Realistic crash: plate-synthesis losses (sigma1 Laplacian term, small eta), firm 6 m/s stroke, rim 0.65 mm, strikes on the bell flank, the bow and the thin edge
9 changed files+36−33
MODEL.mdmodified+12−11View file
@@ -26,7 +26,7 @@ and the bending strains are $\kappa_{\alpha\beta} = w_{\alpha\beta}$.
2626
2727 ## Stress resultants
2828
29-The material is linearly elastic and isotropic, with Young's modulus $E$, Poisson's ratio $\nu$ and density $\rho$, and has a Kelvin-Voigt viscosity of time constant $\eta$. The membrane forces and bending moments per unit length are
29+The material is linearly elastic and isotropic, with Young's modulus $E$, Poisson's ratio $\nu$ and density $\rho$, and has a small Kelvin-Voigt viscosity of time constant $\eta$. The membrane forces and bending moments per unit length are
3030
3131 $$
3232 N_{\alpha\beta} = \frac{E h}{1 - \nu^2} \left[ (1 - \nu)\, \tilde\varepsilon_{\alpha\beta} + \nu\, \tilde\varepsilon_{\gamma\gamma}\, \delta_{\alpha\beta} \right], \qquad
@@ -40,12 +40,12 @@ with $\tilde\varepsilon = \varepsilon + \eta\, \varepsilon_t$, $\tilde\kappa = \
4040 The motion is defined by its weak form. For all test functions $\delta u, \delta v \in H^1(A)$ and $\delta w \in H^2(A)$, without constraint on either edge,
4141
4242 $$
43-\int_A \rho h \left[ (u_{tt} + 2 \sigma_0 u_t)\, \delta u + (v_{tt} + 2 \sigma_0 v_t)\, \delta v + (w_{tt} + 2 \sigma_0 w_t)\, \delta w \right] dA
43+\int_A \rho h \left[ (u_{tt} + 2 \sigma_0 u_t)\, \delta u + (v_{tt} + 2 \sigma_0 v_t)\, \delta v + (w_{tt} + 2 \sigma_0 w_t)\, \delta w + 2 \sigma_1\, \nabla w_t \cdot \nabla \delta w \right] dA
4444 + \int_A \left( N_{\alpha\beta}\, \delta\varepsilon_{\alpha\beta} + M_{\alpha\beta}\, \delta\kappa_{\alpha\beta} \right) dA
4545 = - \sum_{i=1}^{3} F_i(t) \int_A \phi_i\, \delta w\, dA,
4646 $$
4747
48-where $\sigma_0$ is a loss rate, $F_i$ and $\phi_i$ are the force and footprint of the $i$th strike of the next section, and $\delta\varepsilon$, $\delta\kappa$ are the variations of the strains at the current state,
48+where $\sigma_0$ and $\sigma_1$ are loss coefficients, $F_i$ and $\phi_i$ are the force and footprint of the $i$th strike of the next section, and $\delta\varepsilon$, $\delta\kappa$ are the variations of the strains at the current state,
4949
5050 $$
5151 \delta\varepsilon_{xx} = \delta u_x + (Z_x + w_x)\, \delta w_x, \qquad
@@ -62,14 +62,14 @@ $$
6262 $$
6363
6464 $$
65-\rho h\, (w_{tt} + 2 \sigma_0 w_t) = \partial_\alpha \left[ N_{\alpha\beta} (Z_\beta + w_\beta) \right] - \partial_{\alpha\beta} M_{\alpha\beta} - \sum_i F_i(t)\, \phi_i,
65+\rho h\, (w_{tt} + 2 \sigma_0 w_t) - 2 \sigma_1\, \nabla \cdot (\rho h\, \nabla w_t) = \partial_\alpha \left[ N_{\alpha\beta} (Z_\beta + w_\beta) \right] - \partial_{\alpha\beta} M_{\alpha\beta} - \sum_i F_i(t)\, \phi_i,
6666 $$
6767
6868 together with the free-edge conditions that the weak form implies at $r = R$ and $r = r_h$. The rigid motions of the shell are suppressed. Let $\langle \mathbf{a}, \mathbf{b} \rangle = \int_A \rho h\, \mathbf{a} \cdot \mathbf{b}\, dA$ be the mass-weighted inner product of displacement fields $(u, v, w)$, and let $\mathcal{R}$ be the span of the six rigid fields $(1, 0, 0)$, $(0, 1, 0)$, $(0, 0, 1)$, $(0, 0, x)$, $(0, 0, y)$ and $(-y, x, 0)$. The solution is orthogonal to $\mathcal{R}$ at all times, $\langle (u, v, w), \mathbf{r} \rangle = 0$ for every $\mathbf{r} \in \mathcal{R}$, and the weak form above is required only for test functions orthogonal to $\mathcal{R}$. This is the shell held by an ideal support that fixes its position and orientation and constrains nothing else: the reaction is a body force in $\mathcal{R}$, equal to $\rho h$ times the rigid acceleration field that balances the net force and moments of the strikes, and the elastic motion is untouched. The shell is at rest at $t = 0$: $u = v = w = 0$ and $u_t = v_t = w_t = 0$.
