``Piano hammer in flight''
Suppose we wish to model a situation in which a mass of size
kilograms is traveling with a constant velocity. This is an
appropriate model for a piano hammer after its key has been pressed
and before the hammer has reached the string.
Figure F.2 shows the ``wave digital mass'' derived previously.
The derivation consisted of inserting an infinitesimal
waveguideF.3 having (real) impedance
, solving for the force-wave reflectance of the mass as seen from
the waveguide, and then mapping it to the discrete time domain using
the bilinear transform.
We now need to attach the other end of the transmission line to a ``force source'' which applies a force of zero newtons to the mass. In other words, we need to terminate the line in a way that corresponds to zero force.
Let the force-wave components entering and leaving the mass
be denoted and
, respectively (i.e., we are dropping
the subscript `d' in Fig.F.2).
The physical force associated with the mass is
![$\displaystyle f(n) = f^{{+}}(n) + f^{{-}}(n)
= f^{{+}}(n) - f^{{+}}(n-1)
$](http://www.dsprelated.com/josimages_new/pasp/img4914.png)
![$ f^{{+}}(n) = -f^{{-}}(n) =
f^{{+}}(n-1)$](http://www.dsprelated.com/josimages_new/pasp/img4915.png)
Figure F.8a (left portion) illustrates what we derived
by physical reasoning, and as such, it is most appropriate as a
physical model of the constant-velocity mass. However, for actual
implementation, Fig.F.8b would be more typical in
practice. This is because we can always negate the state variable
if needed to convert it from
to
. It is
very common to see final simplifications like this to maximize
efficiency.
Note that Fig.F.8b can be interpreted physically as a wave
digital spring displaced by a constant force
.
Extracting Physical Quantities
Since we are using a force-wave simulation, the state variable
(delay element output) is in units of physical force (newtons).
Specifically,
. (The physical force is, of
course, 0, while its traveling-wave components are not 0 unless the
mass is at rest.) Using the fundamental relations relating traveling
force and velocity waves
![\begin{eqnarray*}
f^{{+}}(n) &\isdef & \quad\! R_0 v^{+}(n)\\
f^{{-}}(n) &\isdef & - R_0 v^{-}(n)\\
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/pasp/img4920.png)
where here, it is easy to convert the state variable
to
other physical units, as we now demonstrate.
The velocity of the mass, for example, is given by
![$\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:
Force Driving a Mass
Previous Section:
General Series Adaptor for Force Waves