It has come to my attention recently that between my rambling pseudo-mathematical dissertations, my clumsy stabs at social commentary, and my occasional attempts to squat on unclaimed geometrical phenomena of engineering significance, I have, periodically, put useful information in this space. This fact revealed itself to me through a rash of recent e-mails that, though each unique, seemed all to follow this basic pattern:
"Sport Compact Car is really great, and you aren't so bad either...for a nerd. Anyway, my dog ate my February 1998 issue, so I lost the 'Technobabble' where you explained how to make an oscillation overthruster out of some dryer vent and a used toaster. Your back issues department hangs up on me every time I say 'overthruster,' so I was hoping you could just explain it to me in an e-mail."
This seemed like a perfect excuse to simply re-examine some of the more useful topics on which I have pontificated in the past, thus avoiding the need to actually come up with a new idea this month. Brilliant, no?
Finding the Ideal Shift Point
Here's a hint: In almost any high-performance car, shifting at the redline will offer the best acceleration. Of course, it's that "almost," that rare exception, which keeps you up at night thinking about your quarter mile times, isn't it? The goal when deciding where to shift for maximum acceleration is simply to have the most force accelerating you forward at any given moment. Accelerative force comes directly from engine torque and gearing, so to figure this out, you need to know the torque curve of your engine and you need to know your gear ratios.
OK, armed with this information, your goal is to make sure that for every moment during your quarter-mile blast to glory, the sum of your engine's torque output at that moment and the gear you are in is higher than it would be in any other gear. Let's say you are at 6500 rpm in second gear in a stock Focus, for example. Our Project Focus made about 80 lb-ft of torque at 6500 rpm, and second gear is 2.136:1, so the torque at the output shaft is about 171 lb-ft. To check torque in the next gear you have to see what rpm a third gear shift would get you. For that, you use this little formula:
Plugging in the third gear ratio of 1.448:1, you get about 4400 rpm. Back to the torque curve (our most recent stock Focus torque curve was printed in "Project Focus, Part I" in the May '01 issue) we see about 120 lb-ft of torque, which ends up being 173 lb-ft after it goes through a 1.448:1 gear ratio. Since the torque is about the same, the 2-3 shift should be at about 6500 rpm. If the torque in third had been significantly higher, you would want to try this whole exercise again with a lower shift point. If it were lower, you could hang onto some more revs in second.
Running through this whole exercise over and over again until you find the right point for every gear change can be a little tedious. If you prefer to play with spreadsheets, and you can get the torque data every 100 rpm or so, you can simply plot torque at the outptut shaft vs. speed for all of the gears. Anywhere the line for one gear crosses the line for the next, that's your shift point. If the lines never cross, take it to the redline.
Rotational Inertia
The rotational inertia topic was so big, it took me two months (June and August '99) to get it sorted out. Here's the deal: There is this rule of thumb among racers that adding weight to something that rotates is far more detrimental to performance than if you add it to the body of the car. This is absolutely true, and by bumbling through some physics, and after slipping and falling on a radian, I managed to get a few formulas figured out that could tell you just how much worse.
Any moving object has kinetic energy, as does an object sitting in place and rotating. An object that is both rotating and moving (like a rolling wheel, for example), has kinetic energy from both, meaning that accelerating or decelerating that rolling object will take more power than one that is just sliding along. How much more power is the question.
The answer, it turns out, depends on how the weight is distributed on the wheel. An extra pound on the tread of a rolling tire has as much kinetic energy as 2 lbs on the floor of the car. As you move toward the center of the wheel, the rotational effect drops until, at the center, a pound is just a pound. The formula I derived to determine the exact relationship between weight on a wheel and weight in the car isn't worth repeating here for one simple reason. It requires that you know the moment of inertia of the wheel, and measuring that is virtually impossible. What you need to know is that changing to tires that are 1 lb heavier will effectively add 8 lbs to the car (four tires, remember) and that adding a pound to the wheels will effectively add somewhere around 6 lbs to the car.
That only considers acceleration and braking; handling is dramatically affected by unsprung weight as well, but no simple formula is going to tell you how big the effect is.
The one useful formula that did come out of my diatribe on inertia was the effect of a light flywheel. Because the flywheel's rotational kinetic energy is directly linked to the car's motion (through the gearbox), you can also calculate the effect of a lighter flywheel. Here's the formula: Next month, we will tell you all about the aluminum FocusSport flywheel we just put in our Project Focus. Until then, know this: The flywheel is 12.5 lbs lighter than stock and has a radius of 5.5-inches; the Focus' first gear is 3.667:1, the final drive is 3.82:1, and the tire radius is 12.1 inches. What's it all mean? In first gear, the car feels more than 250 lbs lighter!
The HydraCharger
Way back in the April '99 issue, I introduced a prototype supercharger that Garrett was working on called the HydraCharger. By mating a turbocharger's compressor to a tiny little turbine driven by high-pressure hydraulic fluid, this little device promised the efficiency of a turbocharger with the response of a supercharger and tunability and packaging benefits all its own. Over the next few months I was besieged with e-mails and phone calls wanting to know more. Unfortunately, I had told all I knew.
Well, now I know more. Apparently, Dave Kapich, a consultant for Garrett, designed the HydraCharger (which will probably go by a different name in the future), at the time.
Garrett ultimately dropped the program, and the rights to the design reverted to him. Kapich Engineering is finishing the development work as we speak, concentrating mainly on boring stuff like long-haul truck engines, but there is one glimmer of performance hope. A two-stage hydraulic pump and a prototype of Kapich's compressor are sitting under the hood of an Infiniti I30 at Jim Wolf Technology. The car is supposed to be drivable by the time this magazine hits the newsstands.