So one day while I wasn't doing my math homework, I realized I almost knew how
to figure this stuff out for real. And, for some strange reason, I tried to.
With the classes I was taking when I came up with this, it's not like I was
hurting for math problems or anything, so maybe I'm just a glutton for
punishment. Or maybe my brain was stimulated by the thought-provoking class
discussions. Or maybe I'm just a misunderstood genius. (Hey, knock that off
over there.) Probably, it was the really good coffee I've been drinking
lately. Some day, maybe, I'll understand how to integrate a multivariable
function, and I'll finish this by hand once and for all. Or maybe someone will
plug my little formula into some big bucks math program which will do it for
me. But for now, I'll punch the numbers into my handy dandy TI-82, and it'll
do a sort of half-assed numerical integration and show me pretty graphs and
stuff when I ask it to do a definite integral.

So first, I needed to figure out piston motion in terms of crank rotation.
That's this one:

To get piston position for theta:

r = stroke radius
c = connecting rod length
b = bore radius
theta = degrees of crankshaft rotation away from TDC
y = piston's distance from TDC

y = - r cos theta - SQRT(c^2 - (r sin theta)^2) + c + r

And to get volume inside the cylinder:

V = y(pi b^2) + volume above piston at TDC

So now we can calculate total cylinder volume for theta, which we need to
figure out nRT (as in, PV=nRT). Figure out a way to mount a pressure
transducer in the combustion chamber of the engine under development. Drill a
hole into the combustion chamber, use some super spiffy spark plug with one
built in, whatever. Then track crankshaft position with a toothed wheel and
pickup or some such thingy. Then run the engine with a known setup (stroke,
bore, con rod length, fuel, temp) and track pressure in the cylinder for
theta. You'll end up with a scatter plot that looks something like this, but
includes pressure data from the intake and exhaust strokes too (from -360 to
360 instead of just -180 to 180 like in this diagram):

The whole cycle looks something like this:

Run this test every 500 rpm or so from the bottom of the range of real power
to just short of where you think stuff will come flying through the engine cases.
Test a couple of different fuels while you're at it in case some burn faster than
others. You might need that data later. So now you have scatter plots showing
pressure at theta for a known stroke/bore/rod, but that doesn't do anything for
you yet, because the piston motion, and therefore volume @ theta, isn't the
same for a different rod length, which is what you're testing. Remember
Pressure times Volume = number of moles times the gas law constant 'R' times
Temperature in degrees Kelvin... PV = nRT. Good luck counting the number
of moles of gas that are in the cylinder, or traclomg the temperature in terms of
theta. Plus with all the reactions going on, there's no way I'm ever in this
lifetime going to be able to figure out how to calculate nRT at every instant
from first principles. But from the scatter plot, we have pressure as a function
of theta. From the piston motion function we have volume as a function of
theta. If you perform a gazillionth degree regression sum of the points of the
plot and find the equation of that curve then divide by volume for every theta,
you have nRT as a function of theta, which for a known rpm (time) through the
various equations noted gives pressure as a function of theta for any connecting
rod length running at that rpm, and since you tested every 500rpm, you also
have a function for nRT based on RPM. There are a lot of factors I'm electing
to neglect, like, oh, friction (such a trivial thing, really...), piston motion's effect
on port flow (number of moles) and charge turbulence (burn rate, probably), but
this should be close enough to get an idea of what's going on in there.

Okay, so now we have a function for pressure in terms of degrees of crankshaft
rotation (theta). Because I forget how to put Greek symbols here, I'm using
the following:

d = theta
pi = pi, not p times i
Pressure = f(theta), so P = f(d), which is really a big crazy function, I'm
just going to call it 'P' and put it in another function which really is just
dying to be integrated symbolically, and you can't stop me!

torque = r*SQRT((P*pi*b^2)^2+(P*pi*b^2)*((r sin d)/SQRT(c^2-(r sin d)^2))^2)sin((90-d)+(cos^-1((r sin d)/c)))

Then find the definite integral of that function from 0 to 720 degrees (four
stroke), and just like magic, we have work performed for one
cycle.

Now take that engine and hook it up to a DC dyno or just spin the damned thing
over somehow (with the ignition off, no fuel in the fuel
system, and the throttle wide open) through the range of rpm and measure the
torque required just to overcome friction and pumping losses every few hundred
rpm. Let's just call that friction, since that's what it is, really. Now
here's the easy part:

rpm(Torque - friction)/(constant for whatever units of measurement you're
using... 5252 for English to find horsepower, or something else to get kilowatt
hours) = power put out, or output (after taxes and stuff)

Repeat for every 500rpm or so, do another regression sum to find the curve, do
the definite integral between the minimum engine speed you think the engine will
see on the track and redline, then do it again for different connecting rod lengths.
The one with the most area under the curve is the fastest engine, in my little
theoretical paper world, anyway. Of course, you could just put to graphs next to
each other and eyeball them too, like everyone already does anyway.

Hey, if the rod fits in the engine, it might even work.

:)

(no, I'm not holding my breath either)
 

Drop me a line.
patrick@cape.com

Using the formulae above, I wrote an Excel spreadsheet with some graphs so you can play around with the bore, stroke, and rod length to get a feel for how the numbers interact. That math is ridiculous for anyone who isn't currently in a college level physics course or a hard core nerd, but pretty much anyone can punch in some numbers and get a feel for how changes to this affects that. It's set up to show two different configurations side by side, which I think helps to get a feel for how it all works.