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07-30-2014, 12:57 AM | #1 |
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All about drive train inertia and mass factors with sample M4 calculations (long...)
Well I suppose this one has been itchin' on me since a long past debate when I was totally ignorant about the importance of mass factors. Thanks to a good beating on by fellow member bruce.augenstein@comcast, I'm a bit wiser on this topic today...
The motivation here is threefold:
Force, Mass, Torque, Inertia. Just like mass resists acceleration when things are rotating it is inertia which resists angular acceleration. When a structure like a car accelerates certain parts are accelerated only linearly and have only translational kinetic energy. Spinning components also posses another form of kinetic energy, rotational kinetic energy. The work done by the engine must go into production of energy in both forms. Since all rotating parts also translate along with the vehicle those parts take more energy to get to a given vehicle velocity than an identical non-rotating component. This is in fact, as you will see below, very significant for a calculation or simulation of vehicle performance. My interest in this topic peaked when I had written a very nice (IMHO) vehicle performance simulation tool from scratch and realized that without this accounting of inertia, gear by gear, the tool was not very accurate. The same effects here are also relevant for braking, discussing wheel tire weights, etc. Drivetrain Inertia: The relevant components that spin in the entire drivetrain must be accounted for in the formulas and simulation for vehicle performance. These components include engine, flywheel, transmission, drive shaft, differential, axles and wheels/tires/brakes. There are some other insignificant components as well like bearing, which in some places are included in my calculations and in other places are not. Following Fundamentals of Vehicle Dynamics by Gillespie, with some obvious extensions to equation 2-9b and using SI units, we can write (for no incline, no hitch load) This is basically a clever application of F=ma and T=Iα with the understanding of imperfect (lossy) transmission through the entire system. Please note inertia and losses are not at all the same mechanism nor the same effect. Losses reduce torque or power and are "lossy" the power is truly lost to friction, thus heat and windage losses, whereas spinning drivetrain components act precisely like additional mass in the vehicle. If you coast with the clutch in, the rotational kinetic energy stored in the spinning components slow the loss of speed (deceleration) to true losses such as aero and tires making the vehicle "feel" heavier. Also on the flip side when power shifting during acceleration or when a DCT does the same, some of this engine/flywheel/clutch inertia is "dumped" inelastically (i.e. angular momentum is conserved but not energy) into the drivetrain providing a short transient bump up in vehicle acceleration. Here are the relevant definitions/descriptions: Equation (1) is simple, vehicle acceleration follows torque to the wheels (after losses) minus wheel rolling drag minus aerodynamic drag force. (2) is pretty much the definition of the mass factor. It is a sum of inertia weighted by gear ratios and gear ratios squared, normalizing the terms to make them dimensionless as is mass factor itself (by r and m). From (1) and (2) if the vehicle has no rotating drive train components the mass factor is 1 and the denominator of the acceleration equation is simply vehicle mass. As the inertia of the drive train components increase (weight and size, more on that later) the vehicle responds exactly like it is simply heavier but indeed more than heavier by just the increased weight of the components. Their is no approximation here. This is exact, simple Newtonian physics. Mass factor is still quite rigorous and exact. The only approximation is that the drivetrain losses can all be represented by power transmitted = power input x efficiency and efficiency does not have to be constant but is always taken as such with this being understood to be a very good approximation. Calculating Inertias and Mass Factor Inertia is not quite as simple as mass since inertia is actually a matrix, it depends not only on the mass of the object but also on the way the mass is arranged (say hollow pipe vs. solid pipe of the same mass). It also depends on how the object is spun. Imagine spinning a long thin cylindrical bar around the axis of the cylinder. It spins easy, i.e. high angular acceleration for a low torque. Now