Newmark average acceleration

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Research, not a design tool.

The paper describes a time-stepping procedure for structural dynamics. Here we exercise its average-acceleration specialization on a deliberately small linear benchmark; we do not claim to reproduce a figure from the paper.

The example

A 1 m axial frame member, fixed at the left, with only axial translation free at the right. E = 10 MPa, A = 0.01 m², synthetic density = 300000 kg/m³. The consistent axial mass gives m = ρAL/3 = 1 t and k = EA/L = 100 kN/m. A 1 kN step acts from t = 0; initial displacement and velocity are zero. This is an exact SDOF finite-element benchmark, not a continuum bar solution.

The production 3D time-history solver runs through the existing WASM wrapper, with β = ¼, γ = ½ and no damping. The replay is a native run of the same Rust solver. No new integrator is used.

Exact reference

u(t) = (F/k)[1 − cos(ωt)], ω = √(k/m) = 10 rad/s. T = 2π/ω. The discrete average-acceleration period is Tₙ = πΔt / atan(ωΔt/2). Periods below are measured from successive upward crossings of u = F/k; interpolation affects coarse-step measurements.

Checks: every computed sample agrees with the discrete closed form within 10⁻¹⁰ m; measured period agrees with the discrete prediction within 0.5%. At Δt/T = 0.025, maximum error against the continuous solution is below 0.5 mm over four periods. Coarser steps deliberately show larger errors.

Period elongation versus time step

Not yet

• Published figures and numerical examples from the paper.
• Damping, harmonic loading, nonlinear response and multi-degree-of-freedom frames.
• Independent reviews (0); this is not engineering certification.

Cite this

Stabileo Research, “Newmark average acceleration”, reproduction of Nathan M. Newmark (1959), doi:10.1061/jmcea3.0000098. Implementation: research/benchmarks/newmark-1959 at commit 8068cd16ea2c. Engine: the Stabileo engine at the same commit. Permanent links to this exact build are planned.