Structural dynamics · modal coordinate

Combining Modal Motion

Construct free response from individual modal contributions.

Approx. 32 min

Learning objectives

By the end of this lesson

Define modal coordinate in physical engineering terms.
Use modal expansion 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 physical displacement vector is reconstructed by summing scaled modal patterns.

u(t) = Φq(t)

Where an engineer uses it

Ring-down response after a controlled pull test: This application turns modal coordinate into a real modelling decision.

Project the measured initial state onto the retained modes.

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

modal coordinate

A time-varying amplitude multiplying one mode shape in the modal expansion of physical response.

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

Build the physical picture

Construct free response from individual modal contributions. 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

Modal expansion

u(t) = Φq(t)

The physical displacement vector is reconstructed by summing scaled modal patterns.

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

Interactive concept

Explore modal coordinate

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

How is the load applied?

How does the structure behave?

LoadTime →

Load⇣

Lower dynamic tendency

A gradual action on a relatively rigid structure is more likely to remain close to static behaviour.

Real-world application

Ring-down response after a controlled pull test

This application turns modal coordinate into a real modelling decision.

Modelling move: Project the measured initial state onto the retained modes.
Engineering question: How many modes are needed to reproduce the observed response?

Calculate, compare, interpret

Editable formula question

Convert period into frequency: Combining Modal Motion

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

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.