# cymbal-bell-v1: bowed bronze disk with a bell and a mounting hole, struck with a drum stick

This is the third challenge of the cymbal tier. It is [cymbal-bow-v1](../cymbal-bow-v1/README.md) with two additions at the center: a bell, a dome 110 mm across and 15 mm high on top of the bow, and the mounting hole, 12 mm across, through its apex. Material, thickness, damping, strike, listener, output and error measures are unchanged, so the difference between the two challenges is the effect of the bell alone. On a real cymbal the bell carries the "ping" of a stick played on it and stiffens the center so that the low modes of the whole cymbal rise; here the strike stays on the bow, so what changes is the body's response, not the excitation. The deliverable is again six seconds of pressure at the listener.

The machine-readable version of this specification is [challenge.json](challenge.json).

## Model

### Reference geometry

With $R$ the disk radius, $H$ the bow height, $r_b$ and $h_b$ the bell radius and height, and $r_h$ the hole radius, the mid-surface is the graph

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

The bell is a cosine dome: its slope vanishes on the axis and at its edge $r = r_b$, where it joins the bow with continuous slope but a jump in curvature, the crease that a lathed bell has. The unit normal, the body $\Omega$ as the normal offset of $S$ by $\pm h/2$, the top face $S^+$ and the bottom face $S^-$ are defined exactly as for cymbal-bow-v1, and the rim is the ruled surface swept by the normal segment at $r = R$. The hole adds an inner edge, the ruled surface swept by the normal segment at $r = r_h$, which is traction-free like the rest of the boundary. The largest slope of the mid-surface is about 0.46, on the flank of the bell at $r \approx r_b / 2$ (the dome contributes 0.43 there and the bow 0.03); the bow alone reaches 0.25, at the rim. The apex of the bell is $H + h_b = 40$ mm above the rim plane.

### Material, equations of motion, strike, and output

All as for cymbal-bow-v1: the Saint Venant-Kirchhoff solid with Kelvin-Voigt viscosity and mass-proportional loss, traction-free everywhere except the strike, at rest at $t = 0$, free (with the rigid motion linearized in the output as defined there). The strike is the same vertical raised-cosine pulse on the top face at $0.85 R$ on the $x$ axis; the bell does not reach that radius, so the normalization constant $C$ is that of cymbal-bow-v1. The output is the same integral of the normal acceleration over $S^+$ with the retarded time and the same wav format, sampling and scale.

## Parameters

Those of cymbal-bow-v1, plus the bell and the hole.

| Symbol | Value | Unit | Meaning |
| --- | --- | --- | --- |
| $R$, $H$, $h$ | 0.2032, 0.025, 0.001 | m | disk radius, bow height, thickness |
| $r_b$ | 0.055 | m | bell radius |
| $h_b$ | 0.015 | m | bell height above the bow |
| $r_h$ | 0.006 | m | hole radius |
| $E$, $\nu$, $\rho$ | 110e9, 0.34, 8800 | Pa, , kg/m^3 | B20 bronze |
| $\sigma_0$, $\eta$ | 1.1, 1.2e-8 | 1/s, s | damping constants |
| $(x_s, y_s)$, $R_s$, $T_s$, $F_{\max}$, $C$ | (0.17272, 0), 0.005, 0.001, 1000, 2.385927e-5 | m, m, s, N, m^2 | strike, as for the bow |
| $\rho_a$, $c_a$, $\mathbf{x}_L$ | 1.204, 343, (0.06, 0.08, 0.5) | kg/m^3, m/s, m | air and listener, as for the bow |

The bell's flank slope of 0.46 is beyond the range where shallow-shell theories are accurate, and its curvature (radius about $r_b^2 / (\pi^2 h_b / 2) \approx 41$ mm at the apex) is far tighter than the bow's. Thin-shell solutions of this challenge therefore carry a modeling error in the bell region that the flat disk and the bow do not have; a converged three-dimensional solution is the arbiter, as always.

## Error measures, reference, field export

As for cymbal-bow-v1. The field export is the normal displacement of the mid-surface above each point of the 61 by 61 grid, NaN outside the disk and inside the hole. Note the caution stated with the first solutions of the flat disk and the bow: at this strike the response is chaotic beyond about 0.15 s, so the waveform and spectrogram errors between any two solutions have a floor of roughly 0.6 and 0.5.

## Appendix: shallow-shell reduction

The Marguerre equations of cymbal-bow-v1 apply with the full profile as the initial shape $w_0 = Z$. For the bell, $\nabla^2 w_0$ is no longer constant: with $Z_b(r) = (h_b / 2)(1 + \cos(\pi r / r_b))$, $Z_b'' = -(h_b / 2)(\pi / r_b)^2 \cos(\pi r / r_b)$ and $Z_b' / r = -(h_b / 2)(\pi / r_b) \sin(\pi r / r_b) / r$, so the coupling operator $\mathcal{L}(w_0, \cdot)$ has the two principal curvatures $Z_{rr}$ and $Z_r / r$ as coefficients on the meridional and circumferential second derivatives. The hole makes the domain of the Airy function an annulus; the in-plane traction-free condition on the inner edge then allows $\Phi$ and $\partial \Phi / \partial n$ to take the values of an arbitrary linear function there, fixed by the single-valuedness of the in-plane displacement (Michell's conditions), rather than zero. A solver that closes the hole neglects the mass and stiffness of a 12 mm disk at the apex, a small effect on a body whose apex hardly moves, but an approximation that should be stated.