QF Solver 0.2.8 WP03B — BEAM2 and DISCRETE dynamic V&V

This addendum records a bounded analytical campaign for the four BEAM2 and DISCRETE dynamic routes left experimental by WP03A. It does not modify 0.2.7 evidence, numerical source code, or the four static/modal decisions recorded by WP03A. It does not claim universal dynamic validation or certification.

The machine-readable record is wp03b_dynamic_vnv.json. Its decision delta is wp03b_maturity_matrix.json, which explicitly preserves the WP03A static/modal decisions. The executable campaign is test_028_beam2_discrete_dynamic_vnv.py.

Frozen campaign contract

The baseline is 6b9c08f91c876bf5136a9da39555149b89dc2b98. Before execution, the machine-readable record froze each route's system, assumptions, analytical oracle, observables and tolerances. Each route is executed twice and must yield identical canonical replay digests. This is reproducible analytical V&V, not a substitution for an unavailable historical external artefact.

Route	Analytical oracle	Decision	Public maturity action
BEAM2 Newmark	exact axial SDOF free vibration	QUALIFIED_BOUNDED	Owner confirmation pending
BEAM2 harmonic	exact damped axial SDOF FRF	QUALIFIED_BOUNDED	Owner confirmation pending
DISCRETE Newmark	exact damped mass-spring response	EXPERIMENTAL	No maturity action
DISCRETE harmonic	exact damped mass-spring FRF	QUALIFIED_BOUNDED	Owner confirmation pending

The analytical oracle is appropriate because each system is deliberately a single declared physical degree of freedom with closed-form coefficients. The oracle is independent of the Newmark time stepping and direct complex solve; it does not extend the result to general beam or discrete assemblies.

BEAM2 Newmark

The case is one straight, constant-section, isotropic BEAM2 with fixed root, consistent axial mass and UX-only initial displacement. It is undamped free vibration for four periods using Newmark average acceleration at 40 and 80 samples per period. The physical reference is u(t) = u0 cos(omega t) with omega = sqrt((EA/L)/(rho A L/3)).

The frozen checks cover displacement and velocity histories, zero-crossing frequency, state-plane phase, amplitude, mechanical-energy conservation, time step refinement and deterministic replay. The scope excludes transverse dynamics, releases, joints, offsets, distributed loads, nonlinear response and multi-element extrapolation.

BEAM2 harmonic

The same axial BEAM2 is driven by a unit UX harmonic load with a declared mass-proportional damping ratio of 0.02. Seven frozen frequency ratios span below resonance, the sampled resonance point and above resonance. The oracle is H(omega) = 1/(k - omega^2 m + i omega alpha m).

Amplitude, wrapped phase, complex response, solver residual, sampled resonance and two off-resonance comparisons are checked. The sampled peak is not a continuous resonance search and does not qualify transverse or general BEAM2 frequency response.

DISCRETE Newmark

The case is one grounded UX spring (1000 N/m) and one 10 kg concentrated mass, with initial displacement 0.02 m, zero initial velocity and declared mass-proportional damping ratio 0.02. The underdamped closed form supplies displacement, velocity, damped frequency and state-plane phase references.

The campaign checks two time steps, deterministic replay, monotone mechanical energy, and the balance of mechanical energy plus trapezoid-integrated damping power. The energy quadrature is only a verification observable; it does not add a new solver output. Coupled stiffness/mass, rotational inertia, local orientations, gaps, nonlinear springs and multi-node systems remain outside the qualified scope.

DISCRETE harmonic

The same scalar spring-mass system uses the same declared damping ratio and seven frozen frequency ratios. The damped SDOF complex FRF is the closed-form oracle. The campaign checks amplitude, phase, complex response, residual, sampled resonance and off-resonance behavior. It does not qualify general MDOF or alternative damping-model harmonic response.

Decision and Owner gate

BEAM2 Newmark, BEAM2 harmonic and DISCRETE harmonic meet their predeclared bounded technical gates. DISCRETE Newmark is intentionally retained as EXPERIMENTAL: its coarse-step phase error is 0.0128952201 rad, above the frozen 0.012 rad limit. The tolerance and time grid were not adjusted after the result.

The technical record retains its pre-Owner state. The final Owner decision is maintained separately in the final WP03 Owner gate, so its frozen scope, tolérances and replay observations are not rewritten.