6969
7070 ## The strikes
7171
72-The cymbal is struck three times by a drum stick, at the times $t_i$ and points $(x_i, 0)$ of the parameter table: just outside the bell, then at half the radius, then near the rim, two seconds apart. The stick is a point mass $m_s$ that moves along $z$ and interacts with the shell through a Hertz contact force. The $i$th strike has the footprint
72+The cymbal is struck three times by a drum stick, at the times $t_i$ and points $(x_i, 0)$ of the parameter table: on the outer flank of the bell, then on the bow at $0.59 R$, then near the rim at $0.91 R$, two seconds apart. The stick is a point mass $m_s$ that moves along $z$ and interacts with the shell through a Hertz contact force. The $i$th strike has the footprint
7373
7474 $$
7575 \phi_i(x, y) = \frac{1}{c_\phi\, \pi R_s^2}\; \phi_0\!\left( \frac{\varrho_i}{R_s} \right), \qquad
@@ -107,7 +107,7 @@ with $w_{tt}$ taken as zero at negative times. This is the field of a planar sou
107107
108108 The rigid constraint is not a convenience. The shallow-shell strains are not invariant under finite rotations: a shell that has tilted by an angle $\theta$ behaves in this model like a shell of slope $\theta$, with a membrane-bending coupling that a real rotated plate does not have. Left free, the shell tilts by about a radian within half a second of the first strike, its partials glide and its treble drains into in-plane motion, none of it physical. Holding the six rigid components at zero removes the problem exactly, and the strikes' momentum and angular momentum go into the support.
109109
110-In the linear regime about the flat state, $\sigma_0$ damps every mode at the rate $\sigma_0$ and $\eta$ damps a mode of angular frequency $\omega$ at the rate $\eta \omega^2 / 2$.
110+The losses are the three of plate synthesis. In the linear regime about a flat uniform state a bending mode of wavenumber $k$ and angular frequency $\omega$ decays at the rate $\sigma_0 + \sigma_1 k^2 + \eta \omega^2 / 2$, with $k^2 = \omega \sqrt{\rho h / D}$. With the values below the sixty-decibel decay time is about 5 s below 500 Hz, 3.5 s at 2 kHz, 2 s at 5 kHz, 1.2 s at 10 kHz and 0.8 s at 15 kHz, roughly the ring of a crash cymbal; the viscous term is small in the audible range and gives the model a definite cutoff above it.
