Tech Tip of the Month: Calculations and Π
While scanning Model Engine World to locate references for the ED Bee (this month's featured engine), I rediscovered a two page article that went to great pains to complicate a simple procedure: calculating engine capacity. Pressed for time, I did not read the text in detail, so if I'm doing the original author a disservice, I apologise in advance, but one thing certainly stuck out like dog's ears, that being problems with pi.
The "capacity" quoted for our little model engines is more accurately, the "swept volume". Under what I'll call normal circumstances, this is quite simply the volume of a cylinder; in other words, the area of the cylinder multiplied by the distance the piston moves from BDC to TDC. Stated in our terms, this is the area of the bore (or piston), times the stroke. So first we need to calculate the area of a circle.
I have the advantage of a 1950's primary school education where the "times tables" and basic formulae (like the area of a circle) were beaten into us by rote--literally! In case you had a more enlightened education and lack the scar tissue that engraves that formulae in my memory, it's "pie-arr-squared" (the radius of the circle squared, times the mathematical constant provided by ratio of circumference to diameter for any circle, known since the 15th century as "pi").
For us: one-half the bore squared, times pi. Then multiply that by the stroke and you have the capacity (in most cases, but I'll come back to that).
So why did the MEW piece make such a big production over this? Possibly it IS a big deal if whatever hand-held calculator you have lying around does not have the constant "pi" hardwired. Being an engineer and scientist, nearly all mine do. But if I'm stuck, I have a couple of approximations to fall back on.
The first is "22 on 7" (more scar tissue). This gives a value that is high by 0.004%, so as a purist, I must scorn it. A far better one is 355/113. I read this one in a wonderful little out-of-print book called Asimov on Numbers, written by the famous writer of science fact and fiction, Isaac Asimov [1]. By coincidence, as I write this, an abomination of a movie usurping the title one of Asimov's most beloved fictional works is premiering world-wide. Namely, "I, Robot" and the less said about that, the better! [mumble, grumble, car chases, explosions... save me from hollywood, etc...]
A chapter in reference [1] charmingly called "A Piece of Pi", gives the history of the symbol π and its uses. The chapter is a wonderful romp, but I won't bore you with it beyond summarizing Asimov's findings on the origin of the symbol. We commonly use the Latin circumference when referring to circles, and perimeter from the Greek perimetron (meaning "around") when talking about polygons. English mathematician William Oughtred is credited with using the Greek symbol "π", called "pi", being the first letter of perimetron. It appears to have stuck. Oddly, we still refer to the "diameter" of a circle, which comes from the Greek diametron meaning "the measurement through" and symbolized as "δ" (lower case "delta") by Oughtred. Go figure.
Back to the topic. I said that bore area times stroke would be mostly correct. The case where it is not is when the axis of the crankshaft and cylinder are displaced. The displacement is called "desax". It has two effects. First, it changes the time the piston will spend on up-stroke and down-stroke from equal to skewed. Second, it reduces slightly the distance the piston will travel (stroke) from a measurement taken as twice the crankpin displacement fom the crankshaft axis.
If we arrange the desax correctly, we can get a fast power stroke, followed by a slower compression stroke that helps transfer. Westbury was especially fond of this arrangement. I could present the math for this, but I believe that I'm running on the limit of your patience by now, so I'll demur and instead present ref [2] for the dedicated. There's a story, perhaps apocryphal, that tells of a noted engine manufacturer who had a whole production run of crankcases cast with the desax the wrong way 'round. Ouch.
Before closing on this topic, a quick word on units. Naturally when calculating displacement, your result will reflect the units used for bore and stroke. So imperial inches will give cubic inches. If you want cc's (cubic centimetres), you'll need to either convert bore and stroke to centimetres, or apply a conversion factor to the cuin value. All through the years that SIC was published, I shook my head in bemusement everytime Bob Washburn admonished readers to convert millimetres to inches by multiplying by 0.03934--a number I doubt I could ever memorize, regardless of painful consequences. Eventually I wrote to SIC and RAW saw fit to publish that 25.4 was much easier to remember, and more accurate when used for conversions. Just divide or multiply depending on the direction. You don't even have to remember when to divide or multiply. The answer will look wrong if you get it wrong. Finally, the ccs to cuin conversion constant is 0.061. Divide by it when going from cuin to cc; multiply for the other direction.
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