Structural Dynamics / What structural dynamics is

Deciding whether a dynamic analysis is needed

A practical procedure: what to estimate, what to compare it against, and what to do with the answer.

Lesson review: expert verification remains open

This lesson has received a full-context editorial review and a check of its technical explanations. Selected arithmetic has been checked separately. This is not independent engineering sign-off.

Checks needed for project use

A dynamics reviewer must confirm excitation models, damping, modal assumptions and any comfort or serviceability interpretation. Small displacement alone is not an acceptance criterion.

These are educational models and exercises. Check their assumptions before using the results. The full technical content and every diagram have not yet been independently verified.

Concrete and steel calculations retain their stated first-generation Eurocode teaching basis. Use the standards and amendments specified for your project; do not mix generations without an appropriate design basis.

Read the lesson register and outstanding review notes

Why this matters

Dynamic analysis is expensive — in time, in modelling effort and in the judgement it demands. Most structures never need it. The skill worth having early is not running the analysis but deciding whether to, and being able to say why not when the answer is no.

That decision rests on two numbers you can usually estimate in a few minutes.

By the end of this lesson you should be able to

Estimate a natural frequency from mass and stiffness without a model
Compare loading timescales against the natural period systematically
Apply the frequency-ratio test to decide whether dynamics matters
Say what to do when the answer is 'possibly'

Natural frequency of a single-degree-of-freedom system

\omega_n = \sqrt{\frac{k}{m}}, \qquad f_n = \frac{1}{2\pi}\sqrt{\frac{k}{m}}, \qquad T_n = 2\pi\sqrt{\frac{m}{k}}

What it calculates: How fast the structure vibrates when displaced and released

ω_n: Natural circular frequency (rad/s)
f_n: Natural frequency (Hz)
T_n: Natural period (s)
k: Stiffness — force per unit displacement (N/m)
m: Mass — NOT weight (kg)

This assumes

A single degree of freedom, or the first mode of a system that has more
Linear elastic stiffness
Mass and stiffness measured at the same point, in the same direction

In plain terms: Stiffness raises the frequency and mass lowers it, both under a square root. Quadrupling the stiffness only doubles the frequency, which is why stiffening a structure out of a vibration problem is expensive and why changing its depth — which raises I sharply — is the effective move.

Take care

The unit error that ends more dynamics calculations than any other

In ω_n = √(k/m), the m is MASS, in kilograms. Not weight, not force, not kilonewtons.

If your loads are in kN and you have a weight W in kN, then

m = W/g = W × 1000/9.81 kg

Getting this wrong changes the frequency by a factor of √9.81 = 3.13 — enough to move a floor from 'fine' to 'unusable' and, worse, enough to look plausible.

The fastest self-check is to work the shortcut form instead. For an equivalent single-degree-of-freedom model whose weight and static deflection δ use the same effective mass and stiffness:

f_n = (1/2π)√(g/δ) ≈ 0.5/√δ Hz, with δ in metres

Under that idealisation, a 10 mm self-weight deflection gives f_n ≈ 0.5/√0.01 = 5 Hz. If your matrix calculation disagrees with that estimate by a factor near 3, check the units and the equivalent-mass/model assumptions.

Assess the relevant time scales

Estimate the natural frequencies using an appropriate model. A self-weight-deflection estimate requires compatible equivalent mass and stiffness. A building estimate such as T ≈ 0.1N is only a rough, model-dependent check.

For a transient force, examine both rise time and total duration relative to the relevant natural periods. A short rise time indicates a rapidly applied load; it does not make a sustained step an impulse. An impulse approximation requires a force pulse short compared with the response period and an appropriate description of its integrated force.

For periodic force excitation of a linear SDOF system, compare the forcing frequency with the natural frequency. Resonance may be important near equality. High frequency can give small displacement without negligible acceleration. Multiple modes, damping and the force history may affect the result.

Choose the response quantity needed for the check: force or stress for strength, acceleration for comfort or equipment, displacement for clearances, and stress range for fatigue. Test sensitivity to uncertain stiffness, mass and damping assumptions.

Worked example

Estimating a natural frequency three ways, and checking them against each other

Given

A steel plant platform: a single mass of 8 tonnes supported on four columns
Each column is a 203×203×46 UC, 3.5 m long, fixed top and bottom
I about the weak axis is 1 548 cm⁴, E = 210 GPa

Find

The natural frequency, by direct calculation and by two independent checks.

Assumptions

The platform is rigid compared with the columns, so this is genuinely a single degree of freedom
Columns are fixed at both ends against rotation, so each contributes 12EI/h³
The columns' own mass is neglected — it is small compared with 8 tonnes

Practice

A tank of mass 12 000 kg sits on a support structure with a lateral stiffness of 4.8 MN/m. What is its natural frequency in Hz?

Practice

A structure has a natural period of 0.5 s. A load is applied over 0.1 s. Give the ratio t_r/T_n, and use it to say which treatment applies.

Practice

An equivalent SDOF floor model deflects 6 mm under a force equal to its effective mass times g. Estimate its natural frequency in Hz using the deflection shortcut.

Check yourself

A pump forces at 8 Hz. A floor has a natural frequency of 25 Hz. What is the appropriate treatment?

Check yourself

Which of these is the clearest sign that a load must be treated as dynamic rather than static?

Worked example

A screening test that takes thirty seconds

Given

A steel-framed plant room floor, 7.5 m span, supporting a fan
The fan runs at 900 rpm
A first estimate of the floor's fundamental frequency is 9 Hz

Find

Whether a dynamic analysis is needed, before building anything

Summary

Estimate the natural frequency first — everything else is a comparison against it
f_n ≈ 0.5/√δ with δ in metres is fast, and fails differently from a matrix calculation
For transients, examine both rise time and total duration relative to the relevant natural periods; rise time alone does not establish an impulse
For periodic loads, compare f/f_n: below 0.3 quasi-static, above 3 inertia-dominated
Check the harmonics of a periodic load, not only its fundamental
Mass in kilograms, never weight in kilonewtons — the error factor is 3.13 and it looks plausible