111111
112112 ## Parameters common to all challenges
113113
@@ -118,19 +118,20 @@ In the linear regime about the flat state, $\sigma_0$ damps every mode at the ra
118118 | $E$ | 110e9 | Pa | Young's modulus (B20 bell bronze) |
119119 | $\nu$ | 0.34 | | Poisson's ratio |
120120 | $\rho$ | 8800 | kg/m^3 | density |
121-| $\sigma_0$ | 1.1 | 1/s | loss rate |
122-| $\eta$ | 1.2e-8 | s | Kelvin-Voigt time constant |
121+| $\sigma_0$ | 1.1 | 1/s | frequency-independent loss rate |
122+| $\sigma_1$ | 8e-5 | m^2/s | loss coefficient of the Laplacian term |
123+| $\eta$ | 1e-9 | s | Kelvin-Voigt time constant |
123124 | $m_s$ | 0.02 | kg | stick tip mass |
124-| $v_0$ | 10 | m/s | stick speed at contact |
125+| $v_0$ | 6 | m/s | stick speed at contact |
125126 | $k_c$ | 1e9 | N/m^(3/2) | Hertz contact stiffness |
126127 | $R_s$ | 0.005 | m | footprint radius |
127128 | $t_1, t_2, t_3$ | 0, 2, 4 | s | strike times |
128-| $x_1, x_2, x_3$ | 0.07, 0.1016, 0.17272 | m | strike points on the $x$ axis: just outside the bell, $0.5 R$, $0.85 R$ |
129+| $x_1, x_2, x_3$ | 0.045, 0.12, 0.185 | m | strike points on the $x$ axis: bell flank, bow, rim |
129130 | $\rho_a$ | 1.204 | kg/m^3 | air density |
130131 | $c_a$ | 343 | m/s | speed of sound in air |
131132 | $\mathbf{x}_L$ | (0.65, 1.125, 0.75) | m | listener point |
132133
133-All values are exact decimal numbers. The listener is 1.5 m from the origin, 30 degrees above the plane $z = 0$, at an azimuth of 60 degrees from the $x$ axis, on which the strike points lie. The strikes are the same in both challenges; the first lands 15 mm outside the bell's edge, clear of its crease, where the bow is nearly flat. The stick values are provisional until the first solutions have been heard; they will be fixed before any solution is published, and a challenge is never changed after that.
134+All values are exact decimal numbers. The listener is 1.5 m from the origin, 30 degrees above the plane $z = 0$, at an azimuth of 60 degrees from the $x$ axis, on which the strike points lie. The strikes are the same in both challenges. On the crash cymbal the first lands on the outer flank of the bell (slope 0.26), the second on the bow, the third 18 mm from the rim on the thin edge; on the flat annulus they land at the same radii. The stick carries 0.36 J, a firm stroke rather than the hardest a drummer gives. The stick values are provisional until the first solutions have been heard; they will be fixed before any solution is published, and a challenge is never changed after that.
134135
135136 ## Deliverable
136137
README.mdmodified+2−2View file
@@ -2,12 +2,12 @@
22
33 This repository defines two simulation challenges on one geometrically nonlinear thin-shell model of a 16 inch crash cymbal struck by a drum stick, and publishes the comparison of every registered solution at [challenges/cymbal2/site](https://vault1.magland.org/challenges/cymbal2/site/). It follows the arrangement of the [string](https://vault1.magland.org/challenges/string), [plate](https://vault1.magland.org/challenges/plate) and [cymbal](https://vault1.magland.org/challenges/cymbal) repositories: the model is defined in the continuum, solutions live in separate registered repositories, and the shared [challenge-tools](https://vault1.magland.org/challenges/challenge-tools) package scores them against a reference chosen from among the submissions.
44
5-The model is defined once, in [MODEL.md](MODEL.md): a shallow shell with in-plane inertia, Saint Venant-Kirchhoff through the thickness, Kelvin-Voigt and mass-proportional damping, free at both edges with its six rigid motions held, struck by a point mass through a Hertz contact, radiating to a listener 1.5 m away through the Rayleigh integral. Each challenge fixes the profile and the thickness. In both, the same stick lands three times, two seconds apart: just outside the bell, at half the radius, and near the rim, so that the difference in sound between a hit near the stiff center and a crash near the rim comes out of the same equations within one seven-second recording.
5+The model is defined once, in [MODEL.md](MODEL.md): a shallow shell with in-plane inertia, Saint Venant-Kirchhoff through the thickness, Kelvin-Voigt and mass-proportional damping, free at both edges with its six rigid motions held, struck by a point mass through a Hertz contact, radiating to a listener 1.5 m away through the Rayleigh integral. Each challenge fixes the profile and the thickness. In both, the same stick lands three times, two seconds apart: on the bell, on the bow, and near the thin rim, so that the difference in sound between the ping of the bell, the tone of the bow and the crash of the edge comes out of the same equations within one seven-second recording.
