Define the output acceleration with the rigid motion linearized
2 changed files+7−3
challenges/cymbal-bow-v1/README.mdmodified+3−1View file
@@ -39,7 +39,7 @@ $$
3939 \mathbf{S} = \lambda\, \mathrm{tr}(\mathbf{E} + \eta\, \dot{\mathbf{E}})\, \mathbf{I} + 2 \mu\, (\mathbf{E} + \eta\, \dot{\mathbf{E}}),
4040 $$
4141
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.
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 momentum and angular momentum; the output is defined with the rigid motion linearized, as for the flat disk.
4343
4444 ### The strike
4545
@@ -89,6 +89,8 @@ $$
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9090 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.
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92+The acceleration in this integral is the total acceleration with the rigid motion linearized. At each instant let $\mathbf{c}(t) \in \mathbb{R}^3$ and $\mathbf{Q}(t) \in SO(3)$ be the mass-weighted best-fit rigid placement of the body, minimizing $\int_\Omega \rho\, |\mathbf{x} + \mathbf{u} - \mathbf{c} - \mathbf{Q} \mathbf{x}|^2\, dV$; let $\boldsymbol{\omega}$ be the angular velocity of $\mathbf{Q}$ and $\boldsymbol{\alpha} = \dot{\boldsymbol{\omega}}$; and let $\tilde{\mathbf{u}} = \mathbf{Q}^{\mathsf{T}} (\mathbf{x} + \mathbf{u} - \mathbf{c}) - \mathbf{x}$ be the displacement relative to that frame. The acceleration used is $\mathbf{a} = \ddot{\mathbf{c}} + \boldsymbol{\alpha} \times \mathbf{x} + \partial^2 \tilde{\mathbf{u}} / \partial t^2$, and $\mathbf{n} \cdot \mathbf{a}$ replaces $\mathbf{n} \cdot \partial^2 \mathbf{u} / \partial t^2$ above. For small motions $\mathbf{a}$ is $\partial^2 \mathbf{u} / \partial t^2$; the two differ only by the centripetal and Coriolis terms of the rigid motion. The body is free and the strike imparts momentum and angular momentum, so in the exact model it keeps turning slowly after the strike, and without this linearization the output would contain the accelerations of that turning, which no thin-plate model represents and which is not part of the sound. The convention keeps the translational and rotational kick of the strike in the output, since $\ddot{\mathbf{c}}$ and $\boldsymbol{\alpha}$ are nonzero while the strike acts, and it coincides with what modal thin-plate solvers compute when they include their rigid modes. The optional field export uses the same linearization: the displacement $\tilde{\mathbf{u}} + \mathbf{c} - \mathbf{x} + \boldsymbol{\theta} \times \mathbf{x}$ with $\boldsymbol{\theta}$ the rotation vector of $\mathbf{Q}$, which for a thin-plate solver is its transverse displacement with the rigid modes included.
93+
9294 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.
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9496 ## Error measures and reference
challenges/cymbal-flat-disk-v1/README.mdmodified+4−2View file
@@ -39,7 +39,7 @@ $$
3939 \mathbf{F} \mathbf{S}\, \mathbf{N} = -\, p(x, y, t)\, \mathbf{e}_z \text{ on } S^+,
4040 $$
4141
42-with $\mathbf{N}$ the outward unit normal of the reference configuration, so the strike is a dead load: a force per unit reference area, always directed along $-z$, whatever the disk's deformation. The disk starts at rest, $\mathbf{u} = \partial \mathbf{u} / \partial t = \mathbf{0}$ at $t = 0$. It is free, so rigid-body translation and rotation are possible and the strike imparts net momentum; this is intended, and the output, which depends on acceleration, is unaffected by uniform drift.
42+with $\mathbf{N}$ the outward unit normal of the reference configuration, so the strike is a dead load: a force per unit reference area, always directed along $-z$, whatever the disk's deformation. The disk starts at rest, $\mathbf{u} = \partial \mathbf{u} / \partial t = \mathbf{0}$ at $t = 0$. It is free, so rigid-body translation and rotation are possible and the strike imparts momentum and angular momentum; this is intended, and the output is defined with the rigid motion linearized (see the output section), so the slow turning of the free disk after the strike does not enter it.
4343
4444 ### The strike
4545
@@ -78,7 +78,7 @@ The constant $c_\phi$ is $\int_0^1 2 s\, \phi(s)\, ds$, so that the total force
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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.
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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.
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 and turning at about 7 rad/s; the linearized rigid motion carries no acceleration after the strike, so it does not enter the output, but it dominates the raw displacement in the field export.
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8383 ## Output signal
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@@ -90,6 +90,8 @@ $$
9090
9191 with $\partial^2 u_z / \partial t^2$ taken as zero for negative arguments and evaluated on $S^+$ in the reference configuration (the deformation is small compared with the listener distance, and the integral is defined over the reference surface). This is the field of a planar radiator in an infinite rigid baffle; it is an idealization of the true radiation from a free disk, adopted because it is an explicit formula that needs no solution of the acoustic wave equation. A later challenge will replace it with the coupled acoustic problem.
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93+The acceleration in this integral is the total acceleration with the rigid motion linearized. At each instant let $\mathbf{c}(t) \in \mathbb{R}^3$ and $\mathbf{Q}(t) \in SO(3)$ be the mass-weighted best-fit rigid placement of the body, minimizing $\int_\Omega \rho\, |\mathbf{x} + \mathbf{u} - \mathbf{c} - \mathbf{Q} \mathbf{x}|^2\, dV$; let $\boldsymbol{\omega}$ be the angular velocity of $\mathbf{Q}$ and $\boldsymbol{\alpha} = \dot{\boldsymbol{\omega}}$; and let $\tilde{\mathbf{u}} = \mathbf{Q}^{\mathsf{T}} (\mathbf{x} + \mathbf{u} - \mathbf{c}) - \mathbf{x}$ be the displacement relative to that frame. The acceleration used is $\mathbf{a} = \ddot{\mathbf{c}} + \boldsymbol{\alpha} \times \mathbf{x} + \partial^2 \tilde{\mathbf{u}} / \partial t^2$, and $a_z$ replaces $\partial^2 u_z / \partial t^2$ above. For small motions $\mathbf{a}$ is $\partial^2 \mathbf{u} / \partial t^2$; the two differ only by the centripetal and Coriolis terms of the rigid motion. The body is free and the strike imparts momentum and angular momentum, so in the exact model it keeps turning slowly after the strike, and without this linearization the output would contain the accelerations of that turning, which no thin-plate model represents and which is not part of the sound. The convention keeps the translational and rotational kick of the strike in the output, since $\ddot{\mathbf{c}}$ and $\boldsymbol{\alpha}$ are nonzero while the strike acts, and it coincides with what modal thin-plate solvers compute when they include their rigid modes. The optional field export uses the same linearization: the displacement $\tilde{\mathbf{u}} + \mathbf{c} - \mathbf{x} + \boldsymbol{\theta} \times \mathbf{x}$ with $\boldsymbol{\theta}$ the rotation vector of $\mathbf{Q}$, which for a thin-plate solver is its transverse displacement with the rigid modes included.
94+
9395 The deliverable is a wav file with the following properties.
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9597 | Property | Value |