooth per complete period.
- The crutch connection is rigid in the illustration.
- Pallet geometry, recoil, lock, drop, impulse energy, friction, wear and real clock accuracy are excluded.
The period calculation includes the finite-amplitude correction. Wheel speed assumes half a tooth per beat. The schematic shows the counting sequence without solving the contact between pallet faces and teeth.
Anchor, pallets, crutch and escape wheel
The anchor's two pallet faces alternately hold and release the escape wheel. A crutch connects anchor motion to the pendulum. The wheel supplies intermittent impulses while the pendulum provides the timing reference.
The animation uses one declared counting convention: each beat advances the wheel half a tooth pitch, so one full oscillation advances one tooth. It does not claim dimensioned pallet faces or solve lock, recoil, drop or impulse contact.
Historical pendulum-clock escapement
The Science Museum Group catalogs anchor escapements in longcase clocks and preserves working models of both anchor and recoil forms. The illustrated topology is useful for seeing how oscillation gates a continuously driven wheel.
This page does not identify every pendulum escapement as the same mechanism: deadbeat, recoil, gravity and other forms have materially different contacts.
Ideal timing and declared indexing
The beat interval is T/2. Under the selected one-tooth-per-period convention, mean wheel speed is 60/(NT) rpm and angular advance per beat is 180/N degrees.
Amplitude delay is one million times [1/AGM(1,cos(alpha/2))-1]. It compares ideal finite-amplitude period with the same pendulum's small-angle period; it is not a real clock rate prediction.
A 30-tooth counting example
At L=0.993621 m, g=9.8066 m/s squared and alpha=3 degrees, the exact ideal period is about 2.00035 s and the beat interval about 1.00017 s. A 30-tooth wheel under the declared indexing turns at about 0.9998 rpm and advances 6 degrees per beat.
The amplitude-delay result compares this ideal period with the small-angle period at the same effective length and gravity.
Timing illustration, not pallet design
Tooth profile, pallet span, lock, drop, recoil, impulse angle, clearances, force and energy require a specific escapement geometry and drive model. They cannot be inferred from pendulum length and tooth count alone.
The timing results alone cannot diagnose a real clock or determine its adjustment and maintenance needs.
Pendulum escapement questions
Does the escapement set the ideal natural period?
The ideal pendulum relation sets the reference period; a real escapement and environment perturb it.
Why is the step half a tooth per beat?
Two beats make one complete pendulum period. Two half-tooth steps therefore advance one tooth per period.
Does the model distinguish recoil from deadbeat?
No. It shows a generic anchor counting topology and explicitly does not solve contact geometry.
Can it diagnose or rate a real clock?
No.
Pendulum escapement references
- Science Museum Group, anchor escapement collection: historic longcase-clock mechanisms and models.
- Anchor escapement technical animation record: named anchor, pallet and wheel topology plus the 30-tooth, one-minute example.
- OpenStax University Physics, Pendulums: ideal pendulum definition and small-angle relation.
- NIST DLMF, AGM algorithms: elliptic-integral evaluation method.
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