The
article was scanned from the August 1971 American Aircraft Modeler (page
29). It has been out of print for decades, and is difficult to access unless
you are fortunate enough to buy one off of eBay. Hopefully the original
author won't mind that I have reproduced it here, but if he does, I will
remove it.
This is a very cleverly written article by Mr. John G.
Burdick, and believe it or not, there were some very unflattering  even
angry  letters to the editor in the following months.
There is one
slightly nuanced feature that depends on the reader being familiar with
the meaning of complex numbers (those with both a real part and an imaginary
part). If you look at the graph in Figure 2, you will see in the caption
that the scales have been multiplied by a factor of √1. According to what
most people have been taught, one cannot take the square root of a negative
number. That is because if you take the result of a square root and multiply
it by itself, you should end up with your original number; e.g., √4 = 2,
so 2 * 2 = 4. That's good.
Now, suppose you try to do √4, what
number can you multiply by itself and end up with 4? 2 * 2 = 4, and 2
* 2 = 4 also. Hmmm... So, in order to devise a way to work with such entities,
mathematicians came up with the "imaginary operator," i (or j in engineering).
We will just use i in the example. By definition, i is set equal to the
square root of 1. Therefore, if you say √1 = i, then correspondingly,
i * i = 1. Going back to the example, then, √4 = √(1 * 4) = √1 * √4
= i * 2 (written as √4 = i2). Now working back, i2 * i2 = (i * i) * (2
* 2) = 1 * 4 = 4. QED
Why go to that trouble to explain imaginary
numbers? Look at the Figure 2 graph again. If the torque and power scales
are multiplied by √1, that means the numbers are all "imaginary;" not real!
As I said, lots of people didn't get that part of it, and wrote to condemn
Mr. Burdick for being ignorant. The irony is great!
The Zero Displacement
Engine
by John G. Burdick


Cutaway view of ZDE engine shows low engine height possible
only with Zero Stroke design. Low compression ratio (0.00 to 1)
permits use of NoLead fuels. 
FIGURE 2  Predicted performance curves of ZDE. Torque is essentially
constant throughout RPM range. Power does not peak as with nonconventional
engines. For convenience in representation, Torque and Power scales
shown have been multiplied by a factor of √1. 

FIGURE 3  Crankshaft Design for ZDE. Shaft proper
and Crankpin are coaxial, yielding zero stroke. Fuel induction is
conventional shaftrotor with hollow crankpin; Large crankpin reduces
crankcase Volume. 
THE SPECIFIC POWER OUTPUT of miniature aircraft engines has shown a
remarkable increase since the days of the Browns and Baby Cyclones. Probably
the invention of the replaceable glow plug was the most important of several
technological factors contributing to this increase, but advances in metallurgy,
fuel chemistry and engine design have also been contributory.
The
author of this paper recently had occasion to plot from published data the
relationship between engine displacement and power output. Interestingly
enough, this turned out to be a nonlinear plot; that is, engine output
was not a simple multiple of engine displacement. There was, rather, a definite
tendency for the smaller engines to have a higher specific power output
than the larger engines. Curvefitting techniques, using the leastsquares
criterion, were applied to this data, using a regression equation of the
form:
Power=Ad + Bd + C (where d equals displacement). A startling
result emerged  C was not equal to zero! In nonmathematical terms, this
simply means that it should be possible to construct a miniature engine
having zero displacement but greater than zero power output. This conclusion,
while certainly surprising, is not altogether so; only a few years ago the
present 01 and 02 engines would have been regarded as impossible.
Given that a ZDE is theoretically possible, the remainder of this paper
will concern itself with the possible configurations and applications of
such an engine.
Design of the ZDE: Since the product of two numbers
is zero if either is zero, zero displacement might be obtained through the
choice either of zero stroke or zero bore. Practical considerations, primarily
the production problems inherent in manufacturing a piston of zero diameter,
suggest the choice of zero stroke rather than zero bore for the ZDE. While
this would definitely be counter to the current practice of square or oversquare
design (that is, the borestroke ratio would be considerably smaller than
usual). it is felt that the advantages to be gained would outweigh this
disadvantage.
Specifically, some of the advantages would be as follows.
(1) Existing small engine designs could be easily modified into the ZDE
by proper crankshaft design: that is, with the crankpin coaxial 'with the
shaft proper. Provision for fuel induction would be similar to current practice,
except that the crankpin itself would be hollow. (2) Engine life should
be relatively great since even at high rpm's linear piston speed would be
low. (3) Very little dynamic crankshaft counterbalancing would be required.
Note that factors (2) and (3) taken together suggest that a very high operating
rpm would be feasible, perhaps in the 55,00065,000 range.
Application
of the ZDE: A reasonable estimate of the power output of the ZDE may be
obtained, if proper units are chosen, from the value of the constant C mentioned
above. This turns out to be about 0.025 hp, although the high operating
rpm mentioned earlier might lead to a somewhat greater power output. Since
power outputs of this order would seem ideal for indoor Re, discussion will
be confined to this application.
Current RC power loading parameters
suggest a weight of about six ounces as optimum for an aircraft utilizing
the ZDE. The recent development of extremely light radios suggest that this
is a practical figure and one which could be attained, or only slightly
exceeded, by careful construction technique. Certainly, the requirement
for highlift surfaces which would lead to good lowspeed performance, a
primary requirement in indoor RC, would also lead to the ability to tolerate
a considerably higher power loading than usual.
A hypothetical design
suitable for indoor ZDEpowered RC includes these design features: a scalelike
jet fuselage, necessary since the very high operating rpm of the ZDE, plus
its predictably low torque, would seem to dictate a shrouded propeller or
ducted fan design. The canopy provides room for the radio gear, since the
cone necessary for ducted fan operation would occupy most of the fuselage
interior. The biplane design, utilizing a high camber airfoil, would provide
lift enough for good lowspeed flight as well as compensate for the possible
high power loading. Finally, the counterbalanced control surfaces were chosen
to reduce actuator loads.
Conclusion: Several useful purposes would
be served by the development of the ZDE, primarily as a power unit for indoor
RC and perhaps for indoor freeflight as well. It is hoped that some manufacturer
will. in the near future, consider making such an engine available to model
builders.