Hp vs Nm - revisited

Bendros diskusijos apie variklius ir ju tuninga.
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spade
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Hp vs Nm - revisited

Postby spade » 18 Nov 2005, 09:12

Jei anksciau buvo madinga vertinti arklius, tai dabar tarp tu, kurie kazka ismano tapo madinga girti torka, o arklius nuvertint... "Horsepower sells cars, torque wins races" ir t.t. ir pan.
As visad sakydavau, kad torkas zinoma neblogai, bet man svarbiausia arkliai (kur nors ~8krpm :wink: ). Daugumai buvo nelabai aisku kodel.

Tai va. Kad taptu aiskiau - sumesiu cia toki straipsni is TigersUnited page'o. Kaip jis anonsuojamas tam paciam puslapyje:

Torque: Useful Concept or Automotive Red Herring?
Dr. Bob Palmer has prepared THE definitive article on that subject that gets mangled on the Tiger List consistently. This straightforward explanation of the relationships between horsepower, r.p.m., torque, and how it relates to transmission gearing, acceleration, and other performance issues is concisely described. This is your Number 1 guide to the mysteries of these issues.
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spade
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Postby spade » 18 Nov 2005, 09:24

TORQUE: USEFUL CONCEPT or AUTOMOTIVE RED HERRING ?
By Bob Palmer
March 4, 2000

Overview:

Time was when the performance of engines was discussed in terms of horsepower. These days it seems more savvy to talk about torque. So, is torque really a better concept than horsepower for evaluating and comparing the performance of automotive engines in various applications, and in particular with respect to Tigers? To answer this question, we will start with some very basic concepts and develop these ideas into some specific recommendations for building engines and gearing them appropriately to suit your individual goals.


Discussion:

Seems like it just gets harder and harder to keep up with this fast-paced, high-tech world we’re living in. Just about the time you think you’ve become an expert at something, they go and introduce a radical new technology and you’re back to square one. One of the things I like most about my Tiger is its inherent simplicity; no fancy electronics or computers. The good part of this is that if something goes wrong, I can probably quickly diagnose and repair it. But, in spite of all the electronic controls and gadgets on modern engines, they are still basically the same as they always were. After almost thirty-five years in production, the venerable Ford Windsor small block is a case in point. With all of this experience with internal combustion engines, you would think we would already have learned pretty much everything that’s worth knowing. However, there’s always more to be learned and even the old lessons need to be retaught to a new and eager audience. Unfortunately, all too often the wisdom of the past has been abandoned in favor of trendy rheotoric inspired by a combination of ignorance and the profit motive. Now, there’s nothing new about this problem. Since even before the time of the chariots it’s been caveat emptor - let the buyer beware. Most of the bogus information is just hype for the purpose of getting a leg up in competition for your hard-earned bucks. Sometimes, however, the errors are of a more fundamental nature and reflect a basic misunderstanding of the issues. While it would be foolish to try and correct all of the misleading and bogus information being purveyed by even so-called experts, some of the more important issues should be confronted from time to time. A case in point are the concepts of torque and horsepower, their relationship to each other and to the performance of your car. Time was when the performance of engines was discussed in terms of horsepower. These days it seems more savvy to talk about torque. So, is torque really a better concept than horsepower for evaluating and comparing the performance of automotive engines in various applications, and in particular with respect to Tigers? To answer this question, we will start with some very basic concepts and develop these ideas into some specific recommendations for building engines and gearing them appropriately to suit your individual goals.

Being a physicist by training as well as by nature, I like to start with the basics and build a good fundamental understanding of a subject. The advantage of this approach is that (1) it has a much better chance of getting the correct answer, and (2) it allows you to apply what you learn to a variety of similar situations, rather than just having specific answers to specific questions. With this philosophy in mind, let’s start at a very fundamental level with the concepts of torque and horsepower and their relation to the performance of automotive engines and then go on to present some rather simple guidelines for building and gearing engines for optimum performance. First of all, what do we really mean when we talk about engine performance? While for some people performance might mean gas mileage, miles between overhauls, or the ability to turn really high rpms, the measure of performance that relates to how quick you can turn a 1/4 mile or how fast you can run an oval or road track is most directly related to acceleration. That good old feeling of acceleration you get on your backside when you push on the accelerator (appropriately named), regardless of how much sturm and drang accompany it, is really the bottom line.


