Default contact stiffness 1e7 N/m^1.5
A softer contact than the Hertz value for a hard wooden tip on metal, standing for the compliance of a real tip; the form's step follows, and the theory note and the tutorial say so.
4 changed files+4−4
THEORY.mdmodified+1−1View file
@@ -91,7 +91,7 @@ $$
9191
9292 with $\sigma_m = \ln(1000)/T_{60}(f_m)$ from the design's decay times, plus the pad term of Section 3 on a stand. The listener signal and the movie are then sums over the modes.
9393
94-**Strike.** The stick tip is a free mass $M_s$ (30 g by default, the effective mass of a stick in a wrist stroke) arriving with speed $v_0$ at a point of the plate. The contact patch consists of the plate nodes within the tip radius (5 mm) plus the nearest one; its contact coordinate is the mass-weighted mean transverse displacement $w_c = \sum_k (m_k/M_p) w_k$ of these nodes, the penetration is $\delta = w_c - z$ with $z$ the tip position, and the force is the Hertz law $F = K\delta^{\alpha}$ for $\delta > 0$ and zero otherwise, with $\alpha = 1.5$ and $K = 5\times10^8\ \mathrm{N/m^{1.5}}$ for wood on metal. The force is spread over the patch in proportion to the node masses, which makes the pair conjugate: the modal force is $-F g_m$ and the contact coordinate is $\sum_m g_m q_m$ with the same gains $g_m = \sum_k (m_k/M_p)\, \phi_m(k)$. The stick obeys $M_s \ddot z = F$. The contact is stepped at 2 microseconds; over each step the force is held constant, and every oscillator is advanced by its exact solution under a constant force, so the step is accurate and unconditionally stable for the plate, and only the stick is integrated explicitly. Within the hit the stick may chatter, releasing and re-engaging as the plate runs ahead of it and comes back; once the contact has been open for 5 ms the stick is withdrawn, so that a plate rocking on its stand cannot strike it again tens of milliseconds later, as a drummer lifts the stick well before that. After that instant $t_1$ every mode decays freely, and its state is one complex amplitude, $q_m(t) = \mathrm{Re}\left[A_m e^{s_m (t - t_1)}\right]$ with $s_m = -\sigma_m + i\sqrt{\omega_m^2 - \sigma_m^2}$. A thin plate is a soft target: its driving-point impedance is $8\sqrt{D\rho h}$, about 60 N s/m for 1 mm bronze, so the contact lasts about a millisecond and the peak force is a few hundred newtons; the excitation is broadband, and the sound is a short bright attack followed by the ring. A hard hit in the linear model is simply a loud one; the model does not know that the plate would have stiffened.
94+**Strike.** The stick tip is a free mass $M_s$ (30 g by default, the effective mass of a stick in a wrist stroke) arriving with speed $v_0$ at a point of the plate. The contact patch consists of the plate nodes within the tip radius (5 mm) plus the nearest one; its contact coordinate is the mass-weighted mean transverse displacement $w_c = \sum_k (m_k/M_p) w_k$ of these nodes, the penetration is $\delta = w_c - z$ with $z$ the tip position, and the force is the Hertz law $F = K\delta^{\alpha}$ for $\delta > 0$ and zero otherwise, with $\alpha = 1.5$ and $K = 10^7\ \mathrm{N/m^{1.5}}$ by default, a softer contact than the Hertz value for a hard wooden tip on metal (of order $10^8$ to $10^9$), which stands for the compliance of a real tip and gives a rounder attack; the value is a parameter of the strike. The force is spread over the patch in proportion to the node masses, which makes the pair conjugate: the modal force is $-F g_m$ and the contact coordinate is $\sum_m g_m q_m$ with the same gains $g_m = \sum_k (m_k/M_p)\, \phi_m(k)$. The stick obeys $M_s \ddot z = F$. The contact is stepped at 2 microseconds; over each step the force is held constant, and every oscillator is advanced by its exact solution under a constant force, so the step is accurate and unconditionally stable for the plate, and only the stick is integrated explicitly. Within the hit the stick may chatter, releasing and re-engaging as the plate runs ahead of it and comes back; once the contact has been open for 5 ms the stick is withdrawn, so that a plate rocking on its stand cannot strike it again tens of milliseconds later, as a drummer lifts the stick well before that. After that instant $t_1$ every mode decays freely, and its state is one complex amplitude, $q_m(t) = \mathrm{Re}\left[A_m e^{s_m (t - t_1)}\right]$ with $s_m = -\sigma_m + i\sqrt{\omega_m^2 - \sigma_m^2}$. A thin plate is a soft target: its driving-point impedance is $8\sqrt{D\rho h}$, about 60 N s/m for 1 mm bronze, so the contact lasts about a millisecond and the peak force is a few hundred newtons; the excitation is broadband, and the sound is a short bright attack followed by the ring. A hard hit in the linear model is simply a loud one; the model does not know that the plate would have stiffened.