imagine spinning it around its center of mass like a balancing beam. Here it really resists and applied torque and gives a low angular acceleration for a relatively higher torque. This is because it's moment around the second case axis is significantly larger than the first case. Fortunately the formulas used for inertia are easy to derive and easy to apply. I only ended up using formulas for a solid cylindrical bar around the axis of its cylinder and a hollow cylindrical tube spun the same way. See the second column in this graphical table. It's worth pointing out that if one knows the material density (trivial to get) and has a precise 3D CAD drawing, most CAD software will spit out precise inertias. Given that we don't have precious BMW CAD drawings and I ain't going to disassemble a cars entire engine and drivetrain and throw my calipers on every single feature on dozens of parts, we are going to have to start approximations to calculate these masses, I's and the mass factors. Let's not turn this into bickering about "you got that gear shaft length or radius wrong by 1/4 inch" type of discussions. This is, from this point on, APPROXIMATIONS. We can later determine if our approximations are good by examining the results in a vehicle simulator that takes mass factor as a key input. I will talk a bit more about error and sensitivity in these estimations later. Now that being said, if you know a weight or size of a component positively and want to point that out, please do and I can update the spreadsheet. I used the following two images for the transmission shaft size estimates. The first image is from the ZF 6MT from the BMW X3 and the second are the input/output and lay shaft from the Audi B7 RS4 transmission. Fig 1: Sample transmission photos used for transmission component size estimations No, these are not the M3's transmissions. I couldn't find relevant pictures of those (MTs) but this level of approximation will find pretty well the same answer for these and the M3/4 anyway. Unfortunately this inspection/estimation method is quite a bit less obvious when attempting to do the same for an automatic transmission or DCT... You can download the entire xlsx file here. I had to ignore the error about a large file and use the File menu to download it. I would be surprised if all of my terminology, approximations or formula are self explanatory on the spreadsheet. Thus I am happy to answer questions about any calculation or term. One paper I found quite useful for both engine and wheel inertia is "Problems of Rotational Mass in Passenger Vehicles, Ubysz. This paper used a CAD method to calculate the engine inertia of a 2.0l inline 4 cylinder engine. For lack of any better information for the S65 V8, I simply doubled this value. Crude, yes, critical, no. This same reference has a very useful empirical study showing that wheel+tire moments of inertia are linear on a log-log plot vs. total tire radius. I actually used a more discretized approach to calculate the vehicles wheel+tire inertia very carefully respecting wheel and tire masses (precisely known) and then approximating their geometry (i.e. no spokes, no dish, perfectly cylindrical barrel, etc.). The regression though gave me a good idea the calculation was very reasonable. For those not wanting to download the entire spreadsheet here are the results of the calculation with the mass factor, gear by gear for both the E9X and F8X M cars and for both M-DCT and MT. I have assumed that other than flywheel weights the transmissions have identical inertia and this is clearly a somewhat crude approximation. More on this too just a bit later. Fig 2: Table of Mass Factors Below the primary table of results are some very simple formulas for mass factors using only gear ratios that I found across a fairly intensive literature search. As you can see none of them capture both the low and high gear results like this detailed approach. Note: I have not yet included brake rotors! This will be a minor revision. Discussion of Mass Factors: As you can see mass factors are very significant! In first gear your M3 will accelerate like to weighs 129-136% of its scale weight due to drivetrain inertia. That can be as much as 1350 lb! About 90% of this is due to engine/flywheel/clutch and the part of the transmission rotating at engine speed. About 80% of that is due to engine, flywheel and clutch with each contributing roughly equally. The results steadily decline across gears such that in the higher/highest gears only the wheel/brake/axle contributions are important and amount to about an extra 5% of total mass. The three (labeled) columns below