Hiscox 1152 shows a double ratchet-wheel escapement with a pendulum. Two spaced wheels share a shaft, and their teeth act alternately on the pendulum pallets. The two wheel planes provide alternating contact with the pendulum pallets.
Double Ratchet-Wheel Escapement · Pendulum Timing
Two spaced wheels share one shaft. Compare ideal pendulum timing and wheel speed using an explicit half-pitch indexing example.
Equation Used
- The two wheels share one shaft.
- Equal tooth counts and half-pitch indexing are selected for the counting example.
- L is effective point-mass pendulum length.
- Amplitude is measured on each side of vertical.
- Gravity is 9.80665 m/s².
- Contact-force and recoil dynamics are not solved.
The calculator gives ideal pendulum timing and selected wheel indexing; it does not establish contact tolerances or timekeeping accuracy.
Two wheels and alternating pendulum pallets
The source presents the wheels edge-on, separated along their common shaft. The pendulum swings between them and carries the alternating pallet contacts. The oblique view exposes both wheel planes so their common arbor and separation can be seen.
The timing example selects equal tooth counts with half-pitch indexing. One pallet holds while the opposite contact is clear; a transfer admits the other wheel to its pallet. Each swing advances the common shaft by half a pitch, giving one complete tooth pitch per full pendulum oscillation.
Hiscox does not supply tooth count, exact indexing, pallet profiles or dimensions. Those details in the illustration are selected explanatory geometry. Its transfer windows and clearances are prescribed; it does not solve contact force, recoil or energy transfer.
Explore the pendulum and counting relationship
The calculator links effective pendulum length and amplitude to ideal oscillation period. Equal tooth counts and the stated half-pitch convention then convert that period into average common-shaft speed.
The effective length is measured from the suspension to the ideal point mass. A real compound pendulum needs its equivalent length from mass distribution and moment of inertia. The drawing fits the viewport and does not rescale the entire apparatus when this timing length is changed.
The entered wheel diameter gives tooth-tip arc spacing. It does not establish tooth strength, clearance, pallet engagement or a suitable clock design.
Pendulum period and selected wheel indexing
For effective length L, amplitude α on each side and g=9.80665 m/s², the ideal pendulum period is T=2π√(L/g)/AGM(1,cos(α/2)). AGM denotes the arithmetic-geometric mean. This includes the finite-amplitude correction to the small-angle period T₀=2π√(L/g).
Two swings occur per full period, so beats per minute are 120/T. With N teeth on each wheel and half-pitch indexing, advance per beat is 180/N degrees and mean common-shaft speed is 60/(NT) rpm.
For tip-circle diameter D, arc pitch is πD/N. Side-to-side bob displacement is 2L sinα, converted to millimetres in the output. Amplitude delay is 10⁶(T/T₀−1) ppm. It describes period change, not a guaranteed daily rate error for a real escapement.
A 1 m pendulum and 30-tooth wheels
At ±5° amplitude, the ideal period is approximately 2.007 s, giving approximately 59.8 alternating beats per minute. With 30 teeth on each wheel, the selected half-pitch step is 6° and mean shaft speed is about 0.996 rpm.
At a 100 mm tip-circle diameter, arc pitch is π×100/30≈10.472 mm. The bob moves approximately 174.3 mm from one extreme to the other.
Doubling pendulum length multiplies period by √2 and divides wheel speed by √2. Doubling tooth count halves shaft speed and angular advance without changing the pendulum period.
Scope of the model
The engraving establishes two spaced wheels, their common shaft and the alternating pendulum pallets. Dimensions and operating clearances must be established separately for a working escapement.
The calculator assumes an ideal pendulum, fixed local gravity, rigid geometry and reliable alternating releases. Air resistance, suspension stiffness, drive torque, pallet friction, wheel inertia and real escapement error are excluded. The animation is slowed to make the transfers visible.
Double ratchet-wheel questions
Does each wheel turn independently?
No. The wheels share the common shaft shown in the source.
Is one wheel only a locking wheel and the other only an impulse wheel?
No. In this model, the pendulum contacts alternate between the two wheel planes.
Does the source specify half-pitch indexing?
No numerical indexing is provided. Half-pitch indexing is the explicit counting convention selected for this demonstration.
Can these results be used as repair tolerances?
No. The period and counting equations do not determine contact dimensions.
Primary mechanism reference
Gardner D. Hiscox, Mechanical Movements, Powers, Devices and Appliances, device 1152, printed page 284, identifies a double ratchet-wheel escapement and pendulum with alternating pallet action. The oblique view, dimensions and half-pitch example are explanatory additions.
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