# Thoughts on swapping rear diff in 330i



## rumratt (Feb 22, 2003)

Anyone out there who is painfully bored can help me work through the pros and cons of getting a new rear diff for my non-ZHP 330i. My car has the 2.93 diff, and although it's not horrible, I can't help but feeling like the gears are too far spaced apart for the power of the engine. Put this together with the fact that I have a 6th gear that I never use, and I can't stop thinking about getting a new diff.

Dilema 1: Do I really want to rip out a perfecty working diff from a 6 month old car.  It seems like I'm taking a nice, stable situation and opening it up for complications.

Dilema 2: Where to get it. It seems that you can get used OEM diffs from a junk yard for $250-$500. This keeps the swap in a price range that makes it likely to actually happen.. but do I really want to put a used part with unknown condition into my new car? 

Dilema 3: LSD vs open. One one hand, I feel stupid replacing the diff with another open diff, but it looks like going for a LSD would increase the cost by at least $1K or more, making the swap unlikely. If I'm going to start tossing big numbers around I might as well sell my car and buy a used M3.

Dilema 4: Which ratio. I know nick will tell me to get something big like a 4.8  but right now I'm thinking 3.23 or 3.38. Cruising speed in 6th would be fine with both of these. My main concern is that I don't make the car annoying to drive with constant shifting. A 3.23 would make my gear ratios almost identical to an E36 M3 (which I like because it means the final result is somewhat of a known quanity). Also, with an open diff, I'm not sure how low it makes sense to go; at some point 1st gear will become the "smoke one tire" gear.

Dilema 5: Auto-X. I plan to auto-x this year. If I end up doing it regularly I might regret not being eligible in stock class. But I think I care more about my daily driving feel then which car's I'm ranked against at the auto-X. :dunno:

Dilema 6: Resale. I might loose several thousand when I sell the car. That is *if* I tell them about it when I sell. (I guess I would have to).


Wow. That's quite a long list of dilemas. I am seriously considering it, but maybe I'm not ready for this leap yet.


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## BradS (Aug 27, 2003)

rumratt,

The argument that 2.93 is too tall for the 6 speed has some considerable merit, IMHO. Couple of points:

1. If you keep your existing diff in the garage, you can always swap it back in. If you're not happy with it or at resale time.

2. If you are going to do it, do it right. Don't be kicking yourself that you didn't go deep enough on gearing or put the LSD in. Do it once and do it right. 3.38 should be a great gearset, I think. Even up in the 3.46 range would be nice. The OD 6th opens a lot of doors for steeper gearing.

3. One thought point on the LSD--if its just for traction of the line, then having an open diff can be a driveline saver at some point, because the tire (singular) will spin before something expensive and hard to fix breaks. The fusible link theory of drivline abuse. Have something cheap and easy to fix designed to break before expensive diffs/driveshafts/clutches etc. A few dollars of Michelin peeled off beats a broken driveshaft every time.


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## rumratt (Feb 22, 2003)

BradS said:


> 3. One thought point on the LSD--if its just for traction of the line, then having an open diff can be a driveline saver at some point


Yeah, this is one thing I'm wondering about. I certainly don't plan to become a stoplight racer and do burnouts all over the place. And there is the other benefits to a LSD like helping snow traction.

But then again, why buy a LSD if you're not going to drive somewhat spirited, so I guess I am putting the drivetrain at some risk. Of course, I believe a higher final drive ratio will actually help relieve pressure from the drivetrain in front of the diff.


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## BradS (Aug 27, 2003)

rumratt said:


> Yeah, this is one thing I'm wondering about. I certainly don't plan to become a stoplight racer and do burnouts all over the place. And there is the other benefits to a LSD like helping snow traction.
> 
> But then again, why buy a LSD if you're not going to drive somewhat spirited, so I guess I am putting the drivetrain at some risk. Of course, I believe a higher final drive ratio will actually help relieve pressure from the drivetrain in front of the diff.


I think an LSD is the way to go. Spirited driving, in the BMW fashion, deserves it. Spirited being defined as not an endless series of 0-60 or 1/4 mile runs.

If I bring mine back to the states, a 3.46 with an LSD will be going in.


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## rumratt (Feb 22, 2003)

BradS said:


> If I bring mine back to the states, a 3.46 with an LSD will be going in.


Yeah, I'm actually getting psyched for a LSD, but I can't find one for under $2K. :yikes:  rennsport advertizes one on their web site for $1600, but the E46fanatics guys say it's vaporware. I got a reply from "Performance Gearing" and they want between $3200-3900 for a rebuilt LSD. :yikes: :bawling:

I certainly can't justify 3-4K on a diff. I'd rather buy a used M3 instead.

Some places sell just the internals of a diff, and then you need to install it. That means i'd need to buy a higher-ratio diff from a junk yard ($300-400), then the LSD internals ($1200) then have the diff put together (semi-educated guess: $400), then have it installed in the car (another $200 maybe). That's $2200, and my understanding is that if they don't do it properly, it can make noise and never be quite right. 

Anyone know if there is a cheaper way to get a LSD into an E46? I might have to give up and accept that my car is doomed to an open diff.


