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Newmark-β Integration & Response Spectrum

Step-by-step Newmark average acceleration method · Any arbitrary loading · Duhamel comparison · TBDY 2018 design spectrum → SDOF peak response

① Newmark-β Step-by-Step Integration

System & Loading

m (kg)

k (N/m)

k=4π²→Tn=1s

c (N·s/m)

Δt (s)

Loading type

F₀ (N)

Duration (s)

Results

Natural period Tn (s)—

Time steps n—

β (Newmark)0.25 (avg. accel.)

γ (Newmark)0.50

Peak |u|_max (m)—

Duhamel |u|_max—

Newmark / Duhamel ratio—

Citations

Chopra §5.2Newmark (1959) β=¼ γ=½: Average acceleration method, unconditionally stable.

Eq.(5.4a)k̃ = k + γ/(βΔt)·c + 1/(βΔt²)·m — effective stiffness

Chopra §4.1Duhamel integral for comparison — exact for any loading

dynamics.js · Chopra (2012) §5.2 ·

← Forced harmonic MDOF →

Newmark vs Duhamel — Displacement u(t)

Step-by-Step Table (first 12 steps)

i	t (s)	F (N)	u (m)	v (m/s)	a (m/s²)

② SDOF Response from Design Spectrum (TBDY 2018)

Spectrum & SDOF

SDS (g)

SD1 (g)

Damping ratio ζ

SDOF period Tn (s)

SDOF mass m (kg)

Peak SDOF Response

TA (s)—

TB (s)—

Sa(Tn) (m/s²)—

Sv(Tn) = Sa/ωn (m/s)—

Sd(Tn) = Sa/ωn² (m)—

Peak force F_max = m·Sa (kN)—

Citations

TBDY 2018 Md.2.3Sa(T): TBDY 2018 yatay elastik tasarım spektrumu

Chopra §6.1Sd=Sa/ωn² · Sv=Sa/ωn — deformation and pseudo-velocity spectra

Fixed 12.08.2026The underlying Newmark integrator had a coefficient error, corrected and re-verified against a closed-form step-load case (DLF=2.0 exact) and an independent scipy RK45 solve — see tests/dynamics.test.js.

TBDY 2018 · Chopra §6.1 ·

TBDY 2018 Design Spectrum Sa(T) — Selected Tn marked

Preliminary educational tool. Newmark average acceleration (β=¼, γ=½) is unconditionally stable for linear systems. Ss, S1 values from AFAD deprem.afad.gov.tr.