Strike 1000 N over 1 ms; signal scale 80 Pa
2 changed files+8−8
challenges/cymbal-flat-disk-v1/README.mdmodified+4−4View file
@@ -55,7 +55,7 @@ $$
5555 c_\phi = \frac{1}{2} - \frac{2}{\pi^2}.
5656 $$
5757
58-The constant $c_\phi$ is $\int_0^1 2 s\, \phi(s)\, ds$, so that the total force $\int_{S^+} p\, dS = F_{\max}\, \psi(t / T_s)$ peaks at $F_{\max}$ at $t = T_s / 2$, and the total impulse is $F_{\max} T_s / 2$. Both factors are continuously differentiable and compactly supported, and the strike footprint lies inside the top face. The footprint radius and the duration are those of a drum stick tip on a cymbal: a contact patch a centimeter across and a contact time of a couple of milliseconds. The pulse is prescribed rather than produced by a contact model, so that the problem has no unknown other than the disk's motion.
58+The constant $c_\phi$ is $\int_0^1 2 s\, \phi(s)\, ds$, so that the total force $\int_{S^+} p\, dS = F_{\max}\, \psi(t / T_s)$ peaks at $F_{\max}$ at $t = T_s / 2$, and the total impulse is $F_{\max} T_s / 2$. Both factors are continuously differentiable and compactly supported, and the strike footprint lies inside the top face. The footprint radius and the duration are those of a drum stick tip on a cymbal: a contact patch a centimeter across and a contact time of a millisecond. The pulse is prescribed rather than produced by a contact model, so that the problem has no unknown other than the disk's motion.
5959
6060 ## Parameters
6161
@@ -70,15 +70,15 @@ The constant $c_\phi$ is $\int_0^1 2 s\, \phi(s)\, ds$, so that the total force
7070 | $\eta$ | 1.2e-8 | s | Kelvin-Voigt viscosity |
7171 | $(x_s, y_s)$ | (0.17272, 0) | m | strike center, at $0.85 R$ from the center |
7272 | $R_s$ | 0.005 | m | strike footprint radius |
73-| $T_s$ | 0.002 | s | strike duration |
74-| $F_{\max}$ | 100 | N | peak total force |
73+| $T_s$ | 0.001 | s | strike duration |
74+| $F_{\max}$ | 1000 | N | peak total force |
7575 | $\rho_a$ | 1.204 | kg/m^3 | air density |
7676 | $c_a$ | 343 | m/s | speed of sound in air |
7777 | $\mathbf{x}_L$ | (0.06, 0.08, $h/2$ + 0.5) | m | listener point |
7878
7979 All values are exact decimal numbers, not rounded measurements. The material constants are representative of the tin bronze (about 80 percent copper, 20 percent tin) that most cymbals are made of; the challenge does not claim to match any particular instrument. The listener is 0.5 m above the top face and 0.1 m from the axis, at an azimuth of about 53 degrees from the strike direction. It is deliberately not on the strike's symmetry axis: the exact solution is symmetric about the $x$ axis and never excites the modes that are antisymmetric in $y$, so a discretization that breaks this symmetry, a Cartesian grid whose staircase boundary is not symmetric about the axis for instance, exposes that error at the listener.
8080
81-With these constants the disk's lowest elastic modes in the linear thin-plate limit are the two-nodal-diameter mode at 21.9 Hz, the first axisymmetric mode at 38.0 Hz, and the three-nodal-diameter mode at 50.9 Hz (see the appendix). The constant $\sigma_0$ gives the lowest modes a sixty-decibel decay time of about 6.3 s, and $\eta$ makes a mode at 5 kHz decay in about 1 s, one at 10 kHz in 0.28 s, and one at 20 kHz in 72 ms. The impulse of the strike is $F_{\max} T_s / 2 = 0.1$ N s, with a contact time of two milliseconds, a stick with a soft tip or a firm rather than a violent stroke. Under the linear model this strike would bend the disk by about four thicknesses under the stick, well into the regime where membrane stiffening matters, and the previews used to choose it showed the pitch glide and the slow build-up of the cascade over the first half second without the metallic slam that harder, shorter pulses produce. The pulse itself feeds mainly the modes below about a kilohertz, so the treble content of the sound is produced by the nonlinear cascade rather than by the strike. The disk is free, so the impulse also sets it drifting at about 0.09 m/s; this rigid motion carries no acceleration after the strike and does not enter the output, but it dominates the raw displacement in the field export.
81+With these constants the disk's lowest elastic modes in the linear thin-plate limit are the two-nodal-diameter mode at 21.9 Hz, the first axisymmetric mode at 38.0 Hz, and the three-nodal-diameter mode at 50.9 Hz (see the appendix). The constant $\sigma_0$ gives the lowest modes a sixty-decibel decay time of about 6.3 s, and $\eta$ makes a mode at 5 kHz decay in about 1 s, one at 10 kHz in 0.28 s, and one at 20 kHz in 72 ms. The impulse of the strike is $F_{\max} T_s / 2 = 0.5$ N s, a hard crash stroke. Under the linear model this strike would bend the disk by about twenty thicknesses under the stick, far into the regime where membrane stiffening dominates. In previews on a coarse grid about a quarter of the radiated energy lies above 3 kHz; the millisecond pulse itself feeds mainly the modes below about 2 kHz, so that treble is produced by the nonlinear cascade, and the sound has the attack and the broadband wash of a struck cymbal. A softer strike of 100 N over 2 ms was tried first and sounded like a tapped plate rather than a stick on a cymbal. The disk is free, so the impulse also sets it drifting at about 0.44 m/s; this rigid motion carries no acceleration after the strike and does not enter the output, but it dominates the raw displacement in the field export.
8282
8383 ## Output signal
8484
challenges/cymbal-flat-disk-v1/challenge.jsonmodified+4−4View file
@@ -10,10 +10,10 @@
1010 "sample_format": "float32",
1111 "signal": "acoustic pressure at the listener point from the Rayleigh integral over the top face",
1212 "signal_unit": "Pa",
13- "signal_scale": 8.0
13+ "signal_scale": 80.0
1414 },
1515 "playback": {
16- "gain_db": 36,
16+ "gain_db": 30,
1717 "note": "Suggested playback gain for listening; presentation only, not part of the deliverable."
1818 },
1919 "parameters": {
@@ -27,8 +27,8 @@
2727 "strike_x": 0.17272,
2828 "strike_y": 0.0,
2929 "strike_radius": 0.005,
30- "strike_duration": 0.002,
31- "strike_peak_force": 100.0,
30+ "strike_duration": 0.001,
31+ "strike_peak_force": 1000.0,
3232 "rho_air": 1.204,
3333 "c_air": 343.0,
3434 "listener_x": 0.06,