Basics:

Acceleration is a basic and well understood concept in physics. In fact, Newton’s second law, F = ma, gives a simple relation for the force (F), mass (m), and acceleration (a) of an object like, for example, an automobile. Written as a=F/m, we express mathematically what we already know instinctively, which is that acceleration increases as we add force and decreases as we add mass. Hence, our interest in little cars with big motors. (For the sake of simplicity, we’ll overlook effects like wind resistance and traction in this discussion.) So now we’ve come to the not so surprising conclusion that, for a given weight of car, we need to apply as big a force as we can in order to get the most acceleration. So what produces this force, torque or horsepower? Now here’s where the discussion gets a little tricky. In Fred Puhn’s book "How to Make Your Car Handle", he states somewhat sarcastically that "torque seems to be that characteristic of a motor people refer to below a certain rpm". In fact, torque is a concept that is best related to zero rpm but we can, if we like, abstract it conceptually from horsepower at finite rpms. Maybe one way to relate torque to what is happening with a motor is to consider a torque wrench and think about using it to apply torque to the motor’s crankshaft. The units of torque reflect its nature; e.g., foot-pounds. If I push on a torque wrench that is one foot long with a force of 100 pounds, the scale should read 100 ft-lbs. Of course, any combination of force and lever arm which, when multiplied, gives the same product produces the same torque. If the motor is kept from turning while this torque is applied then the wheels don’t turn, the car doesn’t move, and no work is done. The important point here is that you can apply all the torque you want, but work is only available when motion occurs in conjunction with this torque.


Discussion:

Seems like it just gets harder and harder to keep up with this fast-paced, high-tech world we’re living in. Just about the time you think you’ve become an expert at something, they go and introduce a radical new technology and you’re back to square one. One of the things I like most about my Tiger is its inherent simplicity; no fancy electronics or computers. The good part of this is that if something goes wrong, I can probably quickly diagnose and repair it. But, in spite of all the electronic controls and gadgets on modern engines, they are still basically the same as they always were. After almost thirty-five years in production, the venerable Ford Windsor small block is a case in point. With all of this experience with internal combustion engines, you would think we would already have learned pretty much everything that’s worth knowing. However, there’s always more to be learned and even the old lessons need to be retaught to a new and eager audience. Unfortunately, all too often the wisdom of the past has been abandoned in favor of trendy rheotoric inspired by a combination of ignorance and the profit motive. Now, there’s nothing new about this problem. Since even before the time of the chariots it’s been caveat emptor - let the buyer beware. Most of the bogus information is just hype for the purpose of getting a leg up in competition for your hard-earned bucks. Sometimes, however, the errors are of a more fundamental nature and reflect a basic misunderstanding of the issues. While it would be foolish to try and correct all of the misleading and bogus information being purveyed by even so-called experts, some of the more important issues should be confronted from time to time. A case in point are the concepts of torque and horsepower, their relationship to each other and to the performance of your car. Time was when the performance of engines was discussed in terms of horsepower. These days it seems more savvy to talk about torque. So, is torque really a better concept than horsepower for evaluating and comparing the performance of automotive engines in various applications, and in particular with respect to Tigers? To answer this question, we will start with some very basic concepts and develop these ideas into some specific recommendations for building engines and gearing them appropriately to suit your individual goals.

