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Newmark average-acceleration method

Analysis method · assumptions, essential equations, full derivation, algorithm, code and verification references.

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Analysis methodROUTE-DEPENDENT · linear dynamics; mixed route experimental

The default β=1/4, γ=1/2 scheme uses a constant effective matrix when M, C, K, and Δt are fixed.

01 Model and scope

Newmark assumes displacement and velocity vary according to a two-parameter time interpolation over each step. β = 1/4 and γ = 1/2 give average acceleration for a linear system; substituting the update into M a + C v + K u = f produces an effective linear system at the new time.

Route assumptions and calculation checks

  1. The linear route assumes constant M, C, K and a declared time grid; direct factorization may be reused.
  2. Rayleigh damping uses C=αR M+βR K. A two-point fit is not independent damping for every mode.
  3. Time-step convergence, energy behaviour, load interpolation, and solve diagnostics are separate checks.

Useful foundations and prerequisites

Linear staticUnits, signs and error measures

02 Essential equations and symbols

un+1=un+Δt vn+Δt2[(12−β)an+βan+1]\mathbf u_{n+1}=\mathbf u_n+\Delta t\,\mathbf v_n+\Delta t^2\left[\left(\frac12-\beta\right)\mathbf a_n+\beta\mathbf a_{n+1}\right]vn+1=vn+Δt[(1−γ)an+γan+1]\mathbf v_{n+1}=\mathbf v_n+\Delta t\left[(1-\gamma)\mathbf a_n+\gamma\mathbf a_{n+1}\right]Keff=K+a0M+a1C\mathbf K_{eff}=\mathbf K+a_0\mathbf M+a_1\mathbf CKeffun+1=feff\mathbf K_{eff}\mathbf u_{n+1}=\mathbf f_{eff}β≥14(γ+12)2,γ≥12\beta\ge\frac14\left(\gamma+\frac12\right)^2,\qquad\gamma\ge\frac12

Definitions and conventions

Δt\Delta{}t
Time-step size.
β,γ\beta{}, \gamma{}
Newmark interpolation parameters.
M, C, K
Mass, damping and stiffness matrices.
Keff
Effective matrix for the new-step solve.
tₙtₙ₊₁ΔtKnown time stateuₙ, vₙ, aₙEffective solveKeff uₙ₊₁ = feffRecover and checkaₙ₊₁, vₙ₊₁
One Newmark time stepKinematics and the effective system connect two time states. Recovery of acceleration and velocity completes the step before the time-history checks.
4 detailed steps

03 Step by step derivation

Follow the assumptions, equations and discrete operations in order. The full explanation stays visible.

1 · Start from the semi-discrete equations of motion

Finite-element assembly gives mass, damping, and stiffness matrices driven by a time-dependent load. Newmark assumes a local interpolation of acceleration over one time step; β and γ select how the new acceleration contributes to displacement and velocity updates.

Ma+Cv+Ku=f(t)\mathbf M\mathbf a+\mathbf C\mathbf v+\mathbf K\mathbf u=\mathbf f(t)

2 · Apply the Newmark kinematic update

At time n+1, displacement and velocity are expressed in terms of their previous values and the unknown new acceleration. For average acceleration, β=1/4 and γ=1/2. With constant linear matrices and a fixed time step, the same effective matrix can be reused.

un+1=un+Δtvn+Δt2[(12−β)an+βan+1]\mathbf u_{n+1}=\mathbf u_n+\Delta t\mathbf v_n+\Delta t^2\left[(\tfrac12-\beta)\mathbf a_n+\beta\mathbf a_{n+1}\right]vn+1=vn+Δt[(1−γ)an+γan+1]\mathbf v_{n+1}=\mathbf v_n+\Delta t\left[(1-\gamma)\mathbf a_n+\gamma\mathbf a_{n+1}\right]

3 · Eliminate acceleration into an effective solve

Substitute the update into equilibrium at n+1. The unknown can be displacement, acceleration, or an equivalent correction; the implementation assembles an effective stiffness from K, M, and C. Rayleigh damping is a model assumption, not a property inferred from the time integrator.

a0=1βΔt2,a1=γβΔta_0=\frac{1}{\beta\Delta t^2},\qquad a_1=\frac{\gamma}{\beta\Delta t}Keff=K+a0M+a1C,Keffun+1=feff\mathbf K_{eff}=\mathbf K+a_0\mathbf M+a_1\mathbf C,\qquad \mathbf K_{eff}\mathbf u_{n+1}=\mathbf f_{eff}C=αRM+βRK\mathbf C=\alpha_R\mathbf M+\beta_R\mathbf K

4 · Verify the time history, not just one endpoint

Check initial conditions, time-grid and load interpolation, algebraic residual at each step, time-step refinement, and energy or damping trends. Compare measured observables with the analytical oscillator or the exact shell fixture. Frequency-domain and transient evidence remain distinct routes.

β≥14(γ+12)2,γ≥12standard linear unconditional-stability region\beta\ge\tfrac14(\gamma+\tfrac12)^2,\qquad \gamma\ge\tfrac12\quad\text{standard linear unconditional-stability region}

04 Algorithm and implementation

This sheet provides explanatory references rather than an inline source excerpt. Use the labeled sources to inspect the precise boundary.

05 Verification and limitations

ROUTE-DEPENDENT · linear dynamics; mixed route experimental

One-DOF oscillator and MITC4 free/damped/forced fixtures have analytic or modal oracles.

The formulation and its explanatory checks do not by themselves establish a qualified application domain. Consult the capability matrix and the applicable recorded case.

Capability matrix →Recorded cases and original sources →Run a maintained example →

06 Sources and related sheets

Newmark equationsMITC4 transient inputMixed dynamics input

Continue the reasoning

Linear harmonic responseBeam and discrete oscillator
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