Structural dynamics · impulse

Response to a Brief Impulse

Use impulse response as a building block for general loading.

Approx. 32 min

Learning objectives

By the end of this lesson

Define impulse in physical engineering terms.
Use impulse–momentum relation 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

A very brief action is often characterized more usefully by its area than by peak force alone.

I = ∫p(t)dt = mΔv

Where an engineer uses it

Impact on a protective barrier: This application turns impulse into a real modelling decision.

Estimate impulse from the force–time history before solving the subsequent free response.

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

impulse

Force integrated over a short time interval, producing a finite change in momentum.

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

Build the physical picture

Use impulse response as a building block for general loading. 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

Impulse–momentum relation

I = ∫p(t)dt = mΔv

A very brief action is often characterized more usefully by its area than by peak force alone.

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

Force-balance lab

Explore impulse

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

Displacement, u 20 mmVelocity, u̇ 0.12 m/sAcceleration, ü 0.60 m/s²Educational force limit 3.0 kN

Illustrative linear SDOF: m = 1000 kg, k = 20 kN/m, c = 5 kN·s/m

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Fᵢ 0.60 kN ←→ Fₛ + Fᴅ

Stiffness forceFₛ = ku = 0.40 kN

Damping forceFᴅ = cu̇ = 0.60 kN

Inertia forceFᵢ = mü = 0.60 kN

Educational demand/limit ratio0.53

Calculated component-force sum is within the limit you entered.

Not a structural safety verdict. The entered limit is only a learning comparison. Real safety requires code-defined load combinations, strength and serviceability checks, geometry, materials, detailing, and professional review.

Real-world application

Impact on a protective barrier

This application turns impulse into a real modelling decision.

Modelling move: Estimate impulse from the force–time history before solving the subsequent free response.
Engineering question: Does duration justify treating the action as impulsive relative to structural period?

Calculate, compare, interpret

Editable formula question

Assemble instantaneous dynamic demand: Response to a Brief Impulse

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.