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BMW 3-Series (E90 E92) Forum > BMW E90/E92/E93 3-series General Forums > General E90 Sedan / E91 Wagon / E92 Coupe / E93 Cabrio > Mathematical Performance Model (328i, 335i, M3)



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      09-08-2009, 11:36 PM   #1
dougman
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Mathematical Performance Model (328i, 335i, M3)

I used to design vehicles; now I design computers for a living. But since I still love cars I got some of it out of my system this weekend with a simple mathematical model in Excel for our cars. Here's how it works and some of the results:

Vehicle Inputs:
- Vehicle weight
- Drag coefficient and frontal area
- Axle and gear ratios
- Tire size
- Torque curve (one point every 1000 rpm)
- Driveline efficiency
- Weight distribution and cg height
- Max tire friction

And also:
- shift RPM (when to shift)
- shift delay (how long to shift; I used 0.5 s)
- RPM for clutch dump

Then the program calculates the torque to the wheels and corresponding force to accelerate the vehicle, checks to make sure it's not friction limited (and reduces as appropriate). Then it subtracts the forces for aerodynamic losses, figures out the change in velocity, increments the time, and calculates a new torque, and so on...

I did three runs using the data for 328i, 335i, and M3. I adjusted driveline efficiencies slightly (but within reasonable values of 80%-90%) to get close to published data for the vehicles. The model delivered these results:
- 328i: 0-60 in 6.1s; 1/4 mile in 14.6s
- 335i: 0-60 in 4.9s; 1/4 mile in 13.5s
- M3: 0-60 in 4.1s; 1/4 mile in 12.5s

If anyone's interested I provide more info (or maybe I should sell it!). But here are some screenshots:

1) Input sheet and some results for M3:


2) speed vs. time graphs for all three:


3) Acceleration vs. speed for 1st three gears: (red=M3; green=335i; blue=328i)


4) Acceleration vs. speed for gears 4 - 6 (red=M3; green=335i; blue=328i)


Interesting to look at these last graphs and think about how and when the M3 can pull on the 335i. Also, it's interesting to see that you should shift the 335i from 4-5 and 5-6 before redline (when the acceleration graphs cross). The 328i and M3 should be shifted at redline.

Enjoy. I make NO claims as to the veracity of these models; I did them purely for the fun of it.

But I'm happy to run some "what if" analyses--i.e. driveline ratio changes, etc. etc. Could make for some fun "virtual" drag races.
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      09-08-2009, 11:45 PM   #2
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Very interesting, thank you for taking the time.
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      09-09-2009, 12:02 AM   #3
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Quote:
Originally Posted by dougman View Post
I used to design vehicles; now I design computers for a living. But since I still love cars I got some of it out of my system this weekend with a simple mathematical model in Excel for our cars. Here's how it works and some of the results:

Vehicle Inputs:
- Vehicle weight
- Drag coefficient and frontal area
- Axle and gear ratios
- Tire size
- Torque curve (one point every 1000 rpm)
- Driveline efficiency
- Weight distribution and cg height
- Max tire friction

And also:
- shift RPM (when to shift)
- shift delay (how long to shift; I used 0.5 s)
- RPM for clutch dump

Then the program calculates the torque to the wheels and corresponding force to accelerate the vehicle, checks to make sure it's not friction limited (and reduces as appropriate). Then it subtracts the forces for aerodynamic losses, figures out the change in velocity, increments the time, and calculates a new torque, and so on...

I did three runs using the data for 328i, 335i, and M3. I adjusted driveline efficiencies slightly (but within reasonable values of 80%-90%) to get close to published data for the vehicles. The model delivered these results:
- 328i: 0-60 in 6.1s; 1/4 mile in 14.6s
- 335i: 0-60 in 4.9s; 1/4 mile in 13.5s
- M3: 0-60 in 4.1s; 1/4 mile in 12.5s

If anyone's interested I provide more info (or maybe I should sell it!). But here are some screenshots:

Interesting to look at these last graphs and think about how and when the M3 can pull on the 335i. Also, it's interesting to see that you should shift the 335i from 4-5 and 5-6 before redline (when the acceleration graphs cross). The 328i and M3 should be shifted at redline.

Enjoy. I make NO claims as to the veracity of these models; I did them purely for the fun of it.

But I'm happy to run some "what if" analyses--i.e. driveline ratio changes, etc. etc. Could make for some fun "virtual" drag races.
Cool exercise!

Yah, with the 335i you don't have to be close to redline to wring out the most power; shifting somewhere just before 6000rpm is good enough. It's because forced induction doesn't work as well at higher rpms, and also because the fat torque curve/plateau from 2000rpm up to 5500rpm.

