Suppose now that we wish to drive
along a frictionless
surface using a variable force
. This is similar to the
previous example, except that we now want the traveling-wave
components of the force on the mass to sum to
instead of 0:
must be computed as
. This is shown in Fig.F.9
The simplified form in Fig.F.9
b can be interpreted as a wave
with applied force
delivered from an infinite
. That is, when the applied force goes to zero, the
termination remains rigid at the current displacement
Above we derived how to handle the external force
by direct physical
reasoning. In this section, we'll derive it using a more general
step-by-step procedure which can be applied systematically to more
gives the physical picture of a free mass
an external force in one dimension. Figure F.11
electrical equivalent circuit
for this scenario in which the external
force is represented by a voltage source emitting volts
and the mass is modeled by an inductor
having the value
Physical diagram of an external force driving a mass
sliding on a frictionless surface.
Electrical equivalent circuit of the force-driven mass in Fig.F.10.
The next step is to convert the voltages and currents in the
electrical equivalent circuit to wave variables
gives an intermediate equivalent circuit in which
an infinitesimal transmission line section with real impedance
has been inserted to facilitate the computation of the wave-variable
, as we did in §F.1.1
to derive Eq.
Intermediate equivalent circuit for the
force-driven mass in which an infinitesimal transmission line section
has been inserted to facilitate conversion of the mass impedance
into a wave-variable reflectance.
Intermediate wave-variable model of the
force-driven mass of Fig.F.11.
depicts a next intermediate equivalent circuit in
which the mass has been replaced by its reflectance (using ``
to denote the continuous-time reflectance
, as derived in
). The infinitesimal transmission-line section is now represented
by a ``resistor'' since, when the voltage source is initially
``switched on'', it only ``sees'' a real resistance having the value
interface). After a short period
determined by the reflectance of the mass,F.4
``return waves'' from the mass result in an ultimately
impedance. This of course must be the case because the
mass does not dissipate energy. Therefore, the ``resistor'' of
Ohms is not a resistor in the usual sense since it does not convert
(the voltage drop across it) into heat
. Instead, it
converts potential energy into propagating waves with 100%
efficiency. Since all of this wave energy is ultimately reflected by
the terminating element (mass, spring
, or any combination of masses
and springs), the net effect is a purely reactive impedance, as we
know it must be.
Interconnection of the wave digital mass
with an ideal force source by means of a two-port parallel adaptor.
The symbol ``'' is used in the WDF literature to signify a
To complete the wave digital
model, we need to connect our wave
digital mass to an ideal force source which asserts the value
each sample time. Since an ideal force source has a zero internal
impedance, we desire a parallel two-port junction which connects the
shown in Fig.F.14
) we have that the common junction force is equal to
from which we conclude that
The outgoing waves are, by Eq.
for this model, the reflection
seen on port 1 is
from port 1 is
. In the opposite
direction, the reflection coefficient
on port 2 is
the transmission coefficient from port 2 is
. The final
result, drawn in Kelly-Lochbaum
form (see §F.2.1
diagrammed in Fig.F.15
, as well as the result of some
elementary simplifications. The final model is the same as in
, as it should be.
Wave digital mass driven by external force .
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