Example Impulse Response Calculation

Let's take the series case:

Suppose the mass is hit with a force impulse:

$$f_{\mbox{ext}}(t)=\delta(t) \;\leftrightarrow\; F_{\mbox{ext}}(s)=1$$

Then the Laplace transform of the force $f_m(t)$ on the mass after time 0 is given, using the ``voltage divider'' formula, by

$$F_m(s) = \frac{ms}{ms+\frac{k}{s}} = \frac{s^2}{s^2+\frac{k}{m}}$$

Define $\omega_0^2\isdef k/m$.

The mass velocity Laplace transform is then

$$\begin{eqnarray*} V_m(s) &=& \frac{F_m(s)}{ms} \;=\; \frac{1}{m} \cdot \frac{s}{... ...}\right]\\ [5pt] &\leftrightarrow& \frac{1}{m} \cos(\omega_0 t). \end{eqnarray*}$$

Thus, the impulse response of the mass oscillates sinusoidally with

• radian frequency $\omega_0=\sqrt{k/m}$
• amplitude $1/m$
• velocity at a maximum at time $t=0$
• momentum $m\,v(0+)$ = 1