Whereas the M3 by design has to be wrung out to just short of its 8300rpm redline in order to get the most power.
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      09-09-2009, 03:15 AM   #4
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Interesting. I have driven the BSR in Virginia, but never in a performance car, so I am not a guru in getting the most out of a performance engine.
I have paddle shifters and have not been able to take the BMW drivers course here in Europe, due to the fact they charge a fortune. I have wondered at what points I was supposed to be shifting and I now know that I have been doing it a bit wrong. Once in a while I like to do a quick 0-60 just to remind myself what's under the hood. I will follow your input the next time and shift before 6000 during higher torque and see if it pulls a little stronger.

Thanks for the time you put into this. You may have just improved my appreciation for this engine.

Gill.
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      09-09-2009, 10:04 AM   #5
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Very cool thread. This gets my vote for one of the more interesting threads we've had in quite some time. Maybe I'm just a geek for data. It would be cool to add the E90 330i and 325i on there while you are at it.

A bit OT, but your meticulous attention to detail and fluency with computers reminded me of another impressive thread I remembered from years ago. Not the same topic, but another guy with a love for these cars and some computer fluency, doing a CAD model of the E90 based solely on photos he'd seen in brochures and online: http://www.e90post.com/forums/showthread.php?t=10074
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      09-09-2009, 08:49 PM   #6
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Thanks for the feedback gentlemen.

Any requests for simulations? New axle ratio on the M3? Race on the wet (half friction)? All cars shift at 5000 RPM; does 335i beat M3? Effects of weight reduction programs? Effects of cutting shift time down (i.e. DCT)?

I can run any of these through--while the model may not be spot-on accurate, it should be pretty good at predicting changes in performance.
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      09-09-2009, 08:58 PM   #7
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good info indeed.
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      09-09-2009, 09:17 PM   #8
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You got some skills.
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      09-09-2009, 09:19 PM   #9
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Wow, awesome information! I love statistics and visuals like these, very cool!

Thanks for taking the time to do this and share with us.

Subscribed for future reference (and in case you add more).
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      09-09-2009, 09:26 PM   #10
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Nice. But would it be hard to add a 335 with a tune to the list. It would be great to see jb+, jb3 and procede all next to eachother on that list.
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      09-09-2009, 09:29 PM   #11
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i'm so ashamed. When I have rare free time, and a computer is involved, i'm usually looking at porn.

j.k.


kind of.
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      09-09-2009, 09:33 PM   #12
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Originally Posted by jeremicium View Post
i'm so ashamed. When I have rare free time, and a computer is involved, i'm usually looking at porn.

j.k.


kind of.
haha
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      09-09-2009, 09:42 PM   #13
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Quote:
Originally Posted by Prowess Symphony View Post
Nice. But would it be hard to add a 335 with a tune to the list. It would be great to see jb+, jb3 and procede all next to eachother on that list.
Sure. All I need is the torque curve.
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      09-09-2009, 09:43 PM   #14
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Originally Posted by jeremicium View Post
i'm so ashamed. When I have rare free time, and a computer is involved, i'm usually looking at porn.

j.k.


kind of.

Both are hard-core... Plus I have the joy of 12-hour flights to China every few months...
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      09-09-2009, 09:44 PM   #15
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Originally Posted by dougman View Post
Sure. All I need is the torque curve.
Well this could get interesting. Who has the tune torque info?
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      09-09-2009, 11:25 PM   #16
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That was interesting!
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      09-10-2009, 12:47 AM   #17
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Damn look at that g difference at first gear!!!
I need the facking M!!!!!!!!!!!!!
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      09-10-2009, 01:20 AM   #18
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Wheres the tuners to give out the necessary info? I wanna see how the tunes stack up against stock and the m3.
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      09-10-2009, 01:24 AM   #19
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sticky!!
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      09-10-2009, 04:50 AM   #20
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the minority of us i think are stock with our 335i...

I agree - add tuned 335i and stock 330i to the list.. I used to own a 330i. It was much quicker than the 328.
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      09-10-2009, 08:51 AM   #21
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Quote:
Originally Posted by dougman View Post
Also, it's interesting to see that you should shift the 335i from 4-5 and 5-6 before redline (when the acceleration graphs cross). The 328i and M3 should be shifted at redline.
Turbocharged cars make more power/torque in lower RPMs, and taper off to redline. It should be no surprise that you should shift before redline because you're losing power..
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      09-10-2009, 11:40 AM   #22
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Thanks OP for the great info. A couple ideas for additional comparisons:

e46 M3, manual transmission
e90 330i, manual transmission
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