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## Kaz (Dec 21, 2001)

BTW, no diff will do anything about gears 'spaced too far apart.' The difference between each gears is a function of the ratios in the tranny.

Now, I don't know the ratio of 1st gear in the 6sp, but as someone who has driven a 3.46 car daily for almost 3 years, a consequence of going too short with the diff is that you turn your 6speed back into a 5, by making 1st pretty useless. If this was a 60s muscle car where I'd have a choice of rear ends, I think I'd personally be happier with a 3.38.


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## BradS (Aug 27, 2003)

rumratt said:


> Anyone know if there is a cheaper way to get a LSD into an E46? I might have to give up and accept that my car is doomed to an open diff.


Convince Auburn or Tractech that there is a market for BMW LSD's that have a street price of $500 or so. Though there probably isn't enough demand to warrant the development costs, unfortunately.

$3500+ for a REBUILT LSD?!?! Is there a smiley for someone smoking crack?


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## Chris90 (Apr 7, 2003)

If you don't like to hang the ass end out a lot, you don't need an LSD right? 
I personally would get it, cause It would make screwing around easier and more fun, but if you don't drive like that, you don't need it.
Trade up for a 330 ZHP? I think buying an M3 would cost a lot more than $3-4k, since you'd have to sell your car.


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## rumratt (Feb 22, 2003)

Kaz said:


> BTW, no diff will do anything about gears 'spaced too far apart.' The difference between each gears is a function of the ratios in the tranny.


Yeah, I worded that poorly. I realize that when I shift from one gear to the next, the RPM's will land in the same place as before.



> Now, I don't know the ratio of 1st gear in the 6sp, but as someone who has driven a 3.46 car daily for almost 3 years, a consequence of going too short with the diff is that you turn your 6speed back into a 5, by making 1st pretty useless.


Yeah, I'm actually worried about this, and I'd be curious to hear your impressions of exactly how annoying the 3.46 is. A lot of e46fanatics guys are happy with their 3.46, but it seems like it would be way too much for me. Particularly with a 3.0 engine, which would get you through the range even faster. I'm concerned that even a 3.38 would be too much as well and I'm considering a 3.23 or 3.15.


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## rumratt (Feb 22, 2003)

BradS said:


> $3500+ for a REBUILT LSD?!?! Is there a smiley for someone smoking crack?


I guess the bottom line is that they don't really want to spend time on E46's, but for that money they're willing. :dunno:



> If you don't like to hang the ass end out a lot, you don't need an LSD right?


There's also accelerating hard while cornering hard (one wheel won't spin so easily first), and also the car will have more traction in snow because it won't be so easy to spin one wheel.


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## rwg (May 10, 2002)

The LSD is easily the feature I enjoy most on the e36 m compared to the e46 330 (and that's saying a lot - I have a big list). It's not about drag racing for me, it's about cornering. If you did decide to switch the differential out, you would be missing an incredible opportunity to improve your car dramatically.

The problem is the brake system (I think). You would have to figure out some way to program it to stop trying to act as a psuedo differential, yet still keep abs going.


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## rumratt (Feb 22, 2003)

rwg said:


> The problem is the brake system (I think). You would have to figure out some way to program it to stop trying to act as a psuedo differential, yet still keep abs going.


Good point, although I'm not sure I see the problem. From my experience, the car applies some braking to the spinning wheel. It doesn't apply very much though because it's still pretty easy to spin just one wheel.

So if I had an LSD, the difference between the rear wheel speeds would be less, thus the less correction the electronic system would try to apply. It still might do SOME braking, but it would be to the spinning wheel only, so this would essentially be "helping out" the LSD. :dunno:

This all assumes that I have pressed the DSC button once. Otherwise, any wheel spin and the engine shuts down.



> Just got to remember to turn off DSC and Dynamic Brakes everytime you start the car.


Yes, this is the other obvious solution. It's easy to shut it all the way off. I just want to make sure that if I *don't* shut it off I'm not dragging my rear brakes around every time I accelerate hard because the computer is confused. I don't see how this would happen though.


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## rumratt (Feb 22, 2003)

ff said:


> I actually like the effective gearing of the non-ZHP better. Seemed to make better use of the engine's torque than my "ZHP" does.


Hmmm. Why do you say that? The difference is tiny anyway (between 2.93 and 3.07) so I'm wondering if it's some other aspect of the ZHP experience that makes you feel that way. :dunno:


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## Moderato (Nov 24, 2003)

ff said:


> Entirely possible. It could be the 75 pound wheels on the ZHP
> 
> My previous 330 was so fluid, so quick, so perfect in every gear. Not so with the ZHP


What year/model/options/wheel/tires was your last 330?


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## rumratt (Feb 22, 2003)

Kaz said:


> BTW, no diff will do anything about gears 'spaced too far apart.' The difference between each gears is a function of the ratios in the tranny.


Here's an update. Kaz's simple statement above made me start thinking about exactly what effect does a shorter rear diff have. After spending half a day thinking about it and filling up some excel spreadsheets, I'm much less excited about this swap.

