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Structural dynamics · mode shape

Natural Shapes of Motion

Find frequencies and deformation patterns a coupled system prefers.

Approx. 32 min

Learning objectives

By the end of this lesson

  • Define mode shape in physical engineering terms.
  • Use eigenvalue equation 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

Nonzero mode shapes exist at eigenvalues ω² that make the dynamic stiffness matrix singular.

(K − ω²M)φ = 0
02

Where an engineer uses it

Modal test of a multi-storey building: This application turns mode shape into a real modelling decision.

Compare measured operating deflection patterns with analytical eigenvectors after consistent normalization.

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

mode shape

A relative deformation pattern in which an undamped linear system can oscillate at one natural frequency.

Why it mattersIt provides the central modelling idea for natural shapes of motion and connects observed motion to a solvable engineering representation.

Build the physical picture

Find frequencies and deformation patterns a coupled system prefers. 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

Eigenvalue equation

(K − ω²M)φ = 0

Nonzero mode shapes exist at eigenvalues ω² that make the dynamic stiffness matrix singular.

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

Application studio

Explore mode shape

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.

19

Real-world application

Modal test of a multi-storey building

This application turns mode shape into a real modelling decision.

Modelling move
Compare measured operating deflection patterns with analytical eigenvectors after consistent normalization.
Engineering question
Which modes contribute to the measured direction and frequency band?

Calculate, compare, interpret

Editable formula question

Convert period into frequency: Natural Shapes of Motion

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.