Structural dynamics · frequency-response curve

Reading Resonance Curves

Interpret amplification across forcing frequencies.

Approx. 32 min

Learning objectives

By the end of this lesson

Define frequency-response curve in physical engineering terms.
Use frequency ratio 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

Values near one place forcing close to the model's natural frequency and usually demand special attention.

r = ω/ωₙ

Where an engineer uses it

Operating-speed sweep for industrial equipment: This application turns frequency-response curve into a real modelling decision.

Plot measured or calculated amplitude over the full operating range rather than at one speed.

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

frequency-response curve

A graph showing steady response amplitude or phase against forcing frequency or frequency ratio.

Why it mattersIt provides the central modelling idea for reading resonance curves and connects observed motion to a solvable engineering representation.

Build the physical picture

Interpret amplification across forcing frequencies. 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

Frequency ratio

r = ω/ωₙ

Values near one place forcing close to the model's natural frequency and usually demand special attention.

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

Application studio

Explore frequency-response curve

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

Operating-speed sweep for industrial equipment

This application turns frequency-response curve into a real modelling decision.

Modelling move: Plot measured or calculated amplitude over the full operating range rather than at one speed.
Engineering question: Which operating band should be avoided or crossed quickly?

Calculate, compare, interpret

Editable formula question

Convert period into frequency: Reading Resonance Curves

Select an oscillator period and calculate its cyclic natural frequency.

fₙ = 1/Tₙ

Natural period Tₙ = 1 sMin 0.1 · Textbook example 1 · Max 5 s

Substitute your current valuesfₙ = 1 ÷ 1Answer format: enter the numerical result only in Hz. 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.