Module 1 · Lesson 1.3
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.
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'
What it calculates: How fast the structure vibrates when displaced and released
- ωn
- Natural circular frequency (rad/s)
- fn
- Natural frequency (Hz)
- Tn
- 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.
A procedure
Step 1 — Estimate the natural frequency. Use whatever is quickest that is honest. The self-weight deflection shortcut is usually enough. For a building, Tn ≈ 0.1N. For a simple beam or column, √(k/m) directly.
Step 2 — Identify the loading's characteristic time or frequency. For a transient, how long does it take to be applied? For a periodic load, at what frequency does it repeat, and what harmonics does it have? For random loading, over what band does it have energy?
Step 3 — Form the ratio.
For a TRANSIENT load, compare the rise time tr against the natural period:
| tr / Tn | Treatment |
|---|---|
| Above about 4 | Static. Inertia never engages. |
| 0.5 to 4 | Dynamic. Compute a dynamic load factor. |
| Below 0.5 | Impulsive. The peak force stops mattering; the impulse governs. |
For a PERIODIC load, compare the forcing frequency against the natural frequency, β = f/fn:
| β | Treatment |
|---|---|
| Below about 0.3 | Quasi-static. Response follows the load. |
| 0.3 to 3 | Dynamic. Resonance is a live concern near β = 1. |
| Above about 3 | Inertia-dominated. The structure barely responds. |
Step 4 — Decide what governs. If the answer is dynamic, ask which quantity matters: peak force (strength), peak acceleration (comfort and equipment), displacement (clearances and cladding) or stress range (fatigue). They do not all peak at the same time and they do not all need the same analysis.
Step 5 — When it is marginal, say so and bound it. Marginal cases are the normal case. The right response is not to guess but to bracket: run the calculation with the most and least favourable plausible values of the uncertain parameter — usually damping — and see whether the conclusion changes. If it does not, you have your answer despite the uncertainty. If it does, you have identified what you need to pin down.
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 tr/Tn, and use it to say which treatment applies.
Practice
A floor deflects 6 mm under its own weight. 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
- fn ≈ 0.5/√δ with δ in metres is fast, and fails differently from a matrix calculation
- For transients, compare rise time against Tn: above 4 static, below 0.5 impulsive
- For periodic loads, compare f/fn: 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
This is educational material. It uses simplified examples to teach principles, and must not be relied on for real design or safety-critical decisions. Module overview and checkpoint