cymbal-hammered-v1: the belled disk with twenty hammer dents at fixed positions
3 changed files+268−1
README.mdmodified+2−1View file
@@ -9,12 +9,13 @@ This repository defines simulation challenges that approach a cymbal one feature
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 |
1111 | [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 |
12+| [cymbal-hammered-v1](challenges/cymbal-hammered-v1/README.md) | The belled disk with twenty hammer dents at fixed positions: the effect of breaking the symmetry | 6 s mono wav, 48 kHz, float32 |
1213
1314 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.
1415
1516 ## Planned challenges
1617
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.
18+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 (cymbal-hammered-v1: twenty dents at fixed positions, which break the axisymmetry and detune the degenerate mode pairs; denser patterns can follow), 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.
1819
1920 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.
2021
challenges/cymbal-hammered-v1/README.mdadded+67−0View file
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1+# cymbal-hammered-v1: bowed and belled bronze disk with twenty hammer dents, struck with a drum stick
2+
3+This is the fourth challenge of the cymbal tier. It is [cymbal-bell-v1](../cymbal-bell-v1/README.md) with twenty hammer dents pressed into the bow: shallow circular depressions, 24 mm across and half a thickness deep, at fixed positions listed below. Everything else is unchanged, so the difference between the two challenges is the effect of the dents alone. A real cymbal carries hundreds of such marks; twenty large ones are a first step that already does the essential thing, which is to break the symmetry of revolution. On an axisymmetric body every mode with nodal diameters comes in a degenerate pair; the dents split the pairs by different amounts and mix modes of different angular order, which is where the shimmer and the beating of a cymbal's ring come from, and they add local curvature that feeds the nonlinear cascade. The dents are a change of the rest shape only; the residual stresses and the work hardening of real hammering are not modeled. The deliverable is again six seconds of pressure at the listener.
4+
5+The machine-readable version of this specification, including the dent list, is [challenge.json](challenge.json).
6+
7+## Model
8+
9+### Reference geometry
10+
11+With $Z_0(r)$ the bow-plus-bell profile of cymbal-bell-v1 (paraboloid of height $H$ and cosine dome of radius $r_b$ and height $h_b$), the mid-surface is the graph over the annulus $r_h \le r \le R$ of
12+
13+$$
14+Z(x, y) = Z_0(r) + \sum_{k=1}^{20} d(\rho_k), \qquad \rho_k = \sqrt{(x - x_k)^2 + (y - y_k)^2}, \qquad
15+d(\rho) = \begin{cases} -\dfrac{\delta}{2} \left( 1 + \cos \dfrac{\pi \rho}{r_d} \right), & \rho < r_d, \\[2ex] 0, & \rho \ge r_d, \end{cases}
16+$$
17+
18+a raised-cosine depression of radius $r_d$ and depth $\delta$ at each center $(x_k, y_k)$, with continuous slope at its edge. The dents lie entirely on the bow (their centers are at least $r_b + r_d$ from the axis and at most $R - r_d$ from it), do not overlap one another (centers at least 30 mm apart), and stay clear of the strike footprint (the nearest center is more than $r_d + R_s$ from the strike point). The body, the top and bottom faces, the rim and the hole edge are the normal offsets of this surface exactly as before, with the normal now depending on both $x$ and $y$.
19+
20+The centers were drawn once, uniformly over the area of the annulus between $r_b + r_d$ and $R - r_d$ with rejection of positions closer than 30 mm to an earlier one or within 22 mm of the strike point, and rounded to 0.1 mm. Only the resulting coordinates are part of the specification.
