The cymbal2 model#

Every challenge in this repository is an instance of the model defined here. A challenge fixes the mid-surface profile $Z(r)$ , the thickness $h(r)$ and the output scale $s_\text{ref}$ ; its README gives those and nothing else changes. The problem is to compute the pressure signal $p(t)$ of the output section for seven seconds, during which the cymbal is struck three times, and write it as a wav file.

Geometry#

The cymbal is a thin shell. Its mid-surface is the graph of a function $Z$ over the annulus

A = \{ (x, y) : r_h \le r \le R \}, \qquad r = \sqrt{x^2 + y^2},

and its thickness is $h$ . Both are functions of $r$ alone and are given per challenge. The rim $r = R$ and the edge of the hole $r = r_h$ are free. Below, subscripts $x$ , $y$ and $t$ denote partial derivatives, Greek indices run over $x, y$ , repeated Greek indices are summed, and $\delta_{\alpha\beta}$ is the Kronecker delta.

Unknowns and strains#

The unknowns are the displacements $u(x, y, t)$ , $v(x, y, t)$ , $w(x, y, t)$ of the mid-surface point $(x, y, Z(x, y))$ along the $x$ , $y$ and $z$ axes. The membrane strains are those of shallow-shell theory,

\varepsilon_{xx} = u_x + Z_x w_x + \tfrac{1}{2} w_x^2, \qquad \varepsilon_{yy} = v_y + Z_y w_y + \tfrac{1}{2} w_y^2, \qquad \varepsilon_{xy} = \tfrac{1}{2} \left( u_y + v_x + Z_x w_y + Z_y w_x + w_x w_y \right),

and the bending strains are $\kappa_{\alpha\beta} = w_{\alpha\beta}$ .

Stress resultants#

The material is linearly elastic and isotropic, with Young's modulus $E$ , Poisson's ratio $\nu$ and density $\rho$ , and has a Kelvin-Voigt viscosity of time constant $\eta$ . The membrane forces and bending moments per unit length are

N_{\alpha\beta} = \frac{E h}{1 - \nu^2} \left[ (1 - \nu)\, \tilde\varepsilon_{\alpha\beta} + \nu\, \tilde\varepsilon_{\gamma\gamma}\, \delta_{\alpha\beta} \right], \qquad M_{\alpha\beta} = D \left[ (1 - \nu)\, \tilde\kappa_{\alpha\beta} + \nu\, \tilde\kappa_{\gamma\gamma}\, \delta_{\alpha\beta} \right],

with $\tilde\varepsilon = \varepsilon + \eta\, \varepsilon_t$ , $\tilde\kappa = \kappa + \eta\, \kappa_t$ and $D = E h^3 / (12 (1 - \nu^2))$ .

Equations of motion#

The motion is defined by its weak form. For all test functions $\delta u, \delta v \in H^1(A)$ and $\delta w \in H^2(A)$ , without constraint on either edge,

\int_A \rho h \left[ (u_{tt} + 2 \sigma_0 u_t)\, \delta u + (v_{tt} + 2 \sigma_0 v_t)\, \delta v + (w_{tt} + 2 \sigma_0 w_t)\, \delta w \right] dA + \int_A \left( N_{\alpha\beta}\, \delta\varepsilon_{\alpha\beta} + M_{\alpha\beta}\, \delta\kappa_{\alpha\beta} \right) dA = - \sum_{i=1}^{3} F_i(t) \int_A \phi_i\, \delta w\, dA,

where $\sigma_0$ is a loss rate, $F_i$ and $\phi_i$ are the force and footprint of the $i$th strike of the next section, and $\delta\varepsilon$ , $\delta\kappa$ are the variations of the strains at the current state,

\delta\varepsilon_{xx} = \delta u_x + (Z_x + w_x)\, \delta w_x, \qquad \delta\varepsilon_{yy} = \delta v_y + (Z_y + w_y)\, \delta w_y, \qquad \delta\varepsilon_{xy} = \tfrac{1}{2} \left( \delta u_y + \delta v_x + (Z_x + w_x)\, \delta w_y + (Z_y + w_y)\, \delta w_x \right), \qquad \delta\kappa_{\alpha\beta} = \delta w_{\alpha\beta}.