The bottom line is that a shorter final drive ratio plays a big roll when starting from a stop, so for 0-60 times or street-light racing, it's a big improvement. But in my case, I'm not that excited about being fast off the line. The scenario I was hoping to improve was when you're driving at some speed (say speed X) and you want to minimize the time needed to accelerate up to another speed (Y). This could be passing a car, accelerating onto a highway, etc. I was willing to do more shifting to get there, as long as it was faster.

So will a shorter final drive ratio improve your time to get from speed X to speed Y? (Assume that speed X is a decent amount above zero). The answer is no, and this is true even if you ignore the shifting delay (I mean assume that you have infinitely fast shifts) . Actually, the real answer is "not necessarily" because either car could be faster and it depends 100% on whether speeds 'X' and 'Y' line up well with the cars effective gearing.

Here's the logic. As Kaz pointed out, it's not squishing the gears together; it's just scaling them all be the same factor, or shifting them down. This has a big impact on 1st gear, because there is no gear below 1st. The higher gears are also effectively lowered, but there is less magic here because you always had the option of being in a lower gear in the first place. If you graph the power available to the ground through the range of speeds from X-Y it will look like two zig-zagging lines, where one is shifted to the left (the lower diff). However, the sum of the area below the lines is the same. (That is, unless you get near the endpoints, meaning you run out of gears on top or bottom).

Here's an example: (based on another of Kaz's comments.) Assume you start with a 6-speed, and imagine that gears 2-6 are all equally spaced apart (so when you upshift you always reduce the gearing by the same percentage). Now, replace the rear diff with a shorter ratio so that gear 3 becomes equal to what gear 2 used to be. You'll now shift through gears 3-6 rather than 2-5 and you have gained absolutely nothing. (except you've got one heck of a low 1st gear).

Bottom line: I'd still love a LSD, and having a shorter gearing might be fun for off the line (or 5mph starts) but if it doesn't help me merge onto a highway faster, I'm not nearly as excited about it. I'm amazed at how confused I was about the effects of changing the final drive ratio (was I the only one  ?) although IMO it is really not obivious.


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## elbert (Mar 28, 2002)

rwg said:


> The problem is the brake system (I think). You would have to figure out some way to program it to stop trying to act as a psuedo differential, yet still keep abs going.


If you hold the DSC button for around 3 seconds, it will shut off the DSC and traction control completely. ABS will still function.

As a side note, I was very surprised to find that Quaife isn't offering an E46 LSD by now :dunno:


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## BradS (Aug 27, 2003)

rumratt said:


> So will a shorter final drive ratio improve your time to get from speed X to speed Y? (Assume that speed X is a decent amount above zero). The answer is no, and this is true even if you ignore the shifting delay (I mean assume that you have infinitely fast shifts) . Actually, the real answer is "not necessarily" because either car could be faster and it depends 100% on whether speeds 'X' and 'Y' line up well with the cars effective gearing.
> 
> Here's the logic. As Kaz pointed out, it's not squishing the gears together; it's just scaling them all be the same factor, or shifting them down. This has a big impact on 1st gear, because there is no gear below 1st. The higher gears are also effectively lowered, but there is less magic here because you always had the option of being in a lower gear in the first place. If you graph the power available to the ground through the range of speeds from X-Y it will look like two zig-zagging lines, where one is shifted to the left (the lower diff). However, the sum of the area below the lines is the same. (That is, unless you get near the endpoints, meaning you run out of gears on top or bottom).
> 
> Here's an example: (based on another of Kaz's comments.) Assume you start with a 6-speed, and imagine that gears 2-6 are all equally spaced apart (so when you upshift you always reduce the gearing by the same percentage). Now, replace the rear diff with a shorter ratio so that gear 3 becomes equal to what gear 2 used to be. You'll now shift through gears 3-6 rather than 2-5 and you have gained absolutely nothing. (except you've got one heck of a low 1st gear).


Are you overlooking the effects of torque multiplication, and looking at it solely as a function of rpm vs. speed. A lower geared car should accelerate harder in ANY gear because you are multiplying the engines torque more while enables the engine to rev faster = better acceleration.

In order to put this into numbers that matter, assume you have an engine that makes a steady 200 ft/lbs of torque throughout the rpm range in question (it doesn't, but it is a constant in this equation), and you are accelerating in 3d gear from a decent speed. A 330 has a 1.66 3d gear and you have a 2.93 rear diff, so you have 973 ft/lbs of torque to accelerate a constant mass (the car). If you change to a 3.46 rear gear, you now have 1148 ft/lbs to accelerate the same mass--some 20% more. Will you be turning more revs when you start if the same speed? Yep. Will you have to shift faster? Sure, because you are closer to the shift point and will get there faster. But you will accelerate harder.

The obvious retort is, I'll start in 2d. Maybe. The 2-3 drop in a 330 is 34%, whereas the diff swap you are talking about is 15-20%. Dropping to 2d may involve too many revs.

Or, to heck with the math and find someone with a lower gear, drive it and see if you like it!


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## Kaz (Dec 21, 2001)

BradS said:


> Are you overlooking the effects of torque multiplication, and looking at it solely as a function of rpm vs. speed. A lower geared car should accelerate harder in ANY gear because you are multiplying the engines torque more while enables the engine to rev faster = better acceleration.


