cymbal-bell-v1: the bowed disk with a cosine-dome bell and a mounting hole
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README.mdmodified+2−1View file
@@ -8,12 +8,13 @@ This repository defines simulation challenges that approach a cymbal one feature
88 | --- | --- | --- |
99 | [cymbal-flat-disk-v1](challenges/cymbal-flat-disk-v1/README.md) | Flat bronze disk struck with a drum stick, geometrically nonlinear (Saint Venant-Kirchhoff): radiated pressure | 6 s mono wav, 48 kHz, float32 |
1010 | [cymbal-bow-v1](challenges/cymbal-bow-v1/README.md) | The same disk bowed into a shallow paraboloid, 25 mm rise: the effect of curvature alone | 6 s mono wav, 48 kHz, float32 |
11+| [cymbal-bell-v1](challenges/cymbal-bell-v1/README.md) | The bowed disk with a 110 mm cosine-dome bell 15 mm high and a 12 mm mounting hole: the effect of the bell alone | 6 s mono wav, 48 kHz, float32 |
1112
1213 The starting point is the simplest object that can sound like a cymbal: a flat, free disk of bell bronze with the diameter and thickness of a 16 inch crash, uniformly thick, struck near the rim by a short stick-like force pulse and radiating to a listener above it through the Rayleigh integral. The material model is Saint Venant-Kirchhoff, so the strain is exact and the response geometrically nonlinear; the material remains elastic. This is the model of the plate tier's second challenge applied to a disk with a harder, shorter strike, and it produces the pitch glide and broadband cascade that distinguish a cymbal from a struck plate. Thin-plate theories (von Kármán with or without in-plane inertia) are approximations to this model, and their error relative to a converged three-dimensional solution includes the modeling error of plate theory. Until such a solution is submitted, the reference is the best thin-plate solution and the reported errors measure only differences within plate theory; the results page says which it is.
1314
1415 ## Planned challenges
1516
16-The features of a real cymbal are added one at a time, each as a new challenge on the same material and strike, so that the effect of each on the sound and on the difficulty of the computation can be seen in isolation. In the intended order: a radial bow (drafted as cymbal-bow-v1: a shallow paraboloid, which couples bending and stretching even at small amplitude), a central bell (a spherical dome around the mounting hole, which stiffens the center and adds the ping), hammering (a pattern of local dents that breaks the axisymmetry and detunes the degenerate mode pairs), and a thickness that varies with radius (thinner at the rim). Curved and dented geometries also make the Cartesian-grid discretizations of the flat disk less natural, and polar or mapped grids are expected to appear among the solutions.
17+The features of a real cymbal are added one at a time, each as a new challenge on the same material and strike, so that the effect of each on the sound and on the difficulty of the computation can be seen in isolation. In the intended order: a radial bow (cymbal-bow-v1: a shallow paraboloid, which couples bending and stretching even at small amplitude), a central bell (cymbal-bell-v1: a cosine dome around the mounting hole, which stiffens the center and adds the ping), hammering (a pattern of local dents that breaks the axisymmetry and detunes the degenerate mode pairs), and a thickness that varies with radius (thinner at the rim). Curved and dented geometries also make the Cartesian-grid discretizations of the flat disk less natural, and polar or mapped grids are expected to appear among the solutions.
1718
1819 Each challenge lives in `challenges/<challenge-id>/` and consists of a `README.md`, which is the authoritative specification, and a `challenge.json` with the parameters, output format, error measures, and the recommended grid for optional field exports in machine-readable form. A challenge is never changed once solutions exist for it; a corrected or modified model gets a new ID.
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challenges/cymbal-bell-v1/README.mdadded+47−0View file
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1+# cymbal-bell-v1: bowed bronze disk with a bell and a mounting hole, struck with a drum stick
2+
3+This is the third challenge of the cymbal tier. It is [cymbal-bow-v1](../cymbal-bow-v1/README.md) with two additions at the center: a bell, a dome 110 mm across and 15 mm high on top of the bow, and the mounting hole, 12 mm across, through its apex. Material, thickness, damping, strike, listener, output and error measures are unchanged, so the difference between the two challenges is the effect of the bell alone. On a real cymbal the bell carries the "ping" of a stick played on it and stiffens the center so that the low modes of the whole cymbal rise; here the strike stays on the bow, so what changes is the body's response, not the excitation. The deliverable is again six seconds of pressure at the listener.
