READMEs for the 150-dent hammering
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README.mdmodified+2−2View file
@@ -9,14 +9,14 @@ 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 |
12+| [cymbal-hammered-v1](challenges/cymbal-hammered-v1/README.md) | The belled disk with 150 hammer dents at fixed positions: the effect of breaking the symmetry | 6 s mono wav, 48 kHz, float32 |
1313 | [cymbal-tapered-v1](challenges/cymbal-tapered-v1/README.md) | The hammered cymbal lathed to a taper, 1.2 mm at the bell to 0.7 mm at the rim: the effect of the thickness profile | 6 s mono wav, 48 kHz, float32 |
1414
1515 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.
1616
1717 ## Planned challenges
1818
19-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 (cymbal-tapered-v1: 1.2 mm at the bell to 0.7 mm 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.
19+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: 150 dents at fixed positions, which break the axisymmetry and detune the degenerate mode pairs), and a thickness that varies with radius (cymbal-tapered-v1: 1.2 mm at the bell to 0.7 mm 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.
2020
2121 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-hammered-v1/README.mdmodified+10−33View file
@@ -1,6 +1,6 @@
1-# cymbal-hammered-v1: bowed and belled bronze disk with twenty hammer dents, struck with a drum stick
1+# cymbal-hammered-v1: bowed and belled bronze disk with 150 hammer dents, struck with a drum stick
22
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.
3+This is the fourth challenge of the cymbal tier. It is [cymbal-bell-v1](../cymbal-bell-v1/README.md) with 150 hammer dents pressed into the bow: circular depressions 20 mm across and a full thickness deep, at fixed positions listed in challenge.json. Everything else is unchanged, so the difference between the two challenges is the effect of the dents alone. This is the hammering of a hand-hammered cymbal, visibly dimpled; a first version with twenty dents half a thickness deep split the degenerate mode pairs by only a couple of hertz and was hard to tell from the bell alone in a crash, so the pattern was made denser and deeper. What the dents do is 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.
44
55 The machine-readable version of this specification, including the dent list, is [challenge.json](challenge.json).
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@@ -11,36 +11,13 @@ The machine-readable version of this specification, including the dent list, is
1111 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
1212
1313 $$
14-Z(x, y) = Z_0(r) + \sum_{k=1}^{20} d(\rho_k), \qquad \rho_k = \sqrt{(x - x_k)^2 + (y - y_k)^2}, \qquad
14+Z(x, y) = Z_0(r) + \sum_{k=1}^{150} d(\rho_k), \qquad \rho_k = \sqrt{(x - x_k)^2 + (y - y_k)^2}, \qquad
1515 d(\rho) = \begin{cases} -\dfrac{\delta}{2} \left( 1 + \cos \dfrac{\pi \rho}{r_d} \right), & \rho < r_d, \\[2ex] 0, & \rho \ge r_d, \end{cases}
1616 $$
1717
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 $y$.
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), overlap only at their edges (centers at least 21 mm apart, so the flat crest between neighbours can vanish), and stay clear of the strike footprint (the nearest center is more than $r_d + R_s + 5$ mm 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 $y$.
1919
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 |
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 21 mm to an earlier one or within 20 mm of the strike point, and rounded to 0.1 mm. Only the resulting coordinates, listed in `geometry.dents` of challenge.json (x, y, radius and depth per dent), are part of the specification; they cover the bow at a density of about 1.2 dents per square decimeter.
4421
4522 ### Material, equations of motion, strike, and output
4623
@@ -52,11 +29,11 @@ Those of cymbal-bell-v1, plus the dents.
5229
5330 | Symbol | Value | Unit | Meaning |
5431 | --- | --- | --- | --- |
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 |
32+| $r_d$ | 0.010 | m | dent radius |
33+| $\delta$ | 0.0010 | m | dent depth (one thickness) |
34+| $(x_k, y_k)$ | challenge.json | m | dent centers, 150 of them |
5835
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.
36+A dent's flank slope is at most $\pi \delta / (2 r_d) \approx 0.16$, and its curvature at the center is $(\delta / 2)(\pi / r_d)^2 \approx 49$ 1/m (a radius of 20 mm), twice as tight as the bell's; on the linear shell modes at 4 mm resolution these dents split the degenerate pairs by about 5 Hz (median over the first 40 modes), against 0.3 Hz for the undented bell on the same grid and 2.5 Hz for the earlier pattern of twenty half-depth dents. 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.
6037
6138 ## Error measures, reference, field export
6239
@@ -64,4 +41,4 @@ As for cymbal-bell-v1, including the caution that at this strike the response is
6441
6542 ## Appendix: shallow-shell reduction
6643
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).
44+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 (10 mm radius against 4 mm or 8 mm cells here, so 8 mm cells barely resolve them).
challenges/cymbal-tapered-v1/README.mdmodified+2−2View file
@@ -1,6 +1,6 @@
11 # cymbal-tapered-v1: the hammered cymbal lathed to a tapered thickness
22
3-This is the fifth challenge of the cymbal tier. It is [cymbal-hammered-v1](../cymbal-hammered-v1/README.md) with one change: the thickness is no longer uniform. It is 1.2 mm over the bell and decreases linearly along the bow to 0.7 mm at the rim, the taper that lathing gives a 16 inch crash. Everything else, the bow, the bell, the hole, the twenty dents, the material, the damping, the strike, the listener, the output and the error measures, is unchanged, so the difference between the two challenges is the effect of the taper alone. A thinner rim lowers the modes that live near the edge and makes them radiate and cascade differently from a uniform sheet; a heavier center raises the low modes of the whole body. The deliverable is again six seconds of pressure at the listener.
3+This is the fifth challenge of the cymbal tier. It is [cymbal-hammered-v1](../cymbal-hammered-v1/README.md) with one change: the thickness is no longer uniform. It is 1.2 mm over the bell and decreases linearly along the bow to 0.7 mm at the rim, the taper that lathing gives a 16 inch crash. Everything else, the bow, the bell, the hole, the 150 dents, the material, the damping, the strike, the listener, the output and the error measures, is unchanged, so the difference between the two challenges is the effect of the taper alone. A thinner rim lowers the modes that live near the edge and makes them radiate and cascade differently from a uniform sheet; a heavier center raises the low modes of the whole body. The deliverable is again six seconds of pressure at the listener.
44
55 The machine-readable version of this specification is [challenge.json](challenge.json).
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@@ -36,7 +36,7 @@ Those of cymbal-hammered-v1 with the uniform thickness $h$ replaced by the two t
3636 | $h_e$ | 0.0007 | m | thickness at the rim |
3737 | $C$ | see challenge.json | m^2 | strike normalization integral |
3838
39-The mass of the body is close to that of the uniform 1 mm disk (the mean thickness over the area is about 0.86 mm), so the drift and the kick of the strike are similar; the bending stiffness, which goes with $h^3$, is 1.7 times the uniform value at the bell and 0.34 times it at the rim. The dents, half a millimeter deep, are now a larger fraction of the local thickness near the rim (0.7 mm) than in cymbal-hammered-v1.
39+The mass of the body is close to that of the uniform 1 mm disk (the mean thickness over the area is about 0.86 mm), so the drift and the kick of the strike are similar; the bending stiffness, which goes with $h^3$, is 1.7 times the uniform value at the bell and 0.34 times it at the rim. The dents, a millimeter deep, are now deeper than the local thickness near the rim (0.7 mm).
4040
4141 ## Error measures, reference, field export
4242