So how do we turn torque into work? This is also very easy to do mathematically. In fact, many of you probably already know that power is equal to the product of torque times angular velocity (i.e., rpms). The way we can derive this is by starting with the fundamental fact that work is equal to force times distance. To extend this idea to power, which is the rate at which work is done, multiply force times distance-per-unit-time, or simply force times velocity. Going back to the idea of applying a torque wrench to the crankshaft, suppose we let the crankshaft make one revolution while applying 100 ft-lbs of torque. Say we’re pushing with a force of 100 lbs at a distance of one foot. The distance we move in one revolution is 2p times one foot, so force times distance is equal to 100*2p*1 = 628 ft-lbs of work. Notice how easy it was to segue from torque to work. We even have the same units, but with torque the units were force at a distance and, with power, it’s force through a distance. Now, if we make a full rotation in one minute, just multiply by one rpm to get 628 ft-lbs/min for the rate at which we are generating power. So how do we relate to these units of ft-lbs/min? Well, the more familiar unit of horsepower is defined as 550 ft-lbs/sec (or 745.7 watt), so let’s convert our units to seconds by dividing by 60 seconds/minute; so 628 ft-lbs/min ÷60 sec/min = 10.5 ft-lbs/sec and then, dividing again by 550 ft-lbs/sec per horsepower, gives 0.02 horsepower as the rate we are doing work for this particular example. If you’d rather not go through this derivation every time you want to convert torque to horsepower, just remember the simple conversion factor which is 550*60/2p = 5252. If you multiply torque times rpm and divide by 5252 you get horsepower. Conversely, if you divide horsepower by rpm and multiply by 5252 you get torque. Now I’ll bet your asking yourself how you managed to get along all these years without this important information; right? Hopefully, at least a few of you gear heads have been sucked into this discussion in the interests of intellectual curiosity and are hoping to emerge with a little better feeling for the concepts. For the rest of you, don’t despair, there’s more practical information at the end of the tunnel, so hang in there. While there is more than one way for an engine to do work, producing acceleration is perhaps the most interesting to Tiger owners and drag racing is probably the purest expression of the quest for sheer acceleration. Consequently, a good question to ask is what characteristics of a motor will get you across the finish line first. Of course this simple question is complicated by factors like traction, weight, aerodynamics, and reaction time. But, keeping all these things equal, what do we want from our motor to get the best elapsed time and speed. Incidentally, most folks familiar with drag racing know that your speed is a better indication of your motor’s horsepower that your e.t. This is because hookup is so critical in drag racing since what happens early in the race is integrated over a relatively longer time. This fact is also not lost on the better track racers who will make adjustments in their driving to let them enter the straight-away at a higher speed, even though they may have to give up a little distance in order to do so. For instance, if you can exit turn nine at Willow Springs 20-30 mph faster than the guy you’re following, you’ll overtake him down the straight-away, even if he’s got somewhat more acceleration than you.


Note 1: Engineers reserve the term motor for devices that convert electrical energy into mechanical energy. However, in common use, the terms motor and engine are equivalent and will be used interchangeably here.

Practicalities:

But getting back to the drag strip, how do we get down the quarter mile the quickest? Given an unlimited amount of either horsepower or torque, the thing that ultimately limits the acceleration of a dragster is adhesion between the tires and the asphalt, and this limit implies a maximum acceleration; i.e., smoke ‘em, but not too much, all the way down the strip. This idea implies that we want a motor that puts out constant torque; i.e., constant acceleration versus rpm. With this ideal motor we would have just one gear and just let it wind up and keep producing constant torque and acceleration, but also higher and higher horsepower from start to finish (remember, power equals force times speed). While an electric motor comes close to such an ideal, flat torque curve, unfortunately internal combustion engines behave quite differently. But we can learn something important by considering this simplified example of an ideal constant torque motor. Suppose we build a very powerful constant torque motor that has the optimum torque for maximum acceleration for a particular car. Then we could use one-to-one gearing and the torque at the wheels would be the same as the torque of the motor. Well, we put this all together and it goes like hell and we instantly start breaking drag race records. But some guy with a motor that has only half the optimum torque has a bright idea. He gears his motor two-to-one and - voilà! Now he’s got the same torque at the rear wheels as you do. The only difference is, his motor runs twice as fast. But since his is also an ideal constant torque motor, he goes just as fast as your motor with twice the torque. Let’s get just a bit more realistic and suppose that the torque falls off some with rpm. If you double the rpm but the torque drops by less than half, you still win. By playing this little gedanken game you quickly come to the realization that, by using the right gears, you get the most torque at the rear wheels by running the motor where its torque times rpm are a maximum. But torque times rpm is just good old horsepower! Horsepower (or watts or whatever other power units you prefer) is just a simple and direct way to indicate the ability of an engine to do work. It is certainly a whole lot simpler than thinking “torque times rpm, torque times rpm, torque times rpm ------.“

