plate-strike-nl-v1: hard-struck free aluminum plate, geometrically nonlinear#

This challenge is plate-strike-v1 with one change in the model and one in the strike. The plate is the same free 12 in by 12 in by 1.04 mm aluminum plate with the same damping constants, but the material is a Saint Venant-Kirchhoff hyperelastic solid, so that strain is measured exactly and stress acts on the deformed configuration, and the strike carries a hundred times the impulse. The material remains elastic; the nonlinearity is purely geometric. It is what produces the sound of a gong or a cymbal when struck hard: a downward pitch glide of the low partials as the amplitude decays, and a cascade of energy from the struck modes into a broadband wash. The strike here is chosen so that the transverse displacement predicted by the linear model would be many times the plate thickness, deep in the regime where these effects are unmistakable.

The deliverable, listener geometry, error measures, and optional field export are identical to plate-strike-v1, except for the duration and scale stated below. The machine-readable version of this specification is challenge.json.

Model#

With $\Omega$ , $S^+$ , $\mathbf{u}$ , $\rho$ , $\sigma_0$ , $\eta$ , $\lambda$ and $\mu$ as in plate-strike-v1, let $\mathbf{F} = \mathbf{I} + \nabla \mathbf{u}$ be the deformation gradient and

\mathbf{E} = \tfrac{1}{2} \left( \nabla \mathbf{u} + \nabla \mathbf{u}^{\mathsf{T}} + \nabla \mathbf{u}^{\mathsf{T}} \nabla \mathbf{u} \right)

the Green-Lagrange strain. The second Piola-Kirchhoff stress is linear in $\mathbf{E}$ and in its rate,

\mathbf{S} = \lambda\, \mathrm{tr}(\mathbf{E} + \eta\, \dot{\mathbf{E}})\, \mathbf{I} + 2 \mu\, (\mathbf{E} + \eta\, \dot{\mathbf{E}}),

and the equation of motion in the reference configuration is

\rho\, \frac{\partial^2 \mathbf{u}}{\partial t^2} = \nabla \cdot (\mathbf{F} \mathbf{S}) - 2 \sigma_0 \rho\, \frac{\partial \mathbf{u}}{\partial t}, \qquad \mathbf{x} \in \Omega, \quad t > 0,

where $\nabla \cdot$ acts on the reference coordinates. For small displacement gradients this reduces to the linear model of plate-strike-v1 with the same constants. The boundary conditions are the reference-configuration tractions

\mathbf{F} \mathbf{S}\, \mathbf{N} = \mathbf{0} \text{ on } \partial\Omega \setminus S^+, \qquad \mathbf{F} \mathbf{S}\, \mathbf{N} = -\, p(x, y, t)\, \mathbf{e}_z \text{ on } S^+,

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 plate's deformation. The plate starts at rest, $\mathbf{u} = \partial \mathbf{u} / \partial t = \mathbf{0}$ at $t = 0$ .

The strike pressure $p$ has the same form as in plate-strike-v1, a raised-cosine bump of radius $R$ at $(x_s, y_s)$ times a Hann pulse of duration $T_s$ , with total force peaking at $F_{\max}$ ; only $F_{\max}$ and $T_s$ differ.

Parameters#

All parameters are those of plate-strike-v1 except the peak force and the duration of the strike. In particular the material constants ( $E$ , $\nu$ , $\rho$ ), the damping constants ( $\sigma_0$ , $\eta$ ), the strike position and radius, the air constants, and the listener point are unchanged.

Symbol	Value	Unit	Meaning
$F_{\max}$	250	N	peak total force of the strike
$T_s$	0.002	s	strike duration

The impulse is $F_{\max} T_s / 2 = 0.25$ N s, that of a mallet rather than a drumstick. For comparison, the 10 N, 0.5 ms tap of plate-strike-v1 carries 0.0025 N s and bends the plate by only a few hundredths of a millimeter (the several millimeters visible in its field export are rigid-body drift); a linear extrapolation of that response to the present impulse gives a bending amplitude of several millimeters, several times the thickness, which the nonlinear model limits by membrane stiffening. The longer pulse feeds mainly the modes below about 1 kHz, so the high-frequency content of the sound is produced by the nonlinear cascade rather than by the strike itself, as in a gong. The plate is free, so the impulse also sets it drifting at about 1 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.

Output signal#

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.

Error measures and reference#

The same two error measures as plate-strike-v1, with the waveform error primary, and the same reference-selection rule. Strong nonlinearity makes long waveform agreement between different methods harder to achieve than in the linear challenges, and the spectrogram error will often be the more informative of the two; both are reported.

Optional field export#

Identical to plate-strike-v1: the normal displacement of the mid-plane, $u_z(x, y, 0, t)$ , on 61 by 61 points at 1 kHz, as HDF5.

Appendix: thin-plate reductions#

Two thin-plate reductions of this model are in common use for gongs and cymbals. The von Kármán equations keep the membrane strain to second order in the transverse displacement $w$ and either neglect in-plane inertia, in which case the in-plane state is an Airy stress function $\Phi$ with

\rho h\, \frac{\partial^2 w}{\partial t^2} + 2 \sigma_0 \rho h\, \frac{\partial w}{\partial t} + D\, \nabla^4 \left( w + \eta\, \frac{\partial w}{\partial t} \right) = \mathcal{L}(w, \Phi) + p, \qquad \nabla^4 \Phi = -\frac{E h}{2} \mathcal{L}(w, w),

with $\mathcal{L}(f, g) = f_{xx} g_{yy} + f_{yy} g_{xx} - 2 f_{xy} g_{xy}$ , or retain it, in which case the in-plane displacements are unknowns with their own wave equations coupled to $w$ through the membrane strain. Both are approximations of the three-dimensional model above, and the form with in-plane inertia is the closer one. Free edges require vanishing bending moment and effective shear force for $w$ and vanishing in-plane tractions.