Piano Hammer Strike in 3D: Felt Contact, Inharmonic Partials and Timbre

What is modelled

One piano string struck by a felt hammer, resolved in time at 44.1 kHz and drawn in slow motion on a Blender-modelled string rig (agraffe, bridge, soundboard, cast-iron plate, hammer with felt head). This is the 3D companion of the 2D page "Piano Hammer: Felt Hardness and String Timbre"; both pages run the same solver and the same numerical checks.

Equations

• Stiff damped string: utt = c²uxx − κ²uxxxx − 2b1ut + 2b2utxx + F(t)δ(x−x0)/μ, with c = √(T/μ), κ² = EI/μ, simply supported ends, explicit finite differences.
• Hammer of mass 3 g: mh uh'' = −F, felt law F = K·ηp with η = uh − u(x0) and p = 2.5 (Chaigne and Askenfelt). A harder felt means larger K, a shorter contact and more energy in the upper partials.
• Stiffness makes the partials inharmonic: fn = n f0√(1 + B n²), B = π²EI/(T L²). A strike at L/n puts the hammer on a node of mode n and suppresses it.
• The sound is the force the string exerts on the bridge, T ux(L).

Checks run in the page

• Fundamental against the closed form f0√(1+B).
• Partial 10 against f0 n √(1+Bn²).
• Energy lost by the hammer equals the string energy at release.
• Soft, medium and hard felt: contact time falls and the spectral centroid rises monotonically.
• A strike at L/4 suppresses partial 4.
• Geometry of the Blender rig: bridge distance, strike point and felt crown follow the solver for the selected note.

Honest simplifications

• One vertical polarisation, one string per note (a real note has up to three), the same 1 mm steel wire at every pitch with a simplified scale length (real bass strings are wound), a rigid bridge so nothing is radiated: no soundboard, no longitudinal waves, no hammer-shank flexibility, no aftersound.
• The motion is drawn in slow motion and with a vertical exaggeration (slider); the real peak displacement is a fraction of a millimetre and the felt compression is drawn squashed in proportion to it. The hammer is shown on a straight vertical path (a real one swings on an arc) and its drawn rebound is clipped at 4.5 cm below the string, as if the backcheck caught it.
• The felt constants K are in the realistic range but are not measurements of one instrument. The rig is a scale illustration, the neighbour strings are decoration.

What to try

• Press "Soft felt" then "Hard felt" and compare contact time (graph 1) and the high partials (graph 2).
• Pin the current spectrum, change the strike point to L/4 and watch partial 4 disappear.
• Choose a high note: the string gets shorter and stiffer, so inharmonicity (B) grows.