0-60 change based on differential ratio change
 

Anyone know a way to calculate a 0-60 time if a different differential ratio is installed? I understand there are many variables. I am looking for a equation where nothing changes except the gear ratio.

As an example. if the 0-60 time is 10 with a 3.00 differential ratio and nothing changes except changing the differential ratio to 4.00 what will be the new 0-60 time? And, is any change a constant such as if the 0-60 time was 10 and the ratio was changed from 3.00 to 3.50 would the same formula still be applicable?

I doubt you could calculate that because of too many variables. It should be possible to take the torque and power curve to make an estimate using the F=ma solved for a: a=F/m.

If you take a plot of the torque curve v RPM that is the same as Force (force*length=torque. Length stays the same so force can interchange with torque)

If you use the ratio of the two different rear ends you will get a value such as 1.2. You would plot a new torque v RPM to determine the increased @ wheel torque (it will be the equivalent of the engine having more torque)

The speed of the car drops by the same ratio. You would have to factor that in.

You could either make plots and do the math to figure out the actual acceleration at say every 1/10 of second. Reverse engineer the actual force applied to get your car to 60 based on actual tests performed.

Once you have the actual force from each 1/10 s figured out, you multiply that by the differential ratio ratio to make a new curve. Then calculate a new set of force values every 1/10 s. You could probably get pretty close.

The easier way is to Google search the results of somebody else that already did the same thing. Try to find a similar engine doesn't need to be X5 but maybe heavier sedan like 740.

You will get say 10-15% more acceleration but the gears have to run 10-15% longer which you would think will cancel out. The whole point off a higher ratio rear end is to shift some of the highest effort acceleration into a more torquey part of the power band. The trade off is less power at higher rpm so it's "all about the bass no treble". (all low-end).

It's quite possible you would end up no faster 0-60 but hella faster 0-30.

You can use an app like dash command for aromatic 0-60 measurements it uses the accelerometer in the phone to determine launch and taps into the car computer to know the speed.

I would be very interested to know the results. Oh you will surely get lower mpg keep that in mind.

I am familiar with what ratios do to speed, rpms, etc and the variables having made many swaps over the years. I've made swaps on many combination of HP/TQ engines. I know the time over any measured distance will be reduced with a lower ratio. I need a formula for dummies. I don't expect it to be spot on but would like a ballpark estimate before I start the project. I don't want to use real life results of someone due to the variables. I have several ways I can accurately measure times that I can use after the lower ratio install. I know it will reduce mileage, don't care. There are many plug the numbers online to calculate time over distance. I need one that includes diff ratio in the calculation. Can't find one.

You can't find because I don't believe it exists. Find an example where someone else has already done a similar ratio change

I'm not surprised if it doesn't exist. I have done lots of searching.

Read this article: http://www.badasscars.com/index.cfm/...rod/prd446.htm

Do a zero to 60 (well 70 or so). Detemine the speed at the shift just near that speed.

If the ratio ratio will move that shift point before 60 you will have a slower 0-60 if you put in a shorter rear end. Hmm. How do you change the rear without the front?

Anyhow since a shift at least one possibly two will be involved there can not be a simple formula involved as your first post, however with some very good info harvested from the linked URL I think you can get somewhere.

Mostly that you can determine a rough idea of the shift points at what mph and rpm possibly and if you can pick a rear end ratio that will keep your shift far enough from 60 to avoid top over rpm before 60 not shift just before 60 either

That info is simple ratios and will give you a go-nogo before you start.

There is a program called cartest (window) that has cars programmed Into it that you can put in different rear ends to get estimates of 0-60 and 1/4 mile.

I did some searching for exact examples of somebody before and after values when changing the rear axle and saw some who said when the shifts happen before the trap you will get a slower time.

I'm sure you can find a chart already done with 0-60 of your car so you can determine the exact shift points.

Simple ratio will move those shift points to the modified speed @ shift. If you don't move a shift point slower than 60 from above you will speed up the process.

That app mentioned should give an idea of how much.

Unfortunately there is just no way it can be a simple formula because the torque curve changes a lot over the RPM range.

Well.. this isn't 0-60 calculator and is for e46, but might want to look at this.

BMW E46 Gear Ratio / Cruising RPM calculator

Thanks. I am familiar with that article. I understand the ins and outs of changing dif ratios from the good ol' trial and error days. I have only one option for a lower gear ratio. I have 3.91s and am going to 4.10s out of a 3.0. I purchased them more than a year ago. With the 3.91s Number of shifts will be the same. It is necessary to change both diffs. Usually the extra shifts referenced in the article are for a manual trans. Not nearly as much of a concern with an automatic. Most of what I do is from stoplights. I want the extra torque for the launch where I have the advantage over 2 wheel drive even if they have launch control. I will check out cartest.

Out of interest are you looking for pure 0 to 60 improvements or other ranges 50 to 90 etc?