This is true... But only if in any given case, engine output at a certain speed in a certain gear is more than it would be with a different final drive. If you drop the ratio so much that you end up being one gear higher than you used to, you've negated all the effort you just put in, not only because you're most likely in a higher gear and turning less RPMs, but the higher gear will have less (or negative, if it's OD) torque multiplication.

So if you drove in the same gear you always did before the swap, you'll see gains in this area. But then again, you could save a lot of money and just drive one gear lower all the time.

I'm planning on getting my car on a Dynojet in a few weeks. I'm really curious to see what comes out of that.


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## BradS (Aug 27, 2003)

Kaz said:


> This is true... But only if in any given case, engine output at a certain speed in a certain gear is more than it would be with a different final drive. If you drop the ratio so much that you end up being one gear higher than you used to, you've negated all the effort you just put in, not only because you're most likely in a higher gear and turning less RPMs, but the higher gear will have less (or negative, if it's OD) torque multiplication.
> 
> So if you drove in the same gear you always did before the swap, you'll see gains in this area. But then again, you could save a lot of money and just drive one gear lower all the time.
> 
> I'm planning on getting my car on a Dynojet in a few weeks. I'm really curious to see what comes out of that.


True. The only difference is that a diff swap in the 15-20% lower range might keep you in the same gear, vice downshifting, which, because of the much greater gear spreads, might be too many RPM.

I think the other thing to keep in mind is that for short bursts of acceleration, a short rear end is much less of an advantage. So if it's quick, 1-3 second bursts of cut and thrust all in one gear, then no real reason to change, since you can set that with the tranny. But the longer your acceleration runs are, the more of advantage a shorter rear will be. So if you're making lots of runs from, say 20-80 mph, I think the gains are much greater and its more worthwhile.


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## rumratt (Feb 22, 2003)

BradS said:


> So if you're making lots of runs from, say 20-80 mph, I think the gains are much greater and its more worthwhile.


No, I am convinced that this is definitely not true. Unfortunately, I don't have a lot of time now to put together a convincing argument.

I do agree with you that you have more torque to the ground when you are in the same gear at the same RPM. The problem is that you need to shift sooner with the shorter diff. Once you shift, the original diff is now putting more torque to the ground than the shorter one. It's a zero-sum game (excluding the benefits in first gear).


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## rumratt (Feb 22, 2003)

ff said:


> The 18"-ers feel like they weight twice what the winter wheels do.


How about getting new wheels? I'm sure someone would do an even swap for a pair of M68's, no? You might even make some money? :dunno:


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## rumratt (Feb 22, 2003)

ff said:


> Certainly worth considering. I actually don't care much for the ZHP wheels. Too many "spokes".
> 
> Now accepting offers.


Cool!

Now get your wheel discussion out of my thread before you hijack it completely.


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## BradS (Aug 27, 2003)

rumratt said:


> No, I am convinced that this is definitely not true. Unfortunately, I don't have a lot of time now to put together a convincing argument.
> 
> I do agree with you that you have more torque to the ground when you are in the same gear at the same RPM. The problem is that you need to shift sooner with the shorter diff. Once you shift, the original diff is now putting more torque to the ground than the shorter one. It's a zero-sum game (excluding the benefits in first gear).


Yes, but at some point you have to shift the original as well, and now its putting considerably less power to the ground as well. It is not a zero sum game--the time factor matters greatly. Try looking it at this way: in any given acceleration run of a set duration measured by speed, the average torque put to the ground by a numerically higher gear will be greater and the car will accelerate quicker. There will be times where it less at particlur points, to wit, right after the shifts. But I think you'll find if you graph it that is the case. (I am not nearly smart enough to graph this)

Now, there will be discreet sequences where it will be slower due to an extra shift. My car is just at 60 mph at the top of second gear. If I changed to a 3.46, my 5-60 time will probably drop, but my 5-50 time will be quicker, as will the 5-70.

Empirically, you might want to go down to your local drag strip and talk to some of the local bracket racers. See what happened to their 1/4 miles runs after gear swaps subtracting out the first 100 feet (many places give you time slips with 100 ft time, so that can be taken out, which is where most of your 1st gear advantage is.)

It also could very well be the case that for your driving, it will not matter a whit. In which case this has been a great discussion, at least for me.


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## rumratt (Feb 22, 2003)

BradS said:


> Yes, but at some point you have to shift the original as well, and now its putting considerably less power to the ground as well.


Agreed. What I am arguing is that they will zig-zag in power and the sum under the curve is equal.



> Empirically, you might want to go down to your local drag strip and talk to some of the local bracket racers. See what happened to their 1/4 miles runs after gear swaps subtracting out the first 100 feet (many places give you time slips with 100 ft time, so that can be taken out, which is where most of your 1st gear advantage is.)


I agree that the gearing helps a lot if you're anywhere near first gear. But if you're well out of first (much more than 100 ft or so), then I'll maintain my argument.



> In which case this has been a great discussion, at least for me.


