Numerical diff modeling

[Jersey Tom]

Here's somethin to think about:

Suppose, hypothetically, you're writing some limit handling code for your 1000cc open-wheeler project (or just as applicably, a FSAE car). Doing trimmed (yaw accel = 0) cornering simulation ain't bad. Think I even have outlined how to do it in my blog.

Braking isn't even bad either.

On-throttle though, requires a diff model. RCVD has some material on it, but not totally in depth and misses a diff type or two. Taylor Race also has a pretty good document, though not all-encompassing. There was a thread on this forum related to diffs, but it was long and not entirely to this point.

There's some set of constraints you'd need to know for each type of diff. Some are easy. Some not. Ex:

1. Spool diff:: RPM_LR_shaft = RPM_RR_shaft
2. Open diff:: Torque_LR_wheel = Torque_RR_wheel (Getting torque from tire model radius data, slip stiffness, slip ratio..)
3. Detroit Locker:: Part of this is easy.. in that one or both of the half shafts are always locked to the diff like a spool. I'm not entirely sure what condition causes the one or other axle to unlock though. Moreover (and this isn't included in the excellent Taylor Race document), when that one axle unlocks is it totally free-rolling? Or does it still have some amount of preload or drag on it?
4. Cam and Pawl:: Locks into a spool at some amount of drive torque. How much? There must be some "deadband" around 0? Is it completely open otherwise? Does it lock under engine braking? (You'd think so) Or is it "directional?"
5. Salisbury:: I have to think about this one, but it shouldn't be too bad to figure out other than ballpark numbers for just how much friction the sets of clutches provide.
6. ATB:: This is where I get hung up. In a straight line, you get 50/50 torque split. Fine. If one tire for some reason started to lose grip and slip, the other (slower moving) tire would get extra torque.

From center of corner through exit is where things get interesting from a modeling perspective.. bearing in mind that apparently the slow-moving axle gets more torque.

Initially, starting at truly neutral throttle (no drive torque or engine braking), the inside tire is at lower RPM assuming everything else with tire radius is equal (which it probably isnt, but hey..). On initial throttle application would the inside rear get the heavy share of drive torque? As the inside rear slips just enough to get the LR and RR at equal shaft speeds, the drive torque split is 50/50. Finally when there's enough torque to really start slipping the inside rear, the outside rear starts getting more torque?

That being the case you'd think that torque bias ratio would be a function of a speed difference between left and rear output shafts. True? If so, how much? How many RPM difference is needed for a 60:40 split? For 70:30? 80:20?

Is that sensitive to the direction of applied drive torque? For example let's say the LR is at 900 rpm and the RR is at 905 rpm. If I give the diff some drive torque, the LR should get more than the RR. If engine braking torque is applied (ie opposite direction) does the LR still get the higher share?

Stuff to consider...

[Jersey Tom]

Here's somethin to think about:

Suppose, hypothetically, you're writing some limit handling code for your 1000cc open-wheeler project (or just as applicably, a FSAE car). Doing trimmed (yaw accel = 0) cornering simulation ain't bad. Think I even have outlined how to do it in my blog.

Braking isn't even bad either.

On-throttle though, requires a diff model. RCVD has some material on it, but not totally in depth and misses a diff type or two. Taylor Race also has a pretty good document, though not all-encompassing. There was a thread on this forum related to diffs, but it was long and not entirely to this point.