4+
5+The machine-readable version of this specification is [challenge.json](challenge.json).
6+
7+## Model
8+
9+### Reference geometry
10+
11+With $R$ the disk radius, $H$ the bow height, $r_b$ and $h_b$ the bell radius and height, and $r_h$ the hole radius, the mid-surface is the graph
12+
13+$$
14+S = \{ (x, y, Z(r)) : r_h \le r \le R \}, \qquad
15+Z(r) = H \left( 1 - \frac{r^2}{R^2} \right) + \begin{cases} \dfrac{h_b}{2} \left( 1 + \cos \dfrac{\pi r}{r_b} \right), & r < r_b, \\[2ex] 0, & r \ge r_b. \end{cases}
16+$$
17+
18+The bell is a cosine dome: its slope vanishes on the axis and at its edge $r = r_b$, where it joins the bow with continuous slope but a jump in curvature, the crease that a lathed bell has. The unit normal, the body $\Omega$ as the normal offset of $S$ by $\pm h/2$, the top face $S^+$ and the bottom face $S^-$ are defined exactly as for cymbal-bow-v1, and the rim is the ruled surface swept by the normal segment at $r = R$. The hole adds an inner edge, the ruled surface swept by the normal segment at $r = r_h$, which is traction-free like the rest of the boundary. The largest slope of the mid-surface is about 0.46, on the flank of the bell at $r \approx r_b / 2$ (the dome contributes 0.43 there and the bow 0.03); the bow alone reaches 0.25, at the rim. The apex of the bell is $H + h_b = 40$ mm above the rim plane.
19+
20+### Material, equations of motion, strike, and output
21+
22+All as for cymbal-bow-v1: the Saint Venant-Kirchhoff solid with Kelvin-Voigt viscosity and mass-proportional loss, traction-free everywhere except the strike, at rest at $t = 0$, free (with the rigid motion linearized in the output as defined there). The strike is the same vertical raised-cosine pulse on the top face at $0.85 R$ on the $x$ axis; the bell does not reach that radius, so the normalization constant $C$ is that of cymbal-bow-v1. The output is the same integral of the normal acceleration over $S^+$ with the retarded time and the same wav format, sampling and scale.
23+
24+## Parameters
25+
26+Those of cymbal-bow-v1, plus the bell and the hole.
27+
28+| Symbol | Value | Unit | Meaning |
29+| --- | --- | --- | --- |
30+| $R$, $H$, $h$ | 0.2032, 0.025, 0.001 | m | disk radius, bow height, thickness |
31+| $r_b$ | 0.055 | m | bell radius |
32+| $h_b$ | 0.015 | m | bell height above the bow |
33+| $r_h$ | 0.006 | m | hole radius |
34+| $E$, $\nu$, $\rho$ | 110e9, 0.34, 8800 | Pa, , kg/m^3 | B20 bronze |
35+| $\sigma_0$, $\eta$ | 1.1, 1.2e-8 | 1/s, s | damping constants |
36+| $(x_s, y_s)$, $R_s$, $T_s$, $F_{\max}$, $C$ | (0.17272, 0), 0.005, 0.001, 1000, 2.385927e-5 | m, m, s, N, m^2 | strike, as for the bow |
37+| $\rho_a$, $c_a$, $\mathbf{x}_L$ | 1.204, 343, (0.06, 0.08, 0.5) | kg/m^3, m/s, m | air and listener, as for the bow |
38+
39+The bell's flank slope of 0.46 is beyond the range where shallow-shell theories are accurate, and its curvature (radius about $r_b^2 / (\pi^2 h_b / 2) \approx 41$ mm at the apex) is far tighter than the bow's. Thin-shell solutions of this challenge therefore carry a modeling error in the bell region that the flat disk and the bow do not have; a converged three-dimensional solution is the arbiter, as always.
40+
41+## Error measures, reference, field export
42+
43+As for cymbal-bow-v1. The field export is the normal displacement of the mid-surface above each point of the 61 by 61 grid, NaN outside the disk and inside the hole. Note the caution stated with the first solutions of the flat disk and the bow: at this strike the response is chaotic beyond about 0.15 s, so the waveform and spectrogram errors between any two solutions have a floor of roughly 0.6 and 0.5.