the main table of mass factors are again analytical formulas for mass factors (quite obviously semi-empirical) to avoid a requirement for the level of detail shown in this spreadsheet. The literature sources are provided in the spreadsheet. The second column, Bernd Heißing and Metin Ersoy does the closest job in comparison to this detailed calculation in low gears whereas the third column from author J.Y. Wong does a much better job in the higher gears. Use of These Mass Factors in E92 M-DCT Performance Simulation: Using my own simulation software (as the other two I use do not allow for explicitly putting in mass factors) provides a reasonably good sanity check that these numbers are not terribly far from reality. Item, Results with mass factors all equal to 1.0, Results with these mass factors 1/4 mile time: 11.7 s, 12.5 s 1/4 mi trap: 118.9, 115.3 mph 0-100 mph: 8.10 s, 9.45 s 0-60 mph: 3.35 s, 4.25 s 0-60 feet: 1.85 s, 2.20 s 60-130 mph: 10.8 s, 11.7 s Again this is E92 not F82. Obviously all of the values on the left are for a car much more powerful or with much less mass. In fact less mass as a function of gear... The results on the right are pretty nicely in line with measured data for the car. See that data here. Here is an overlay of results for acceleration vs. time across 5-6 gears from the commercial CarTest software vs. my spreadsheet WITH my calculated mass factors. CarTest is in purple and my results are in blue. he dynamic similarity here is really right on the money. You can certainly see some minor discrepancies in the launch and wheelspin, but the overall predictions above are certainly right on the money (simulation vs. simulation). Also positive is that unrealistically high (based on test data) 1st gear accelerations predicted by CarTest are reduced significantly, strongly indicating a more accurate mass factor here as opposed to CarTest. Unfortunately, despite access to an amazing amount of inputs and outputs, mass factors are handled 100% "behind the scenes" in CarTest. Fig 3: Comparison of Authors Vehicle Simulation using the mass factors in Fig 2 vs. CarTest E92 M3 M-DCT Uncertainties: Using this as a live spreadsheet we can answer a huge number of what if, sensitivity and accuracy questions about our calculated mass factors. What if the E92 estimated engine inertia is off by a full factor of 1/2? The first and second gear F8X M-DCT mass factors change to 1.361 and 1.135 respectively. 1st gear only changes by 2%. All other mass factors are the same within 1%. What if we underestimated the F8X DCT flywheel radius by 25%? That "whopper" changes the 1st gear mass factor (mf) by nearly 4%. The transmission values were some of the "rougher" estimates I made. What if the transmission has twice this total inertia? This also has about a 4% change in the 1st gear mf and less than a 1% change in gears 3 and above. What is the wheel inertia is actually 25% larger than my estimate? This only makes less than a 1% change in all mf. What if the axles were truly weightless? This would not change any mf out to 3 decimal places. Clearly I have underestimated some masses/radii/inertias and overestimated others. The lucky thing about having so many inputs is that much of these errors cancel. This is no excuse for sloppy work but again without 3D CAD this is really tough estimation type of work. Based on this very simple sensitivity analysis I'm reasonably comfortable putting a rough +/- 5% error on these calculations. What would an across the board 5% increase in the E9X M-DCT mfs do to the simulated performance results? This: 1/4 mile time: 12.75 s 1/4 mi trap: 113.5 mph 0-100 mph: 9.95 s 0-60 mph: 4.50 s 0-60 feet: 2.30 s 60-130 mph: 12.20 s It makes them just about on the outer edges of most observed results. Still not terrible predictions! Conclusions:
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07-30-2014, 02:00 AM | #2 | |
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Very well written and informative! Contributions like these (not to mention how much time you spent preparing this) are a great asset to these forums. We should all be thankful that we have members that are bothered to contribute with this level of expertise! And all for free... BTW, found this slightly related piece of info on M-Power.com: http://www.m-power.com/_open/s/close...lang=en&x=3157 Quote:
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07-30-2014, 08:31 AM | #3 | |
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God damn that was good stuff, swamp. Elucidating and well analyzed. Anyone who completed high-school math courses should find this accessible. Great job.
