Tourbillon Lab (2D)

Why a balance wheel runs differently in different positions

A perfect balance wheel has its centre of mass on its axis, so gravity cannot affect its period. A real one has a small poise error: an unbalance m e (mass times offset of its centre of gravity). In a position where the balance plane is vertical, gravity adds a torque −m e g sinχ cos(θ + δ + Φ + W), where θ is the balance angle, δ the angular position of the heavy spot, Φ the cage angle, W the roll of the watch about its dial normal and χ the tilt of the dial from horizontal (χ = 0 dial up or down, no effect; χ = 90° vertical, full effect).

1. Rate error for a fixed position

For a large swing θ = A cosψ, first-order averaging gives the frequency ωeff2 = ω02 − 2 Γ sin(a0) J1(A)/A with Γ = m e g sinχ / I, a₀ = δ + Φ + W and the Bessel function J₁. For small A, J₁(A)/A → 1/2 and this is just a change of the effective stiffness. The rate is 86400 (ωeff/ω0 − 1) s/day: it depends on sin(a₀), so it is plus in one vertical position, minus in the opposite one, and zero when the heavy spot is level or the dial is horizontal.

2. What the tourbillon does

The tourbillon carries balance, lever and escape wheel in a cage that turns slowly (usually once a minute) so that Φ sweeps through a full circle. Because the rate error is proportional to sin(a₀), its average over one cage revolution is (1/2π) ∫ sin(δ + Φ + W) dΦ = 0: the positional errors cancel to first order. The cage is driven by a fixed fourth wheel (60 teeth) that meshes with the 6-leaf pinion of the escape wheel, so the escape wheel turns 10 times for each turn of the cage, matching 30 beats per escape revolution at 5 beats per second and a 60-second cage.

3. Drive

The escapement gives the balance Ebeat = π I ω02 Aset2 / (2Q) per beat over ±20°, which replaces the damping loss at the set amplitude.

Model checks on this page

• RK4 simulation of the balance in five fixed positions against the Bessel-function formula (first-order theory; second-order effects are a few per cent).
• RK4 simulation with the cage turning: the measured mean rate is a small fraction of the worst fixed-position error.
• The mean of the formula over a whole cage turn is zero, and its peak equals 86400 Γ |J₁(A)/A| / ω₀².

What to try

• Pick Crown up, then Crown down: the rate flips sign. Dial up gives about zero.
• Switch the tourbillon on and watch the accumulated error stay close to zero while the stopped watch drifts.
• Raise the poise error or the amplitude: errors grow; at some amplitudes J₁(A) is zero and the error vanishes (about 220°).
• Slow the cage to 600 s: the error no longer averages out within a minute but does over ten minutes.

Honest simplifications

• One poise error only; real watches also have hairspring weight, pivot friction and escapement effects that differ by position.
• The cage is treated as massless; its inertia and the extra drag of the cage pinion are ignored. A real tourbillon also costs amplitude and is itself a source of error.
• Averaging works only to first order and when the cage period is much longer than the balance period; the page shows the residual rate measured in the simulation.
• The movement is generic and not a copy of any maker's calibre; the escape wheel advance is a kinematic approximation of a Swiss lever.

• Pick Crown up, then Crown down: the rate flips sign.
• Switch the tourbillon on and watch the accumulated error stay near zero.
• Raise the poise error and compare the stopped and running cases.