Structural dynamics · time integration

Computing Response Through Time

Advance displacement and velocity when a closed-form solution is unavailable.

Approx. 32 min

Learning objectives

By the end of this lesson

Define time integration in physical engineering terms.
Use discrete acceleration with consistent quantities and units.
Change model inputs and explain the resulting trend.
Separate an educational model result from a code-based safety decision.

Before you calculate

What this means in the real world

What the formula is for

Equilibrium at each time step supplies acceleration; the selected algorithm then updates velocity and displacement.

aᵢ = [pᵢ − cvᵢ − kuᵢ]/m

Where an engineer uses it

Recorded earthquake acceleration applied to a model: This application turns time integration into a real modelling decision.

Choose a time step small enough to represent both input and important structural periods.

What a wrong value can do

A wrong unit, sign, scale, boundary condition, or property can produce a believable number with the wrong physical meaning. It may understate demand, exaggerate it, or lead to the wrong failure check.

Definition

time integration

A step-by-step numerical process for updating structural response through a discretized loading history.

Why it mattersIt provides the central modelling idea for computing response through time and connects observed motion to a solvable engineering representation.

Build the physical picture

Advance displacement and velocity when a closed-form solution is unavailable. Start by identifying the system boundary, response coordinates, applied action, and the properties that store or dissipate energy. A useful model is the simplest one that preserves the behaviour needed for the engineering question.

Formula

Discrete acceleration

aᵢ = [pᵢ − cvᵢ − kuᵢ]/m

Equilibrium at each time step supplies acceleration; the selected algorithm then updates velocity and displacement.

u: response displacement or displacement vectorm
t: times
ω: angular frequency where usedrad/s

Formula experiment

Explore time integration

Change the available model inputs, watch the motion or force relationship, and describe the trend before reading the numerical result.

Mass, m 10 tStiffness, k 800 kN/m

equilibrium

/\/\/\/\/

One full left-right cycle represents the calculated natural period.

Natural periodTₙ = 0.702 s

Natural frequencyfₙ = 1.424 Hz

Circular frequencyωₙ = 8.944 rad/s

Idealized undamped linear SDOF result. Changing mass or stiffness updates the model, not the safety of a real structure.

Real-world application

Recorded earthquake acceleration applied to a model

This application turns time integration into a real modelling decision.

Modelling move: Choose a time step small enough to represent both input and important structural periods.
Engineering question: Is the computed response stable and converged as the time step is refined?

Calculate, compare, interpret

Editable formula question

Assemble instantaneous dynamic demand: Computing Response Through Time

Edit the model state and calculate the force needed to balance inertia, damping, and stiffness.

p(t) = mü + cu̇ + ku

Mass m = 1000 kgMin 100 · Textbook example 1000 · Max 10000 kgAcceleration ü = 0.5 m/s²Min 0 · Textbook example 0.5 · Max 3 m/s²Damping c = 3000 N·s/mMin 0 · Textbook example 3000 · Max 20000 N·s/mVelocity u̇ = 0.1 m/sMin 0 · Textbook example 0.1 · Max 1 m/sStiffness k = 20000 N/mMin 1000 · Textbook example 20000 · Max 100000 N/mDisplacement u = 20 mmMin 0 · Textbook example 20 · Max 100 mm

Substitute your current valuesp = (1000×0.5) + (3000×0.1) + (20000×0.02)Answer format: enter the numerical result only in kN. Example: 12, not “answer = 12”.

Your calculated answer

Round only at the final step. The accepted tolerance for this exercise is ±0.03.

The displayed minimum, maximum, and textbook values define this learning exercise only. They are not universal design limits.

Editable formula question

Compare a model demand with your entered criterion

Enter a demand and a comparison value chosen for this exercise, then interpret the ratio.

utilization indicator = model demand / entered criterion

Model demand D = 4 unitsMin 0.1 · Textbook example 4 · Max 20 unitsEntered criterion L = 5 unitsMin 0.1 · Textbook example 5 · Max 20 units

Substitute your current valuesratio = 4 ÷ 5Answer format: enter the numerical result only (no unit for this dimensionless result). Example: 12, not “answer = 12”.

Your calculated answer

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Round only at the final step. The accepted tolerance for this exercise is ±0.01.

The displayed minimum, maximum, and textbook values define this learning exercise only. They are not universal design limits.