A bell on a wheel swings as a large-angle physical pendulum; the clapper is a second, coupled pendulum that strikes only when the swing is big enough. Drag the 3D view to orbit, scroll or pinch to zoom.
| Partial | ratio | Hz | note | τ s |
|---|
Last 10 s: bell angle θ (gold), clapper angle relative to the bell ψ−θ (blue, dashed = wall at ±34°). Vertical bars = strikes, height = impact speed.
Period T/T0 against amplitude: exact elliptic-integral curve (gold), small-angle line (grey), measured RK4 points (blue) and the live bell (white).
Run an experiment: the simulation is repeated headlessly with the current bell and compared with theory.
Physical pendulum. The bell, headstock and wheel rotate together about the gudgeon axle. With moment of inertia I about the axle and the centre of mass a distance h below it: I·θ'' = −M g h sinθ − cθ' + F·Rw (rope force F on the wheel of radius Rw). Small swings have T0 = 2π√(I/(M g h)). For large amplitude θ0 the period grows: T = T0·(2/π)·K(k), k = sin(θ0/2), with K the complete elliptic integral of the first kind (computed here by the arithmetic-geometric mean). At 90° the period is 18% longer, at 170° about 2.4 times longer.
Clapper. The clapper (a point mass, 3% of the bell mass, on a rod hinged near the crown) is a second pendulum. The two angles obey the Lagrange equations of a double pendulum whose first link carries the bell's inertia; both are integrated with RK4 at 1 ms. When the clapper reaches the wall (|ψ−θ| = φmax) an impulse along the constraint reverses the relative velocity with restitution e = 0.3, conserving momentum of the whole system. For the default 0.9 m bell, below about 50° of swing the clapper never reaches the wall (see the Strike threshold experiment): the bell is silent. This is why ringers must pull the bell high.
Partials. A cast bell vibrates in modes that are not harmonics of one fundamental. A tuned bell has hum ½, prime 1, tierce 1.2 (minor third), quint 1.5 and nominal 2 times the prime, plus upper partials near 2.5, 3 and 4. Each decays as e−t/τ; the model uses τ ∝ √D·r−0.8. Scaling a bell geometrically by D scales every frequency by 1/D: here the prime is 450 Hz·m / D (the order of magnitude of real tuned bells; a large London bell of D = 2.7 m has its strike note near 165 Hz).
Beating. Real bells are slightly asymmetric, so each mode splits into two close frequencies; the pair beats at their difference. The "beat" slider detunes each pair by Δf·r.
Strike note. The ear assigns the bell a pitch ("strike note") that is not a loud spectral line: prime, quint, nominal and the upper partials sit near harmonics 2, 3, 4, 5, 6, 8 of a low fundamental (½ prime, the hum frequency). A harmonic-sieve search finds that fundamental even when you mute hum and prime (the "missing fundamental"). Listeners usually name the strike note near the prime, an octave above the template fundamental. The textbook value, strike note ≈ nominal/2, is shown beside the sieve result.
Simplifications. Bell geometry is fixed: centre of mass h = 0.55 D, clapper hinge 0.12 D and length 0.62 D below the axle, bell + headstock + wheel inertia I = M(h² + 0.16 D²). Wall thickness is not tied to the mass slider (in a real bell heavier at the same diameter would be thicker and higher-pitched); the mass changes the swing dynamics only. The sound is additive synthesis, not a finite-element model; amplitudes and decay times are plausible, not measured. The rope is drawn kinematically.
3D models generated with Meshy AI and processed in Blender (bell); belfry, wheel, headstock and clapper modelled in Blender. Three.js r160 + WebAudio.