21+
22+| dent | $x_k$ (mm) | $y_k$ (mm) | radius from axis (mm) |
23+| --- | --- | --- | --- |
24+| 1 | 82.5 | 136.4 | 159.4 |
25+| 2 | -95.4 | -90.7 | 131.6 |
26+| 3 | 39.1 | -70.9 | 81.0 |
27+| 4 | -123.4 | -104.7 | 161.8 |
28+| 5 | 63.5 | 83.5 | 104.9 |
29+| 6 | 135.7 | 22.2 | 137.5 |
30+| 7 | 139.7 | -55.4 | 150.3 |
31+| 8 | 157.3 | 44.2 | 163.4 |
32+| 9 | -82.6 | -12.3 | 83.5 |
33+| 10 | -174.6 | 37.2 | 178.5 |
34+| 11 | -29.0 | 153.9 | 156.6 |
35+| 12 | 33.8 | 97.8 | 103.5 |
36+| 13 | 17.2 | -100.3 | 101.8 |
37+| 14 | 99.6 | -121.0 | 156.7 |
38+| 15 | 163.8 | 86.4 | 185.2 |
39+| 16 | -24.1 | -183.7 | 185.3 |
40+| 17 | 174.6 | -30.3 | 177.2 |
41+| 18 | -125.1 | 69.5 | 143.1 |
42+| 19 | 14.7 | 131.7 | 132.5 |
43+| 20 | -152.8 | 112.2 | 189.6 |
44+
45+### Material, equations of motion, strike, and output
46+
47+All as for cymbal-bell-v1 and 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. The strike is the same vertical pulse at $0.85 R$ on the $x$ axis; no dent reaches its footprint, so the normalization constant $C$ is unchanged. The output is the same integral of the normal acceleration over the top face, with the same wav format and scale.
48+
49+## Parameters
50+
51+Those of cymbal-bell-v1, plus the dents.
52+
53+| Symbol | Value | Unit | Meaning |
54+| --- | --- | --- | --- |
55+| $r_d$ | 0.012 | m | dent radius |
56+| $\delta$ | 0.0005 | m | dent depth (half the thickness) |
57+| $(x_k, y_k)$ | table above | m | dent centers, 20 of them |
58+
59+A dent's flank slope is at most $\pi \delta / (2 r_d) \approx 0.065$, small compared with the bell's, and its curvature at the center is $(\delta / 2)(\pi / r_d)^2 \approx 17$ 1/m (a radius of 58 mm), comparable to the bell's. The exact solution is no longer symmetric about the $x$ axis, so solvers cannot halve their work by symmetry, and the modes antisymmetric in $y$ are now excited and heard.
60+
61+## Error measures, reference, field export
62+
63+As for cymbal-bell-v1, including the caution that at this strike the response is chaotic beyond about 0.15 s.
64+
65+## Appendix: shallow-shell reduction
66+
67+The Marguerre equations apply with the full dented profile as the initial shape $w_0 = Z(x, y)$. The dents enter $\mathcal{L}(w_0, \cdot)$ through their second derivatives, which are no longer functions of $r$ alone; a solver that expands in the modes of the undented shell needs enough modes to represent the localized curvature of each dent, and one working on a grid needs several cells across a dent (12 mm radius against 4 mm or 8 mm cells here).
challenges/cymbal-hammered-v1/challenge.jsonadded+199−0View file
@@ -0,0 +1,199 @@
1+{
2+ "challenge_id": "cymbal-hammered-v1",
3+ "title": "Bowed bronze disk with a bell, a mounting hole and twenty hammer dents, 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": 160.0
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) + sum over dents of -depth/2 * (1 + cos(pi rho / radius)) * [rho < radius], with Z(r) the bow-plus-bell profile of cymbal-bell-v1 and rho the distance to the dent center",
22+ "thickness": "uniform, measured along the normal to the mid-surface",