Agreed. :thumbup: I wish I had time right now (and the ability) to better explain my argument. For now, think about the example I gave with gears 3-6 become lowered to the point that they are equal to the previous gears 2-5. You agree there's absolutely no benefit there, right (once you're into the relevant speed range). This seems to contradict several of the things you've argued (I previously agreed with your arguments, but this simple example is what made me think I was wrong). And even if you lower it only half as much, the same principle applies. The shift points would occur at different spots (so the drivers would see-saw in terms of who had the lead) but one would never pull away from the other).


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## rwg (May 10, 2002)

elbert said:


> If you hold the DSC button for around 3 seconds, it will shut off the DSC and traction control completely. ABS will still function.


Sure, and you might be comfortable that you would do so every time you got in the car (I am not sure I would be happy with that if potential damage might occur). But what about when your wife drives the car? Anything that requires human intervention like that makes me nervous.

It's possible that it wouldn't be an issue. The concept that an LSD would actually keep the electronic nannies from working is something I hadn't considered.


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## BradS (Aug 27, 2003)

rumratt said:


> Agreed. :thumbup: I wish I had time right now (and the ability) to better explain my argument. For now, think about the example I gave with gears 3-6 become lowered to the point that they are equal to the previous gears 2-5. You agree there's absolutely no benefit there, right (once you're into the relevant speed range). This seems to contradict several of the things you've argued (I previously agreed with your arguments, but this simple example is what made me think I was wrong). And even if you lower it only half as much, the same principle applies. The shift points would occur at different spots (so the drivers would see-saw in terms of who had the lead) but one would never pull away from the other).


This may be where we'll have to agree to disagree. I think that if you and I are in identical 330i's, save that I swapped in a 3.46 rear gear, and we ran from 20 to 80, I would bet you a fine malted adult beverage that I would get there quicker than you. If that didn't happen, I would happily pay up, buy you a second, and rethink all that I currently think about gearing!

(And, if Uncle Sam sends me back to the states next year, I get to put my diff-swapping money where my mounth is.)


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## rumratt (Feb 22, 2003)

BradS said:


> This may be where we'll have to agree to disagree. I think that if you and I are in identical 330i's, save that I swapped in a 3.46 rear gear, and we ran from 20 to 80, I would bet you a fine malted adult beverage that I would get there quicker than you. If that didn't happen, I would happily pay up, buy you a second, and rethink all that I currently think about gearing!
> 
> (And, if Uncle Sam sends me back to the states next year, I get to put my diff-swapping money where my mounth is.)


I'm willing to agree to disagree. But I was hoping you would explain the aparent contradiction in the simple scenario I proposed (reducing by one whole gear, where 2-5 becomes 3-6 and you have no benefit). To maintain your stance you need to be able to explain that somehow. My inability to explain it is what changed my mind on the issue.

And you're welcome to proove me wrong by coming over, swaping in a 3.46 and kick my ass.  If you do, beer's on me, and I'll get a 3.46 myself.


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## adc (Apr 1, 2003)

*Eh, stop arguing*



rumratt said:


> I'm willing to agree to disagree. But I was hoping you would explain the aparent contradiction in the simple scenario I proposed (reducing by one whole gear, where 2-5 becomes 3-6 and you have no benefit). To maintain your stance you need to be able to explain that somehow. My inability to explain it is what changed my mind on the issue.


When BMW dropped a 3:07 instead of the regular 2:97 in the ZHP, acceleration times were quoted half a second faster. So altering your final drive absolutely helps in most acceleration scenarios. Does it help on the track? Who knows, I bet it heavily depends on the track. Most top race teams run different final drives for different tracks.

If there was a reasonable choice for an E46 LSD out there, it would be the first mod this year. As it is, I think I'll put in some rear sunshades for my son's benefit :dunno: .

adc
03 330 ZHP


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## BradS (Aug 27, 2003)

rumratt said:


> I'm willing to agree to disagree. But I was hoping you would explain the aparent contradiction in the simple scenario I proposed (reducing by one whole gear, where 2-5 becomes 3-6 and you have no benefit). To maintain your stance you need to be able to explain that somehow. My inability to explain it is what changed my mind on the issue.
> 
> And you're welcome to proove me wrong by coming over, swaping in a 3.46 and kick my ass.  If you do, beer's on me, and I'll get a 3.46 myself.


Hmm. Given that set of facts, it would be the same. But I am not sure your scenario is realistic because the ratio spreads are not, in fact, constant. The drops are 1-2: 43%, 2-3: 34%, 3-4: 26%, 4-5: 19%, and 5-6: 15%. So, you can't swap in a rear gear that makes 2-5 the same as 3-6, because of the ratio differences.

A couple of additional discussion points:

1. I am not sure that your disregard of 1st gear advantage is fair--currently you can pull all the way to 37mph in 1st--still a valid gear for street driving. That only drops to 31 with a 3.46, btw.

2. A quick survey of the actual numbers indicates that you likely won't be in that lower gear. Going all the way to a 3.46 is only an 18% drop. The range of situations you are likely to find yourself in where that will kick you into a higher gear a fairly small, I would think. And the disadvantage would be more than made up for the longer the run.

