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Structural dynamics · harmonic excitation

Response to Repeating Loads

Build the undamped response to sinusoidal excitation.

Approx. 32 min

Learning objectives

By the end of this lesson

  • Define harmonic excitation in physical engineering terms.
  • Use undamped dynamic magnification 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

01

What the formula is for

The frequency ratio controls amplification in the ideal undamped steady-state solution.

R = 1 / |1 − r²|, r = ω/ωₙ
02

Where an engineer uses it

Rotating machinery on a supporting frame: This application turns harmonic excitation into a real modelling decision.

Represent the dominant periodic force by its harmonic amplitude and frequency.

03

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

harmonic excitation

A load that repeats sinusoidally with a defined amplitude and forcing frequency.

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

Build the physical picture

Build the undamped response to sinusoidal excitation. 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 dynamic magnification

R = 1 / |1 − r²|, r = ω/ωₙ

The frequency ratio controls amplification in the ideal undamped steady-state solution.

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

Formula experiment

Explore harmonic excitation

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

equilibrium
m
/\/\/\/\/

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.

05

Real-world application

Rotating machinery on a supporting frame

This application turns harmonic excitation into a real modelling decision.

Modelling move
Represent the dominant periodic force by its harmonic amplitude and frequency.
Engineering question
How close is operating speed to the support's natural frequency?

Calculate, compare, interpret

Editable formula question

Convert period into frequency: Response to Repeating Loads

Select an oscillator period and calculate its cyclic natural frequency.

fₙ = 1/Tₙ
Substitute your current valuesfₙ = 1 ÷ 1Answer format: enter the numerical result only in Hz. Example: 12, not “answer = 12”.
Hz
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
Substitute your current valuesratio = 4 ÷ 5Answer format: enter the numerical result only (no unit for this dimensionless result). Example: 12, not “answer = 12”.
—
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.