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Duhamel Integral — SDOF General Loading

u(t) = 1/(mωd) · ∫₀ᵗ F(τ)·e^(−ζωn(t−τ))·sin[ωd(t−τ)]dτ · Numerical evaluation · Comparison with Newmark-β · Chopra §4.1

System

m (kg)

k (N/m)

c (N·s/m)

Δt (s)

Loading type

F₀ (N)

Duration (s)

Results

ωn (rad/s)—

Tn (s)—

ζ—

|u|_max (m) — Duhamel—

|u|_max — Newmark-β—

Agreement—

References

Chopra §4.1u(t)=1/(mωd)·∫F(τ)·e^(−ζωn(t−τ))·sin[ωd(t−τ)]dτ

Chopra §5.2Newmark β=¼ γ=½ used for comparison

Fixed 12.08.2026The Newmark engine had a coefficient error (m/(2β) term was written as m·(1/(2β)−1)) causing up to 69% disagreement with Duhamel on this page's own default case. Verified via symbolic algebra and an independent scipy RK45 solve; corrected — agreement is now within 0.2%.

← Newmark Response Spectrum →

Duhamel Integral — SDOF General Loading

System

Results

References

Duhamel vs Newmark