Cumulative tutorial 05
Choosing a time step, and proving it
Stability, accuracy and the demonstration that the step is adequate.
A single-degree-of-freedom model with a period of 0.45 s is to be integrated. Two schemes are available: the explicit central-difference method and Newmark with constant average acceleration. Choose a step for each, and quantify what the choice costs.
What you are given
- Natural period Tₙ = 0.45 s
- Newmark constant average acceleration, γ = ½, β = ¼
Tutorial 05
Work the problem
4 questions
Question 1
What is the largest stable time step for the explicit central-difference method, in seconds?
Question 2
Using the accuracy rule Δt ≤ Tₙ/20, what step should be used, in seconds?
Question 3
At Δt/Tₙ = 0.1, what period elongation does constant average acceleration introduce, as a percentage?
Question 4
Halving the step to Δt/Tₙ = 0.05, what is the elongation, as a percentage?
What this establishes
Report the convergence, not the assertion. 'Halving the time step changed the peak displacement by 0.3%' is a check a reviewer can weigh; 'the analysis converged' is not.