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Jeremy Magland Hold the shell's six rigid motions: the shallow-shell strains are not rotation invariant and a free, tilting shell is unphysical b9146b9 4 Commits
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challenge.jsonHold the shell's six rigid motions: the shallow-shell strains are not rotation invariant and a free, tilting shell is unphysical
README.mdTwo challenges, each with three strikes at 0, 2 and 4 s; 7 s output
scene.json3D scenes of both geometries for the results site (scripts/make_scenes.py)
README.md

cymbal2-crash-v1: the crash cymbal struck on the bell, at mid-radius and near the rim#

The cymbal of this tier: a paraboloidal bow with a cosine-dome bell around the mounting hole and a thickness that tapers from 1.2 mm at the bell to 0.7 mm at the rim, struck three times by the stick, on the flank of the bell, at half the radius and at 0.85 R, two seconds apart. The three hits are identical sticks; what differs is the local stiffness they meet, so the ping of the bell and the crash of the rim come out of the same equations. The model, the strikes, the listener, the output, the error measures and the common parameters are defined in MODEL.md; this page fixes what is specific to this challenge. The machine-readable version is challenge.json.

Geometry#

Z(r)=H(1−r2R2)+{hb2(1+cos⁡πrrb),r<rb,0,r≥rb,h(r)={hc,r≤rb,hc+(he−hc) r−rbR−rb,rb<r≤R,Z(r) = H \left( 1 - \frac{r^2}{R^2} \right) + \begin{cases} \dfrac{h_b}{2} \left( 1 + \cos \dfrac{\pi r}{r_b} \right), & r < r_b, \\[2ex] 0, & r \ge r_b, \end{cases} \qquad h(r) = \begin{cases} h_c, & r \le r_b, \\[1ex] h_c + (h_e - h_c)\, \dfrac{r - r_b}{R - r_b}, & r_b < r \le R, \end{cases}

for rh≤r≤Rr_h \le r \le R. The bell joins the bow with continuous slope and a jump in curvature at r=rbr = r_b; the thickness is continuous there.

Symbol Value Unit Meaning
HH 0.025 m bow height
rbr_b 0.055 m bell radius
hbh_b 0.015 m bell height above the bow
hch_c 0.0012 m thickness over the bell
heh_e 0.0007 m thickness at the rim
srefs_\text{ref} 16 (provisional) Pa output scale

At the three strike points the surface is 29, 19 and 7 mm above the rim plane, its slope is 0.43, 0.12 and 0.21, and the thickness is 1.20, 1.04 and 0.80 mm; the first footprint spans 30 to 40 mm from the axis, clear of the hole. The largest slope of the profile is about 0.46, on the flank of the bell, where shallow-shell theory is a coarse approximation of a real cymbal; the model is nevertheless defined as written.

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