Where the solution is smooth this is the strong form

\rho h\, (u_{tt} + 2 \sigma_0 u_t) = \partial_x N_{xx} + \partial_y N_{xy}, \qquad \rho h\, (v_{tt} + 2 \sigma_0 v_t) = \partial_x N_{xy} + \partial_y N_{yy},

\rho h\, (w_{tt} + 2 \sigma_0 w_t) = \partial_\alpha \left[ N_{\alpha\beta} (Z_\beta + w_\beta) \right] - \partial_{\alpha\beta} M_{\alpha\beta} - \sum_i F_i(t)\, \phi_i,

together with the free-edge conditions that the weak form implies at $r = R$ and $r = r_h$ . The shell is at rest at $t = 0$ : $u = v = w = 0$ and $u_t = v_t = w_t = 0$ .

The strikes#

The cymbal is struck three times by a drum stick, at the times $t_i$ and points $(x_i, 0)$ of the parameter table: on the bell, then at half the radius, then near the rim, two seconds apart. The stick is a point mass $m_s$ that moves along $z$ and interacts with the shell through a Hertz contact force. The $i$th strike has the footprint

\phi_i(x, y) = \frac{1}{c_\phi\, \pi R_s^2}\; \phi_0\!\left( \frac{\varrho_i}{R_s} \right), \qquad \varrho_i = \sqrt{(x - x_i)^2 + y^2}, \qquad \phi_0(s) = \begin{cases} \tfrac{1}{2} (1 + \cos \pi s), & 0 \le s \le 1, \\ 0, & s > 1, \end{cases} \qquad c_\phi = \tfrac{1}{2} - \tfrac{2}{\pi^2},

which lies inside $A$ and integrates to one over $A$ , and the surface height under it is the footprint average $\bar w_i(t) = \int_A \phi_i\, w\, dA$ . Let $z_i(t)$ , defined for $t \ge t_i$ , be the height of the tip of the $i$th stick. It meets the surface at $t_i$ with relative speed $v_0$ , wherever the motion of the shell has taken that surface,

z_i(t_i) = \bar w_i(t_i), \qquad \dot z_i(t_i) = \partial_t \bar w_i(t_i) - v_0.

The penetration is $\delta_i(t) = \bar w_i(t) - z_i(t)$ and the contact force is

F_i(t) = k_c\, \max(\delta_i(t), 0)^{3/2} \quad \text{for } t \ge t_i, \qquad F_i(t) = 0 \quad \text{for } t < t_i.

It acts on the stick upward, $m_s \ddot z_i = F_i$ , and on the shell downward, as the traction $-F_i \phi_i$ along $z$ in the equations of motion. Each stick is otherwise free: there is no gravity and no hand, it stays in the model after its rebound, and contact occurs whenever $\delta_i > 0$ , including any later contact.

Output#

The output is the acoustic pressure at the listener point $\mathbf{x}_L$ given by the Rayleigh integral of the transverse acceleration over the shell,

p(t) = \frac{\rho_a}{2\pi} \int_A \frac{1}{d(x, y)}\; w_{tt}\!\left( x, y,\; t - \frac{d(x, y)}{c_a} \right) dA, \qquad d(x, y) = \left| \mathbf{x}_L - (x, y, Z(x, y)) \right|,

with $w_{tt}$ taken as zero at negative times. This is the field of a planar source in a rigid baffle; the acoustic problem is not solved, and the air exerts no load on the shell.