66
77 | ID | Geometry | Deliverable |
88 | --- | --- | --- |
99 | [cymbal2-flat-v1](challenges/cymbal2-flat-v1/README.md) | flat annulus, 1.0 mm | 7 s mono wav, 48 kHz, float32 |
10-| [cymbal2-crash-v1](challenges/cymbal2-crash-v1/README.md) | bow, cosine bell, taper 1.2 to 0.7 mm | 7 s mono wav, 48 kHz, float32 |
10+| [cymbal2-crash-v1](challenges/cymbal2-crash-v1/README.md) | bow, cosine bell, taper 1.2 to 0.65 mm | 7 s mono wav, 48 kHz, float32 |
1111
1212 Each challenge lives in `challenges/<challenge-id>/` as a `README.md`, which together with MODEL.md is the specification, and a `challenge.json` with the parameters, output format, error measures and field-export grid in machine-readable form. The stick parameters and the output scales are provisional until the first solutions have been heard; once a solution is published, a challenge is never changed, and a corrected model gets a new ID.
1313
challenges/cymbal2-crash-v1/README.mdmodified+4−4View file
@@ -1,6 +1,6 @@
1-# cymbal2-crash-v1: the crash cymbal struck near the bell, at mid-radius and near the rim
1+# cymbal2-crash-v1: the crash cymbal struck on the bell, on the bow and near the rim
22
3-The cymbal of this tier: a paraboloidal bow with a cosine-dome bell around the mounting hole and a thickness that tapers from 1.2 mm at the bell to 0.7 mm at the rim, struck three times by the stick, just outside the bell, at half the radius and at 0.85 R, two seconds apart. The three hits are identical sticks; what differs is the local stiffness they meet, so the tighter sound near the stiff center and the crash of the rim come out of the same equations. The model, the strikes, the listener, the output, the error measures and the common parameters are defined in [MODEL.md](../../MODEL.md); this page fixes what is specific to this challenge. The machine-readable version is [challenge.json](challenge.json).
3+The cymbal of this tier: a paraboloidal bow with a cosine-dome bell around the mounting hole and a thickness that tapers from 1.2 mm at the bell to 0.65 mm at the rim, struck three times by the stick, on the outer flank of the bell, on the bow at 0.59 R and near the rim at 0.91 R, two seconds apart. The three hits are identical sticks; what differs is the local stiffness they meet, so the ping of the bell, the tone of the bow and the crash of the thin edge come out of the same equations. The model, the strikes, the listener, the output, the error measures and the common parameters are defined in [MODEL.md](../../MODEL.md); this page fixes what is specific to this challenge. The machine-readable version is [challenge.json](challenge.json).
44
55 ## Geometry
66
@@ -18,7 +18,7 @@ for $r_h \le r \le R$. The bell joins the bow with continuous slope and a jump i
1818 | $r_b$ | 0.055 | m | bell radius |
1919 | $h_b$ | 0.015 | m | bell height above the bow |
2020 | $h_c$ | 0.0012 | m | thickness over the bell |
21-| $h_e$ | 0.0007 | m | thickness at the rim |
21+| $h_e$ | 0.00065 | m | thickness at the rim |
2222 | $s_\text{ref}$ | 16 (provisional) | Pa | output scale |
2323
24-At the three strike points the surface is 22, 19 and 7 mm above the rim plane, its slope is 0.08, 0.12 and 0.21, and the thickness is 1.15, 1.04 and 0.80 mm; the first footprint spans 65 to 75 mm from the axis, outside the bell's edge at 55 mm. The largest slope of the profile is about 0.46, on the flank of the bell, where shallow-shell theory is a coarse approximation of a real cymbal; the model is nevertheless defined as written.