So, let’s get this really clear; an engine’s potential for producing acceleration is directly related to horsepower, so you get maximum acceleration when your motor is putting out its maximum horsepower; period! Where it happens to put out its maximum torque has no relevance whatsoever relative to maximizing acceleration. The point of maximum torque might be an indication of the rpm range where the motor is most efficient, but that’s another story. Regardless of whether you have a little tiny motor with small torque that can turn lots of rpms, or a great big motor with lots of torque, but limited rpm, if they both put out the same horsepower then they both produce exactly the same acceleration. In fact, little tiny motors have a distinct advantage in terms of weight which is why the highest performance cars like F1’s use little motors turning 12-14,000 rpms and making great gobs of power. That’s not to say there’s no drawbacks to this philosophy, but in F1 style racing the advantages outweigh the drawbacks. Let’s press on a little further with the main point here. I’ve tried to convince you by building logically from the fundamentals that it’s really horsepower, not torque that counts in measuring what a motor can do. But like all stories, its a little more complicated than that because in practice, a motor must operate over a range of rpms. So the more complete story is that the average acceleration is directly related to the average horsepower between the shift points. Now I know it’s inconvenient to have to visualize a curve instead of just remembering a number. Wouldn’t it be nice if you could just say motor A makes 325 hp and motor B makes 350 hp so B is faster than A. But we are just quoting the maximum horsepower at a particular rpm and in a real situation the rpms of the motor keep changing through the gears. If I can gear my 325 hp motor to get its average horsepower between shifts higher than your 350 hp motor, I win. So the whole story involves both the horsepower curve of the motor and how well the gears are matched to this curve. In general, the closer the gears the better up to the point where you’re losing too much time shifting.

Based on our discussion so far, we should now be able to make some pretty good choices relative to what we’re trying to achieve. If we want to build a drag motor for a Tiger we can start by assuming you are going to be using either a close or wide ratio top loader four speed. At the starting line you only need enough torque at the wheels to keep the tires on the edge of adhesion, assuming you’re not running big wide gummy slicks that work best with a certain amount of slipping. In any case, as you go down the track you will need to build horsepower output from the motor in order to maintain constant torque at the wheels to hold them near the breaking point. At some point a few hundred feet or so down the track you will no longer have as much power available from the motor as you could use and from there on you will want to keep the motor making its highest possible average horsepower. Ideally, you will be somewhat past the peak rpm in fourth gear as you cross the finish. How much past depends on how fast your power falls off after the peak. The ideal, as I said above, is to maximize the average power between the shift points. I read somewhere that you should be making your peak horsepower just as you cross the finish line. That’s wrong. Gear a little lower (higher numerically) so as to get the highest average horsepower. The same idea holds in track racing. In general, you should be past your peak horsepower rpm at the end of the straight-away. Let’s get down to some brass tacks by starting with the transmission ratios, which is what determines the rpm spread between shifts. For Ford toploader four-speeds, the following gears are standard:

So, let’s get this really clear; an engine’s potential for producing acceleration is directly related to horsepower, so you get maximum acceleration when your motor is putting out its maximum horsepower; period! Where it happens to put out its maximum torque has no relevance whatsoever relative to maximizing acceleration. The point of maximum torque might be an indication of the rpm range where the motor is most efficient, but that’s another story. Regardless of whether you have a little tiny motor with small torque that can turn lots of rpms, or a great big motor with lots of torque, but limited rpm, if they both put out the same horsepower then they both produce exactly the same acceleration. In fact, little tiny motors have a distinct advantage in terms of weight which is why the highest performance cars like F1’s use little motors turning 12-14,000 rpms and making great gobs of power. That’s not to say there’s no drawbacks to this philosophy, but in F1 style racing the advantages outweigh the drawbacks. Let’s press on a little further with the main point here. I’ve tried to convince you by building logically from the fundamentals that it’s really horsepower, not torque that counts in measuring what a motor can do. But like all stories, its a little more complicated than that because in practice, a motor must operate over a range of rpms. So the more complete story is that the average acceleration is directly related to the average horsepower between the shift points. Now I know it’s inconvenient to have to visualize a curve instead of just remembering a number. Wouldn’t it be nice if you could just say motor A makes 325 hp and motor B makes 350 hp so B is faster than A. But we are just quoting the maximum horsepower at a particular rpm and in a real situation the rpms of the motor keep changing through the gears. If I can gear my 325 hp motor to get its average horsepower between shifts higher than your 350 hp motor, I win. So the whole story involves both the horsepower curve of the motor and how well the gears are matched to this curve. In general, the closer the gears the better up to the point where you’re losing too much time shifting.

