BMW M3 M4 RPM Redline Visual, Finally

[swamp2]

Originally Posted by solstice

The power delivery of the S55 seems to be very close to most modern turbo engines. More specific the curves and power levels seems very close to the N63. The N63 has a lot of power and able to propel the 550s, 650 and even the 750 with gusto but what it's not IMO is an exciting engine. I drove several 650s ( GC, Coupe, Vert, rwds and Xis) and 750s ( 750i and 750 il ) on track and drag strip. While strong there is definately turbo lag and the engine feel like it's losing breath above 5500 rpm. You can go very fast, smooth and effortless but there is no traditional race character to it at all. The power is undeniable though if that's the main concern, I put down a 13.3s at 109.55 mph strip time with the 750iL, just flooring it from idle rpms. This power in a 3500 lbs car with some M fairy dust should make for quite a beast, but will it be nearly as exciting, responsive and predictable as the S65, cause the N63 sure isn't even in the same realm. We'll see.

Good post as well. Despite not having anywhere near the experience of driving all of these cars the math/physics tell me what those cars and the new M4 will be like in character. Probably a somewhat mundane point to those with more formal training and heresy to those who feel the only way to know any particular car is to drive it. The latter has some element of truth but overall certainly isn't.

Are you losing faith that the M4 will be all you hope and will be a suitable next car for you, personally?

[swamp2]

Originally Posted by hotsoup

Honestly, I feel there are just too many variables when you try to theoretically, accurately model the acceleration of the car over the course of several gear shifts. I am curious about the approximations and assumptions made from user input parameters, and differential equations used in their (CarTest's) numerical method (assuming they're using something like Runge-Kutta to solve the ODEs). Also would like to see the differences between the experimental and theoretical data. I'm no mechanical engineer (chemical engineer), but off the top of my head there are a crazy amount of not-so-insignificant variables besides gear ratios, torque specs, and horsepower - there's drivetrain losses as a function of angular velocity, anti-parallel drag forces as a function of velocity, static coefficient of friction of the tire rubber, normal (down)forces as a function of velocity, etc.

To achieve what we want (find optimal shift points), it might be best to go with empirical data (as another person has mentioned in this thread) which would be much more reliable, accurate, and easy to gather. I imagine something like CarTest (or any other simulation engine) would be better if one were planning to build out an engine without having one handy in order to make some design decisions before committing on specs/parts.
...

Good questions. I hope to address all of them. Many are partially or fully addressed in a large number of previous posts here on the forum on this and related topics.

The proof is pretty well in the pudding though... Although one can get bogged down in the large amount of physics involved one can at the same time distill this down to its essence.

On a slightly more abstract level - what it say THE 1/4 mile time for any given vehicle? Real world testing involves both controllable and random variables as well as systematic errors. Thus there isn't really a true, single value for such a time. The discrepancies associated with the results from CarTest have been proven across a fairly wide range of vehicles and performance levels to be as small or smaller than the natural variation we observe in testing. Similary it's predictive power to capture a change in performance given a certain isolated change in inputs has also been fairly well demonstrated. The results are certainly immensely more accurate and consistent than say dyno tests...

Distilling down to the key inputs they are simply an rpm dependent torque or power curve (or in many cases just peak values and some reasonable interpolation for other rpms) and the vehicle weight. Although gear ratios are important for some individual contests they are not that important for contests across multiple gears such as a 1/4 mi time or trap.

Have a look at the possibilities from just a super simple (peak!) power to weight prediction here. This should give you some comfort that most of the other factors you mention are simply higher order corrections to the dominant physics of power to weight.

Now specifically I will list which of your concerns are addressed by CarTest. There are plenty more that are included and are critical such as tire losses (see link below). Some are indeed addressed more accurately than others (one specific example is gear ratios and shift times which are certainly handled more accurately than tire friction and tire losses). I certainly appreciate the skepticism here. I was myself initially quite skeptical of the tools ability to capture all of the relevant physics. I even posted about that quite some time ago. Since then I've completely satisfied my skepticism as to the tool satisfying the level of accuracy mentioned above. Despite very small unknowns that remain such as the one that surfaced here - wheel weight vs. wheel inertia, my overall confidence remains quite firm.

