Structural dynamics · damped free vibration

How Vibration Fades

See how damping changes amplitude, frequency, and return to rest.

Approx. 32 min

Learning objectives

By the end of this lesson

Define damped free vibration in physical engineering terms.
Use underdamped response envelope 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

The exponential envelope decays faster when damping ratio or natural frequency increases.

u(t) ∝ e^(−ζωₙt)

Where an engineer uses it

A footbridge after pedestrians stop: This application turns damped free vibration into a real modelling decision.

Use an equivalent viscous damper to represent several small energy-loss mechanisms.

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

damped free vibration

Free motion in which energy dissipation causes the response envelope to decrease with time.

Why it mattersIt provides the central modelling idea for how vibration fades and connects observed motion to a solvable engineering representation.

Build the physical picture

See how damping changes amplitude, frequency, and return to rest. 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

Underdamped response envelope

u(t) ∝ e^(−ζωₙt)

The exponential envelope decays faster when damping ratio or natural frequency increases.

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

Application studio

Explore damped free vibration

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

Participating mass 20 tEffective stiffness 1800 kN/mRepeated action frequency 2.0 Hz

m: Floor and roof massk: Lateral frame system

Frequencies are separatedf / fₙ = 1.32

The selected repeated action is not close to this model's natural frequency. Other modes and load effects may still govern.

Natural period0.662 s

Natural frequency1.510 Hz

Frequency ratio1.32

Preset values are intentionally illustrative. The motion graphic shows frequency proximity, not predicted displacement or structural safety.

Real-world application

A footbridge after pedestrians stop

This application turns damped free vibration into a real modelling decision.

Modelling move: Use an equivalent viscous damper to represent several small energy-loss mechanisms.
Engineering question: How many noticeable cycles remain after excitation stops?

Calculate, compare, interpret

Editable formula question

Explore velocity-dependent resistance: How Vibration Fades

Change damping and velocity, then calculate the viscous damping-force magnitude.

Fᴅ = c u̇

Damping coefficient c = 500 kN·s/mMin 1 · Textbook example 500 · Max 5000 kN·s/mVelocity u̇ = 0.2 m/sMin 0.01 · Textbook example 0.2 · Max 2 m/s

Substitute your current valuesFᴅ = 500 × 0.2Answer 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 ±0.02.

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.