I have a full Quaife set up on my E53 and that allowed me just over a second improvement in the 0 to 60 range simply by letting me floor the accelerator without losing any traction. This made up for not having any fancy modern computer controlled launch control.

Of course having a few extra horses under the hood also helps so maybe some engine upgrades?

SB

I've gone as far as I can go with engine mods and power adders. Don't need limited slip as traction is not an issue. Only interested in 0-60.

I recall reading you were planning on changing to the 4.10 diff in a different thread. I had thought you installed them by then... please report back the results when you have completed the work :D I am also quite curious..

One of the difficulties when calculating vehicle performance during hard acceleration is that the resultant propulsive force (i.e. the tractive force less the drag force) is not just accelerating the mass of the vehicle but also the inertia of the roadwheels, transmission and engine. In low gears, a good proportion of the torque produced by the engine is used up in simply accelerating its own inertia. The vehicle gets whatever is left.

The aerodynamic drag force is easy to calculate from air density, drag coefficient and frontal area. Rolling resistance drag can reasonably assumed to be 1.5% of vehicle weight. The torque converter K-factor characteristics can be matched to those of the engine to determine the turbine torque vs. speed characteristic at the transmission input shaft. The gear ratios of the 5HP24 transmission are known (1st = 3.57143, 2nd = 2.20000, 3rd = 1.50467, 4th = unity, 5th = 0.80374). Efficiency values would need to be assumed for each ratio.

The standard formula can then be applied to calculate the converter output torque under acceleration (Toc) and hence vehicle performance :

http://i260.photobucket.com/albums/i...ps6tkxktmu.jpg

where :

ToA = steady state torque from converter matching
TE = engine torque
IE = engine inertia
Fd = aerodynamic drag + rolling resistance drag
me = equivalent vehicle mass (taking driveline inertias into account)
DAR = drive axle ratio (the thing that we’re interested in here)
GR = transmission gear ratio
rr = tyre rolling radius
No = converter turbine speed
NE = engine speed
ηT = transmission efficiency

The issue you have is not having a K-factor and torque ratio curve for the SACHS WA4/W280 S-2GWK converter as this will be a trade secret, unless someone has measured one and kindly posted the results online somewhere.

Phil

I was never able to find out if they are a direct swap. I pulled the rear diff anyway and found it fits but the side shafts are not the same. Rather than use the side shafts on the original diffs not knowing about the front fitment I put the original back in. I did that because I didn't want to pull the front and find out it was not a direct swap. I was hopeful I could find out if it was or not before moving forward.

Got very side tracked with other stuff and did other mods and never got back to it. Current very flexible plan is to order side shafts and see if diff has same splines. If it does then pull the front to see if it fits. That is if I finish up working on my daughter's Porsche this week.

Sounds like you have some projects ahead of you.. best of luck!

Could you show us your 5.0 E53 ??? :2thumbs:

What I need is a formula for dummies that assumes all factors are the same other than the drive axle ratio change. If the 0-60 elapsed time is X with a 3.91 front/rear drive axle ratio and nothing changes other than the drive axle ratio then the theoretical 0-60 elapsed time with a 4.10 front/rear drive is reduced by X seconds.

Sounds like a theoretical estimate is not possible, nor is any comparison even if it were possible to find another 4.6 with exactly the mods I have done? From my experience it should be between one and two tenths. That would be motivation that I obviously need since I've had the diffs for over a year, maybe two.

I thought about this but I have no idea if it correlates: 4.10 minus 3.90 equals .20., 3.91 divided by .20 equals 4.9 percent, the percentage difference of the two ratios. Therefore .049 times the current 0-60 elapsed time equals the theoretically possible reduction in the current 0-60 elapsed time possible going from a 3.91 diff ratio to a 4.10 ratio.

The 0-60 time for a stock 2002 4.6 is about 6.2 seconds. 6.2 seconds times .049 equals .3 seconds. Conclusion is that by only changing the differential ratio from 3.91 to 4.10 The 0-60 elapsed time would decrease by .3 seconds. The revised 0-60 time would then be 5.9 seconds. Based on ancient racing experience that's a little more than I remember but it was real world when traction was always a variable. Traction is not a variable now and I don't expect it to be with the ratio change.

3.9 to 4.1 is a 5% difference.

The initial launch will be very close to a 5% difference but the engine has to Rev 5% more and will get winded sooner it will however have 5% less effort so it can sweep through the curve faster.

Shift points are similar auto or manual look at any 0-60 plot the speed is flat during the shift as there is no acceleration.

The force available to the wheels will not be the full 5% gain but if it was half or 2.5% that would get you from 6.1 to 5.95 0-60 which is substantial gain.

If the 4.1 was used in the 3.0 it was clearly chosen by BMW to give the weaker motor a boost in acceleration and basically return some of the pep zapped when dropping 100 HP. It makes perfect sense that ratio will give a nice boost to the low speed performance on the v8!

The concern I have is are those diff capable of the higher torque and power? I suspect they are over designed to handle off road and when used on road they will be more than capable.