-Gear ratios and final drive ratio - check
-Wheel size (even exact based on rev/mi numbers if you have them) - check
-Drivetrain losses including transmission, differential and axles - check
-Static and dynamic friction coefficients - check
-Shift times - check
-Down force - not handled and not required
-Center of mass shift with acceleration resulting in changes to peak sustainable wheel force without spin - check
-AWD, FWD and RWD - check

The full equation CarTest uses was partially quoted above this was the drive force equation. The other forces such as road slope and drag are relatively trivial additions. I've actually modeled a very similar equation simply with V = Vinitial + a * t (i.e. no formal solving of a differential equation) and iterating over a small time interval (inside the context of a huge spreadsheet with a half dozen sheets and thousands of lines...). It's less accurate, less generic and requires the approximation (perhaps even "fudge") discussed herein of using mass factors, but can capture the same complex curves of acceleration vs. time that match very closely with those from CarTest.


This post (not mine) shows a variety of screen captures from the software and shows the impressively large number of inputs and outputs one can address with CarTest.

Beyond that - buy it, it's cheap and fairly easy to use and I would be happy to let you know which default parameters I have found to need some modifications.

[swamp2]

Originally Posted by Never Convicted

Over/under on how many errors Swamp finds in this is 7. Place your bets.

Thanks for the "atta boy", however, I think "errors" is a bit of a harsh way to characterize the fairly smart skepticism that hotsoup has voiced.

[swamp2]

Originally Posted by catpat8000

I've used CarTest a lot in the past to model some mods I did to a C6 corvette I used to own and I found it to be pretty accurate, once all the parameters were accurate. I don't think we know all the M3/M4 variables we need to know yet.

I also think the accuracy depends what you are trying to model. If all you want are 1/4 mile trap speeds and ETs, I think CarTest does this well. Yes there are lots of variable but their contribution to the final result varies significantly.

In my view, Swamp knows what he is talking about and I have familiarity with the tool so I have some confidence in the results.

Thanks. Specifically with regards to the bold statement this is why my most recent predictions of the overall performance of the new M4 uses a simple bracketing approach estimating unknowns like weight of CSIC brakes, other potential weight unknowns or optimism on BMWs behalf as well as trying to capture likely underrating. Link here.

Any level of error I've made with regards to gearing should not have a major impact on any simulation results that cover range of gears. Some specific in gear results could though be quite a way off due to this still unknown.

[swamp2]

Originally Posted by CanAutM3

Since the inertia losses are proportional to acceleration, Iw has an impact of optimal shift points. And so does vehicle mass, road slope, wind direction, etc... which is something I never anticipated.
...
BTW, thanks for entertaining the discussion in this thread, it is much appreciated on my part. There are not that many people out there that have the ability to dive in such details

Still thinking about this. One quick observation though is that shift points as determined by CarTest absolutely do depend on losses (true losses not inertial "losses") but do not depend on vehicle mass, road slope nor wheel weight... In addition doubling the FD ratio and halving the gear ratios also has no effect. Perhaps some signficant and some insignificant approximations.

Me too, good discussion. Also rewarding that others are interested in this stuff...

[catpat8000]

Originally Posted by swamp2

It sort of comes down to reconciling these two facts:

1. Peak acceleration in any gear occurs at the point(s) where the engine crank torque is peak*. As a result the acceleration curve basically mirrors the torque curve NOT the power curve.

2. Comparing two vehicles identical in all aspects except engines, at a given speed the car that can produce more power will out accelerate the other (WOT of course). Engine torque is truly 100% irrelevant in this point. No not 95% or 99% irrelevant, 100% irrelevant.

Originally Posted by johns

Thanks for the explanation.

I guess I'm still having a little trouble reconciling those two points above because they don't seem to be mutually inclusive.

For example, at a given speed, if car 1 had more HP than car 2 but less torque then according to your second point car 1 should out accelerate car 2. But your first point says the car with the higher torque should have a higher acceleration. They seem to be contradicting each other. What am I missing?

Here is one way to think about it. Peak acceleration in a given gear occurs when the crank torque is highest, true. But the higher the rpm at which you make peak torque, and the wider the rpm range you make it over, the more you can use gearing to increase the instantaneous torque seen at the rear wheels.

