f723da2cymbal2: four challenges on one thin-shell cymbal model with a stick strikeJeremy Magland 1# cymbal2-rim-v1: the cymbal struck near the rim
3The 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 by the stick at 0.85 R on the $x$ axis, the crash stroke. The model, the strike, the listener, the output, the error measures and the common parameters are defined in [MODEL.md](../../MODEL.md); this page fixes what is specific to this challenge. The machine-readable version is [challenge.json](challenge.json). The challenges cymbal2-mid-v1 and cymbal2-bell-v1 are the same cymbal struck elsewhere.
5## Geometry and strike
7$$
8Z(r) = H \left( 1 - \frac{r^2}{R^2} \right) + \begin{cases} \dfrac{h_b}{2} \left( 1 + \cos \dfrac{\pi r}{r_b} \right), & r < r_b, \\[2ex] 0, & r \ge r_b, \end{cases}
9\qquad
10h(r) = \begin{cases} h_c, & r \le r_b, \\[1ex] h_c + (h_e - h_c)\, \dfrac{r - r_b}{R - r_b}, & r_b < r \le R, \end{cases}
11$$
13for $r_h \le r \le R$. The bell joins the bow with continuous slope and a jump in curvature at $r = r_b$; the thickness is continuous there.
15| Symbol | Value | Unit | Meaning |
16| --- | --- | --- | --- |
17| $H$ | 0.025 | m | bow height |
18| $r_b$ | 0.055 | m | bell radius |
19| $h_b$ | 0.015 | m | bell height above the bow |
20| $h_c$ | 0.0012 | m | thickness over the bell |
21| $h_e$ | 0.0007 | m | thickness at the rim |
22| $(x_s, y_s)$ | (0.17272, 0) | m | strike point, $0.85 R$ |
23| $s_\text{ref}$ | 16 (provisional) | Pa | output scale |
25At the strike point the surface is 7 mm above the rim plane, its slope is 0.21, and the thickness is 0.80 mm. 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.