Suppose we wish to model a situation in which a mass
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.
shows the ``wave digital mass
'' derived previously.
The derivation consisted of inserting an infinitesimal
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
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
, respectively (i.e.
, we are dropping
the subscript `d' in Fig.F.2
The physical force associated with the mass is
The zero-force case is therefore obtained when
. This is illustrated in Fig.F.8
Wave digital mass in flight at a constant velocity.
a (left portion) illustrates what we derived
by physical reasoning, and as such, it is most appropriate as a
of the constant-velocity mass. However, for actual
b would be more typical in
practice. This is because we can always negate the state variable
if needed to convert it from
. It is
very common to see final simplifications like this to maximize
Note that Fig.F.8
b can be interpreted physically as a wave
digital spring displaced
by a constant force
Since we are using a force-wave
simulation, the state variable
(delay element output) is in units of physical force
. (The physical force is, of
course, 0, while its traveling-wave
components are not 0 unless the
is at rest.) Using the fundamental relations relating traveling
force and velocity
here, it is easy to convert the state variable
other physical units, as we now demonstrate.
The velocity of the mass, for example, is given by
Thus, the state variable
can be scaled by
to ``read out''
the mass velocity.
The kinetic energy
of the mass is given by
, the square of the state variable
can be scaled by
to produce the physical kinetic energy associated with the mass.
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