3. Example: a run from 30 to 80 (and assuming a flat torque band, and discounting aerodynamic losses etc, etc). Chances are you won't be in 1st with either gear as you'll be near the top of the rev range. So, you start in 2d. With the 2.93, you'll be turning a hair over 3000 rpm, with the 3.46, right at 3500. Shifiting at 6500, you'll catch 3d at 64mph now, and at 55mph with the 3.46. I think you will agree that the 3.46 will accelerate harder to 55. Between 55 and 64, the 3.46 will not accelerate as hard, because now we are in 3d, but at 64, the 3.46 gets its advantage back and holds it all the way since it can pull over 80 at 6500.

So, in our hypothetical run, over 50mph of spread, the 2.93 should have the advantage during 9mph of the run, while the 3.46 has an advantage over 41 mph of the run.

I think this is a simplification, but also a valid comparison model. Please feel free to plug your own numbers in to see how it works in different scenarios.

Cheers!


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## adc (Apr 1, 2003)

*Great analysis*

Bravo! Excellent analysis!

Of course, it assumes the acceleration is constant throughout the rev range. It isn't so, especially in the ZHP. I think that will complicate matters somewhat .

To my understanding at least, it is simple: lowering the final drive ratio increases the effective torque to the rear wheels, which means harder acceleration in every gear. It also means having to swap gears a little more frequent, although this is harder to quantify and greatly depends on the acceleration/track secenario. I understand your post to be exactly that: an acceleration scenario...

Peace,

adc
03 330 ZHP



BradS said:


> Hmm. Given that set of facts, it would be the same. But I am not sure your scenario is realistic because the ratio spreads are not, in fact, constant. The drops are 1-2: 43%, 2-3: 34%, 3-4: 26%, 4-5: 19%, and 5-6: 15%. So, you can't swap in a rear gear that makes 2-5 the same as 3-6, because of the ratio differences.
> 
> A couple of additional discussion points:
> 
> ...


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## rumratt (Feb 22, 2003)

BradS said:


> Hmm. Given that set of facts, it would be the same. But I am not sure your scenario is realistic because the ratio spreads are not, in fact, constant. The drops are 1-2: 43%, 2-3: 34%, 3-4: 26%, 4-5: 19%, and 5-6: 15%. So, you can't swap in a rear gear that makes 2-5 the same as 3-6, because of the ratio differences.


Ahah! I agree. But realizing this alone is a big point in understanding what changing the rear diff does, and does not do. My point is that changing the rear diff does two things. 1) (major) puts more torque to the ground in first gear, 2) (minor) since the gears are not equally spaced, it allows you to get to the closer spaced gears at lower speeds.



> 1. I am not sure that your disregard of 1st gear advantage is fair--currently you can pull all the way to 37mph in 1st--still a valid gear for street driving. That only drops to 31 with a 3.46, btw.


Agreed. In reality, first gear is a large part of 0-60 or 0-80 or whatever. But it's not relevant in 40-100.



> 2. A quick survey of the actual numbers indicates that you likely won't be in that lower gear. Going all the way to a 3.46 is only an 18% drop. The range of situations you are likely to find yourself in where that will kick you into a higher gear a fairly small, I would think. And the disadvantage would be more than made up for the longer the run.


Don't agree. If you assume my constant gear ratios assumption above (which I agree is not true in practice) than you don't gain magical torque at the wheels by moving the shift points. The sum under the curves are the same.

That said, there may by a psycological factor that you are more likely to try to keep it in the same gear that you did before. You're used to shifting into 3rd at X MPH, so you keep doing so (and thus rev higher). But you could just as easily change your driving philosophy and shift later with the existing diff



> Between 55 and 64, the 3.46 will not accelerate as hard, because now we are in 3d, but at 64, the 3.46 gets its advantage back and holds it all the way since it can pull over 80 at 6500.


I can come up with scenarios where you're better of with the 3.46 and also for the 2.93. It just depends on whether the range lines up well with the shift points. I've got lots of scenarios in a spreadsheet. Scenarios like "start at speed X, and always shift at RPM Y, trying to get to speed Z". Then plot the torque that gets to the ground throughout the full speed range. I could post them if you like.


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## rumratt (Feb 22, 2003)

adc said:


> Of course, it assumes the acceleration is constant throughout the rev range. It isn't so, especially in the ZHP. I think that will complicate matters somewhat .


I agree this is a factor we are ignoring for now, but I claim that it is not relevant to the points being argued. We should agree first assuming a flat torque curve, then bring in that extra factor and debate whether it changes anything (which I claim it doesn't).



> To my understanding at least, it is simple: lowering the final drive ratio increases the effective torque to the rear wheels, which means harder acceleration in every gear.


I agree with this statement *if you compare the same gear and same RPM for both situation* then the 3.46 puts more torque to the ground for sure. But that is a meaningless comparison. You need to compare if both cars are travelling at the same SPEED, who has more torque to the ground. And the cars will not always be in the same gear.


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## BradS (Aug 27, 2003)

rumratt said:


> I can come up with scenarios where you're better of with the 3.46 and also for the 2.93. It just depends on whether the range lines up well with the shift points. I've got lots of scenarios in a spreadsheet. Scenarios like "start at speed X, and always shift at RPM Y, trying to get to speed Z". Then plot the torque that gets to the ground throughout the full speed range. I could post them if you like.


