Structural dynamics · free vibration

Motion After Release

Connect initial displacement and velocity to natural oscillation.

Approx. 32 min

Learning objectives

By the end of this lesson

Define free vibration in physical engineering terms.
Use undamped free response 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

Initial displacement and velocity set the amplitude and phase; mass and stiffness set the natural frequency.

u(t) = u₀ cos(ωₙt) + (v₀/ωₙ) sin(ωₙt)

Where an engineer uses it

A floor released after a short disturbance: This application turns free vibration into a real modelling decision.

Idealize the dominant floor motion as one coordinate and estimate its effective mass and stiffness.

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

free vibration

Motion governed by the system's initial state after the external dynamic force has been removed.

Why it mattersIt provides the central modelling idea for motion after release and connects observed motion to a solvable engineering representation.

Build the physical picture

Connect initial displacement and velocity to natural oscillation. 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

Undamped free response

u(t) = u₀ cos(ωₙt) + (v₀/ωₙ) sin(ωₙt)

Initial displacement and velocity set the amplitude and phase; mass and stiffness set the natural frequency.

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

Force-balance lab

Explore free vibration

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

/\/\/\/

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

A floor released after a short disturbance

This application turns free vibration into a real modelling decision.

Modelling move: Idealize the dominant floor motion as one coordinate and estimate its effective mass and stiffness.
Engineering question: How quickly will the motion repeat after the disturbance ends?

Calculate, compare, interpret

Editable formula question

Experiment with period: Motion After Release

Change mass and stiffness, predict the period, and check your result.

Tₙ = 2π√(m/k)

Generalized mass m = 1000 kgMin 100 · Textbook example 1000 · Max 10000 kgLateral stiffness k = 40000 N/mMin 5000 · Textbook example 40000 · Max 200000 N/m

Substitute your current valuesTₙ = 2π√(1000 ÷ 40000)Answer format: enter the numerical result only in s. Example: 12, not “answer = 12”.

Your calculated answer

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.

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.