9595
9696 **Radiation.** The listener model is the Rayleigh integral for a baffled plate,
9797
src/main.tsmodified+1−1View file
@@ -267,7 +267,7 @@ class App {
267267 { key: "strikeFracY", label: "Position y (fraction of height)", step: 0.05, min: 0, max: 1 },
268268 { key: "strikeVelocity", label: "Stick speed", unit: "m/s", step: 0.5, min: 0.01 },
269269 { key: "stickMass", label: "Stick mass", unit: "g", scale: 1e-3, step: 5, min: 1 },
270- { key: "contactStiffness", label: "Contact stiffness K", unit: "N/m^a", step: 1e8, min: 1e5 },
270+ { key: "contactStiffness", label: "Contact stiffness K", unit: "N/m^a", step: 1e6, min: 1e5 },
271271 { key: "contactExponent", label: "Contact exponent a", step: 0.1, min: 1, max: 3 },
272272 { key: "contactRadius", label: "Tip radius", unit: "mm", scale: 1e-3, step: 1, min: 0.5 },
273273 ],
src/params.tsmodified+1−1View file
@@ -30,7 +30,7 @@ export const defaultParams: SimParams = {
3030 strikeFracY: 0.5,
3131 strikeVelocity: 6,
3232 stickMass: 0.03,
33- contactStiffness: 5e8,
33+ contactStiffness: 1e7,
3434 contactExponent: 1.5,
3535 contactRadius: 5e-3,
3636 listenerDist: 0.5,
src/tutorial/steps.tsmodified+1−1View file
@@ -464,7 +464,7 @@ ${loss ? live(`This design: $T_{60}$ = ${t60At(loss, 100).toFixed(2)} s at 100 H
464464 return `
465465 <p>The stick tip is a free mass $M_s$ (30 g by default, the effective mass of a stick in a wrist stroke) arriving at the speed $v_0$. While it is pressed into the plate by a penetration $\\delta$, the wood and the metal deform locally and push back with the Hertz contact force</p>
466466 $$ F = K\\delta^{\\alpha}, \\qquad \\alpha = \\tfrac32, $$
467-<p>the law for two curved elastic bodies pressed together, with $K$ set by the tip radius and the elastic constants of wood and metal. The stick obeys $M_s\\ddot z = F$, the plate is pushed by $-F$ over the small patch of nodes under the tip, and the force is zero once the two separate. The stick may bounce and hit again within a millisecond; after 5 ms of free flight it is withdrawn, as a drummer lifts it.</p>
467+<p>the law for two curved elastic bodies pressed together, with $K$ set by the tip radius and the elastic constants of wood and metal; the default of $10^7$ N/m<sup>1.5</sup> is softer than a hard wooden tip on metal would give, and stands for the compliance of a real tip. The stick obeys $M_s\\ddot z = F$, the plate is pushed by $-F$ over the small patch of nodes under the tip, and the force is zero once the two separate. The stick may bounce and hit again within a millisecond; after 5 ms of free flight it is withdrawn, as a drummer lifts it.</p>
468468 <p>How long does the contact last? A large thin plate is a soft target. When a force $F$ is applied at a point, the point starts moving at once at the velocity $F/Z$, where</p>
469469 $$ Z = 8\\sqrt{D\\rho h} $$
470470 <p>is the driving-point impedance of an infinite plate, a real constant that does not depend on frequency, one of the few exact results of plate theory. The stick, decelerated by the reaction, gives up its momentum in a time of order $M_s/Z$, and the peak force is at most about $Zv_0$; the Hertz compliance rounds off the start of the pulse and the finite plate's modes ring under it. A pulse of duration $\\tau$ carries energy up to about $1/\\tau$ in frequency, so a millisecond of contact excites the modes below a kilohertz strongly and the higher ones progressively less; a harder tip or a lighter stick shortens the contact and brightens the sound. In the linear model a harder hit is only a louder one, since nothing in the plate stiffens; the full model knows better (last chapter).</p>