44+
45+## Appendix: shallow-shell reduction
46+
47+The Marguerre equations of cymbal-bow-v1 apply with the full profile as the initial shape $w_0 = Z$. For the bell, $\nabla^2 w_0$ is no longer constant: with $Z_b(r) = (h_b / 2)(1 + \cos(\pi r / r_b))$, $Z_b'' = -(h_b / 2)(\pi / r_b)^2 \cos(\pi r / r_b)$ and $Z_b' / r = -(h_b / 2)(\pi / r_b) \sin(\pi r / r_b) / r$, so the coupling operator $\mathcal{L}(w_0, \cdot)$ has the two principal curvatures $Z_{rr}$ and $Z_r / r$ as coefficients on the meridional and circumferential second derivatives. The hole makes the domain of the Airy function an annulus; the in-plane traction-free condition on the inner edge then allows $\Phi$ and $\partial \Phi / \partial n$ to take the values of an arbitrary linear function there, fixed by the single-valuedness of the in-plane displacement (Michell's conditions), rather than zero. A solver that closes the hole neglects the mass and stiffness of a 12 mm disk at the apex, a small effect on a body whose apex hardly moves, but an approximation that should be stated.
challenges/cymbal-bell-v1/challenge.jsonadded+77−0View file
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1+{
2+ "challenge_id": "cymbal-bell-v1",
3+ "title": "Bowed bronze disk with a bell and a mounting hole, struck with a drum stick, geometrically nonlinear: radiated pressure",
4+ "version": 1,
5+ "output": {
6+ "type": "wav",
7+ "sample_rate_hz": 48000,
8+ "num_samples": 288000,
9+ "channels": 1,
10+ "sample_format": "float32",
11+ "signal": "acoustic pressure at the listener point from the Rayleigh-type integral over the top face of the normal acceleration, with the rigid motion linearized",
12+ "signal_unit": "Pa",
13+ "signal_scale": 80.0
14+ },
15+ "playback": {
16+ "gain_db": 30,
17+ "note": "Formerly a suggested playback gain; the results sites now loudness-normalize playback and this field is unused."
18+ },
19+ "geometry": {
20+ "type": "shell-of-revolution",
21+ "mid_surface": "z = Z(r) = bow_height * (1 - r^2 / disk_radius^2) + (bell_height / 2) * (1 + cos(pi r / bell_radius)) * [r < bell_radius], hole_radius <= r <= disk_radius",
22+ "thickness": "uniform, measured along the normal to the mid-surface",
23+ "hole": "circular, radius hole_radius, centered on the axis, free edge"
24+ },
25+ "parameters": {
26+ "disk_radius": 0.2032,
27+ "bow_height": 0.025,
28+ "bell_radius": 0.055,
29+ "bell_height": 0.015,
30+ "hole_radius": 0.006,
31+ "h": 0.001,
32+ "E": 110000000000.0,
33+ "nu": 0.34,
34+ "rho": 8800.0,
35+ "sigma0": 1.1,
36+ "eta": 1.2e-08,
37+ "strike_x": 0.17272,
38+ "strike_y": 0.0,
39+ "strike_radius": 0.005,
40+ "strike_duration": 0.001,
41+ "strike_peak_force": 1000.0,
42+ "strike_normalization_m2": 2.385927e-05,
43+ "rho_air": 1.204,
44+ "c_air": 343.0,
45+ "listener_x": 0.06,
46+ "listener_y": 0.08,
47+ "listener_z": 0.5
48+ },
49+ "metrics": {
50+ "primary": "waveform_l2",
51+ "waveform_l2": {},
52+ "spectrogram_l2": {
53+ "window": "hann_periodic",
54+ "n_fft": 2048,
55+ "hop": 512
56+ }
57+ },
58+ "fields": {
59+ "format": "hdf5",
60+ "quantity": "normal displacement of the mid-surface, u . n, at the mid-surface point above each grid point; NaN outside the disk and inside the hole",
61+ "units": "m",
62+ "axes": [
63+ "t",
64+ "y",
65+ "x"
66+ ],
67+ "shape": [
68+ 3001,
69+ 61,
70+ 61
71+ ],
72+ "frame_rate_hz": 1000,
73+ "dtype": "float32"
74+ },
75+ "reference_solution": null,
76+ "model": "saint-venant-kirchhoff"
77+}