Striking a Structure with a Hammer

A hammer blow is a huge force over a tiny instant. The Dirac delta captures it as an idealized kick, and the structure answers by ringing at its own frequency. Engineers read that ringing to learn how the structure vibrates.

How it works

The Dirac delta function

In one sentence

The Dirac delta satisfies $\int f(t)\delta(t - a)\,dt = f(a)$ and $\mathcal{L}\{\delta(t - a)\} = e^{-as}$ ; the response to $P\delta(t - a)$ is $P$ times the impulse response, shifted to start at $a$ .

General formula

y'' + 2{\htmlData{sym=3}{\zeta}}{\htmlData{sym=2}{\omega}}\,y' + {\htmlData{sym=2}{\omega}}^2y = {\htmlData{sym=1}{P}}\,{\htmlData{sym=0}{\delta(t - a)}} \;\Rightarrow\; y = \frac{\htmlData{sym=1}{{P}}}{{\htmlData{sym=4}{\omega_d}}}\,e^{-{\htmlData{sym=3}{\zeta}}{\htmlData{sym=2}{\omega}}(t - a)}\sin\big({\htmlData{sym=4}{\omega_d}}(t - a)\big)\,{\htmlData{sym=5}{u(t - a)}}

Integrate the equation across the instant of the blow: everything stays finite except $y''$ , so the blow changes the velocity by exactly $P$ and leaves the position alone.

What the symbols mean

$\delta(t - a)$
Impulse
An instant kick at time a with total area 1.
$P$
Impulse strength
How hard the hammer hits.
$\omega$
Natural frequency
How fast the structure would vibrate undamped.
$\zeta$
Damping ratio
How quickly the ringing dies out.
$\omega_d$
Damped frequency
The frequency you actually hear: a little below ω.
$u(t - a)$
Step
Nothing happens before the blow.

The Dirac delta $\delta(t - a)$ is zero except at $t = a$ , with total area 1: an instant kick.
Its transform is just $e^{-as}$ , so the response is the system's impulse response shifted to $t = a$ .
The blow gives the structure a sudden velocity $P$ with no displacement; afterwards it rings freely at $\omega_d = \omega\sqrt{1 - \zeta^2}$ .
Reading the ringing frequency and how fast it fades gives $\omega$ and $\zeta$ .

Learn the math

The Dirac Delta FunctionDiff Eq · Lesson 28, Laplace Transforms