So for your example, yes, car 1 will out accelerate car 2. Let's put real world examples to these names. Let's make car 1 an E90 M3 and car 2 an E90 3 series diesel. Pretty similar cars and let's pretend for now they weigh the same (I haven't checked).

Car 1 makes ~300 lb/ft of torque and car 2 makes ~425 lb/ft of torque. Which car accelerates harder? Car 1 of course. Why? Because it makes torque at a much higher rpm therefore is able to use higher gearing to more than overcome the engine torque deficit.

This is why BMW used to say they loved high rpm motors. With a high rpm motor you can make lots of HP without needing to make tons of torque. This allows lighter smaller drivetrains but requires higher tech machinery. Once you move towards lower rpm turbo motors making tons of HP, you'll end up with much heavier, bulkier transmissions and differentials to handle the extra torque.

Pat

[CanAutM3]

Originally Posted by catpat8000

This is why BMW used to say they loved high rpm motors. With a high rpm motor you can make lots of HP without needing to make tons of torque. This allows lighter smaller drivetrains but requires higher tech machinery. Once you move towards lower rpm turbo motors making tons of HP, you'll end up with much heavier, bulkier transmissions and differentials to handle the extra torque.

Pat

IMO, while the statement is true for the input portion of the transmission, it is not entirely correct. The size and required robustness of the differential is mostly proportional to the max wheel torque produced.

Originally Posted by swamp2

2. Comparing two vehicles identical in all aspects except engines, at a given speed the car that can produce more power will out accelerate the other (WOT of course). Engine torque is truly 100% irrelevant in this point. No not 95% or 99% irrelevant, 100% irrelevant.

From what I have discovered in this thread, the statement above from Swamp2, that I also took for granted all these years, is not absolutely true.

I was using exactly this presumption to establish my "theoretical" optimal gearing for the S55 and was proven wrong by Swamp2 himself .

The engine's moment of inertia, gearing and at what engine RPM the power is produced have an impact on how much of that power will be robbed to angularly accelerate the engine instead of accelerating the whole car forward.

[CanAutM3]

Originally Posted by swamp2

One quick observation though is that shift points as determined by CarTest absolutely do depend on losses (true losses not inertial "losses") but do not depend on vehicle mass, road slope nor wheel weight... In addition doubling the FD ratio and halving the gear ratios also has no effect. Perhaps some signficant and some insignificant approximations.

This is probably the most likely explanation. True losses are very complex and difficult to model. The equations we have quoted so far all consider drivetrain efficiency as a constant applied to transmitted torque, which is far from reality. A portion of the efficiency varies with speed and another varies with torque. For example, the viscous losses in a transmission are still present even if no torque is transmitted.

Maybe through tweaking of the input parameters, the errors from the losses in the drivetrain efficiency and the errors in inertia happily cancel themselves out in the CarTest acceleration model. Maybe not so in the optimal shift point calculation model .

I would not have expected that doubling the FD while halving the gear ratios to have an impact on optimal shift points. The resulting relationship between the car acceleration and engine angular acceleration remains the same and the ratio between gear ratios also stays the same.

[hotsoup]

Originally Posted by Never Convicted

Over/under on how many errors Swamp finds in this is 7. Place your bets.

Terrible post. Be more confident in elaborating your posts. Thanks buddy.

Swamp2, thanks for the reply. Some good points to think over. I think we might be talking about two different things if I'm not mistaken. I was focusing on how to identify optimal shift points, and it seems as you're focus is on the macro stuff - predicting 0-60/quarter-mile/etc based on various inputs. For some reason, I was under the impression that shift points were under discussion but if that wasn't the case... oops.

Not so crazy about the stuff on the link. Essentially boils down to fitting a curve using regression. It's practical, fairly accurate, easy, and used ALL the time in engineering, but just so inelegant.

[swamp2]

Hmmm heading slightly in circles...

Originally Posted by CanAutM3

From what I have discovered in this thread, the statement above from Swamp2, that I also took for granted all these years, is not absolutely true.

And I posted long ago about why the statement is not quite exactly true. It fairly close and it's correct in principle if drivetrain losses, inertial terms, drag and tire losses were magically zero. That is certainly a terrible approximation to make to determine overall vehicle performance but again it is all about context. One also is not off by an amount of say an equivalent mass factor in this statement (speed predicted by statement is 20% off).

