Beam README: 1:2 internal resonance between modes 1 and 3 (Kirchhoff, clamped, h = 0.38)
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1616 - Stretching: "local", k3 d^4 / 4 per spring (FPUT); or "Kirchhoff", k3 S^2 / (4 nS) with S the sum of d^2 over the nS springs, which acts as one tension k3 S / nS shared by all springs. They agree when all springs are stretched alike.
1717 - Curvature (toward gongs and cymbals): rise h (m, default 0 = straight) of a stress-free rest shape w0, a half sine between two supports, 1 - cos(pi x / 2L) for a cantilever (flat at the clamp). Stretching is measured from it: d^2 becomes e = d^2 + 2 D0 d (D0 the spring's rest offset) in both forms; bending is unchanged (linear). This adds linear stiffness and a quadratic nonlinearity (softening) to the quartic one (hardening). The animation draws the beam at w0 + u, with the rest shape as the faint line through the rails, both scaled to fit the largest |w0 + u| (so at rest the arch fills the upper half; small vibrations on a large arch look small). The nonlinear scale in the derived values counts only the quartic part. Checked (Node): h = 0 bit-identical to before (all ends, both forms, k = 0 and 10, 50000 damped steps); forces = -grad V to 5e-11 and the linear stiffness = the Hessian at rest to 7e-9 for h = 0.05 and 0.3; energy error <= 7e-5 over 200 T1 up to 100 J.
1818 - Pitch glide (mode-1 frequency from zero crossings, clamped, local, τ 540 s, cents from the linear mode 1): straight beam hardening, pitch falls as the note decays (mode-1 strike 2.51 J: +1237 c at 2 J, +352 at 0.1 J, +25 at 4 mJ). Curvature flips it: h = 0.1 m hardening at high energy, softening below ~0.1 J (pitch falls, then rises slightly); h = 0.2 m +302 c at 2 J, -232 at 0.5 J, -27 at 20 mJ (mode 2 at 1.98 x mode 1, but antisymmetric, so by symmetry it has no quadratic coupling to mode 1: not a 1:2 resonance); h = 0.3 m softening throughout (-358 c at 2 J, -49 at 0.1 J), pitch rises as the note decays. Kirchhoff is similar. Default mass-2 strike, h = 0.3: lowest line rises from ~78 to 100 Hz over ~3 s of sound (17 s note, x219: mode 1 is 0.457 Hz, so the same τ covers fewer periods per second of sound), with steady difference tones (e.g. mode 2 - mode 1 at 79 Hz) from the quadratic term.
19+- 1:2 internal resonance (no code): curvature couples modes quadratically, but only modes of equal symmetry (Q1^2 Q3, not Q1^2 Q2, for a symmetric rest shape). The two lowest symmetric modes, 1 and 3, reach a 2:1 ratio only with Kirchhoff stretching: clamped at h = 0.3795 m (pinned ~0.29 m); with local stretching the ratio stays above 2.15 (h up to 2 m). Settings: Kirchhoff, clamped, h = 0.38, strike mode 3 at 50 mJ, τ 540 s, Modes and Mode history on. Mode 3 (200 Hz in the sound) and mode 1 (100 Hz) trade the energy back and forth, every 30 to 40 mode-1 periods at first (~0.3 to 0.7 s of sound), more slowly as the note decays, settling after ~6 s of the 18.5 s note; on the spectrogram the 200 Hz line pulses and the 100 Hz line fills the gaps. At h = 0.37 (ratio 2.04) or 0.40 (1.92), mode 1 stays near zero: one steady 200 Hz line. Offline (Node, damped): at 50 mJ the exchange happens for h = 0.375 to 0.395 (ratios 2.02 to 1.94); at 0.25 J the band widens to ~0.375 to 0.42, as a parametric resonance should. Stronger strikes (2.5 J) spread the energy over all modes at any h.
1920 - Normal modes of the linear beam from the stiffness matrix (assembled from the force routine) by Jacobi; mode energies as in fput-chain. Checked: pinned frequencies equal the closed form omega_n^2 = (4 k s^2 + 16 κ s^4) / m, s = sin(pi n / (2 (N + 1))), to 1e-14; N = 32 ratios clamped 2.75, 5.35, 8.78 (beam theory 2.76, 5.40, 8.93), cantilever 6.26, 17.48, 34.09 (6.27, 17.55, 34.39). At N = 8: clamped 1, 2.59, 4.68, 6.97, top 13.2; pinned 1, 3.88, 8.29, top 32.2; cantilever 1, 6.13, 16.4, top 75.
2021 - Nonlinear scale: the energy of mode 1 alone at the amplitude where its stretching energy equals its linear energy (N = 8, local: clamped 176 mJ, pinned 8.7 mJ, cantilever 1.1 mJ).
2122 - Damping: drag proportional to mass, every mode's amplitude decays as exp(-t / τ); applied exactly in half steps around each Verlet step, dissipated energy in W.