Fusee and Chain Lab (3D)

The problem a fusee solves

A mainspring pushes hard when it is fully wound and weakly when it is nearly run down; to a first approximation its torque falls linearly with the angle still wound: Ts = Tmin + (Tmax − Tmin) φr/φtot. A balance wheel fed with a changing torque swings with a changing amplitude, and amplitude changes the rate. The fusee and chain (used in marine chronometers and many English watches) levels the force: the spring barrel pulls a chain, the chain unwinds from a cone-shaped, grooved fusee, and the lever arm r_f grows as the spring weakens.

1. Lever arm and drive torque

The chain tension is F = Ts / rb (barrel radius r_b); the torque on the fusee is Tf = F rf = Ts rf/rb. Chain length is conserved, so the remaining wound angle of the spring is the chain length still on the fusee divided by r_b: φr = Lf/rb with dLf/ds = 2π rf(s) along the groove (s = turns from the large end).

2. The ideal profile in closed form

Requiring T_f to be constant and writing ρ = T_max/T_min − 1 gives, after integrating, rf(s) = rmax / √(1 + ρ(2+ρ) s/N) for N turns, with the small end r_max/(1+ρ), the chain length L = 2π rmax N/(1+ρ/2) and a constant output Tout = Tmin rmax/rb. A straight cone matches only at the two ends and over-compensates between them; the page tabulates every profile numerically and compares the ideal one with the formula.

3. From torque to rate

The balance obeys I θ'' = −κ(θ + εθ3) − (ω0 I/Q) θ' + impulse. Each beat the escapement delivers Ebeat = η Tf Δψ (Δψ = fusee angle per beat, set by the train ratio 5670:1), so the amplitude settles at A = √(2 Q Ebeat / (π I ω0²)) and is proportional to √T_f. A softening spring (ε < 0, ε = −8b/3) makes the period T = T0(1 + b A²), so the rate is 86400 (Tref/T − 1) s/day relative to a watch regulated at the reference amplitude. Without a fusee the torque falls by a factor 1.8 over the run and the rate drifts accordingly; with the ideal fusee it stays flat.

Model checks on this page

• Closed-form ideal profile (radius, chain length) against the numerically tabulated one, and the output torque of the ideal fusee constant along the whole run.
• Energy bookkeeping: the work delivered to the fusee equals the work taken from the spring (independent quadratures).
• RK4 simulation of the balance: period against T₀(1 + bA²) and steady amplitude against the energy balance.

What to try

• Switch the profile between Ideal, Straight cone and No fusee and watch the torque and rate graphs.
• Raise the spring ratio T_max/T_min to 2.5: the ideal cone gets steeper and still holds the torque.
• Drag the run position or use time speed 30000x to run the whole 61 hours in about 7 seconds.
• Use the oblique camera to see the grooved cone and the chain moving across to the barrel.

Honest simplifications

• Mainspring torque is linear in the remaining wind; a real spring is closer to flat in the middle and steeper at the ends, so real fusees are cut from measured torque curves.
• The chain is inextensible and its tension is uniform; friction at the groove and in the train is folded into η; the slack in the free span is ignored.
• The escapement is a constant impulse over ±20° of the balance swing; the isochronism defect is a single coefficient b chosen to give a few seconds per day, not a measured one.
• The train shows four stages of 90:8, 72:8, 64:8 and 56:8 teeth and a 15-tooth escape wheel; the lever and hairspring motion are kinematic approximations.
• Generic movement, not a copy of any maker's design.