The statement in itself is accurate in the context of the completely incorrect claim that torque is what matters in vehicle performance. Have a look here for my older ramblings on this.

[swamp2]

Originally Posted by hotsoup

Swamp2, thanks for the reply. Some good points to think over. I think we might be talking about two different things if I'm not mistaken. I was focusing on how to identify optimal shift points, and it seems as you're focus is on the macro stuff - predicting 0-60/quarter-mile/etc based on various inputs. For some reason, I was under the impression that shift points were under discussion but if that wasn't the case... oops.

I did mostly misunderstand your post. When you brought up all of the particular variable/inputs you questioned many were more relevant to overall vehicle performance rather than shift points.

Originally Posted by hotsoup

Not so crazy about the stuff on the link. Essentially boils down to fitting a curve using regression. It's practical, fairly accurate, easy, and used ALL the time in engineering, but just so inelegant.

Not so fast. You might want to dig just a bit deeper.

Curve fitting can be dangerous especially when it leads to a conclusion confounded by correlation not being the same as causation. However, in regards to the scaling of ET and times as power to weight ratio to the 1/3rd power there is actually some great basic physics buried in this. Assuming constant power one can derive a close form expression for trap times with (power/weight)^1/3. Thus here we have correlation (curve fitting) and causation.

[swamp2]

Originally Posted by CanAutM3

This is probably the most likely explanation. True losses are very complex and difficult to model. The equations we have quoted so far all consider drivetrain efficiency as a constant applied to transmitted torque, which is far from reality. A portion of the efficiency varies with speed and another varies with torque. For example, the viscous losses in a transmission are still present even if no torque is transmitted.

I don't fully agree. Losses are both in the real world, in CarTest and in my spreadsheet simulator, speed dependent. Even the most basic ideal damping loss we learn about in Differential Equations is a speed dependent linear loss (sliding friction, damper losses, etc). Furthermore manufacturers even quote things like transmission and diff losses as a percentage. Yes losses in general are from a lot of small complex effects but the overall approximation of them being a percentage and scaling with speed or rpm as required is a pretty good model. No it's not accurate that all trannies or all diffs have the same loss but again the errors from such an assumption are not critical in an overall accurate performance prediction. That approximation will cause some vehicles to be slightly under predicted and others over depending on if the efficiencies of the components are all in total better or worse than the nominal values.

You certainly don't need to be reminded of this but all of these things being user controlled inputs, along with CarTests ability to do sensitivity studies are overall points that make simulation incredibly insightful and useful.

Originally Posted by CanAutM3

I would not have expected that doubling the FD while halving the gear ratios to have an impact on optimal shift points. The resulting relationship between the car acceleration and engine angular acceleration remains the same and the ratio between gear ratios also stays the same.

I would. Look at the basic tractive force equation again here are both Nf and Ntf terms. Again probably not a large real world change in the rpm predicted but it is an effect and it is simply missing in CarTest.

[CanAutM3]

Originally Posted by swamp2

I don't fully agree. Losses are both in the real world, in CarTest and in my spreadsheet simulator, speed dependent. Even the most basic ideal damping loss we learn about in Differential Equations is a speed dependent linear loss (sliding friction, damper losses, etc). Furthermore manufacturers even quote things like transmission and diff losses as a percentage. Yes losses in general are from a lot of small complex effects but the overall approximation of them being a percentage and scaling with speed or rpm as required is a pretty good model. No it's not accurate that all trannies or all diffs have the same loss but again the errors from such an assumption are not critical in an overall accurate performance prediction. That approximation will cause some vehicles to be slightly under predicted and others over depending on if the efficiencies of the components are all in total better or worse than the nominal values.

You certainly don't need to be reminded of this but all of these things being user controlled inputs, along with CarTests ability to do sensitivity studies are overall points that make simulation incredibly insightful and useful.

My comment regarding efficiencies applied to torque was directed at Gillespie's equation on tractive force that we have been quoting so far. I cannot comment on the equations used in CarTest and your spreadsheets since I have not seen the equations.

