Structural Dynamics / module 8
Numerical time integration
How the equation of motion is actually solved — what Newmark's method assumes, why stability and accuracy are different questions, and how to demonstrate that a time step is small enough.
What you will understand
- Explain why closed-form solutions run out and what replaces them
- Derive the Newmark update equations from an assumed acceleration variation
- Distinguish stability from accuracy, and say why an answer can be stable and wrong
- State the stability limits of the common schemes and where they come from
- Recognise period elongation and amplitude decay as the two distinct error modes
- Demonstrate that a time step is adequate rather than asserting it
Lessons
Newmark's method
One assumption about how acceleration varies inside a step, one insistence that equilibrium holds at its end — and everything else is algebra.
Stability, accuracy and demonstrating convergence
Two different failure modes, two different limits — and the one procedure that establishes a time step is small enough rather than asserting it.
Module summary
- Newmark assumes how acceleration varies within a step, then enforces equilibrium at its end
- γ controls numerical damping; β controls the assumed acceleration variation
- γ = 1/2, β = 1/4 (average acceleration) is unconditionally stable with no numerical damping
- γ = 1/2, β = 1/6 (linear acceleration) is more accurate but limited to Δt/T ≤ 0.551
- Central difference is explicit and cheap, and requires Δt/T < 1/π ≈ 0.318
- Unconditional stability means bounded, NOT accurate — a stable answer can be badly wrong
- Average acceleration elongates the period by about (π²/3)(Δt/T)² and preserves amplitude
- γ > 1/2 bleeds amplitude deliberately, and it damps the modes you wanted as well
- A step is adequate when halving it stops changing the quantity you will use
Teaching basis and review status
These are educational models and exercises. Check their assumptions before using the results. The full technical content and every diagram have not yet been independently verified.
Concrete and steel calculations retain their stated first-generation Eurocode teaching basis. Use the standards and amendments specified for your project; do not mix generations without an appropriate design basis.
Read the lesson register and outstanding review notesCheck your understanding
Practice
A structure has m = 5 000 kg, c = 20 kN·s/m and k = 2 MN/m. Using γ = ½, β = ¼ and Δt = 0.01 s, what is the effective stiffness k̂, in MN/m?
Practice
At Δt/T = 0.05, what is the approximate period elongation for constant average acceleration, as a percentage?
Check yourself
Why does the central-difference method have a stability limit when constant average acceleration does not?
Practice
A record is sampled at 0.01 s. An analysis needs Δt = 0.002 s. By what integer factor must the record be sub-sampled?
Check yourself
An analyst reports: 'Δt = T₁/100, which is well within normal practice, so the analysis is converged.' What is wrong with this statement?