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Example Impulse Response Calculation

Let's take the series case:

\epsfig{file=eps/lseriesid.eps}
Suppose the mass is hit with a force impulse:

$\displaystyle f_{\mbox{\normalsize ext}}(t)=\delta(t) \;\leftrightarrow\; F_{\mbox{\normalsize 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

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

Define $ \omega_0^2\mathrel{\stackrel{\mathrm{\Delta}}{=}}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}{s^2+\omega_0^2}\\ [5pt]
&=& \frac{1}{m} \left[\frac{1/2}{s+j\omega_0} + \frac{1/2}{s-j\omega_0}\right]\\ [5pt]
&\leftrightarrow& \frac{1}{m} \cos(\omega_0 t).
\end{eqnarray*}

Thus, the impulse response of the mass oscillates sinusoidally with


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``Lumped Elements, One-Ports, and Passive Impedances'', by Julius O. Smith III, (From Lecture Overheads, Music 420).
Copyright © 2014-03-24 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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