I found a very detailed investigation into final drive ratio as applied to the M3; much lighter car but it will get the idea across.

In this particular case with a shift needed at 59.5 mph it's actually slower to 60 with the 4.1 vs 3.91. The time to 59.5 is quite faster than the 3.91, so… since i'm assuming you aren't aiming for a literal 0 to 60 time, I would expect to see very similar performance gains; look at the fact at higher speeds you loose the advantage.

I found the chart HERE

The guy put a lot of effort into making a detailed equation to estimate values.

https://www.bimmerforums.com/forum/a...5&d=1060547127

Here is a chart of the 3 different ratios; these are for the M3, but seeing the chart you can get a good idea of what to expect.

Oopsie, you are correct. It is a 5% difference. I will correct the post.

Based on the benefit of economies of scale and since the size of cases are the same, my guess is that the gear set, bearings etc. are also the same other than ratio.

Yes, with the lower ratio I will travel less distance before engine reaches 6500 RPM shift point. Engine doesn't have to rev higher to get the most out of the ratio change nor will get "winded" sooner. There should be no difference in the torque curve other than being higher across all rpms. The transmission already shifts before reaching 60 so any impact from an extra shift does not have to be considered.

3.0 isn't a high torque engine. I think the 4.10 in the 3.0 was to get it going faster by increasing the torque and running at a slightly higher RPM at road speed was more efficient.

Very interesting chart. Thanks! 0-60 is the benchmark that I can easily compare with other stock vehicles and in the vast majority of the time I am off the throttle before reaching 60. Even with the mods I have done it is getting quite hard to be an impressive sleeper. The reason we recently purchased an X5M is that rather than starting at 340 HP doing mods as I did I am now starting at 555 HP with tons of torque. With proven aftermarket mods to get to over 700, that's back to a lots of fun sleeper.

Here's an idea that might get you a relatively cheap data point before going through the full swap to see what you get ...

Put on a set of wheels/tires that have a 5% (or whatever exactly you're aiming for) smaller diameter, and take some test runs on those. I can guarantee swapping tires is easier than swapping differentials. And the results will probably be more relevant and accurate that anything you could get using a formula.

0-60 change based on differential ratio change
 

That is a perfect test case as long as can find a set that will fit over the bigger rotors. 5% smaller radius will get 5% higher torque-arm force to the road.

Re: "getting winded": the engine of course will get winded 5% sooner. The car will go over the power hill and hit redline 5% sooner in every gear. Part of the mystery of will you actually achieve faster 0-60.

When the torque band is wide enough the engine may do better with taller rear end. The smart money is on trying a tire with 5% smaller rolling radius. Get some take-off end of life tires or such.

My OBD app does automatic 0-60 and quarter mile timing you just have to stomp on the gas or does all the timing using the accelerometer to start and OBD to measure the speed. ( Of course you would have to account for the 5% wheel error).

That reminds me: both of our X5s have significant Speedo error they read 3-4 mph high at 70mph. Eg at indicated 73 I'm actually going 69-70mph.

The OBD knows the correct speed it's not from changed wheel size etc so I recommend learning on your car what Speedo error you have and read the obd data and confirm it is correct. (I usually clock a measured mile on the freeway with cruise to get my measured mile) GPS will give you a very good estimate of actual speed as well.

I'm going to make the change. I am positive it will improve 0-60 time, it's just a question of how much. If I have problems and the change is worthwhile I'll sort them out. I have done several mods on my 4.6 where I have had no benchmark for results or nothing to base it off that someone else has done. If it is a mistake it certainly won't be the first time I was wrong about something intended to improve performance.

This has been very worthwhile to help me sort out why I have been procrastinating. A major part of it was the beyond terrible experience I had installing the Seicane audio system. It still won't play DVDs. Going to audio shop this week to see if they can sort it out and get all behind me. Thanks for all the input.

My ¢¢: very much agree with your decision I would definitely do the same. Even if I was forced to drive 72 vs 75 to have the same RPM as I used to have (I have misophonia and can not tolerate certain sounds) it would totally be worth it to have the jump on somebody in a Corvette from the late 90s. (or really 99% of any car on the road: hell even sub 1L motorcycles).

Are you talking about your M sport or the other (you have two X5 now right?)

5% works out to 25' in a block. If you lose half of the torque gain to drive train variables you'd still be nearly an additional car length farther in a block vs your start point. Think of recent red light showdowns where it was neck n neck or the other guy was just barely sneaking away this upgrade will net you something like a car length per block. I didn't look at time to distance chart but I suspect you'll have already passed 60 mph by that point.

"Pics or it didn't happen". We would love to see a dash cam video

Diffs are for 2002 4.6.

All I have done on the X5M is buff and wax it, add low restriction air filters and made it so exhaust flaps are open all the time. We are taking it on vacation soon so once we get back I will add a tune. After that I will decide if I want to go further with maybe meth injection and downpipes. Not going to do anything radical as it is my wife's daily driver. But, I can't do stock.