2004981Plate dynamics lab 2: a hammer on a hung plate by Rayleigh-Ritz and by Mindlin-Reissner finite elementsJeremy Magland 1import type { MethodDefinition } from "./types";
2import { ALL_GROUPS } from "./common";
4export const RR_METHOD: MethodDefinition = {
5 id: "rr",
6 name: "Rayleigh-Ritz, Kirchhoff-Love",
7 shortName: "Rayleigh-Ritz",
8 summary:
9 "Thin-plate bending energy on a tensor-Legendre basis, a dense eigensolve, and a prescribed Hertz pulse as the strike.",
10 modesFunction: "plate_rr_modes",
11 strikeFunction: "plate_rr_strike",
12 planFunction: "plate_rr_plan",
13 readingOrder: [
14 "plate_rr_modes.m",
15 "plate_rr_strike.m",
16 "plate_rr_contact.m",
17 "plate_rr_basis.m",
18 "plate_rr_legendre.m",
19 "plate_rr_plan.m",
20 "plate_setup.m",
21 "plate_hertz.m",
22 "plate_damping.m",
23 "plate_radiation.m",
24 "plate_modal_free.m",
25 "plate_synthesize.m",
26 "plate_stress.m",
27 "plate_spectrum.m",
28 "plate_gauss_legendre.m",
29 "plate_plan_common.m",
30 "plate_result_schema.m",
31 ],
32 equationGroups: ALL_GROUPS.filter((g) => !g.methods || g.methods.includes("rr")),
33 validity: [
34 "Kirchhoff-Love thin-plate theory: no shear deformation or rotary inertia, so the highest modes kept (tens of kHz on a 1 mm plate) are a little high, and a thick plate is wrong.",
35 "The strike is a prescribed force pulse, that of a Hertz contact on a rigid target. The plate does not push back on the hammer, so the peak force and the impulse are those of a rigid target: much larger than the coupled contact of the finite-element method predicts for this very light plate.",
36 "The hole enters as a point mass correction and the thread as a point spring; neither perturbs the stiffness. The rigid rotations and the pendulum mode are dropped, so nothing of the plate's whole-body motion after the strike is in this result.",
37 "Modal damping is a three-term model (material Q, air loading, the thread's dashpot) with the constants of the reference simulator, chosen, not measured, and capped at a damping ratio of 0.08.",
38 "Radiated sound is a compact-source estimate with a smooth front/back cancellation below coincidence: meaningful in timbre and decay, approximate in level.",
39 "The Legendre basis is complete on the rectangle: the first ten modes were stable to better than 0.01 Hz between orders 12 and 14. Stresses are second derivatives of the field and converge more slowly than the frequencies.",
40 "The model is linear: no denting, no chatter of the hammer, no thread buzz, no room.",
41 ],
42 numericsOf: (spec) => ({ ...spec.rr }),
43 cost: { msPerWork: 2e-4, msPerSampleMode: 4e-4 },
44};