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:
- A better understanding of this somewhat confusing topic.
- Improving vehicle performance simulation efforts.
- Helping understand dyno performance and potential under rating.
I'll try to cover some of the basic physics here and I'm also providing a very detailed approximate calculation for the mass factors for the E9X M3 and F82 M4. I'll also discuss a bit of simulation results and sensitivity and accuracy.
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:
- Mass factors rigorously represent the power used in accelerating all rotating drive train components dynamically as added mass that varies with gear.
- Mass factors can be calculated very precisely from nothing more than material densities and 3D CAD drawings. Since this typically is not available, a simple geometric estimation approach can still yield reasonable accuracy.
- Mass factors are critical for accurate vehicle acceleration simulation.
- More accurate semi-empirical formula for how mass factors might scale with vehicle mass, vehicle power or torque and wheel/tire sizes would be highly valuable toward making simulation more a-priori accurate across a diverse range of vehicles. No existing mass factor formulas appear accurate enough for a desired level of simulation. However, the Bernd Heißing and Metin Ersoy are by far the closest for most common vehicle performance metrics.
- Additional insight to how to calculate inertias of DCT transmissions and clutches would be valuable. Hollow shafts with shafts spinning inside shafts and the multiple clutches confound the process significantly.
- The equations and principles here could be extended to include the effect of a large radius inertial dyno hub wheel and might provide insight into the investigation of M4 dyno "over achievement".