24+At the three strike points the surface is 25, 16 and 4 mm above the rim plane, its slope is 0.29, 0.15 and 0.22, and the thickness is 1.20, 0.96 and 0.72 mm; the first footprint spans 40 to 50 mm from the axis, on the bell, and the third 180 to 190 mm, 13 mm from the rim. The largest slope of the profile is about 0.46, on the flank of the bell, where shallow-shell theory is a coarse approximation of a real cymbal; the model is nevertheless defined as written.
challenges/cymbal2-crash-v1/challenge.jsonmodified+7−6View file
@@ -27,9 +27,10 @@
2727 "nu": 0.34,
2828 "rho": 8800.0,
2929 "sigma0": 1.1,
30- "eta": 1.2e-08,
30+ "sigma1": 8e-05,
31+ "eta": 1e-09,
3132 "stick_mass": 0.02,
32- "stick_speed": 10.0,
33+ "stick_speed": 6.0,
3334 "contact_stiffness": 1000000000.0,
3435 "contact_exponent": 1.5,
3536 "strike_radius": 0.005,
@@ -42,22 +43,22 @@
4243 "bell_radius": 0.055,
4344 "bell_height": 0.015,
4445 "thickness_center": 0.0012,
45- "thickness_rim": 0.0007
46+ "thickness_rim": 0.00065
4647 },
4748 "strikes": [
4849 {
4950 "t": 0.0,
50- "x": 0.07,
51+ "x": 0.045,
5152 "y": 0.0
5253 },
5354 {
5455 "t": 2.0,
55- "x": 0.1016,
56+ "x": 0.12,
5657 "y": 0.0
5758 },
5859 {
5960 "t": 4.0,
60- "x": 0.17272,
61+ "x": 0.185,
6162 "y": 0.0
6263 }
6364 ],
challenges/cymbal2-crash-v1/scene.jsonmodified+1−1View file
@@ -1 +1 @@
1-{"units":"m","note":"The mid-surface, colored by thickness; the three strike footprints; the listener 1.5 m away (far point). The z axis points up; the hole is at the 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1 (bell), t = 0 s","position":[0.045,0.0,0.02496],"radius":0.005,"color":[0.85,0.2,0.15]},{"name":"strike 2 (bow), t = 2 s","position":[0.12,0.0,0.01628],"radius":0.005,"color":[0.85,0.2,0.15]},{"name":"strike 3 (rim), t = 4 s","position":[0.185,0.0,0.00428],"radius":0.005,"color":[0.85,0.2,0.15]},{"name":"listener","position":[0.65,1.125,0.75],"radius":0.02,"color":[0.15,0.35,0.85],"far":true}],"lines":[{"from":[0,0,0],"to":[0.65,1.125,0.75]}],"up":"z"}
challenges/cymbal2-flat-v1/README.mdmodified+2−2View file
@@ -1,6 +1,6 @@
11 # cymbal2-flat-v1: flat annulus struck three times
22
3-The baseline of the cymbal2 tier: a flat annulus of bell bronze with the diameter and the mounting hole of a 16 inch crash and a uniform thickness of 1 mm, struck three times by the stick, at 70 mm from the axis, at half the radius and at 0.85 R, two seconds apart. It has no bow, bell or taper, so every position on it has the same local stiffness, and the three hits differ only through the mode shapes at the strike points; cymbal2-crash-v1 adds the geometry that makes the position matter. The model, the strikes, the listener, the output, the error measures and the common parameters are defined in [MODEL.md](../../MODEL.md); this page fixes what is specific to this challenge. The machine-readable version is [challenge.json](challenge.json).
3+The baseline of the cymbal2 tier: a flat annulus of bell bronze with the diameter and the mounting hole of a 16 inch crash and a uniform thickness of 1 mm, struck three times by the stick, at 45, 120 and 185 mm from the axis, two seconds apart. It has no bow, bell or taper, so every position on it has the same local stiffness, and the three hits differ only through the mode shapes at the strike points; cymbal2-crash-v1 adds the geometry that makes the position matter. The model, the strikes, the listener, the output, the error measures and the common parameters are defined in [MODEL.md](../../MODEL.md); this page fixes what is specific to this challenge. The machine-readable version is [challenge.json](challenge.json).
