Draft of cymbal-bow-v1: the flat disk bowed into a shallow paraboloid
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README.mdmodified+2−1View file
@@ -7,12 +7,13 @@ This repository defines simulation challenges that approach a cymbal one feature
77 | ID | Title | Deliverable |
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 |
10+| [cymbal-bow-v1](challenges/cymbal-bow-v1/README.md) (draft) | The same disk bowed into a shallow paraboloid, 25 mm rise: the effect of curvature alone | 6 s mono wav, 48 kHz, float32 |
1011
1112 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.
1213
1314 ## Planned challenges
1415
15-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 central bell (a spherical dome around the mounting hole, which stiffens the center and raises the low modes), a radial bow (the shallow conical or curved profile of the main body, which couples bending and stretching even at small amplitude), 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.
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.
1617
1718 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-bow-v1/README.mdadded+111−0View file
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1+# cymbal-bow-v1: bowed bronze disk struck with a drum stick, geometrically nonlinear
2+
3+Draft: this specification is under discussion and has no solutions yet; the signal scale is not fixed.
4+
5+This is the second challenge of the cymbal tier. It is [cymbal-flat-disk-v1](../cymbal-flat-disk-v1/README.md) with one change: the disk is no longer flat. Its mid-surface is a shallow paraboloid rising 25 mm from the rim to the center, the bow of a cymbal without its bell, and the thickness is uniform along the normal. Material, damping, strike, listener, output format and error measures are unchanged, so the difference between the two challenges is the effect of curvature alone. Curvature couples stretching and bending already in the linear regime, which raises the low modes and changes their radiation, and it adds quadratic terms to the nonlinearity, so the pitch glide and the cascade of the flat disk take a different form. The deliverable is again six seconds of pressure at the listener from the surface acceleration.
6+
7+The machine-readable version of this specification is [challenge.json](challenge.json).
8+
9+## Model
10+
11+### Reference geometry
12+
13+Let $R$ be the disk radius and $H$ the bow height. The mid-surface is the graph
14+
15+$$
16+S = \{ (x, y, Z(r)) : r = \sqrt{x^2 + y^2} \le R \}, \qquad Z(r) = H \left( 1 - \frac{r^2}{R^2} \right),
17+$$
18+
19+a paraboloid with its apex at height $H$ on the axis and its rim in the plane $z = 0$. Its unit normal, chosen with positive $z$ component, is
20+
21+$$
22+\mathbf{n}(x, y) = \frac{(2 H x / R^2,\; 2 H y / R^2,\; 1)}{\sqrt{1 + 4 H^2 r^2 / R^4}}.
23+$$
24+
25+The body is the set of points within half a thickness of the mid-surface along its normal,
26+
27+$$
28+\Omega = \{ \mathbf{X} + s\, \mathbf{n}(\mathbf{X}) : \mathbf{X} \in S,\ |s| < h/2 \},
29+$$
30+
31+so the thickness is $h$ everywhere when measured normal to the surface. Its boundary consists of the top face $S^+ = \{ \mathbf{X} + (h/2) \mathbf{n} \}$ (the convex side, facing $+z$), the bottom face $S^- = \{ \mathbf{X} - (h/2) \mathbf{n} \}$, and the rim, the ruled surface swept by the normal segment at $r = R$. The maximum slope of the mid-surface, at the rim, is $2H/R \approx 0.246$, and its radius of curvature at the apex is $R^2 / (2H) \approx 0.826$ m. The greatest departure from the plane $z = 0$ is $H + h/2$ at the apex.
32+
33+### Material and equations of motion
34+
35+The material, the strain measure, the Saint Venant-Kirchhoff stress with Kelvin-Voigt viscosity, the mass-proportional loss, and the equation of motion in the reference configuration are exactly those of cymbal-flat-disk-v1:
36+
37+$$
38+\rho\, \frac{\partial^2 \mathbf{u}}{\partial t^2} = \nabla \cdot (\mathbf{F} \mathbf{S}) - 2 \sigma_0 \rho\, \frac{\partial \mathbf{u}}{\partial t}, \qquad
39+\mathbf{S} = \lambda\, \mathrm{tr}(\mathbf{E} + \eta\, \dot{\mathbf{E}})\, \mathbf{I} + 2 \mu\, (\mathbf{E} + \eta\, \dot{\mathbf{E}}),
40+$$
41+
42+with $\mathbf{E}$ the Green-Lagrange strain and the same constants. All of $\partial\Omega$ is traction-free except for the strike on $S^+$, the body starts at rest, and it is free, so the strike imparts net momentum and the body drifts.