23+ "hole": "circular, radius hole_radius, centered on the axis, free edge",
24+ "dents": [
25+ {
26+ "x": 0.0825,
27+ "y": 0.1364,
28+ "radius": 0.012,
29+ "depth": 0.0005
30+ },
31+ {
32+ "x": -0.0954,
33+ "y": -0.0907,
34+ "radius": 0.012,
35+ "depth": 0.0005
36+ },
37+ {
38+ "x": 0.0391,
39+ "y": -0.0709,
40+ "radius": 0.012,
41+ "depth": 0.0005
42+ },
43+ {
44+ "x": -0.1234,
45+ "y": -0.1047,
46+ "radius": 0.012,
47+ "depth": 0.0005
48+ },
49+ {
50+ "x": 0.0635,
51+ "y": 0.0835,
52+ "radius": 0.012,
53+ "depth": 0.0005
54+ },
55+ {
56+ "x": 0.1357,
57+ "y": 0.0222,
58+ "radius": 0.012,
59+ "depth": 0.0005
60+ },
61+ {
62+ "x": 0.1397,
63+ "y": -0.0554,
64+ "radius": 0.012,
65+ "depth": 0.0005
66+ },
67+ {
68+ "x": 0.1573,
69+ "y": 0.0442,
70+ "radius": 0.012,
71+ "depth": 0.0005
72+ },
73+ {
74+ "x": -0.0826,
75+ "y": -0.0123,
76+ "radius": 0.012,
77+ "depth": 0.0005
78+ },
79+ {
80+ "x": -0.1746,
81+ "y": 0.0372,
82+ "radius": 0.012,
83+ "depth": 0.0005
84+ },
85+ {
86+ "x": -0.029,
87+ "y": 0.1539,
88+ "radius": 0.012,
89+ "depth": 0.0005
90+ },
91+ {
92+ "x": 0.0338,
93+ "y": 0.0978,
94+ "radius": 0.012,
95+ "depth": 0.0005
96+ },
97+ {
98+ "x": 0.0172,
99+ "y": -0.1003,
100+ "radius": 0.012,
101+ "depth": 0.0005
102+ },
103+ {
104+ "x": 0.0996,
105+ "y": -0.121,
106+ "radius": 0.012,
107+ "depth": 0.0005
108+ },
109+ {
110+ "x": 0.1638,
111+ "y": 0.0864,
112+ "radius": 0.012,
113+ "depth": 0.0005
114+ },
115+ {
116+ "x": -0.0241,
117+ "y": -0.1837,
118+ "radius": 0.012,
119+ "depth": 0.0005
120+ },
121+ {
122+ "x": 0.1746,
123+ "y": -0.0303,
124+ "radius": 0.012,
125+ "depth": 0.0005
126+ },
127+ {
128+ "x": -0.1251,
129+ "y": 0.0695,
130+ "radius": 0.012,
131+ "depth": 0.0005
132+ },
133+ {
134+ "x": 0.0147,
135+ "y": 0.1317,
136+ "radius": 0.012,
137+ "depth": 0.0005
138+ },
139+ {
140+ "x": -0.1528,
141+ "y": 0.1122,
142+ "radius": 0.012,
143+ "depth": 0.0005
144+ }
145+ ]
146+ },
147+ "parameters": {
148+ "disk_radius": 0.2032,
149+ "bow_height": 0.025,
150+ "bell_radius": 0.055,
151+ "bell_height": 0.015,
152+ "hole_radius": 0.006,
153+ "h": 0.001,
154+ "E": 110000000000.0,
155+ "nu": 0.34,
156+ "rho": 8800.0,
157+ "sigma0": 1.1,
158+ "eta": 1.2e-08,
159+ "strike_x": 0.17272,
160+ "strike_y": 0.0,
161+ "strike_radius": 0.005,
162+ "strike_duration": 0.001,
163+ "strike_peak_force": 1000.0,
164+ "strike_normalization_m2": 2.385927e-05,
165+ "rho_air": 1.204,
166+ "c_air": 343.0,
167+ "listener_x": 0.06,
168+ "listener_y": 0.08,
169+ "listener_z": 0.5
170+ },
171+ "metrics": {
172+ "primary": "waveform_l2",
173+ "waveform_l2": {},
174+ "spectrogram_l2": {
175+ "window": "hann_periodic",
176+ "n_fft": 2048,
177+ "hop": 512
178+ }
179+ },
180+ "fields": {
181+ "format": "hdf5",
182+ "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",
183+ "units": "m",
184+ "axes": [
185+ "t",
186+ "y",
187+ "x"
188+ ],
189+ "shape": [
190+ 3001,
191+ 61,
192+ 61
193+ ],
194+ "frame_rate_hz": 1000,
195+ "dtype": "float32"
196+ },
197+ "reference_solution": null,
198+ "model": "saint-venant-kirchhoff"
199+}