There's some set of constraints you'd need to know for each type of diff. Some are easy. Some not. Ex:

1. Spool diff:: RPM_LR_shaft = RPM_RR_shaft
2. Open diff:: Torque_LR_wheel = Torque_RR_wheel (Getting torque from tire model radius data, slip stiffness, slip ratio..)
3. Detroit Locker:: Part of this is easy.. in that one or both of the half shafts are always locked to the diff like a spool. I'm not entirely sure what condition causes the one or other axle to unlock though. Moreover (and this isn't included in the excellent Taylor Race document), when that one axle unlocks is it totally free-rolling? Or does it still have some amount of preload or drag on it?
4. Cam and Pawl:: Locks into a spool at some amount of drive torque. How much? There must be some "deadband" around 0? Is it completely open otherwise? Does it lock under engine braking? (You'd think so) Or is it "directional?"
5. Salisbury:: I have to think about this one, but it shouldn't be too bad to figure out other than ballpark numbers for just how much friction the sets of clutches provide.
6. ATB:: This is where I get hung up. In a straight line, you get 50/50 torque split. Fine. If one tire for some reason started to lose grip and slip, the other (slower moving) tire would get extra torque.

From center of corner through exit is where things get interesting from a modeling perspective.. bearing in mind that apparently the slow-moving axle gets more torque.

Initially, starting at truly neutral throttle (no drive torque or engine braking), the inside tire is at lower RPM assuming everything else with tire radius is equal (which it probably isnt, but hey..). On initial throttle application would the inside rear get the heavy share of drive torque? As the inside rear slips just enough to get the LR and RR at equal shaft speeds, the drive torque split is 50/50. Finally when there's enough torque to really start slipping the inside rear, the outside rear starts getting more torque?

That being the case you'd think that torque bias ratio would be a function of a speed difference between left and rear output shafts. True? If so, how much? How many RPM difference is needed for a 60:40 split? For 70:30? 80:20?

Is that sensitive to the direction of applied drive torque? For example let's say the LR is at 900 rpm and the RR is at 905 rpm. If I give the diff some drive torque, the LR should get more than the RR. If engine braking torque is applied (ie opposite direction) does the LR still get the higher share?

Stuff to consider...

[Marlin]

Isn't RPM a minor factor in ATBs? Torque is what makes them bias.

[Jersey Tom]

<BLOCKQUOTE class="ip-ubbcode-quote"><div class="ip-ubbcode-quote-title">quote:</div><div class="ip-ubbcode-quote-content">Originally posted by Marlin:
Isn't RPM a minor factor in ATBs? Torque is what makes them bias. </div></BLOCKQUOTE>

The ATB is just a dumb piece of metal. It doesn't know how much load is on either tire or how much torque capacity each has. I don't see how torque could "make it bias." It gets some amount of input torque.. and something must make it decide how much is split left to right.

Taylor Race has a pdf that kinda describes its operation: http://www.taylor-race.com/pdf...ng_differentials.pdf (http://www.taylor-race.com/pdf/understanding_differentials.pdf)

<BLOCKQUOTE class="ip-ubbcode-quote"><div class="ip-ubbcode-quote-title">quote:</div><div class="ip-ubbcode-quote-content">"The three, five, or six pairs of floating planet gears provide the differentiating capability in a manner similar to the open differential. It splits the torque between the wheels equally, 50% to each if the torque applied does not exceed the grip level of the “weaker” tire." </div></BLOCKQUOTE>

<BLOCKQUOTE class="ip-ubbcode-quote"><div class="ip-ubbcode-quote-title">quote:</div><div class="ip-ubbcode-quote-content">"When one wheel has better traction than the other, and that wheel begins to slip, the end thrusts of the helical gears become unbalanced, and the steep helix angle causes the planet gears to be forced towards the end of their cylindrical chamber. This causes a slight wedging action, biasing a greater proportion of the torque to the wheel with better grip." </div></BLOCKQUOTE>

To me, that sounds like there needs to be some small amount of slip, some difference in shaft speed, to get the torque to bias.

That's in contrast to the Salisbury which progressively locks the two axles together more and more with applied input torque.. or a viscous LSD which requires the inside rear to spin up a lot to start transferring torque to the loaded tire.

[Drew Price]

<BLOCKQUOTE class="ip-ubbcode-quote"><div class="ip-ubbcode-quote-title">quote:</div><div class="ip-ubbcode-quote-content">Originally posted by Jersey Tom:

To me, that sounds like there needs to be some small amount of slip, some difference in shaft speed, to get the torque to bias. </div></BLOCKQUOTE>


This is exactly how I always visualized it working. Being able to take our Torsen apart and re-assemble and actually see the friction washer between the side gears made it much more obvious as well.