I bet another fine malted beverage more of them are better with the 3.46.

If my model is accurate, then here is the full range of shift points in speed, from which we can determine which gear will be provide harder acceleration at any given speed, with a 6500 rpm shift point, 25.2" diameter tires, and a flat torque curve:

Gear 2.93 3.46

1 37.1 31.5
2 64.6 54.7
3 97.3 82.4
4 131.4 111.2

So, that leads to this table, showing which rear gear is better at any given speed:

0-31.5 3.46
31.6-37.1 2.93
37.2-54.7 3.46
54.8-64.6 2.93
64.7-82.4 3.46
82.5-97.3 2.93
97.4-111.2 3.46
111.3-131.4 2.93

I think you have to look at this in light of your drving style. As I look at this, for my driving style, most of my acceleration events will occur in lower end of the speed scale, say starting from and ranging within 0-80 mph. In this range of possible starting speeds, the 3.46 has a vast edge over the 2.93. Up to 54.7 mph, the only speed range where the 2.93 is better is the 5.5 mph at the top of 1st. Out of that 80 mph spread under which I do most of my acceleration, the 3.46 is the clear, indisputable winner, over the course of the driving events in the life of the car. After 80, I think the 2.93 is better. If all you do is 30-40 mph roll-ons, then the 2.93 is better, because the 3.46 has to shift right at 31 mph.

Additional points:

1. That extra 18% torque multiplaction has to get to the ground somewhere, and its down in lower 1/2 of the speed scale, which pputs it right where it does the most good.

2. Granted, you may not always be in the optimum gear when you start, but I think that is a variable that is probably constant in this debate.

3. And, of course, changing from a 2.93 to say, a 3.15 provides a much slimmer argument, especially for the $$$ involved.

4. What about the counter to your argument that there is no roll-on advantage to a lower gear: by that logic, there should be no acceleration disadvantage in dropping down to a, say, 2.46?

5. If someone was really good, had all the figures, they could map the acceleration curves with engine's torque band. (That is not me--I can't even figure out how to put smileys in these posts) The E46 engine has a fairly flat band as I recall, but that would be the truly scientific way to calculate this. With peakier engines (the ultimate street expression being MC engines--they need to be at 5 grand before they produce any torque), being higher up in the torque band is far more important.

I will say that this discussion should clear up confusion that gearing may not, in and of itself, be a panacea to acceleration woes. The right gearing for your driving is not simple to figure out. Plus, its been fun!


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## rumratt (Feb 22, 2003)

BradS said:


> I bet another fine malted beverage more of them are better with the 3.46.


Deal! (not that I think we will ever resolve this.  )



> If my model is accurate, then here is the full range of shift points in speed, from which we can determine which gear will be provide harder acceleration at any given speed


Ah-ha. We are getting somewhere! I actually agree with your numbers. But what you left out is by HOW MUCH they are winning.

For simplicity, go back to my "equally spaced gears" but assume that one gear is shifted only slightly lower than the other (10% say). At most speeds, the lower gearing will have a 10% advantage, but then at the moment you shift gears, the taller gearing will have substantially more torque to the ground. You have to admit that the sum under the curve in that example DEFINITELY adds up to the same in this example (again, excluding first gear, and assuming equally spaced gears.)



> 1. That extra 18% torque multiplaction has to get to the ground somewhere,


I violently disagree with this. It exactly the misconception I am trying to dispell. Not to beat a dead hourse, but in my 2-5 -> 3-6 example, where does the extra 20-30% extra torque go?



> to your argument that there is no roll-on advantage to a lower gear: by that logic, there should be no acceleration disadvantage in dropping down to a, say, 2.46?


With the caviat that you ignore the lower gear, and you assume equally spaced gears, then yes, I agree. It goes out the window when you get near 1st gear, and you reconsider that the gears are actually not spaced apart equally. But these are the only two reasons it has an effect.


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## BradS (Aug 27, 2003)

rumratt said:


> Ah-ha. We are getting somewhere! I actually agree with your numbers. But what you left out is by HOW MUCH they are winning.


Okay, lets plug the numbers in, in ft/lbs, assuming a flat 220 ft/lb of engine torque throughout the relevant range, over the 3 gear 0-82.4mph spread:

Torque Mult
Gear 2.93 3.46 Advantage
1 2549 3010 461
2 1465 1730 265
3 973 1149 176

Going back to the hypothetical, we can see by how much, over what spread. To show the effects over the entirety of the run, I will just multiply torque advantage by mph, which kind of shows to my non-mathematical mind the sum of the advantage. I have no idea if its mathematically accurate, but its the best my liberal arts brain can come up with at 0700:

Speed Better gear torque + torque * mph
0-31.5 3.46 461 14522
31.6-37.1 2.93 849 4670
37.2-54.7 3.46 265 4638
54.8-64.6 2.93 316 3097
64.7-82.4 3.46 176 3115

Total torque * mph advantage 0-82.4 mph:

2.93: 7767
3.46: 22275

Let's clear up a couple of other points, because I think we are arguing two different things:

1. "I violently disagree with this. It exactly the misconception I am trying to dispell. Not to beat a dead hourse, but in my 2-5 -> 3-6 example, where does the extra 20-30% extra torque go?"

