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Structural dynamics · modal truncation

Modal Superposition in Practice

Use selected modes to efficiently estimate response.

Approx. 32 min

Learning objectives

By the end of this lesson

  • Define modal truncation in physical engineering terms.
  • Use truncated 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

01

What the formula is for

Retained modes should capture the mass participation, frequency range, and local response quantities relevant to the problem.

u(t) ≈ Σᵣ₌₁ᴺ φᵣqᵣ(t)
02

Where an engineer uses it

Efficient dynamic analysis of a large finite-element model: This application turns modal truncation into a real modelling decision.

Retain modes based on participation and sensitivity of the required demand, not a fixed count alone.

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

modal truncation

Approximating response with a limited set of modes instead of every mathematical mode in the model.

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

Build the physical picture

Use selected modes to efficiently estimate response. 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

Truncated expansion

u(t) ≈ Σᵣ₌₁ᴺ φᵣqᵣ(t)

Retained modes should capture the mass participation, frequency range, and local response quantities relevant to the problem.

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

Application studio

Explore modal truncation

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

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.

23

Real-world application

Efficient dynamic analysis of a large finite-element model

This application turns modal truncation into a real modelling decision.

Modelling move
Retain modes based on participation and sensitivity of the required demand, not a fixed count alone.
Engineering question
Has the response of interest converged as modes are added?

Calculate, compare, interpret

Editable formula question

Convert period into frequency: Modal Superposition in Practice

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.