My Simulator England / Living Traditions

English Bell
Ringing Simulator

Pull the rope. Find the rhythm.

Gudgeon axis

Wheel and rope

Free clapper

Interactive 3D • Bell Tower

Orbit
camera

Virtual belfry with illustrative model values. Not a guide to ringing a real bell: real ringing needs training, a safe tower and a teacher.

Lab & science

Drag to orbit, wheel / pinch to zoom, tap the striped sally to pull. Space = pull, R = reset, M = sound.

Setup

Bell (mass, centre of mass and swing period all change)

Stay limit (the slider stops the bell here)

Time scale

Volume

Start state (used by Reset)

Camera

Graphics (High on every device; Low = 1x pixel ratio, smaller shadows)

Live readout (model values)

Gold: bell angle (deg). Blue: clapper angle relative to the bell, delta (deg). Dots: strikes.

Experiment: period vs swing angle

A bell is not a harmonic oscillator: the bigger the swing, the longer the period. Run the model at 10 to 90 degrees (no friction, no pulls) and compare with the exact elliptic-integral result.

Missions

Mission 6: the Middle bell has I = 162 kg m², m = 450 kg and d = 0.30 m. What is its small-angle period T = 2 pi sqrt(I / (m g d)), in seconds?

Numerical checks

The science

What you see

A bell hung for English full-circle style ringing, shown here in the chiming range (0 to 90 degrees). The bell, its wooden headstock and the big wheel are one rigid body turning on the gudgeon pins. The rope runs round the wheel, so pulling it turns the whole assembly. Inside, a heavy iron clapper hangs freely from the crown; it lags behind the bell and hits the sound bow near the end of each swing. You can only push the bell at the right moment: a pull at the wrong time works against the swing.

Each control explained

Pull strength sets the peak rope force (100% = 450 N). The rope pulls on the wheel at radius R = 0.79 m, so the peak torque is F R.
Pull rope applies that torque for 0.5 s as a half-sine pulse, in the direction that carries the bell back from its current side. Pull too early and the pulse brakes the bell.
Swing angle is the peak angle of the last half-swing; the white dot on the dial is the live angle.
Timing dots grade your last five pulls by how close each came to the best moment (about 0.3 s after the turning point, found by the work sweep in the tests).
Stay limit is the wooden stay and slider that stop a chimed bell; the bell bounces off it with a restitution of 0.25.
Show mechanism adds labels, torque arrows and the side-view diagram of the bell, wheel, rope and clapper.

Equations

Bell (physical pendulum):  I th'' = - m g d sin(th) - tau_fric + F(t) R
  I = m k^2 (about the gudgeon axis),  d = centre of mass below the axis
  small swing:  T0 = 2 pi sqrt(I / (m g d))
  any swing:    T = T0 (2/pi) K(sin(th0/2))    (K = complete elliptic integral, via the AGM)
Rope pull:      F(t) = s Fmax sin(pi t / 0.5 s),  Fmax = 450 N,  R = 0.79 m
Friction:       tau_fric = 2.0 N m tanh(w/0.03) + 1.2 w
Clapper (pendulum on a moving pivot a = 0.41 m from the axis, length L = 0.95 m):
  L ps'' = - g sin(ps) - a th'' cos(th - ps) + a th'^2 sin(th - ps)
  strike when |ps - th| reaches 0.43 rad; ps' -> th' - e (ps' - th'),  e = 0.35
Work of the rope:  W = integral of tau th' dt   (checked against the energy gain)
Sally height:   y = y0 + R th     (the rope length that runs off the wheel)
Strike note:    f ~ m^(-1/3);  partials 0.5, 1, 1.2, 1.5, 2 x the strike note (hum, prime, tierce, quint, nominal)

Assumptions & honest simplifications

Masses, centres of mass, radii of gyration, rope force and friction are illustrative round numbers typical of a medium ring bell, not data for a real bell.
This is the chiming range (stay at 90 degrees or less). Real full-circle ringing swings through about 360 degrees; the ringer then controls the bell at the balance. That is not modelled.
The clapper's reaction on the bell, rope stretch and rope mass are ignored. The clapper is a point mass on a rigid rod.
The bell sound is synthesised: five decaying sine partials at the usual ratios of a tuned bell, with loudness from the impact speed. It does not model real bell acoustics (see the church-bell partial-tones lab for that).
The "ideal ringer" and the timing grade use the sweet spot found numerically for this model; people ring by feel and by sight.

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English BellRinging Simulator

English Bell
Ringing Simulator