To make this accurate, my statement should have said "in the same gear."

2. As for your 2-5/3-6 example, for the sake of this discussion, I agree, in that hypothetical, it remains the same, since there are no net changes. So, your point there is well-taken, and inarguable.

IMO, what you have done by throwing out 1st gear and creating a hypothetical that is admitedly not real-world is to creat a set of assumptions to fit your conclusion. 1st gear is in play, and the gears are not equally spaced, thus you can't have a rear diff ratio changes that simply bumps you into a higher/lower gear. But, you are correct under your set of conditions--no net change. You win that point.

3. But, let's go back to looking at what would happen on your car if you did change--back to the real world. I am trying to put real numbers on this so you know if you do change, what the net effect is.

Personally, I like the big advantage the lower gear brings, and I have managed to convince myself that a 3.46 is the way to go when routine triple-digits cruising speeds are no longer in my future. Plus, I am also convinced that 6th is useless in the US, and I can afford the steeper rear gear, bringing an otherwise unused 6th gear back in the pattern. You may disagree, and for you, the advantages are not worthwhile or worth the cost.

I also think this discussion concludes that minor gear changes, say 2.93 to 3.07, are going to have a pretty cost-benefit ratio. If you are going to change, bigger may be better.

Cheers!


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## rumratt (Feb 22, 2003)

BradS said:


> Speed Better gear torque + torque * mph
> 0-31.5 3.46 461 14522
> 31.6-37.1 2.93 849 4670
> 37.2-54.7 3.46 265 4638
> ...


Beautiful. But look where most of that advantage is. Subtract out the values from the range 0-31.5, and you get the following:

2.93: 7306
3.46: 7753

This is MUCH closer. And, you just happen to stop at 82.4 mph, whis is when the 3.46 needed to shift. Try going up to 98 MPH (2.93 can stay in 3rd but the 3.46 needs 4th) and these will reverse.

> To make this accurate, my statement should have said "in the same gear."

Then I agree it's correct. It's just not a particularly useful statement, and in fact it's amazingly misleading because there is no clear cut relation from "in the same gear" to "at the same speed" which is what matters. I guess we've beat this one to death though. :eeps:

> IMO, what you have done by throwing out 1st gear and creating a hypothetical 
> that is admitedly not real-world is to creat a set of assumptions to fit your 
> conclusion.

Well, yes and no. The goal was to eliminate as many external factors as possible, come to an agreement, then add them back in. Since the rear diff does nothing in my hypothetical example, then it's fair to conclude that in the real world, any changes you feel from the rear diff must be related to the factors I excluced: ie, 1) 1st gear, and 2) the fact that the ratios are not constant. If we're in agreement with this, then I'm very happy. 



> Personally, I like the big advantage the lower gear brings,


I have no objection to that. If you are starting from a stop (or even low speeds, like 5-10 MPH) then I think it's a no-brainer. And I wasn't trying to overlook the importance of this. I was just very suprised to learn that it doesn't have a very big effect on the rest of the speed range. Very, very, interesting IMO.

By the way, one negative to the lower diff is that the 1-2 shift will come sooner. I still haven't been able to get the 1-2 shift to be consistently smooth (during a fast shift) so the lower diff would mean that my jerky shifts have even more torque!!!  :tsk:


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## BradS (Aug 27, 2003)

rumratt said:


> By the way, one negative to the lower diff is that the 1-2 shift will come sooner. I still haven't been able to get the 1-2 shift to be consistently smooth (during a fast shift) so the lower diff would mean that my jerky shifts have even more torque!!!  :tsk:


Which gets back to thread on CDV's, I think. 

Good discussion. I think we laid out (in excrutiating detail) a model by which people can determine the net effects of a gear swap before dropping the cabbage to do it. I actually kind of like the torque advantage * mph (which I hereby christen TAM) model--I wonder if it has any validity?

So who owes how many beers here? Oh no, more math!


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## rumratt (Feb 22, 2003)

BradS said:


> So who owes how many beers here? Oh no, more math!


I say we each drink 3 and call it even. :drink:


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## RKT BMR (Sep 7, 2002)

rumratt said:


> I say we each drink 3 and call it even. :drink:


Not so fast guys 

I've been following your discussion, and decided to take the plunge and do a complete modeling of the physics and mechanical scenario in Excel, to see if I could help. You can grab the spreadsheet here (you will need Excel to run it).

There are two sheets in the workbook: The first is the analysis, the second is a torque table for the M54 motor. The analysis sheet is locked, with only the intended user-input cells editable. The torque table is completely unlocked. The workbook is not passworded, however, so the whole thing can be unlocked and messed with if you like.

I loaded the gear ratios, curb weight (330ci), and diff ratio from the the BMW site tech specs. Have fun plugging in different values for target speed, shift point (rpm), final ratio, etc. I've verified the results against basic sanity, so I think the whole thing is yielding correct results. The analysis assumed perfect, zero-time shifts, which introduces a bit of error, but this isn't relevant really to that being discussed.

Enjoy.


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## BradS (Aug 27, 2003)

rumratt said:


> I say we each drink 3 and call it even. :drink:


Done!


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