Let's look again at the mass-spring oscillator of §F.3.4, but this time without the driving force (which effectively decouples the mass and spring into separate first-order systems). The physical diagram and equivalent circuit are shown in Fig.F.32 and Fig.F.33, respectively.
Note that the mass and spring can be regarded as being in parallel or in series. Under the parallel interpretation, we have the WDF shown in Fig.F.34 and Fig.F.35.F.5 The reflection coefficient can be computed, as usual, from the first alpha parameter:
(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 :
The resonance frequency of the mass-spring oscillator is therefore
Since the poles are on the axis, there is no damping, as we expect.
Considering the dc case first (), we see from Fig.F.35 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
The loop on the right in Fig.F.35 receives and adds to that. Since , we see it is linearly growing in amplitude. For example, if (with ), we obtain , or
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:
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 (F.41-F.43) verifies this to be the case.
Under the bilinear transform, the maps to (half the sampling rate). It is therefore no surprise that given (), inspection of Fig.F.35 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
which agrees with the spring, as it must.
It may seem disturbing that such a simple, passive, physically rigorous simulation of a mass-spring oscillator should have to make use of state variables which grow without bound for the limiting cases of simple harmonic motion at frequencies zero and half the sampling rate. This is obviously a valid concern in practice as well. However, it is easy to show that this only happens at dc and , and that there is a true degeneracy at these frequencies, even in the physics. For all frequencies in the audio range (e.g., for typical sampling rates), such state variable growth cannot occur. Let's take closer look at this phenomenon, first from a signal processing point of view, and second from a physical point of view.
Going back to the poles of the mass-spring system in Eq.(F.39), we see that, as the imaginary part of the two poles, , approach zero, they come together at to create a repeated pole. The same thing happens at since both poles go to ``the point at infinity''.
It is a well known fact from linear systems theory that two poles at the same point in the plane can correspond to an impulse-response component of the form , in addition to the component produced by a single pole at . In the discrete-time case, a double pole at can give rise to an impulse-response component of the form . This is the fundamental source of the linearly growing internal states of the wave digital sine oscillator at dc and . It is interesting to note, however, that such modes are always unobservable at any physical output such as the mass force or spring force that is not actually linearly growing.
In the physical system, dc and infinite frequency are in fact strange cases. In the case of dc, for example, a nonzero constant force implies that the mass is under constant acceleration. It is therefore the case that its velocity is linearly growing. Our simulation predicts this, since, using Eq.(F.43) and Eq.(F.42),
The dc term is therefore accompanied by a linearly growing term in the physical mass velocity. It is therefore unavoidable that we have some means of producing an unbounded, linearly growing output variable.
In the case of , the constant force must approach zero, and we are left with at most a constant mass velocity in the limit (not a linearly growing one, since there can be no dc force at the limit). When the spring force reaches zero, , so that only zeros will feed into the loop on the right in Fig.F.35, thus avoiding a linearly growing velocity, as demanded by the physics. (A constant velocity is free to circulate in the loop on the right, but the loop on the left must be zeroed out in the limit.)
In the case of , the mass becomes unaffected by the spring force, so its final velocity must be zero. Otherwise, the attached spring would keep compressing or stretching forever, and this would take infinite energy. (Another way to arrive at this conclusion is to note that the final kinetic energy of the mass would be .) Since the total energy in an undriven mass-spring oscillator is always constant, the infinite-mass limit must be accompanied by a zero-velocity limit.F.6 This means the mass's state variable in Fig.F.35 must be forced to zero in the limit so that there will be no linearly growing solution at dc.
In summary, when two or more system poles approach each other to form a repeated pole, care must be taken to ensure that the limit is approached in a physically meaningful way. In the case of the mass-spring oscillator, for example, any change in the spring constant or mass must be accompanied by the physically appropriate change in the state variables and/or . It is obviously incorrect, for example, to suddenly set in the simulation without simultaneously clearing the spring's state variable , since the force across an infinitely compliant spring can only be zero.
Similar remarks apply to repeated poles corresponding to . In this case, the mass and spring basically change places.
If the change in or is deemed to be ``internal'', that is, involving no external interactions, the appropriate accompanying change in the internal state variables is that which conserves energy. For the mass and its velocity, for example, we must have
- Comparing digital and analog frequency formulas.
This first exercise verifies that the elementary ``tank circuit''
always resonates at exactly the frequency it should, according to the
bilinear transform frequency mapping
, where denotes ``analog frequency'' and
denotes ``digital frequency''.
- Find the poles of Fig.F.35 in terms of .
- Show that the resonance frequency is given by
- Recall that the mass-spring oscillator resonates at
. Relate these two resonance frequency formulas
via the analog-digital frequency map
- Show that the trig identity you discovered in this way is true.
I.e., show that
- Find the poles of Fig.F.35 in terms of .
Mass and Dashpot in Series