Presets
Playback
Tuning scale and sound
The scale sets the pitch of mode 1 of each tube; the tube length follows from the tuning formula.
Tube material
Stiffness / density and the loss factor decide the pitch and how long it rings.
Tubes
Striker and strings
Wind
A gusty wind with a slowly wandering direction.
Layers
Model checks
The science
Why a tube rings at a definite pitch
A chime tube hung from strings at its vibration nodes is a free-free beam. Its bending modes follow Euler-Bernoulli theory, f_n = (beta_n L)^2 / (2 pi L^2) * sqrt(E I / (rho A)), with beta_n L = 4.730, 7.853, 10.996, 14.137 (roots of cos x cosh x = 1). For a thin tube sqrt(I / A) = sqrt(r_o^2 + r_i^2) / 2, so the pitch depends on the material only through sqrt(E / rho), grows with the wall radius and falls as 1 / L^2. To tune a tube to the pitch f_1 you cut it to L = sqrt((4.730)^2 sqrt(I/A) sqrt(E/rho) / (2 pi f_1)). The overtones are not harmonic: f_2 / f_1 = 2.756, f_3 / f_1 = 5.404, which is why a chime sounds like a bell and not like a flute.
Where the strings and the strike go
The strings are tied at the nodes of mode 1 (0.224 L from each end) so they do not damp the fundamental. A strike at the middle of the tube excites only the symmetric modes 1 and 3, because the mode 2 shape has a node there; the strike height slider moves the strike and the last graph shows the shape factor |phi_n(x)| that scales each overtone.
Wind, striker and tubes
Each of the striker and the tubes is a horizontal pendulum, x'' = -(g / l) x - 2 zeta w x' + F / m, pushed by the quadratic wind drag F = 1/2 rho_air C_d A |u - v| (u - v). The striker has a large light sail, so it moves more than the heavy tubes and swings into the ring. The wind speed is a mean plus an Ornstein-Uhlenbeck gust process with correlation time tau, and its direction wanders slowly. A hit is a collision with restitution e along the line of centres: impulse J = (1 + e) mu v_n, mu = m_s M / (m_s + M). Soft contacts below 4 cm/s count as pushing, not as a strike.
Sound
A strike of impulse J at height x gives mode n the amplitude J phi_n(x) / sqrt(M), and the mode then decays as exp(-pi f_n eta t), where eta is the material loss factor (the 60 dB decay time is T60 = 6.9 / (pi f eta)). Aluminium and steel ring for many seconds, brass and copper less, hollow bamboo only a fraction of a second. Switch the sound on to hear the four modes of the struck tube.
What is simplified
The tubes are thin Euler-Bernoulli beams, so shear and rotary inertia (a few per cent at mode 3 and above) are neglected; the strings, the top plate and the sail motion in the vertical direction are not modelled; tubes do not strike each other; the loss factors are typical orders of magnitude, and the radiated sound is a sum of damped sines, without the room.
What to try
- Hear the physics: switch the sound on, choose Calm breeze and compare Aluminium with Bamboo (hollow): the same wind gives a long ring against a dull knock.
- Press Detuned cluster: neighbouring tubes with almost equal pitch beat against each other.
- Move the strike height to 50 %: the overtone graph loses mode 2 (node of the mode shape).
- Reduce the gap or enlarge the sail: the hits per minute rise; at very small gaps the striker only leans on the tubes.
- Rotate the wind direction and see which tubes are struck: those on the downwind side.