The stand as a rigid clamp at the mounting radius (20 mm), replacing the hole and the rigid-motion constraint
9 changed files+21−21
MODEL.mdmodified+6−6View file
@@ -7,10 +7,10 @@ Every challenge in this repository is an instance of the model defined here. A c
77 The cymbal is a thin shell. Its mid-surface is the graph of a function $Z$ over the annulus
88
99 $$
10-A = \{ (x, y) : r_h \le r \le R \}, \qquad r = \sqrt{x^2 + y^2},
10+A = \{ (x, y) : r_m \le r \le R \}, \qquad r = \sqrt{x^2 + y^2},
1111 $$
1212
13-and its thickness is $h$. Both are functions of $r$ alone and are given per challenge. The rim $r = R$ and the edge of the hole $r = r_h$ are free. Below, subscripts $x$, $y$ and $t$ denote partial derivatives, Greek indices run over $x, y$, repeated Greek indices are summed, and $\delta_{\alpha\beta}$ is the Kronecker delta.
13+and its thickness is $h$. Both are functions of $r$ alone and are given per challenge. The inner edge $r = r_m$ is where the stand holds the cymbal: the felt washers clamp it there, and the model takes the clamp as rigid, so the displacement and the slope of $w$ vanish on that circle. The rim $r = R$ is free. The mounting hole and the metal under the felt, inside $r_m$, are not part of the model. Below, subscripts $x$, $y$ and $t$ denote partial derivatives, Greek indices run over $x, y$, repeated Greek indices are summed, and $\delta_{\alpha\beta}$ is the Kronecker delta.
1414
1515 ## Unknowns and strains
1616
@@ -37,7 +37,7 @@ with $\tilde\varepsilon = \varepsilon + \eta\, \varepsilon_t$, $\tilde\kappa = \
3737
3838 ## Equations of motion
3939
40-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,
40+The motion is defined by its weak form. The solution satisfies $u = v = w = 0$ and $\partial w / \partial r = 0$ on the inner edge $r = r_m$, and so do the test functions; the rim carries no condition. For all such $\delta u, \delta v \in H^1(A)$ and $\delta w \in H^2(A)$,
4141
4242 $$
4343 \int_A \rho h \left[ (u_{tt} + 2 \sigma_0 u_t)\, \delta u + (v_{tt} + 2 \sigma_0 v_t)\, \delta v + (w_{tt} + 2 \sigma_0 w_t)\, \delta w + 2 \sigma_1\, \nabla w_t \cdot \nabla \delta w \right] dA
@@ -65,7 +65,7 @@ $$
6565 \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
68-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$.
68+together with the clamped conditions at $r = r_m$ and the free-edge conditions that the weak form implies at $r = R$. 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
@@ -105,7 +105,7 @@ with $w_{tt}$ taken as zero at negative times. This is the field of a planar sou
105105
106106 ## Remarks
107107
108-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.
108+The clamp at the mount is what makes the strike position matter. Struck on the bell, a shell whose center is free would move its whole bell as a piston on the bow and swallow the stick's energy into the bow's modes like any other hit; held at the mount, the bell hit loads the bell against the support, the stick rebounds, and mainly the bell's own high modes ring. The clamp also removes the rigid motions, so nothing needs to be said about drift. The felt's compliance and the slow swing of a real cymbal on its stand are neglected.
109109
110110 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
@@ -114,7 +114,7 @@ The losses are the three of plate synthesis. In the linear regime about a flat u
114114 | Symbol | Value | Unit | Meaning |
115115 | --- | --- | --- | --- |
116116 | $R$ | 0.2032 | m | outer radius (16 in diameter) |
117-| $r_h$ | 0.006 | m | hole radius |
117+| $r_m$ | 0.02 | m | mounting radius (clamped inner edge) |
118118 | $E$ | 110e9 | Pa | Young's modulus (B20 bell bronze) |
119119 | $\nu$ | 0.34 | | Poisson's ratio |
120120 | $\rho$ | 8800 | kg/m^3 | density |
README.mdmodified+1−1View file
@@ -2,7 +2,7 @@
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: 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.
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, clamped at the mounting radius and free at the rim, 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 | --- | --- | --- |
challenges/cymbal2-crash-v1/README.mdmodified+3−3View file
@@ -1,6 +1,6 @@
11 # 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 0.65 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).