43+
44+### The strike
45+
46+The strike acts on the top face and pushes along $-z$, as a stick coming down vertically:
47+
48+$$
49+\mathbf{F} \mathbf{S}\, \mathbf{N} = -\, p(\mathbf{y}, t)\, \mathbf{e}_z \quad \text{on } S^+, \qquad
50+p(\mathbf{y}, t) = \frac{F_{\max}}{C}\; \phi\!\left( \frac{r'(\mathbf{y})}{R_s} \right) \psi\!\left( \frac{t}{T_s} \right),
51+$$
52+
53+where for a point $\mathbf{y} = (y_1, y_2, y_3)$ of the top face $r'(\mathbf{y}) = \sqrt{(y_1 - x_s)^2 + (y_2 - y_s)^2}$ is its horizontal distance from the strike center, $\phi$ and $\psi$ are the raised-cosine bump and the Hann pulse of the flat-disk challenge, and
54+
55+$$
56+C = \int_{S^+} \phi\!\left( \frac{r'(\mathbf{y})}{R_s} \right) dS(\mathbf{y})
57+$$
58+
59+normalizes the pressure so that the total downward force is $F_{\max}\, \psi(t / T_s)$ and the impulse is $F_{\max} T_s / 2$, exactly as before. The footprint is the part of the top face whose horizontal projection lies within $R_s$ of $(x_s, y_s)$; since $dS = dA / (\mathbf{n} \cdot \mathbf{e}_z)$, $C$ is $c_\phi \pi R_s^2$ divided by a local cosine close to $0.98$, and its value to six digits is given in the parameter table for convenience. Solvers may compute it themselves.
60+
61+## Parameters
62+
63+All parameters of cymbal-flat-disk-v1 are unchanged; the bow height is new and the listener point is restated relative to the rim plane.
64+
65+| Symbol | Value | Unit | Meaning |
66+| --- | --- | --- | --- |
67+| $R$ | 0.2032 | m | disk radius (16 in diameter) |
68+| $H$ | 0.025 | m | bow height, apex above the rim plane |
69+| $h$ | 0.001 | m | thickness, normal to the mid-surface |
70+| $E$, $\nu$, $\rho$ | 110e9, 0.34, 8800 | Pa, , kg/m^3 | B20 bronze |
71+| $\sigma_0$, $\eta$ | 1.1, 1.2e-8 | 1/s, s | damping constants |
72+| $(x_s, y_s)$ | (0.17272, 0) | m | strike center (horizontal), $0.85 R$ |
73+| $R_s$, $T_s$, $F_{\max}$ | 0.005, 0.001, 1000 | m, s, N | strike footprint radius, duration, peak force |
74+| $C$ | see challenge.json | m^2 | strike normalization integral |
75+| $\rho_a$, $c_a$ | 1.204, 343 | kg/m^3, m/s | air |
76+| $\mathbf{x}_L$ | (0.06, 0.08, 0.5) | m | listener point, above the rim plane |
77+
78+The listener is 0.5 m above the rim plane, and therefore about 0.475 m above the apex; it is the same point as in the flat-disk challenge up to the half thickness, which was measured from the mid-plane there. The strike point on the mid-surface is at height $Z(0.85 R) \approx 6.9$ mm and the local slope is about 0.21.
79+
80+With these constants the shell's membrane (ring) frequency is $c_L / (2 \pi R_c) \approx 725$ Hz with $c_L = \sqrt{E / (\rho (1 - \nu^2))} \approx 3760$ m/s and $R_c$ the apex radius of curvature. Below it, modes that stretch the mid-surface are stiffened by the membrane strain and their frequencies rise above those of the flat disk, while the lowest modes of a free shallow shell, which are nearly inextensional, rise only modestly; the appendix gives the shallow-shell equations from which solvers can estimate the linear spectrum before computing it.
81+
82+## Output signal
83+
84+The output is the pressure at the listener point defined by the Rayleigh-type integral over the top face of the acceleration component along the reference normal,
85+
86+$$
87+s(t) = \frac{\rho_a}{2\pi} \int_{S^+} \frac{1}{|\mathbf{x}_L - \mathbf{y}|}\; \mathbf{n}(\mathbf{y}) \cdot \frac{\partial^2 \mathbf{u}}{\partial t^2}\!\left( \mathbf{y},\; t - \frac{|\mathbf{x}_L - \mathbf{y}|}{c_a} \right) dS(\mathbf{y}), \qquad \text{in Pa},
88+$$
89+
90+with $\mathbf{n}$ the unit normal of the reference top face and the integrand zero for negative retarded times. For the flat disk this is the Rayleigh integral; for the curved surface it is no longer the exact field of a baffled radiator, and it is adopted as a defined output with the same form so that the two challenges are directly comparable. A later challenge will replace it with the coupled acoustic problem.
91+
92+The wav has the same format as before: 48 kHz, mono, 32-bit float, 288000 samples, sample $k$ equal to $s(k / f_s) / s_\text{ref}$, no normalization. The scale $s_\text{ref}$ (`output.signal_scale`) will be fixed from the first solutions so that the peak sample is near 0.5, before any solution is registered.
93+
94+## Error measures and reference
95+
96+The same two error measures as for cymbal-flat-disk-v1 (relative waveform and spectrogram $\ell^2$ errors, the waveform error primary) and the same reference-selection rule. With a strike of this strength the response is strongly nonlinear and long waveform agreement between different methods is not expected; the spectrogram error is the more informative measure.
