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Physical Perspective on Repeated Poles in Mass-Spring System

In the physical system, dc and infinite frequency are in fact strange cases. In the case of dc, for example, a nonzero constant force implies that the mass $ m$ is under constant acceleration. It is therefore the case that its velocity is linearly growing. Our simulation predicts this, since, using Eq.$ \,$ (F.43) and Eq.$ \,$ (F.42),

\begin{eqnarray*}
v_m(n) &=& \frac{f^{{+}}_m(n)}{m} - \frac{f^{{-}}_m(n)}{m}
= \frac{1}{m} \left[x_2(n+1) + x_2(n)\right] \\
&=& \frac{1}{m} \left[2(n+1) + 2n\right]x_0
= \frac{1}{m} (4 n x_0 + 2 x_0).
\end{eqnarray*}

The dc term $ 2x_0/m$ is therefore accompanied by a linearly growing term $ 2nx_0/m$ in the physical mass velocity. It is therefore unavoidable that we have some means of producing an unbounded, linearly growing output variable.


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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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