Mechanical Watch Movement (2D)

The balance wheel is a torsion oscillator

A balance wheel of moment of inertia I on a hairspring of torsional stiffness κ obeys I θ'' = -κ θ, a harmonic oscillator with period T = 2π sqrt(I / κ). A watch rate is quoted in vibrations (half-periods) per hour: vph = 7200 / T, so 28,800 vph means T = 0.25 s. The escapement lets the escape wheel advance one tooth per oscillation and gives the balance a small push; the gear train turns these beats into the second, minute and hour hands.

Temperature

The spring stiffness is proportional to Young's modulus E, which changes with temperature by a coefficient e (per kelvin), and the wheel's inertia changes with thermal expansion (ΔI / I = 2α ΔT). Hence Δf / f = 1/2 (e - 2α) ΔT, about -10 seconds per day per kelvin for a plain steel spring on a brass wheel and close to zero for compensated alloys.

Amplitude and isochronism

A linear spring would be perfectly isochronous (period independent of amplitude). Real escapements and springs are not, so the rate depends on amplitude; this lab models that with a user-set slope in seconds per day per 10 degrees. As the mainspring unwinds over the power reserve the amplitude falls, which shifts the rate.

Power

The balance stores E = 1/2 κ θ0^2. With quality factor Q the loss per cycle is 2π E / Q, and the barrel must supply this divided by the escapement efficiency. Dividing the mainspring energy by that power gives the power reserve.

Simplifications

Illustrative component values, no position (gravity) error, no shock, a single constant Q and escapement efficiency, rate error modelled by the two terms above only.

• Switch the spring material from plain steel to compensated alloy and watch the temperature curve flatten.
• Raise the day-night temperature swing and see the deviation line wander.
• Pick 36,000 vph: the stiffness needed rises with the square of the frequency, and so does the power use.
• Reduce the mainspring energy and see the power reserve shrink and the amplitude fall.