I think we are saying the same thing, while some losses are torque dependant, some of the losses are speed dependant. Applying a fixed efficiency number that is only torque dependant, as is done in Gillespie's equation, is not really accurate (Gillespie even mentions it himself in his text). Drivetrain losses also vary with other parameters such as temperature, this is why I am saying that they are complex to accurately model. But yes, there are decent enough ways to approximate their impact.

Thanks for clarifying that efficiencies are also speed dependent in CarTest, it certainly brings more accuracy.

Originally Posted by swamp2

I would. Look at the basic tractive force equation again here are both Nf and Ntf terms. Again probably not a large real world change in the rpm predicted but it is an effect and it is simply missing in CarTest.

I don't see a "direct" impact (see equations below). However, since the drive shaft would be spinning twice as fast, more power is consumed to accelerate it angularly. This would would slightly reduce the acceleration parameter (ax) in the equation below. But as you said, the impact would be so minimal on optimal shift points, that it becomes negligible in real life. This is why I said I did not expect it to make a difference.

[CanAutM3]

Originally Posted by swamp2

Hmmm heading slightly in circles...


And I posted long ago about why the statement is not quite exactly true.

Yep, "closing the loop" does imply going in a circle in some way

However, I never intended for my comment to imply that you did not understand this. I know you have played long enough with all this to have an in depth comprehension of it all .

Originally Posted by swamp2

One also is not off by an amount of say an equivalent mass factor in this statement (speed predicted by statement is 20% off).

IMO, one would be off by the ratio of mass factors in the same way as for optimal shift points (see post#263).

[swamp2]

CanAutM3:

I think the flaw I'm seeing in this additional derivation is that ηtf, the combined efficiency of the transmission and final drive is rpm dependent. You should also use the 1 and 2 subscripts on ηtf, i.e. η1f and η2f. Although it may not affect your final result, you also have subscript problems in your original derivation. The first t subscript is for transmission gear and the second f is for the final drive ratio.

Although your derived equation may determine a condition that is met at the ideal shift point it does not explicitly solve for the shift point. Even ignoring the rpm dependency of ηtf one can't just pick out some terms and say they don't change on both sides and then conclude the shift point is also unchanged.

Let me share the way both CarTest and I have implemented this. The value one enters as a fixed percent, say 15% (loss) is scaled according to rpm and redline. As an aside I believe that both correlation with test as well as published loss figures indicate 11% is more accurate for the E92 M3.

Let's say we call the 15% number the loss, ℓ

Thus ηtf = (1 - rpm x ℓ/redline) (efficiency = 1 - loss)

And examining the state of affairs just before and just after any shift ideal or not, the rpm has changed and thus so has ηtf.

Originally in my observation and tests in CarTest I just looked at the Nf^2 term in the original equation and realized it isn't invariant under the aforementioned doubling and halfing process. Yes the other primary Ntf term is invariant but there is simply no way everything else in the equation will cancel out this Nf term change precisely. The drive power curves change in both gear n and gear n+1 and there is no way the curves will just happen to intersect at the exact same rpm. The doubling and halfing process will change the ideal shift points, however, again, probably by a fairly small amount.

Also as another side note I think this loss is handled a bit differently by different simulators. Some may choose redline or perhaps redline/2, etc. for the constant in the formula above. Unfortunately, this level of detail about CarTest is not available. My spreadsheet simulator uses redline and the formula above.

Anyway these differences are significant. They are part of the dominant engine torque term, not part of any inertial (nor equivalently mass factor) terms.

[CanAutM3]

Originally Posted by swamp2

one can't just pick out some terms and say they don't change on both sides and then conclude the shift point is also unchanged.

I might have erred, I am not an expert mathematician . However going over the math again, I do not see any errors in my deductions. Can you be more specific about the above point.

Originally Posted by swamp2

I think the flaw I'm seeing in this additional derivation is that ηtf, the combined efficiency of the transmission and final drive is rpm dependent. You should also use the 1 and 2 subscripts on ηtf, i.e. η1f and η2f. Although it may not affect your final result, you also have subscript problems in your original derivation. The first t subscript is for transmission gear and the second f is for the final drive ratio.

Although your derived equation may determine a condition that is met at the ideal shift point it does not explicitly solve for the shift point. Even ignoring the rpm dependency of ηtf one can't just pick out some terms and say they don't change on both sides and then conclude the shift point is also unchanged.