3+The cymbal of this tier: a paraboloidal bow with a cosine-dome bell, held at the mount 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
@@ -10,7 +10,7 @@ Z(r) = H \left( 1 - \frac{r^2}{R^2} \right) + \begin{cases} \dfrac{h_b}{2} \left
1010 h(r) = \begin{cases} h_c, & r \le r_b, \\[1ex] h_c + (h_e - h_c)\, \dfrac{r - r_b}{R - r_b}, & r_b < r \le R, \end{cases}
1111 $$
1212
13-for $r_h \le r \le R$. The bell joins the bow with continuous slope and a jump in curvature at $r = r_b$; the thickness is continuous there.
13+for $r_m \le r \le R$. The bell joins the bow with continuous slope and a jump in curvature at $r = r_b$; the thickness is continuous there. The clamped inner edge at $r_m = 20$ mm sits on the bell, 35 mm inside its crease, where the dome is 21 mm above the bow.
1414
1515 | Symbol | Value | Unit | Meaning |
1616 | --- | --- | --- | --- |
@@ -21,4 +21,4 @@ for $r_h \le r \le R$. The bell joins the bow with continuous slope and a jump i
2121 | $h_e$ | 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 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.
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 between the mount and the crease, 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+2−2View file
@@ -18,11 +18,11 @@
1818 "type": "shell-of-revolution",
1919 "mid_surface": "Z(r) = bow_height (1 - r^2/disk_radius^2) + bell_height/2 (1 + cos(pi r / bell_radius)) [r < bell_radius]",
2020 "thickness": "h(r) = thickness_center for r <= bell_radius, linear from thickness_center at bell_radius to thickness_rim at disk_radius",
21- "domain": "annulus hole_radius <= r <= disk_radius, both edges free"
21+ "domain": "annulus mount_radius <= r <= disk_radius; the inner edge is clamped (the stand), the rim is free"
2222 },
2323 "parameters": {
2424 "disk_radius": 0.2032,
25- "hole_radius": 0.006,
25+ "mount_radius": 0.02,
2626 "E": 110000000000.0,
2727 "nu": 0.34,
2828 "rho": 8800.0,
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+3−3View file
@@ -1,11 +1,11 @@
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 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).
3+The baseline of the cymbal2 tier: a flat annulus of bell bronze with the diameter of a 16 inch crash and a uniform thickness of 1 mm, clamped at the mounting radius, 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
77 $$
8-Z(r) = 0, \qquad h(r) = h_0, \qquad r_h \le r \le R.
8+Z(r) = 0, \qquad h(r) = h_0, \qquad r_m \le r \le R.
99 $$
1010
1111 | Symbol | Value | Unit | Meaning |
@@ -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 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.
16+The clamp at the center raises the lowest modes of the free disk (22, 38 and 51 Hz) and turns its rigid motions into elastic ones. 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+2−2View file
@@ -18,11 +18,11 @@
1818 "type": "flat",
1919 "mid_surface": "Z(r) = 0",
2020 "thickness": "h(r) = thickness",
21- "domain": "annulus hole_radius <= r <= disk_radius, both edges free"
21+ "domain": "annulus mount_radius <= r <= disk_radius; the inner edge is clamped (the stand), the rim is free"
2222 },
2323 "parameters": {
2424 "disk_radius": 0.2032,
25- "hole_radius": 0.006,
25+ "mount_radius": 0.02,
2626 "E": 110000000000.0,
2727 "nu": 0.34,
2828 "rho": 8800.0,
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+2−2View file
@@ -31,7 +31,7 @@ def main():
3131 ch = json.load(open(cj))
3232 p = ch["parameters"]
3333 Z, h = profile(p)
34- R, rh = p["disk_radius"], p["hole_radius"]
34+ R, rh = p["disk_radius"], p["mount_radius"]
3535 verts, thick = [], []
3636 for i in range(NR):
3737 r = rh + (R - rh) * i / (NR - 1)
@@ -46,7 +46,7 @@ def main():
4646 points.append({"name": "listener", "position": L, "radius": 0.02, "color": [0.15, 0.35, 0.85], "far": True})
4747 scene = {
4848 "units": "m",
49- "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 center.",
49+ "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 inner edge is the clamped mount.",
5050 "surfaces": [{"name": "mid-surface", "vertices": verts, "grid": [NR, NT], "periodic": True, "scalar": {"name": "thickness (mm)", "values": thick}}],
5151 "points": points,
5252 "lines": [{"from": [0, 0, 0], "to": L}],