Extracting Physical Quantities
Since we are using a force-wave simulation, the state variable
(delay element output) is in units of physical force (newtons).
Specifically,
. (The physical force is, of
course, 0, while its traveling-wave components are not 0 unless the
mass is at rest.) Using the fundamental relations relating traveling
force and velocity waves
![\begin{eqnarray*}
f^{{+}}(n) &\isdef & \quad\! R_0 v^{+}(n)\\
f^{{-}}(n) &\isdef & - R_0 v^{-}(n)\\
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/pasp/img4920.png)
where here, it is easy to convert the state variable
to
other physical units, as we now demonstrate.
The velocity of the mass, for example, is given by
![$\displaystyle v(n) = v^{+}(n) + v^{-}(n) =
\frac{f^{{+}}(n)}{m} - \frac{f^{{-}}(n)}{m} = \frac{2f^{{+}}(n)}{m} = \frac{2}{m}x(n)
$](http://www.dsprelated.com/josimages_new/pasp/img4921.png)
![$ x(n)$](http://www.dsprelated.com/josimages_new/pasp/img404.png)
![$ 2/m$](http://www.dsprelated.com/josimages_new/pasp/img4922.png)
The kinetic energy of the mass is given by
![$\displaystyle {\cal E}_m = \frac{1}{2}mv^2(n) = \frac{2}{m}x^2(n)
$](http://www.dsprelated.com/josimages_new/pasp/img4923.png)
![$ x(n)$](http://www.dsprelated.com/josimages_new/pasp/img404.png)
![$ 2/m$](http://www.dsprelated.com/josimages_new/pasp/img4922.png)
Next Section:
A More Formal Derivation of the Wave Digital Force-Driven Mass
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Series Reflection Free Port