1# cymbal2-crash-v1: the crash cymbal struck near the bell, at mid-radius and 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 three times by the stick, just outside 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 tighter sound near the stiff center 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](../../MODEL.md); this page fixes what is specific to this challenge. The machine-readable version is [challenge.json](challenge.json).
5## Geometry
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| $s_\text{ref}$ | 16 (provisional) | Pa | output scale |
24At the three strike points the surface is 22, 19 and 7 mm above the rim plane, its slope is 0.08, 0.12 and 0.21, and the thickness is 1.15, 1.04 and 0.80 mm; the first footprint spans 65 to 75 mm from the axis, outside the bell's edge at 55 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.