Structural Dynamics / Impulse response, arbitrary loading and convolution

Pulses, and why duration is everything

Step, ramp, rectangular, triangular and half-sine — and the single ratio that decides which of them you need to care about.

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.
Independent validation of the associated interactive implementation, input domain and project applicability remains required; this review checks lesson context and selected calculations, not design certification.
Validate impact/blast load history, contact duration, nonlinear capacity and impulse approximation; do not use the short-pulse comparisons as impact design.

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

For an undamped linear oscillator, a suddenly applied sustained load reaches twice its static displacement. A finite pulse gives a different peak depending on its shape and duration. The rectangular-pulse shock spectrum starts at zero as duration tends to zero at fixed peak force, reaches 2 at half a natural period and stays at 2 for longer durations. Other pulse shapes have different curves.

By the end of this lesson you should be able to

Compute the response to the standard idealised pulses
Explain the two branches of a shock spectrum and what divides them
Say why the peak response can occur after the load has gone
Show that pulse SHAPE stops mattering once the pulse is short enough

Compare the load timescales

The ratio t_d/T_n compares pulse duration with the natural period, but duration alone does not define the response. Rise time, fall time, shape and damping also matter.

An impulse approximation is useful when the structure moves little during the pulse: t_d/T_n must be much smaller than 1, with the approximation checked for the response being used. Equal-area pulses then give approximately equal global responses. They need not give equal local stresses, contact forces or nonlinear damage.

A quasi-static approximation requires sufficiently slow changes throughout the load history. An abrupt start can create a large transient even when the load remains on for many periods.

Rectangular-pulse shock spectrum

For an undamped SDOF system initially at rest, a rectangular pulse of peak F₀ gives:

Duration ratio	Peak displacement divided by F₀/k
t_d/T_n < 0.5	2 sin(π t_d/T_n)
t_d/T_n ≥ 0.5	2

The peak after a short rectangular pulse occurs during free vibration. At half a period or longer, the response reaches the sustained-step peak while the load is still acting. The factor 0.5 separates these exact rectangular-pulse branches; it is not a general boundary for impulse accuracy.

For example, the ideal impulse approximation overestimates the short rectangular-pulse peak by about 1.7% at t_d/T_n=0.1 and about 11% at 0.25. Choose the approximation according to the required accuracy, and compute the finite history when necessary.

Shock spectrum, rectangular pulse, undamped

\text{DLF} = \begin{cases} 2\sin\!\left(\pi\dfrac{t_d}{T_n}\right) & t_d/T_n < 0.5 \\[6pt] 2 & t_d/T_n \ge 0.5 \end{cases}

What it calculates: Peak response as a multiple of the static deflection under the peak force

t_d: Pulse duration (s)
T_n: Natural period of the structure (s)
DLF: Dynamic load factor, x_max/(F₀/k) (—)

This assumes

Rectangular pulse of constant magnitude F₀
No damping — this is the upper bound
Linear elastic single-degree-of-freedom response

In plain terms: The factor never exceeds 2, whatever the pulse duration. A rectangular pulse cannot do worse than a step, because a step is what it becomes when it lasts long enough. Short pulses do progressively less, and the transition is at exactly half a natural period.

Try it

Build a force history and see what it does

Choose a shape and change how long it lasts, measured against the structure's own natural period.

Pulse shape

Pulse duration t_d / T_n

The single most important parameter on this panel.

Natural period T_ns

Damping ratio ζ

Peak force F₀kN

Force against Time. Applied force reaches a peak magnitude of 60 kN.

Response
Static deflection F₀/k

Displacement against Time. Response reaches a peak magnitude of 57.4 mm. Static deflection F₀/k reaches a peak magnitude of 36.5 mm.

Show the numbers behind this plot

t = 0.00 s

Animation time — drag to examine any instant

Static displacement: 36.5 mm
Peak displacement: 57.4 mm
Dynamic load factor: 1.57
Peak occurs at: 0.24 s
Total impulse ∫F dt: 10.9 kN·s
Impulsive estimate I/(mω): 69.1 mm
Rectangular shock spectrum: 1.62

Around t_d ≈ T_n/2 the pulse and the structure are in step, and this is where the dynamic load factor is largest. It is 1.57 here.

What this shows: What decides the response to a pulse is not its magnitude or even its shape, but its DURATION measured against the natural period. Below about half a period only the area under the curve matters.

Worked example

A blast-resistant panel, checked three ways

Given

A façade panel spanning 3.5 m, idealised as SDOF
Natural frequency 14 Hz
A pressure pulse falling linearly from 45 kPa at t=0 to zero at 18 ms
Panel area 3.5 × 1.2 m; damping neglected, which is conservative

Find

The peak response, treated as static, as impulsive, and properly.

Assumptions

The panel stays elastic — a real blast design would not assume this, and the point here is the comparison of methods
Damping neglected: over a fraction of one cycle it removes almost nothing

Predict first

A rectangular pulse and a triangular pulse have the same peak force and the same duration, which is very short compared with the natural period. Which produces the larger response?

Practice

A rectangular pulse lasts 0.06 s on a structure with a natural period of 0.4 s. Using the undamped shock spectrum, what is the dynamic load factor?

Practice

For the same structure, what pulse duration in seconds would give the maximum dynamic load factor of 2?

Practice

A triangular pulse has a peak of 80 kN and lasts 0.025 s. What is its impulse, in kN·s?

Check yourself

For a pulse with t_d/T_n = 0.2, when does the peak response occur?

Worked example

The same impulse, delivered three ways

Given

A structure with a natural period of 0.5 s
A rectangular pulse of the same total impulse, delivered over 0.05 s, 0.25 s and 2.0 s

Find

How the peak response differs

Summary

t_d/T_n is the single ratio that decides the treatment
Use t_d/T_n≪1 for an impulse approximation, and slow rise/fall times for quasi-static response; check accuracy against the actual history
Rectangular shock spectrum: DLF = 2 sin(π t_d/T_n) below 0.5, and 2 above it
In the short-pulse limit, impulse controls the leading-order global response
The impulsive DLF can be written neatly as Iω_n/F₀
For a short pulse the peak occurs AFTER the load has gone
'DLF = 2 to be safe' is neither safe nor conservative outside the step-load case