Pick a scenario; every slider stays editable.
Gold text: depth per peck, number of pecks and the hole depth so far. The beak is drawn at its real position during the peck cycle; the larva turns green when the hole reaches it.
Time scale of the animation (a real peck lasts only ~1 ms). "Impact 1/500" loops just the contact.
For the wedge the force grows linearly with depth, so the head performs a quarter of a harmonic oscillation: x(t) = (v/ω) sin ωt, with ω = √(σk/m).
The science
Impact model. A head+beak of effective mass m hits wood at speed v. The tip resists with a force that grows with penetration x: for a chisel of width w and half-angle φ, F = σ (2 w tanφ) x (linear, like a spring of stiffness k); for a cone, F = σ π tan²φ x². σ is an effective penetration resistance (crushing plus cutting of fibres), larger than the plain crushing strength. Newton’s law m x″ = −F(x) is integrated with RK4 at 2 µs until the head stops.
Closed forms. Wedge: x(t) = (v/ω) sin ωt, ω = √(k/m), so depth = v√(m/k), stopping time = π/2ω, peak force = v√(mk), peak deceleration = v√(k/m). Cone: depth = (3mv²/2σπtan²φ)1/3. Work done = ½mv² and impulse = mv are checked numerically. With 50 g, 6.5 m/s and 25 MPa the stop takes about 1 ms at about 1000 g, the order of magnitude reported for woodpecker heads in high-speed video studies.
Cavity and food. Crater volume per peck (wedge) = w d² tanφ, multiplied by a chip factor for flaking; pecks = volume / chip volume. Muscle energy per peck = ½mv² / efficiency; the daily budget uses an illustrative allometric field metabolic rate 4.8 m0.68 kJ/day. A larva meal costs (h/d + 25 search taps) pecks against its energy content. The wood resistances, chip factor, muscle efficiency, larva energy and budget are illustrative, not measured values for a species; real birds also spend time moving, resting and choosing trees, so the days shown are an idealised lower bound. The head deceleration is also not directly comparable to human concussion limits (~80-100 g for ~10 ms): the pulse is ten times shorter and the skull, beak and tissue are specialised.
Checks. (1) wedge RK4 vs closed form; (2) cone RK4 vs cube-root law; (3) ∫F dx = ½mv²; (4) ∫F dt = mv; (5) wedge crater volume by integration. Simplifications: rigid head, no rebound or elastic wood, no bark, grain or temperature, the head moves along a straight line (the real motion is a rotation about the shoulder) and the beak is a perfect wedge or cone.
The tree, ground, foliage, woodpecker, larva and wood chips are models made in Blender; the physics runs in the page.