97+
98+## Optional field export
99+
100+As for the flat disk, but the quantity is the normal displacement of the mid-surface, $\mathbf{u} \cdot \mathbf{n}$, evaluated at the mid-surface point $(x_i, y_j, Z(r_{ij}))$ above each grid point: datasets `x`, `y` (61 points from $-R$ to $R$), `t` (3001 frames at 1 kHz) and `u` of shape (3001, 61, 61) in meters, NaN outside the disk.
101+
102+## Appendix: shallow-shell reduction
103+
104+For a shallow shell with initial mid-surface height $w_0(x, y) = Z(r)$ and uniform thickness, the Marguerre (Donnell-Mushtari-Vlasov) equations extend the von Kármán system of the flat disk by the initial shape:
105+
106+$$
107+\rho h\, \frac{\partial^2 w}{\partial t^2} + 2 \sigma_0 \rho h\, \frac{\partial w}{\partial t} + D\, \nabla^4 \left( w + \eta\, \frac{\partial w}{\partial t} \right) = \mathcal{L}(w + w_0, \Phi) + p, \qquad
108+\nabla^4 \Phi = -\frac{E h}{2} \left[ \mathcal{L}(w, w) + 2\, \mathcal{L}(w_0, w) \right],
109+$$
110+
111+with $w$ the additional normal (here vertical) displacement, $\Phi$ the Airy stress function, $\mathcal{L}(f, g) = f_{xx} g_{yy} + f_{yy} g_{xx} - 2 f_{xy} g_{xy}$, and $D = E h^3 / (12 (1 - \nu^2))$. For the paraboloid $\nabla^2 w_0 = -4H/R^2$ is constant and $\mathcal{L}(w_0, g) = -(2H/R^2) \nabla^2 g$. The terms linear in $w$ couple bending and stretching and are what raise the low modes; the quadratic terms give the shell a nonlinearity with a preferred direction, so the pitch glide need not be downward. These equations assume small slopes and are an approximation of the three-dimensional model above; at a maximum slope of 0.25 the approximation is reasonable but not exact, and the difference is part of what a converged three-dimensional solution will measure. Free edges require vanishing bending moment and effective shear for $w$ and vanishing in-plane tractions, which for the Airy function on a simply connected domain means $\Phi = \partial \Phi / \partial n = 0$ on the rim.
challenges/cymbal-bow-v1/challenge.jsonadded+75−0View file
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1+{
2+ "challenge_id": "cymbal-bow-v1",
3+ "title": "Bowed bronze disk (paraboloidal shell) struck with a drum stick, geometrically nonlinear: radiated pressure",
4+ "version": 1,
5+ "status": "draft",
6+ "output": {
7+ "type": "wav",
8+ "sample_rate_hz": 48000,
9+ "num_samples": 288000,
10+ "channels": 1,
11+ "sample_format": "float32",
12+ "signal": "acoustic pressure at the listener point from the Rayleigh-type integral of the normal acceleration over the top face",
13+ "signal_unit": "Pa",
14+ "signal_scale": null
15+ },
16+ "playback": {
17+ "gain_db": 30,
18+ "note": "Suggested playback gain for listening; presentation only, not part of the deliverable."
19+ },
20+ "geometry": {
21+ "type": "shell-of-revolution",
22+ "mid_surface": "z = Z(r) = bow_height * (1 - r^2 / disk_radius^2), 0 <= r <= disk_radius",
23+ "thickness": "uniform, measured along the normal to the mid-surface",
24+ "hole": null
25+ },
26+ "parameters": {
27+ "disk_radius": 0.2032,
28+ "bow_height": 0.025,
29+ "h": 0.001,
30+ "E": 110000000000.0,
31+ "nu": 0.34,
32+ "rho": 8800.0,
33+ "sigma0": 1.1,
34+ "eta": 1.2e-08,
35+ "strike_x": 0.17272,
36+ "strike_y": 0.0,
37+ "strike_radius": 0.005,
38+ "strike_duration": 0.001,
39+ "strike_peak_force": 1000.0,
40+ "rho_air": 1.204,
41+ "c_air": 343.0,
42+ "listener_x": 0.06,
43+ "listener_y": 0.08,
44+ "listener_z": 0.5,
45+ "strike_normalization_m2": 2.385927e-05
46+ },
47+ "metrics": {
48+ "primary": "waveform_l2",
49+ "waveform_l2": {},
50+ "spectrogram_l2": {
51+ "window": "hann_periodic",
52+ "n_fft": 2048,
53+ "hop": 512
54+ }
55+ },
56+ "fields": {
57+ "format": "hdf5",
58+ "quantity": "normal displacement of the mid-surface, u . n, at the mid-surface point above each grid point; NaN outside the disk",
59+ "units": "m",
60+ "axes": [
61+ "t",
62+ "y",
63+ "x"
64+ ],
65+ "shape": [
66+ 3001,
67+ 61,
68+ 61
69+ ],
70+ "frame_rate_hz": 1000,
71+ "dtype": "float32"
72+ },
73+ "reference_solution": null,
74+ "model": "saint-venant-kirchhoff"
75+}