Structural Dynamics / module 6
Impulse response, arbitrary loading and convolution
Response to a load history of any shape, built from the response to a single blow — and the discovery that for a short enough pulse the shape stops mattering altogether.
What you will understand
- Explain what an impulse does to a structure, in terms of momentum rather than force
- Derive the unit-impulse response function and say why it starts at zero
- Predict the response to a step, ramp, rectangular, triangular and half-sine pulse
- Use the ratio of pulse duration to natural period to choose the right treatment
- Derive the convolution integral as superposition of impulses
- Say when a short pulse can be described by its impulse alone
Lessons
What a single blow does
An impulse changes how fast a structure is moving, not where it is — and everything in this module is built from that one response.
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.
The convolution integral
Cut any load history into slices, treat each slice as an impulse, add up the free vibrations. That is the whole idea, and the integral is just notation for it.
Module summary
- An impulse changes momentum, not position — the struck structure starts at rest and moving
- h(t) = (1/mωd)e^(−ζωn t)sin ωd t, and h(0) = 0 with slope 1/m
- The convolution integral is superposition applied to a load cut into slices
- Duration, rise time, pulse shape and damping determine whether an impulse or quasi-static approximation is suitable
- When td/Tn is much smaller than 1, impulse can approximate the global response; verify finite-duration error
- For an impulsive pulse the peak occurs AFTER the load has gone, in free vibration
- The dynamic load factor peaks near td ≈ Tn/2, at about 1.8 for a half-sine
- A heavier structure is less affected by a blow — the opposite of static intuition
Teaching basis and review status
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 notesCheck your understanding
Practice
A structure of mass 18 000 kg and natural frequency 9 rad/s is struck by an impulse of 24 kN·s. Ignoring damping, what is the peak displacement in mm?
Practice
A rectangular pulse acts for 0.09 s on a structure whose natural period is 0.5 s. What is the undamped dynamic load factor?
Check yourself
Two pulses have the same area but different shapes, and both are very short compared with the natural period. What can be said about the responses?
Practice
A half-sine pulse of peak 120 kN lasts 0.04 s. What is its impulse in kN·s? The area under a half sine is 2F₀td/π.
Check yourself
Why is the convolution integral rarely used to compute earthquake response in practice?