Remarks#

The shell is free. The strains are invariant under the small rigid motions $w \to w + a + \theta_\alpha x_\alpha$ , $u_\alpha \to u_\alpha - \theta_\alpha (Z + w) - \tfrac{1}{2} \theta_\alpha \theta_\beta x_\beta$ (with $u_x = u$ , $u_y = v$ ), so the momentum and angular momentum that the strikes impart appear as a drift of $w$ that is linear in $t$ between contacts, has no transverse acceleration, and does not enter $p$ ; the in-plane inertia of that drift is of relative order $\rho \Omega^2 R^2 / E$ , about $10^{-8}$ . The transverse displacement itself does contain the drift.

In the linear regime about the flat state, $\sigma_0$ damps every mode at the rate $\sigma_0$ and $\eta$ damps a mode of angular frequency $\omega$ at the rate $\eta \omega^2 / 2$ .

Parameters common to all challenges#

Symbol	Value	Unit	Meaning
$R$	0.2032	m	outer radius (16 in diameter)
$r_h$	0.006	m	hole radius
$E$	110e9	Pa	Young's modulus (B20 bell bronze)
$\nu$	0.34		Poisson's ratio
$\rho$	8800	kg/m^3	density
$\sigma_0$	1.1	1/s	loss rate
$\eta$	1.2e-8	s	Kelvin-Voigt time constant
$m_s$	0.02	kg	stick tip mass
$v_0$	10	m/s	stick speed at contact
$k_c$	1e9	N/m^(3/2)	Hertz contact stiffness
$R_s$	0.005	m	footprint radius
$t_1, t_2, t_3$	0, 2, 4	s	strike times
$x_1, x_2, x_3$	0.035, 0.1016, 0.17272	m	strike points on the $x$ axis: on the bell, $0.5 R$ , $0.85 R$
$\rho_a$	1.204	kg/m^3	air density
$c_a$	343	m/s	speed of sound in air
$\mathbf{x}_L$	(0.65, 1.125, 0.75)	m	listener point

All values are exact decimal numbers. The listener is 1.5 m from the origin, 30 degrees above the plane $z = 0$ , at an azimuth of 60 degrees from the $x$ axis, on which the strike points lie. The strikes are the same in both challenges; on the flat annulus the first lands where the bell would be. The stick values are provisional until the first solutions have been heard; they will be fixed before any solution is published, and a challenge is never changed after that.

Deliverable#

Property	Value
channels	1
sample rate $f_s$	48000 Hz
number of samples $N$	336000 (7.0 s)
sample format	32-bit IEEE float (WAVE_FORMAT_IEEE_FLOAT)
sample $k$ , for $k = 0, \dots, N-1$	$p(k / f_s) / s_\text{ref}$
$s_\text{ref}$	`output.signal_scale` in the challenge's challenge.json, in Pa

The first sample is at $t = 0$ , the instant the first stick touches the shell. Samples are pointwise values of $p$ with no anti-aliasing filter. Do not normalize the signal; absolute amplitude is compared.

Error measures#

Two relative errors against the reference solution are computed over the $N$ samples: the spectrogram $\ell^2$ error (primary; periodic Hann window of length 2048, hop 512, no padding) and the waveform $\ell^2$ error. The formulas are those of the string challenge. The spectrogram error ranks because the response to this strike is chaotic: solutions that agree in every band still differ in waveform after the first tenth of a second, and the waveform error is then near $\sqrt{2}$ for any two of them.

Reference solution#

The reference is chosen automatically among the submitted solutions by the rule described in the repository README, unless reference_solution in challenge.json names one.

Optional field export#

A submission may provide the transverse displacement for visualization; it is not scored. The file is HDF5 with datasets x and y, float64, shape (61,), $x_i = -R + 2R\, i / 60$ and the same for $y$ ; t, float64, shape (3501,), $t_k = k / 500$ ; and u, float32, shape (3501, 61, 61), with u[k, j, i] $= w(x_i, y_j, t_k)$ in meters for grid points in $A$ and NaN elsewhere. The drift of the free shell is part of $w$ . Root attributes challenge_id, solution_id, quantity and units are recommended. The file is about 50 MB and belongs in git LFS or at an external URL.