Module 08
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 this module covers
- 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
One assumption about how acceleration varies inside a step, one insistence that equilibrium holds at its end — and everything else is algebra.
Start lesson →Two different failure modes, two different limits — and the one procedure that establishes a time step is small enough rather than asserting it.
Start lesson →
Module checkpoint
Check what you have taken in
5 questions
Question 1
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?
Question 2
At Δt/T = 0.05, what is the approximate period elongation for constant average acceleration, as a percentage?
Question 3
Why does the central-difference method have a stability limit when constant average acceleration does not?
Question 4
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?
Question 5
An analyst reports: 'Δt = T₁/100, which is well within normal practice, so the analysis is converged.' What is wrong with this statement?