Gear---Close_Ratio---Wide_Ratio
First------ 2.32 ----------- 2.78
Second-- 1.69 ----------- 1.93
Third----- 1.29 ----------- 1.38
Fourth---- 1.00 ----------- 1.00

To see what these numbers mean in terms of engine rpm changes, let’s be somewhat arbitrary and see what happens if we make all our shifts at 6,400 rpm. A little simple math gives the following:

Shift ------------ Close Ratio ------ Avg.rpm ------ Wide Ratio ------ Diff.rpm
First-Second --- 6,400-4,662 --- 5,531(2nd) --- 6,400-4,443 --------- 219
Second-Third -- 6,400-4,885 --- 5,642 (3rd) --- 6,400-4,576 -------- 209
Third-Fourth --- 6,400-4,961 --- 5,680 (4th) --- 6,400-4,638 -------- 323


In the first column we have the rpms at each shift point for a close ratio and in the third column are the corresponding numbers for the wide ratio tranny. The second column gives the average rpm in second, third, and fourth gears assuming we also run out to 6,400 in fourth (first gear is unspecified since we haven’t said what rpm we start at). We can see by this example that the motor must operate over a 1,500-2,000 rpm range in each gear. The fourth column are simply the differences between the wide ratio and the close ratio rpms at the shift points, which are only about 10-20%. In fact, the gear changes on a wide ratio toploader are actually even closer that on the newer Mustang five-speeds; especially between first and second gears and between fourth and fifth gears.

Let’s get back to the main issue here which is about getting the most out of the motor. If you have dynamometer data for your engine (or at least a similar engine) that shows horsepower versus rpm, you can then compare the dyno curve with the shift point numbers and see how well you are doing relative to maximizing the power between the shift points. We see in this particular example that while we are running in second gear our average rpm is 5,531, in third gear it is 5,642, and in fourth gear it is 5,680 (assuming we run all the way to 6,400 again). Given this information, obviously the motor should be making its best horsepower around the average rpm we’re running which, in this case, is a little over 5,600 rpm. The optimum average rpm will actually fall a little bit lower than the peak horsepower rpm because the horsepower curve is not symmetric and typically falls off faster past the peak horsepower point than before it.

Before delving into further details, let’s take stock of what we’ve learned from this hypothetical (but not unrealistic) exercise. Imagine building a real engine that is matched to the above example in terms of its horsepower curve. This motor would probably make its peak horsepower at around 5800 rpm. Then, to get the most out of this motor, we need to run it about 600 rpm past the horsepower peak in each gear. This is really a pretty general conclusion, although the exact numbers will depend on the exact characteristics of the motor, and whether you have a close ratio, wide ratio, five-speed, etc.

Perhaps it is also important to point out what is not relevant in the foregoing discussion; e.g., the torque curve, the rear end ratio, the weight of the car, etc., etc. Of these, the most irrelevant is the torque curve. In fact, my advice to those of you who may have both torque and horsepower dyno data is to tear that torque curve sheet up and toss it in the trash. Failing the courage to take this bold step, perhaps you could at least tape it to the under side of a drawer or some similarly obscure place where, hopefully, over time you will learn to live without it. Remember, if you ever get really desperate for the torque data, you can just divide the horsepower curve by the rpms and multiply by our handy-dandy, easy-to-remember conversion factor (5252).

At this point, the reader probably falls into one of three categories: (1) those of you who have fallen asleep; (2) those of you who disagree violently and are busy thinking of arguments to the contrary; and (3) those of you who already knew all of this and are wondering why I’m wasting your time. To the former and particularly to the latter, I apologize. To those of you who may remain unconvinced, but who are still with me, read on. I think I can anticipate at least one or two of your objections. Also, I may be able to add just a little more practical advice.

Some of you may be thinking that, even if torque doesn’t relate to acceleration, it must at least be an important characteristic of truck engines. At least that’s what common wisdom seems to hold. But I could just as easily have made the same argument and come to the same conclusions by merely substituting work in terms of pulling a weight up an incline, just another form of force times distance, instead of work in terms of acceleration. It just so happens that the practical attributes of truck engines favor lower rpms. In particular, diesel engines have good efficiency and long life. They also have a limited rpm capability, so running at high rpms is simply not an option. A high rpm, high efficiency motor (e.g., a turbine) could easily be a good truck power plant. Although gearing might present some problems in the extreme case, fundamentally there’s no advantage to a low rpm, high torque motor versus a high rpm, low torque motor if both make the same horsepower (although there may a valid issue comparing longevity). Certainly, I’m not suggesting that it would be practical to replace a diesel truck motor with a F1 motor with the same horsepower. For one thing, the cost of a F1 motor would be prohibitive. But if we were to have a hill climb contest between two identically loaded trucks, the truck with the most horsepower between the shift points would win, regardless of the supposed torque advantage of a typical truck engine.

