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05022011, 04:27 AM  #1 
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Peak in gear acceleration is not quite at peak torque...
OK if you are not interested in the math and engineering subtleties of vehicle acceleration performance stop reading here. If you have participated in any of the very lively discussions and debates about hp vs. torque read on...
There have been many threads here on this forum discussing one or more of two key points about vehicle acceleration. I just wanted to point out a minor correction to one of them that many here, myself included, embraced as absolutely true. I'm not so sure why I preiously missed all of the contributing corrections. Probably because it is true for all practical purposes but ultimately just not 100% correct and accurate. Just in review those statements are:
Much discussion has been had as to the seeming contradiction among these statements. However, they are absolutely consistent if you think them through carefully. Or make some spreadsheets... Statement #2 is the one that needs some clarification and refinement. When you write down the long equivalent of F=ma for a car, #2 seems somewhat obvious. The shape of the acceleration curve basically follows the shape of the engine torque curve (gearing just multiplies the torque with some scale factors). However, as written it is only strictly true for a car with lossless tires and in a vacuum. Wow talk about a useless physics problem... lossless tires and driving in a vacuum... Granted these effects are fairly minor but they do affect the results. The equation to use here is again just basically F=ma: a ~ T  Lt  Ld I've left out some constants (mass terms) and units to keep it simple but basically the acceleration is proportional to the engine torque minus loss terms for both tires and drag. Do note that this is in any given gear  and of course if you "get" point #1 above you will know that there is a slightly modified version of this equation for each gear  and that is exactly why you can't claim that "torque is everything" for acceleration. In each gear the speed vs. rpm curve is different and thus these "corrections" are also gear dependent. Furthermore drag varies with the square of vehicle speed. In other words statement #2 is actually closest to being entirely correct in 1st gear where speeds are low enough that tire and aero losses are very small. In 1st gear in any normal car the corrections are indeed truly small enough to ignore completely (on the order of 1%). Just running some off the cuff real numbers in 7th gear in the E9X MDCT the correction for the drag term alone amounts to only about 500 rpm (i.e. the true peak acceleration is about 500 rpm lower than that predicted by the simpler statement, 3000 vs. 3500 rpm). The effect of tire losses is significantly smaller as those losses are only about 1/3rd  1/4th of the aero losses at this speed (around 80 mph). So again when thinking about acceleration performance this "rule" #2 is accurate enough to still make it very useful just not quite 100% accurate down to the finest level of detail and getting the exact true speed/rpm where you will have peak acceleration. Statement #1 could also use a minor correction but not quite for the same reasons, i.e. not due to the power lost to tires or drag. At any given (fixed) speed the only choice becomes which gear. For cars with very linear power curves each lower gear will result in a higher engine power. Here the tire and drag power losses are constant and the drivetrain losses are never high enough at higher rpms to overcome the large extra amount of power made by a downshift. However, for more typical power curves that climb linearly, taper off, peak and then decline (think just for example 2008 LS3 Corvette) you can generally find a speed where two gears will produce the same engine power (this is the same as having drive power curves cross). Here the lower gear will obviously provide more wheel torque and thus more acceleration. Thus "rule" #1 actually does not quite fully cover this somewhat ambiguous case. Well, hopefully some will find my Sunday night (uh Monday morning...) ramblings interesting or useful...
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05022011, 05:14 AM  #2 
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You should've seen the look on my face after I finished reading your post. Well the nausea and dizziness didn't help
I'm interested in this subject however, I'll wait for the English translation.
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05022011, 08:39 AM  #3 
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Good post swamp!
I realize that both 1 and 2 practical have value, but as of late I have really come to embrace #1 as the one with the most eyeopening impact. One thing it really hammers home is that there are only 6 (with 6MT) or 7 (with DCT) different speeds whereby an M3 can actually accelerate with all the power potential of the S65. This of course applies to all other cars (and powered vehicles for that matter) too obviously. Now, if we had a CVT, we could get all the power at any speed (well, any speed in the CVT's range). 
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05022011, 02:20 PM  #4 
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This wording doesn't sit quite right with me: "Peak acceleration at any given speed occurs at peak power."
