cymbal2-mid-v1: the cymbal struck at half the radius#

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 by the stick at 0.5 R on the $x$ axis, between the bell and the rim. The model, the strike, 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. The challenges cymbal2-rim-v1 and cymbal2-bell-v1 are the same cymbal struck elsewhere.

Geometry and strike#

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 $r_h \le r \le R$ . The bell joins the bow with continuous slope and a jump in curvature at $r = r_b$ ; the thickness is continuous there.

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
$h_e$	0.0007	m	thickness at the rim
$(x_s, y_s)$	(0.1016, 0)	m	strike point, $0.5 R$
$s_\text{ref}$	16 (provisional)	Pa	output scale

At the strike point the surface is 19 mm above the rim plane, its slope is 0.12, and the thickness is 1.04 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.