What does all of this mean in terms of building an engine for your car? Using the best information available, try to build your engine with the end result in mind. Do you really want a motor that develops its best horsepower at 7,400 rpm, especially if it means a loss of horsepower at lower rpms? And, how long will it last at those higher rpms? Remember, to get the most out of the 7,400 rpm motor, you would need to be shifting it at over 8,000 rpm! It only makes sense to build the heads and cam to work best at high rpms if you are also willing build the bottom end to be reliable in the same range (plus some safety margin for over-revving). Resist the temptation to go into uncharted territory with bitchin, go-fast parts that happen to fit your budget. You won’t know until it’s all together how it’s going to work. Instead, use the same parts as a previously built and tested engine that most closely gives the horsepower curve you are trying to achieve. This will also save you the cost of a dyno test, although a dyno test is the only way to confirm that you are getting what you expect.

I disagree with statements to the effect that the motor should operate between the torque peak and the horsepower peak, at least not when optimum performance is the goal. This advice appears to be the result of some kind of confused compromise between two competing independent parameters, which is certainly not the case. In fact, torque is simpy a component of horsepower, but since horsepower directly relates to performance, just focus on trying to maximize the average power between the shift points and simply ignore the torque curve. (Am I repeating myself?)

I’ve read in more than one place that, in drag racing, you should gear your car so as to just reach the maximum horsepower at the end of the quarter mile. Wrong! Whether it’s drag racing or optimizing your gears for a particular road track, in general you need to be somewhat beyond your peak horsepower rpm at the end of the straight. The exception to this would be the case of a very long straight-away where you reach your top speed well before the end. In this case the optimum converges to the peak horsepower point. But certainly in the case of a drag race, you are still accelerating at the end of the quarter mile and probably the optimum rpm is still pretty close to your shift point; e.g., 6,400 versus 5,800 rpm in the above example. With some real data and a little calculus, we could solve these problems exactly, but I think that’s a little more detail than we need to get into here. But, to at least give you an idea how to gear your car for the quarter mile, let’s assume we have the motor we discussed above where, to get maximum horsepower through the gears, we need to shift at 6,400 rpm. What rear end ratio should we use? To achieve this rpm at the end of the quarter mile, we need to know our speed, which we can only find out exactly by actual experience. But, knowing our engine’s horsepower and the weight of the car compared with other similar motors and cars, we might expect to be able to get to about 110 mph, a rather ambitious goal, but definitely achievable in a Tiger with a good performing engine. To select the correct rear end ratio to achieve this particular speed at the end of the quarter mile, measure the circumference of your rear tires. With a Tiger, the tires will be right around six feet in circumference. At 110 mph you are going 9,680 feet per minute. Divide 9,680 ft/in by the tire circumference of six feet (which equals one revolution) and we find that the rear wheels are turning 1,613 rpm at 110 mph. So, we have 1,613 rpm at the rear wheels and 6,400 rpm at the motor; divide 6,400 by 1,613 and we get a rear end ratio of 3.97:1, or maybe slightly less to account for tire slipping. (assuming 1:1 fourth gear like top loader four speeds). A Tiger geared like this wouldn’t be very practical on the street, but it would sure go like hell in the quarter mile! (Remember, this example is for illustrative purposes only. Your results may vary!)


Well race fans, so long and remember; when the green flag drops, the b.s. stops!

Bob Palmer

Editors Note: For further delving into the mathematical formulas that control and define our car’s design and performance, there is a site worth visiting, prepared by from Jim Martindale, University of California, Irvine Campus. Jim Martindales Useful Automobile Formulas
DelSolSohcVtec BlueberryBeeProject
e91
IbizaTDI [abandoned]
SV650S [crashed]
WRXwagon [stolen]
LTSR.LT [dead]

Why waste time learning when ignorance is instantaneous?


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