Define the output acceleration with the rigid motion linearized
2 changed files+5−3
challenges/plate-strike-nl-v1/README.mdmodified+2−2View file
@@ -44,11 +44,11 @@ All parameters are those of plate-strike-v1 except the peak force and the durati
4444 | $F_{\max}$ | 40 | N | peak total force of the strike |
4545 | $T_s$ | 0.005 | s | strike duration |
4646
47-The impulse is $F_{\max} T_s / 2 = 0.1$ N s, forty times that of the 10 N, 0.5 ms tap of plate-strike-v1, and the contact is ten times longer: a soft mallet rather than a stick. That tap bends the plate by only a few hundredths of a millimeter (the several millimeters visible in its field export are rigid-body drift); the present strike bends it by about four thicknesses, where membrane stiffening is strong. The 5 ms pulse feeds mainly the modes below a few hundred hertz, so the high-frequency content of the sound is produced by the nonlinear cascade rather than by the strike itself, as in a gong struck with a soft beater. Harder strikes of the same plate were tried and sound like a slammed sheet of metal rather than an instrument. The plate is free, so the impulse also sets it drifting at about 0.4 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, and a viewer will want to remove the best-fit rigid motion from each frame.
47+The impulse is $F_{\max} T_s / 2 = 0.1$ N s, forty times that of the 10 N, 0.5 ms tap of plate-strike-v1, and the contact is ten times longer: a soft mallet rather than a stick. That tap bends the plate by only a few hundredths of a millimeter (the several millimeters visible in its field export are rigid-body drift); the present strike bends it by about four thicknesses, where membrane stiffening is strong. The 5 ms pulse feeds mainly the modes below a few hundred hertz, so the high-frequency content of the sound is produced by the nonlinear cascade rather than by the strike itself, as in a gong struck with a soft beater. Harder strikes of the same plate were tried and sound like a slammed sheet of metal rather than an instrument. The plate is free, so the impulse also sets it drifting at about 0.4 m/s and turning; the linearized rigid motion (see plate-strike-v1) carries no acceleration after the strike, so it does not enter the output, but it dominates the raw displacement in the field export, and a viewer will want to remove the best-fit rigid motion from each frame.
4848
4949 ## Output signal
5050
51-The output is the pressure at the listener point from the Rayleigh integral over the top face, exactly as in plate-strike-v1, with the normal acceleration taken as $\partial^2 u_z / \partial t^2$ on $S^+$ in the reference configuration (the deformation is small compared with the listener distance, and the integral is defined over the reference surface). The wav has the same format: 48 kHz, mono, 32-bit float, 144000 samples (3.0 s), sample $k$ equal to $s(k / f_s) / s_\text{ref}$ with $s_\text{ref}$ given by `output.signal_scale` in challenge.json. Do not normalize; absolute amplitude is compared.
51+The output is the pressure at the listener point from the Rayleigh integral over the top face, exactly as in plate-strike-v1, with the normal acceleration taken on $S^+$ in the reference configuration with the rigid motion linearized as defined there (the deformation is small compared with the listener distance, and the integral is defined over the reference surface). The wav has the same format: 48 kHz, mono, 32-bit float, 144000 samples (3.0 s), sample $k$ equal to $s(k / f_s) / s_\text{ref}$ with $s_\text{ref}$ given by `output.signal_scale` in challenge.json. Do not normalize; absolute amplitude is compared.
5252
5353 ## Error measures and reference
5454
challenges/plate-strike-v1/README.mdmodified+3−1View file
@@ -35,7 +35,7 @@ $$
3535 \mathbf{u}(\mathbf{x}, 0) = \mathbf{0}, \qquad \frac{\partial \mathbf{u}}{\partial t}(\mathbf{x}, 0) = \mathbf{0}.
3636 $$
3737
38-The plate is free, so rigid-body translation and rotation are possible and the strike imparts net momentum. This is intended; the output depends on acceleration and is unaffected by uniform drift.
38+The plate is free, so rigid-body translation and rotation are possible and the strike imparts momentum and angular momentum. This is intended; the output is defined with the rigid motion linearized (see the output section), so the slow turning of the free plate after the strike does not enter it.
3939
4040 ### The strike
4141
@@ -86,6 +86,8 @@ $$
8686
8787 with $\partial^2 u_z / \partial t^2$ taken as zero for negative arguments. This is the field of a planar radiator in an infinite rigid baffle; it is an idealization of the true radiation from a free plate, adopted here 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.
8888
89+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.
90+
8991 The deliverable is a wav file with the following properties.
9092
9193 | Property | Value |