And examining the state of affairs just before and just after any shift ideal or not, the rpm has changed and thus so has ηtf.

Anyway these differences are significant. They are part of the dominant engine torque term, not part of any inertial (nor equivalently mass factor) terms.

Agreed, more precision needs to be brought into the efficiency terms. To achieve this, I believe that the ηt and ηf efficiencies need to be separated and, as you mention, the ηt needs to be made gear dependant with ηt1 and ηt2. I did not want to do it in the first derivation because it means re-arranging Gillespie's tractive force equation (I did not want to be accused of heresy ).

See the new derivation below. The resulting equation does make good sense. When the torque at the transmission output is equal for the two gears, it is the optimal shift point.

The key point is that acceleration is still part of the equation. I am still trying to solve the complete equation ax1=ax2 when v1=v2 to remove acceleration from the equation, but the math gets very messy and I have not yet found a path to an elegant and simplified end equation .

[CanAutM3]

@ Swamp2

On a different note, I am trying to estimate the various I (Ie, It, Id and Iw) for a DCT E9X M3. In all the simulations you have done, would you have the resulting mass factors for the 7 gears of the DCT? I would also need the assumed weight of the car for those mass factors.

[swamp2]

^ A much longer reply by PM, getting waaaaay OT...

Anyway for my own spreadsheet simulator I am using

mass car = 3537 lb
mass driver = 160 lb
shift time = 50 ms

Mass factors by gear

1 1.19
2 1.17
3 1.15
4 1.12
5 1.11
6 1.10
7 1.10

Although I've done very little validation of acceleration in gears 5-7

Comparing my spreadsheet (light blue) vs. CarTest (purple) for acceleration vs. time I get a pretty nice agreement. Note likely mistreatment of wheelspin after 1-2 shift....

[swamp2]

Originally Posted by CanAutM3

I might have erred, I am not an expert mathematician . However going over the math again, I do not see any errors in my deductions. Can you be more specific about the above point.
...

Again...

Originally Posted by swamp2

I think the flaw I'm seeing in this additional derivation is that ηtf, the combined efficiency of the transmission and final drive is rpm dependent. You should also use the 1 and 2 subscripts on ηtf, i.e. η1f and η2f. Although it may not affect your final result, you also have subscript problems in your original derivation. The first t subscript is for transmission gear and the second f is for the final drive ratio.

Originally Posted by swamp2

Although your derived equation may determine a condition that is met at the ideal shift point it does not explicitly solve for the shift point. Even ignoring the rpm dependency of ηtf one can't just pick out some terms and say they don't change on both sides and then conclude the shift point is also unchanged.

[CanAutM3]

Originally Posted by swamp2

Again...

^ I agree on the efficiency. See my new derivation above.

However, for all the other terms that were simplified, they were not simply arbitrarily picked and removed. If, for the condition to be solved, terms are demonstrated equal, they can be simplified. That is mathematically valid.

To determine the theoretical optimal shift point, we need to find the intesection point between the two curves,

Solve f1(x1)=f2(x2) when x1=x2

So yes, IMO, it does solve for the optimal shift point.

[swamp2]

Originally Posted by CanAutM3

^ I agree on the efficiency. See my new derivation above.

However, for all the other terms that were simplified, they were not simply arbitrarily picked and removed. If, for the condition to be solved, terms are demonstrated equal, they can be simplified. That is mathematically valid.

To determine the theoretical optimal shift point, we need to find the intesection point between the two curves,

Solve f1(x1)=f2(x2) when x1=x2

So yes, IMO, it does solve for the optimal shift point.

I still disagree. SOLVING FOR would be an equation where you have rpm = function(torque, losses, moments (or mass factors), etc.) AS AN EXPLICT function where also the terms do not depend on derived quanitites such as the acceleration. Such implicit dependencies is not revealing the true relationships. Again an expression that must be satisfied at an optimal shift point vs. a formula providing the rpm of the optimal shift point absolutely are not the same thing.

I guess we've probably beat this horse about to death at this point...

[mkoesel]

Originally Posted by ASAP

mkoesel? tie? still not a definitive answer

So, ready to concede that bet now? As you may have seen in the thread stickied at the top of the forum, the official rev limit is 7600 RPM.