One request: could you label each of the embedded images as fig 1, fig 2, fig 3, etc? That would make them easier to reference. There are some very good jumping off points for the discussion in the underrating thread, especially considering that many of the dyno results in question are inertia dynos, for which the proportional contribution of inertial losses will be extremely relevant when comparing the E9x and F0x M-cars. Speaking of which, here's a question. This might be better suited for the underrating thread, but I'll pose it here before I launch it in to that discussion. It pertains to these conclusions: Quote:
In more detail, on an inertial dyno, m isn't accelerated at all. Basically, you're dealing 100% with rotational inertia. This leads us to the question of the dynamometer inertial mass. Obviously, the dynamometer can't change the weight of its rollers for every vehicle. So the rollers inaccurately represent m for every vehicle. When deriving HP from the acceleration of a roller by a static vehicle, this would seem to increase the impact of drivetrain inertia on dynamometer outcomes significantly. I'm assuming that the roller mass is less than that of the vehicle, which I guess also varies from dyno to dyno. Again, emphasizing the point that comparing dyno to dyno is a bad idea.
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07-30-2014, 11:03 AM | #4 | |||
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Quote:
Quote:
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Where α is the SI (radians/sec^2) angular acceleration of the dyno drum and Ir is the moment of inertia of the drum. Note the factor of 4 changes to 2 in the "wheel term" (in the case of 2WD) because the only two wheels are being accelerated. Multiply through by α by Ir (again T=Iα) and this is the torque the dyno should be measuring (again SI unit, Nm). Theoretically the roller inertia doesn't matter, however, as posted prior it matters because we have seen the direct empirical evidence that an engines power output depends quite strongly on its rate of increase of rpm (just how fast it is accelerating). Again this is why engine dynos are very carefully run at a precisely fixed rpm for each measurement point - no engine inertia is being "fought". Now where it gets more interesting is that dyno operators typically use ηtf = .85 and certainly know their Ir but they basically ignore all drive train inertia terms to the right of Ir. I believe they are supposed to be captured approximately by using ηtf = .85. Also since Ir is >> all of the other I drive train terms that is justification for dropping those terms. This is one reason to use heavy (high I) drums.
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07-30-2014, 11:18 AM | #5 | |
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FWIW, definitely agree on the steady state power measurement thing. The whole "inertial sweep test" approach is so incredibly flawed in my mind, I'm amazed that anyone uses it.
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07-30-2014, 11:19 AM | #6 |
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Yeah, that is exactly at the crux of the issue I see. They assume the same inertia for all vehicles, which clearly isn't the case. The ones that do measure it use a questionable methodology.
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07-31-2014, 01:55 AM | #7 | |
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In practice though, this formula above is NOT used and the drive train inertia is not factored in in anyway and thus this does lead to errors, absolutely. This also shows very explicitly why an intertial dyno will give different results in different gears and why higher gear numbers will exhibit less error (drivetrain inertia is significantly lower).
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07-31-2014, 09:24 AM | #8 | ||
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So my point is really: on inertial dynamometers -- which assume a fixed drivetrain mass for all measurements -- wouldn't the actual difference in drivetrain mass have a more statistically significant effect than we would see in real world performance and simulation? To make a claim, rather than asking a rhetorical question, it would appear to me that the error in measurement on inertia dynamometers (which use fixed assumptions) would be greater than the figures stated in the paragraph below: Quote:
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07-31-2014, 11:16 AM | #9 | ||
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I don't think I can fully answer your questions but I can say some relevant things.
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Then there are the "simpler" corrections for environmental factors and conversion from wheel to crank values. The former is just a simple formula such as: ref: SAE J1349 Aug 2004 revision This factor, only valid up to a 7% correction, simply multiplies the entire torque/power curve by this correction factor (cf) to account for the difference between a reference environment (air density) and the actual one at the time of measurement. The latter, wheel to crank, is also just a fixed correction, most often dividing by 0.85 (a "15% loss" as folks commonly say). I believe the intent of the latter is to capture both drivetrain inertia and true parasitic losses. Luckily, even the most unskilled dyno operators know that this 15% value is a crude estimate. Quote:
Now if you want to compare "apples to oranges" I certainly believe that the errors, be them in the dyno measurement process itself, or in the use of either correction factor mentioned above are much higher than the relatively small (single low digit %) errors I mention for my calculated mass factors. After all we have the simple empirical evidence that dynos vary about 15% on the E92 M3 regardless of dyno type.
