# cymbal2-crash-v1: the crash cymbal struck on the bell, on the bow and near the rim

The cymbal of this tier: a paraboloidal bow with a cosine-dome bell, held at the mount and a thickness that tapers from 1.2 mm at the bell to 0.65 mm at the rim, struck three times by the stick, on the outer flank of the bell, on the bow at 0.59 R and near the rim at 0.91 R, two seconds apart. The three hits are identical sticks; what differs is the local stiffness they meet, so the ping of the bell, the tone of the bow and the crash of the thin edge 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).

## Geometry

$$
Z(r) = H \left( 1 - \frac{r^2}{R^2} \right) + \begin{cases} \dfrac{h_b}{2} \left( 1 + \cos \dfrac{\pi r}{r_b} \right), & r < r_b, \\[2ex] 0, & r \ge r_b, \end{cases}
\qquad
h(r) = \begin{cases} h_c, & r \le r_b, \\[1ex] h_c + (h_e - h_c)\, \dfrac{r - r_b}{R - r_b}, & r_b < r \le R, \end{cases}
$$

for $r_m \le r \le R$. The bell joins the bow with continuous slope and a jump in curvature at $r = r_b$; the thickness is continuous there. The clamped inner edge at $r_m = 20$ mm sits on the bell, 35 mm inside its crease, where the dome is 21 mm above the bow.

| Symbol | Value | Unit | Meaning |
| --- | --- | --- | --- |
| $H$ | 0.025 | m | bow height |
| $r_b$ | 0.055 | m | bell radius |
| $h_b$ | 0.015 | m | bell height above the bow |
| $h_c$ | 0.0012 | m | thickness over the bell |
| $h_e$ | 0.00065 | m | thickness at the rim |
| $s_\text{ref}$ | 16 (provisional) | Pa | output scale |

At the three strike points the surface is 25, 16 and 4 mm above the rim plane, its slope is 0.29, 0.15 and 0.22, and the thickness is 1.20, 0.96 and 0.72 mm; the first footprint spans 40 to 50 mm from the axis, on the bell between the mount and the crease, and the third 180 to 190 mm, 13 mm from the rim. 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.