I would say: Peak acceleration occurs in whatever gear that gets you to peak power (i.e. RPM) in the shortest amount of time. For example, if the M3 makes peak power at 8000RPM, and your car is at a steady 8000RPM, then you're not accelerating anymore. Your speed is what ever it is, but Acceleration is 0. Does this make sense to you guys? 
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05022011, 02:27 PM  #5  
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However, the conclusion that for each gear there is one speed at which you get maximum in gear acceleration follows directly from statement #2 not #1 . The other interesting result from this is that for pretty much all cars, when in a WOT run through the gears, you are never at this "magic rpm" in each gear (except in 1st gear if starting from a low or zero speed). Again in the M3 it is about 30003500 rpm (again slightly speed/gear dependent). You actually do not want to ever hit peak in gear acceleration. Why  you are too busy "hanging out" at or near max power (Point #1)! Instantaneous acceleration is only important for that very small instant of the maximum push in the back feeling. Staying at or near peak power over an extended period of time (at the lowest possible gear for the longest period of time) is what leads to maximum overall acceleration. The regular caveats apply to the important of a high peak torque for 060 (or 0less than 60) times. Is surely is important here. In such contests you spend an appreciable amount of time at and near the rpm of peak torque and you actually use peak torque. Either way power/weight ratio as opposed to torque/weight (and 2WD vs 4WD) still gives you a reasonable estimate of 060.
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05022011, 02:34 PM  #6  
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I used as few formulas as needed with all of the gory detail left out. Tried to keep it all in English rather than formulas. Just think more about both points, #1 and #2. If you understand both and why they are not inconsistent you definitely get it. "It" meaning what really matters in automobile performance, torque vs. power, instantaneous acceleration vs. overall acceleration, torque multiplication by gearing, etc. i.e. all of the key basics.
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05022011, 02:54 PM  #7  
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Also your statement is not quite true. I think what you mean to say is "peak overall acceleration". That is a key distinction. At a fixed speed the statement I have given is correct. It is basically equivalent to simply saying one should downshift as far as possible without an over rev. However, for cars that have power curves shaped much differently than the M3, say a 2008 LS3 Corvette, things are a bit different. With such a power curve it is not linear or nearly linear up to redline like in the M3. It is linear at low to mid rpm, then drops off, peaks out and turns downward. For such a power curve you should actually use the highest rpm portion of it actually operating below peak power for the best overall acceleration. Even though your power is lower than peak power the acceleration in the lower gear is so much higher than in the higher gear you win out overall. In this way your statement does not cover such cases. And actually most cars power curves look more like an LS3 than the M3. Statement #1 (maybe not entirely obvious) includes the idea that you are in a rolling start drag race and got fully WOT from the given fixed speed in question. Statement #2 sort of assumes a WOT trip all the way across the tach range in one particular gear. You record the acceleration as a function of time then pick out the rpm of the peak acceleration. If instead starting from a rolling condition the engine and or turbo spin up will affect the results.
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05022011, 02:58 PM  #8 
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Hmm. The problem with statement #1 is that unless we change the gearbox, acceleration at a defined speed will stay the same, since peak acceleration will always be done in the lowest gear possible.
I would say your analysis of statement #1 is correct, however I'd like to add to the statement that it is conditional on how much traction is available. You are also correct that friction plays a role, but as speeds increase, drag becomes exponentially greater, and therefore a real factor as speeds climb.
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05022011, 08:39 PM  #9 
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Swamp,
I agree with your statements, But your title conflicts with #2, correct? Assuming you are in the right gear for peak acceleration. Title: Peak in gear acceleration is not quite at peak torque... #2 Peak acceleration in any given gear occurs at the rpm/speed where the engine is producing peak torque.
