Structural dynamics · generalized coordinate

From Continuous Structure to Generalized Coordinate

Approximate a distributed system using a selected deformation shape.

Approx. 32 min

Learning objectives

By the end of this lesson

Define generalized coordinate in physical engineering terms.
Use assumed-shape motion 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 shape function describes spatial deformation and the generalized coordinate describes how its amplitude changes with time.

u(x,t) = φ(x)q(t)

Where an engineer uses it

First-mode vibration of a slender chimney: This application turns generalized coordinate into a real modelling decision.

Choose a shape satisfying essential support conditions and representative of dominant motion.

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

generalized coordinate

A single amplitude parameter multiplying an assumed spatial shape to approximate distributed motion.

Why it mattersIt provides the central modelling idea for from continuous structure to generalized coordinate and connects observed motion to a solvable engineering representation.

Build the physical picture

Approximate a distributed system using a selected deformation shape. 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

Assumed-shape motion

u(x,t) = φ(x)q(t)

The shape function describes spatial deformation and the generalized coordinate describes how its amplitude changes with time.

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

Application studio

Explore generalized coordinate

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

First-mode vibration of a slender chimney

This application turns generalized coordinate into a real modelling decision.

Modelling move: Choose a shape satisfying essential support conditions and representative of dominant motion.
Engineering question: Does the assumed shape capture the response quantity of interest?

Calculate, compare, interpret

Editable formula question

Convert period into frequency: From Continuous Structure to Generalized Coordinate

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.