1# cymbal-tapered-v1: the hammered cymbal lathed to a tapered thickness
3This is the fifth challenge of the cymbal tier. It is [cymbal-hammered-v1](../cymbal-hammered-v1/README.md) with one change: the thickness is no longer uniform. It is 1.2 mm over the bell and decreases linearly along the bow to 0.7 mm at the rim, the taper that lathing gives a 16 inch crash. Everything else, the bow, the bell, the hole, the twenty dents, the material, the damping, the strike, the listener, the output and the error measures, is unchanged, so the difference between the two challenges is the effect of the taper alone. A thinner rim lowers the modes that live near the edge and makes them radiate and cascade differently from a uniform sheet; a heavier center raises the low modes of the whole body. The deliverable is again six seconds of pressure at the listener.
5The machine-readable version of this specification is [challenge.json](challenge.json).
7## Model
9### Reference geometry
11The mid-surface $S$ is that of cymbal-hammered-v1: the graph $Z(x, y)$ of the bow, the bell and the dents over the annulus $r_h \le r \le R$, with unit normal $\mathbf{n}(x, y)$. The thickness is a function of the distance from the axis,
13$$
14h(r) = \begin{cases} h_c, & r \le r_b, \\[1ex] h_c + (h_e - h_c)\, \dfrac{r - r_b}{R - r_b}, & r_b < r \le R, \end{cases}
15$$
17constant over the bell and linear on the bow, continuous at the bell edge. The body is the set of points within half the local thickness of the mid-surface along its normal,
19$$
20\Omega = \{ \mathbf{X} + s\, \mathbf{n}(\mathbf{X}) : \mathbf{X} \in S,\ |s| < h(r(\mathbf{X})) / 2 \},
21$$
23so the top face is $S^+ = \{ \mathbf{X} + (h/2)\, \mathbf{n} \}$ and the bottom face $S^- = \{ \mathbf{X} - (h/2)\, \mathbf{n} \}$ with the local $h$, and the rim and the hole edge are the ruled surfaces swept by the normal segment of length $h(R)$ at $r = R$ and $h(r_h) = h_c$ at $r = r_h$. Since $h$ varies, the faces are not parallel to the mid-surface, by a slope of $|h'| / 2 = (h_c - h_e) / (2 (R - r_b)) \approx 0.0017$, negligible against the slopes of the bow.
25### Material, equations of motion, strike, and output
27All as before: 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. The strike is the same vertical raised-cosine pulse at $0.85 R$ on the $x$ axis; its normalization constant $C$ is the surface integral of the footprint bump over the top face, whose position now depends on the local thickness ($h \approx 0.80$ mm at the strike), and is given in the parameter table. The output is the same integral of the normal acceleration over $S^+$ with the retarded time and the same wav format and scale.
29## Parameters
31Those of cymbal-hammered-v1 with the uniform thickness $h$ replaced by the two thickness values.
33| Symbol | Value | Unit | Meaning |
34| --- | --- | --- | --- |
35| $h_c$ | 0.0012 | m | thickness over the bell and at its edge |
36| $h_e$ | 0.0007 | m | thickness at the rim |
37| $C$ | see challenge.json | m^2 | strike normalization integral |
39The mass of the body is close to that of the uniform 1 mm disk (the mean thickness over the area is about 0.86 mm), so the drift and the kick of the strike are similar; the bending stiffness, which goes with $h^3$, is 1.7 times the uniform value at the bell and 0.34 times it at the rim. The dents, half a millimeter deep, are now a larger fraction of the local thickness near the rim (0.7 mm) than in cymbal-hammered-v1.
41## Error measures, reference, field export
43As for cymbal-hammered-v1, including the caution that at this strike the response is chaotic beyond about 0.15 s.
45## Appendix: shallow-shell reduction
47The Marguerre equations apply with variable coefficients: the mass per unit area $\rho h$, the bending stiffness $D = E h^3 / (12 (1 - \nu^2))$ and the membrane stiffness $E h$ all vary with $r$. The bending operator becomes $\partial_{xx}[D (w_{xx} + \nu w_{yy})] + \partial_{yy}[D (w_{yy} + \nu w_{xx})] + 2 (1 - \nu) \partial_{xy}[D w_{xy}]$, and the compatibility equation for the Airy function becomes
49$$
50\partial_{yy} \frac{\Phi_{yy} - \nu \Phi_{xx}}{E h} + \partial_{xx} \frac{\Phi_{xx} - \nu \Phi_{yy}}{E h} - 2\, \partial_{xy} \frac{(1 + \nu) \Phi_{xy}}{E h} = -\tfrac{1}{2} \mathcal{L}(w, w) - \mathcal{L}(w_0, w),
51$$
53which for constant $h$ reduces to $\nabla^4 \Phi / (E h)$ on the left. A modal solver that uses the eigenfunctions of this variable-coefficient operator (with $\Phi = \partial \Phi / \partial n = 0$ on the rim) in place of the clamped biharmonic modes recovers the same structure of equations as for the uniform shell.