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05032011, 01:24 AM  #10  
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Quote:
Quote:
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05032011, 01:26 AM  #11  
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05032011, 07:15 AM  #12  
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In any case, speed is a necessary side effect of acceleration, so when accelerating, the constraint "any given speed" will cover every speed from where you start accelerating until where you stop accelerating. My point in stating (what I suppose is obvious) is that, for me, it makes the most sense to think about power at speed and then pick the right gear to achieve the best result for your current speed. I.e. you don't do the opposite  pick a gear hoping to get to that perfect RPM to achieve the best acceleration for that particular gear. I.e. this part: "...you do not downshift for more engine torque, you downshift for more engine power which in turn means maximizing wheel torque." Is what is most important. 

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05032011, 09:58 AM  #13  
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Of course you're correct, but the "small correction" is obviously gear dependent, as you've stated, but also may be notsosmall based on the shape of the torque curve. I'd postulate that an S2000 in fifth would have a significant difference between theoretical and actual, based on the fact that the peak is so "soft", with gentle ramps before and after. None of this actually matters, of course (hence the smile), since we're all searching for the power (and not the torque) at the drag strip or on a road course, but it's still interesting. The other (and tiny) modifier is that as you make more torque, wheel slip increases, meaning some of the thrust is lost to heat. OK, that's a dimple on a pimple on a flea's left nut, but hey, just thought I'd bring it up. Bruce 

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05032011, 10:49 AM  #14 
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Love these knowledgeable threads.

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05032011, 11:54 AM  #15  
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I think a more accurate statement would be "Peak acceleration occurs at the point where a given powerplant produces peak power (torque) while utilizing the shortest (numerically highest) possible gear ratio" Speed is an afterthought in this discussion since it is a simple byproduct of engine rpm and gear ratios. Quote:
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05032011, 03:49 PM  #16  
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Good!
Quote:
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05032011, 03:54 PM  #17  
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Uh, no. Completely wrong. Torque is not force they differ by the radius of application (tire radius here). Also peak hp is always simply given by the dyno it is independent of gearing and vehicle speed. That is part of why it is more useful. Read points #1 and #2 again and think about them more to understand when peak torque, peak instantaneous acceleration and peak hp occur.
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05032011, 04:01 PM  #18  
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Now peak OVERALL acceleration occurs simply by maximizing power at every point in time. You can do that typically by staying in any given lower gear as long as possible. Think about it some more. When driving aggressively (i.e. for maximum OVERALL acceleration) in an M3 you are almost always above 5000 rpm. The peak INSTANTANEOUS acceleration in any gear is way down around 30003500 rpm! In most sporty engines you simply never "use" peak torque nor do you hit peak instantaneous acceleration. Odd but true.
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05032011, 06:08 PM  #19  
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05032011, 09:27 PM  #20  
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In all other gears, 3000rpm in that gear (for example 3rd) will be inferior to a higher rpm in a lower gear (2nd) simply due to the multiplication factor of the shorter gear (numerically higher) delivering a greater ammount of resultant torque to the wheels. The alternative is partial throttle acceleration, meaning the torque curve is artificially limited by the driver to match the thresholds of traction, thereby changing the point of peak power. In the end, we are saying the same thing. I just want to make things clear so others don't get the wrong idea. Lots of people get the relationship between HP, torque and gearing all wrong. Remember folks, torque is multiplied through the gearbox, horsepower is not.
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05032011, 09:39 PM  #21  
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Wheel torque is what moves a car. PERIOD. Engine TQ x gearing = wheel torque. Whether an engine does this at 500 rpm or 5000 rpm, as long as the resultant wheel torque figure is the same, the (point) acceleration will be identical. Small detail with this is, due to gearing, the speed at which this acceleration is happening may be different.
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05042011, 01:04 AM  #22  
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The issue of traction is an obvious exception and one that brings enough variability to the problem that there becomes no real definitive answer. However, let's discuss just a bit. Though is does depend on temperature of the tires, surface and generally the maximum friction available in the surface. If you launch carelessly or in a show boating fashion sure in 1st you can have your tires spinning well through 3000 rpm and above. On the other hand it can also be quite the opposite with a good full hookup around that rpm. Also do note that the torque curve for the M3 is VERY flat  about 90% of peak torque available from 2500 rpm all the way to about 8000. Quote:
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Cheers.
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