44
55 ## Geometry
66
@@ -13,4 +13,4 @@ $$
1313 | $h_0$ | 0.001 | m | thickness |
1414 | $s_\text{ref}$ | 16 (provisional) | Pa | output scale |
1515
16-The lowest elastic frequencies of the annulus in the linear regime are near 22 Hz (two nodal diameters), 38 Hz (axisymmetric) and 51 Hz (three nodal diameters); the hole changes them by less than a percent from those of the full disk. Each stick brings about 1 J and a peak force of a few hundred newtons over about half a millisecond, which bends the annulus by many thicknesses and puts it well into the regime of membrane stiffening.
16+The lowest elastic frequencies of the annulus in the linear regime are near 22 Hz (two nodal diameters), 38 Hz (axisymmetric) and 51 Hz (three nodal diameters); the hole changes them by less than a percent from those of the full disk. Each stick brings 0.36 J and a peak force of several hundred newtons over about a millisecond, which bends the annulus by several thicknesses and puts it into the regime of membrane stiffening.
challenges/cymbal2-flat-v1/challenge.jsonmodified+6−5View file
@@ -27,9 +27,10 @@
2727 "nu": 0.34,
2828 "rho": 8800.0,
2929 "sigma0": 1.1,
30- "eta": 1.2e-08,
30+ "sigma1": 8e-05,
31+ "eta": 1e-09,
3132 "stick_mass": 0.02,
32- "stick_speed": 10.0,
33+ "stick_speed": 6.0,
3334 "contact_stiffness": 1000000000.0,
3435 "contact_exponent": 1.5,
3536 "strike_radius": 0.005,
@@ -43,17 +44,17 @@
4344 "strikes": [
4445 {
4546 "t": 0.0,
46- "x": 0.07,
47+ "x": 0.045,
4748 "y": 0.0
4849 },
4950 {
5051 "t": 2.0,
51- "x": 0.1016,
52+ "x": 0.12,
5253 "y": 0.0
5354 },
5455 {
5556 "t": 4.0,
56- "x": 0.17272,
57+ "x": 0.185,
5758 "y": 0.0
5859 }
5960 ],
challenges/cymbal2-flat-v1/scene.jsonmodified+1−1View file
@@ -1 +1 @@
1-{"units":"m","note":"The mid-surface, colored by thickness; the three strike footprints; the listener 1.5 m away (far point). The z axis points up; the hole is at the 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1, t = 0 s","position":[0.045,0.0,0.0],"radius":0.005,"color":[0.85,0.2,0.15]},{"name":"strike 2, t = 2 s","position":[0.12,0.0,0.0],"radius":0.005,"color":[0.85,0.2,0.15]},{"name":"strike 3, t = 4 s","position":[0.185,0.0,0.0],"radius":0.005,"color":[0.85,0.2,0.15]},{"name":"listener","position":[0.65,1.125,0.75],"radius":0.02,"color":[0.15,0.35,0.85],"far":true}],"lines":[{"from":[0,0,0],"to":[0.65,1.125,0.75]}],"up":"z"}
scripts/make_scenes.pymodified+1−1View file
@@ -39,7 +39,7 @@ def main():
3939 th = 2 * math.pi * j / NT
4040 verts.append([round(r * math.cos(th), 5), round(r * math.sin(th), 5), round(Z(r), 5)])
4141 thick.append(round(1e3 * h(r), 4))
42- names = ["strike 1 (near the bell), t = 0 s", "strike 2 (mid), t = 2 s", "strike 3 (rim), t = 4 s"] if ch.get("geometry", {}).get("type") != "flat" else ["strike 1, t = 0 s", "strike 2, t = 2 s", "strike 3, t = 4 s"]
42+ names = ["strike 1 (bell), t = 0 s", "strike 2 (bow), t = 2 s", "strike 3 (rim), t = 4 s"] if ch.get("geometry", {}).get("type") != "flat" else ["strike 1, t = 0 s", "strike 2, t = 2 s", "strike 3, t = 4 s"]
4343 points = [{"name": names[k], "position": [s["x"], s["y"], round(Z(math.hypot(s["x"], s["y"])), 5)], "radius": p["strike_radius"], "color": [0.85, 0.2, 0.15]}
4444 for k, s in enumerate(ch["strikes"])]
4545 L = [p["listener_x"], p["listener_y"], p["listener_z"]]