Module 06
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 this module covers
- 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
An impulse changes how fast a structure is moving, not where it is — and everything in this module is built from that one response.
Start lesson →Step, ramp, rectangular, triangular and half-sine — and the single ratio that decides which of them you need to care about.
Start lesson →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.
Start lesson →
Module checkpoint
Check what you have taken in
5 questions
Question 1
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?
Question 2
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?
Question 3
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?
Question 4
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.
Question 5
Why is the convolution integral rarely used to compute earthquake response in practice?