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From Fig. N.32, we can see that the impedance of the parallel
combination of the mass and spring is given by
|
(N.38) |
(using the product-over-sum rule for combining impedances in
parallel). The poles of this impedance are given by the roots of the
denominator polynomial in :
|
(N.39) |
The resonance frequency of the mass-spring oscillator is therefore
|
(N.40) |
Since the poles
are on the axis, there is
no damping, as we expect.
We can now write reflection coefficient (see Fig. N.34) as
We see that dc (
) corresponds to , and
corresponds to .
DC Analysis.
Considering the dc case first (), we see from Fig. N.34
that the state variable will circulate unchanged in the
isolated loop on the left. Let's call this value
. Then the physical force on the spring is always equal to
|
(N.41) |
The loop on the right in Fig. N.34 receives and adds
to that. Since
, we see it is
linearly growing in amplitude. For example, if
(with
), we obtain
, or
|
(N.42) |
At first, this result might appear to contradict conservation of
energy, since the state amplitude seems to be growing without bound.
However, the physical force is fortunately better behaved:
|
(N.43) |
Since the spring and mass are connected in parallel, it must be the
true that they are subjected to the same physical force at all times.
Comparing Equations (N.41-N.43) verifies this to be the case.
Analysis at Half the Sampling Rate.
Under the bilinear transform, the maps to (half the
sampling rate). It is therefore no surprise that given
(), inspection of Fig. N.34 reveals
that any alternating sequence (sinusoid sampled at half the sampling
rate) will circulate unchanged in the loop on the right, which is now
isolated. Let
denote this alternating sequence.
The loop on the left receives
and adds
to
it, i.e.,
.
If we start out with and
, we obtain
, or
However, the physical spring force is well behaved, since
As a check, the mass force is found to be
which agrees with the spring, as it must.
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