cymbal-hammered-v1: bowed and belled bronze disk with twenty hammer dents, struck with a drum stick#

This is the fourth challenge of the cymbal tier. It is cymbal-bell-v1 with twenty hammer dents pressed into the bow: shallow circular depressions, 24 mm across and half a thickness deep, at fixed positions listed below. Everything else is unchanged, so the difference between the two challenges is the effect of the dents alone. A real cymbal carries hundreds of such marks; twenty large ones are a first step that already does the essential thing, which is to break the symmetry of revolution. On an axisymmetric body every mode with nodal diameters comes in a degenerate pair; the dents split the pairs by different amounts and mix modes of different angular order, which is where the shimmer and the beating of a cymbal's ring come from, and they add local curvature that feeds the nonlinear cascade. The dents are a change of the rest shape only; the residual stresses and the work hardening of real hammering are not modeled. The deliverable is again six seconds of pressure at the listener.

The machine-readable version of this specification, including the dent list, is challenge.json.

Model#

Reference geometry#

With $Z_0(r)$ the bow-plus-bell profile of cymbal-bell-v1 (paraboloid of height $H$ and cosine dome of radius $r_b$ and height $h_b$ ), the mid-surface is the graph over the annulus $r_h \le r \le R$ of

Z(x, y) = Z_0(r) + \sum_{k=1}^{20} d(\rho_k), \qquad \rho_k = \sqrt{(x - x_k)^2 + (y - y_k)^2}, \qquad d(\rho) = \begin{cases} -\dfrac{\delta}{2} \left( 1 + \cos \dfrac{\pi \rho}{r_d} \right), & \rho < r_d, \\[2ex] 0, & \rho \ge r_d, \end{cases}

a raised-cosine depression of radius $r_d$ and depth $\delta$ at each center $(x_k, y_k)$ , with continuous slope at its edge. The dents lie entirely on the bow (their centers are at least $r_b + r_d$ from the axis and at most $R - r_d$ from it), do not overlap one another (centers at least 30 mm apart), and stay clear of the strike footprint (the nearest center is more than $r_d + R_s$ from the strike point). The body, the top and bottom faces, the rim and the hole edge are the normal offsets of this surface exactly as before, with the normal now depending on both $x$ and $y$ .

The centers were drawn once, uniformly over the area of the annulus between $r_b + r_d$ and $R - r_d$ with rejection of positions closer than 30 mm to an earlier one or within 22 mm of the strike point, and rounded to 0.1 mm. Only the resulting coordinates are part of the specification.

dent	$x_k$ (mm)	$y_k$ (mm)	radius from axis (mm)
1	82.5	136.4	159.4
2	-95.4	-90.7	131.6
3	39.1	-70.9	81.0
4	-123.4	-104.7	161.8
5	63.5	83.5	104.9
6	135.7	22.2	137.5
7	139.7	-55.4	150.3
8	157.3	44.2	163.4
9	-82.6	-12.3	83.5
10	-174.6	37.2	178.5
11	-29.0	153.9	156.6
12	33.8	97.8	103.5
13	17.2	-100.3	101.8
14	99.6	-121.0	156.7
15	163.8	86.4	185.2
16	-24.1	-183.7	185.3
17	174.6	-30.3	177.2
18	-125.1	69.5	143.1
19	14.7	131.7	132.5
20	-152.8	112.2	189.6

Material, equations of motion, strike, and output#

All as for cymbal-bell-v1 and cymbal-bow-v1: the Saint Venant-Kirchhoff solid with Kelvin-Voigt viscosity and mass-proportional loss, traction-free everywhere except the strike, at rest at $t = 0$ , free, with the rigid motion linearized in the output. The strike is the same vertical pulse at $0.85 R$ on the $x$ axis; no dent reaches its footprint, so the normalization constant $C$ is unchanged. The output is the same integral of the normal acceleration over the top face, with the same wav format and scale.

Parameters#

Those of cymbal-bell-v1, plus the dents.

Symbol	Value	Unit	Meaning
$r_d$	0.012	m	dent radius
$\delta$	0.0005	m	dent depth (half the thickness)
$(x_k, y_k)$	table above	m	dent centers, 20 of them

A dent's flank slope is at most $\pi \delta / (2 r_d) \approx 0.065$ , small compared with the bell's, and its curvature at the center is $(\delta / 2)(\pi / r_d)^2 \approx 17$ 1/m (a radius of 58 mm), comparable to the bell's. The exact solution is no longer symmetric about the $x$ axis, so solvers cannot halve their work by symmetry, and the modes antisymmetric in $y$ are now excited and heard.

Error measures, reference, field export#

As for cymbal-bell-v1, including the caution that at this strike the response is chaotic beyond about 0.15 s.

Appendix: shallow-shell reduction#

The Marguerre equations apply with the full dented profile as the initial shape $w_0 = Z(x, y)$ . The dents enter $\mathcal{L}(w_0, \cdot)$ through their second derivatives, which are no longer functions of $r$ alone; a solver that expands in the modes of the undented shell needs enough modes to represent the localized curvature of each dent, and one working on a grid needs several cells across a dent (12 mm radius against 4 mm or 8 mm cells here).