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Structural dynamics · energy balance

Energy In, Energy Out

Relate work input, stored energy, and dissipated energy.

Approx. 32 min

Learning objectives

By the end of this lesson

  • Define energy balance in physical engineering terms.
  • Use maximum spring energy 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

A linear elastic element stores energy quadratically with deformation and can return it during motion.

Eₛ = ½ku²
02

Where an engineer uses it

Cyclic movement of a braced bay: This application turns energy balance into a real modelling decision.

Use a force–deformation loop to separate recoverable and dissipated energy.

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

energy balance

Accounting for input work as kinetic energy, strain energy, dissipated energy, and any energy returned by the system.

Why it mattersIt provides the central modelling idea for energy in, energy out and connects observed motion to a solvable engineering representation.

Build the physical picture

Relate work input, stored energy, and dissipated energy. 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

Maximum spring energy

Eₛ = ½ku²

A linear elastic element stores energy quadratically with deformation and can return it during motion.

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

Formula experiment

Explore energy balance

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

equilibrium
m
/\/\/\/\/

One full left-right cycle represents the calculated natural period.

Natural periodTₙ = 0.702 s
Natural frequencyfₙ = 1.424 Hz
Circular frequencyωₙ = 8.944 rad/s

Idealized undamped linear SDOF result. Changing mass or stiffness updates the model, not the safety of a real structure.

09

Real-world application

Cyclic movement of a braced bay

This application turns energy balance into a real modelling decision.

Modelling move
Use a force–deformation loop to separate recoverable and dissipated energy.
Engineering question
Where does the input energy go during repeated loading?

Calculate, compare, interpret

Editable formula question

Relate deformation to restoring force: Energy In, Energy Out

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.