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07-31-2014, 05:26 PM | #10 | |
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Great thread Swamp, as usual
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When running a powertrain on a chassis dyno, power is assessed over a given engine RPM band, not a given road speed. The higher the gear, the greater the RPM and acceleration rate of the drivetrain (downstream of the transmission) relative to the engine. Since Power=Inertia x Angular velocity x Angular acceleration, for a given engine acceleration rate, the higher the gear, the more power will be consumed to accelerate the drivetrain. However, the engine acceleration rate decreases with higher gears (less wheel torque to accelerate the dyno drum). So it is not obvious which gear will yield the lowest inertial effects. It depends on the engine, drivetrain and dyno characteristics. |
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07-31-2014, 05:41 PM | #11 | |
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Because of this, IMO, you are correct in saying that one simply cannot compare dyno results from different powertrains. Further, depending on which gear is used and the weight and size of the dyno rollers, results may vary significantly dyno to dyno for the same powertrain. As I mentioned in the previous post, the power consumed to accelerate each rotating components is equal to the product of the component's inertia with its angular velocity and with its angular acceleration (this is valid on a dyno and on the street/track). Usually, on most chassis dynos, the powertrain is accelerated at a faster rate than an actual acceleration run, so the inertial effects are exaggerated. |
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07-31-2014, 08:12 PM | #12 | |
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07-31-2014, 08:21 PM | #13 |
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You've won the M3Post award for most free time spent on thread. Ever.
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07-31-2014, 10:26 PM | #14 | |
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Then of course there is the S65 bearing wiki thread. The OP, regular guy clearly spent waaaay more time on that thread than I did on this one. Then again if you count the time I spend writing my vehicle simulator (did that quite some time ago) we might be back on par. But really, a kick in the nads for spending too much time on a post. Ouch, that hurts!
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07-31-2014, 10:31 PM | #15 | |
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Thanks.
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Either way, higher gears (number on the shift lever) will always be better both due to less inertial "losses" and lower drivetrain inertia (which is generally unaccounted for in dyno testing).
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08-01-2014, 02:18 AM | #17 | |
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Last edited by CanAutM3; 08-01-2014 at 01:40 PM.. |
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08-01-2014, 02:47 AM | #18 |
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08-01-2014, 10:27 AM | #19 |
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That's exactly what I said... twice...
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08-01-2014, 01:39 PM | #20 |
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Typo on my part (that is what you get trying to post in hotel room with two kids running around ) , revised my OP:
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08-01-2014, 04:46 PM | #21 | |
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(torque actual) = (torque measured dyno) * Ir/(Ir+Id) Id being all drive train inertia in post 4, in the parenthesis, with gear weighting, except Ir. Since Id is not zero the actual torque is not the measured torque and is always less. Also if Id is larger (in a lower labeled gear) the actual torque will decrease. Now granted, Ir/(Ir+Id) is very close to 1 since Ir>>Id. The inertia of the drum is always going to be much larger than the inertia of the drivetrain but in low gears the actual ineria are weighted by the gear factors so maybe Ir>>Id is not always rigorously correct... Yes the relative inertial terms vary by gear with the wheels become more important in high gears, but that doesn't take away the fact that the Ie, If and Iet terms along with the Ntf^2 weighting dominate in low gears making the Id very large. Traction absolutely is not the only reason folks dyno test on numerically higher labeled gears. This gives higher measured power with less consumed in accelerating the drive train.
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08-02-2014, 08:17 AM | #22 | |
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For a given engine acceleration rate, the inertial impact of the engine itself will be the same in all gears, but the impact of the drivetrain and wheel inertias will be greater the higher the gear (the lower the ratio). |
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