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Structural dynamics · lumped-mass model

Building a Lumped-Mass Model

Place mass and stiffness where they preserve important motion.

Approx. 32 min

Learning objectives

By the end of this lesson

  • Define lumped-mass model in physical engineering terms.
  • Use matrix equilibrium 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

Vectors contain coordinate responses, while matrices describe how inertia, damping, and restoring actions are coupled.

Mü + Cu̇ + Ku = p(t)
02

Where an engineer uses it

Multi-storey shear-building idealization: This application turns lumped-mass model into a real modelling decision.

Concentrate floor mass at diaphragm levels and represent storey lateral stiffness between them.

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

lumped-mass model

An idealization concentrating distributed mass at selected coordinates connected by stiffness elements.

Why it mattersIt provides the central modelling idea for building a lumped-mass model and connects observed motion to a solvable engineering representation.

Build the physical picture

Place mass and stiffness where they preserve important motion. 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

Matrix equilibrium

Mü + Cu̇ + Ku = p(t)

Vectors contain coordinate responses, while matrices describe how inertia, damping, and restoring actions are coupled.

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

Interactive concept

Explore lumped-mass model

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.

16

Real-world application

Multi-storey shear-building idealization

This application turns lumped-mass model into a real modelling decision.

Modelling move
Concentrate floor mass at diaphragm levels and represent storey lateral stiffness between them.
Engineering question
Are enough coordinates included to capture the modes relevant to demand?

Calculate, compare, interpret

Editable formula question

Relate deformation to restoring force: Building a Lumped-Mass Model

Change stiffness and displacement, then calculate the linear restoring-force magnitude.

Fₛ = ku
Substitute your current valuesFₛ = 800 × (20 ÷ 1000)Answer format: enter the numerical result only in kN. Example: 12, not “answer = 12”.
kN
Round only at the final step. The accepted tolerance for this exercise is ±0.02.

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.