I'm gonna go pick up my Thai food, and mull this over. (Tom, don't you, like..... have things you should be doing?)

Best,
Drew

[Jersey Tom]

<BLOCKQUOTE class="ip-ubbcode-quote"><div class="ip-ubbcode-quote-title">quote:</div><div class="ip-ubbcode-quote-content">Originally posted by Drew Price:
<BLOCKQUOTE class="ip-ubbcode-quote"><div class="ip-ubbcode-quote-title">quote:</div><div class="ip-ubbcode-quote-content">Originally posted by Jersey Tom:

To me, that sounds like there needs to be some small amount of slip, some difference in shaft speed, to get the torque to bias. </div></BLOCKQUOTE>


This is exactly how I always visualized it working. Being able to take our Torsen apart and re-assemble and actually see the friction washer between the side gears made it much more obvious as well.

I'm gonna go pick up my Thai food, and mull this over. (Tom, don't you, like..... have things you should be doing?)

Best,
Drew </div></BLOCKQUOTE>

Dude I multitask. Got a lot done today.

Hopefully Scotty or someone can shed some light on this.

[Jersey Tom]

I'll add that there are apparently a couple potentially interesting SAE papers on this subject...

2007-01-1583 "Integrated Vehicle and Driveline Modeling"

2006-01-0821 "On the Powertrain Dynamics Influence on Vehicle Performance: The Differentials"

May have to see if I can get my hands on these on Monday.

Since this subject has come up several times in the past it would be nice to get some definitive answers and numbers all in once place. Diff definitely has a big impact on handling that I think a lot of folks overlook in FSAE.

Come to think of it, if on an ATB the slower wheel gets the most torque, you'd think that would have some big implications for off-throttle oversteer and for trailbraking with a single rear disc acting on the diff housing. Especially as the inside rear starts to lock up...

[Drew Price]

I've almost got myself un-confused!

There's two mechanisms that I am still working out how they relate to each other.

In the Torsen the element gears' axis are perpendicular to the axis of the side gears, and in the Quaiffe they're co-axial, which I never noticed until now, so I think the mechanism they operate by is slightly different.

In the Torsen some of the torque split is through a friction washer between the side gears. Under applied torque the side gears thrust to one side of the case, one way for accel, one way for decel. There are needled thrust bearings on the ends of the case, and a brass or steel friction washer between the gears, so that under drive torque the wedging pushed the gears together, and partially locks the axles together, and can transfer torque from one axle to the other through the friction washer.


http://www.torsen.com/images/u.special1.jpg



The element gears on the side of the housing are also geared to each other with spur teeth, so with differentiating action the element gears in each pair rotate opposite each other, and drive torque is sent from the slower side gear, to it's element gear, through the spur teeth to the other element gear, and into the other side gear to add torque to the faster turning wheel.

In the Quaiffe it's the overlap between the paired element gears in the center of the diff that accomplishes it, and I don't think the Quaiffe has a friction surface between the side gears? It's all through the gearing.

Best,
Drew

[Jersey Tom]

Interesting. You wouldn't think you could get much "lock" from the wedging action without actual clutch plates.

It sounds like you're saying torque on the Torsen gets put to the faster turning wheel. You'd think that isn't what you want.

In any event, it still comes down to numbers. If I want to drop some different differential models in a FSAE vehicle handling mapper and see the results.. gotta know how much locking, as a function of either differential wheel speed or input torque or both.

Maybe I'll email up Craig and Scotty in a bit.

[Drew Price]

<BLOCKQUOTE class="ip-ubbcode-quote"><div class="ip-ubbcode-quote-title">quote:</div><div class="ip-ubbcode-quote-content">It sounds like you're saying torque on the Torsen gets put to the faster turning wheel. You'd think that isn't what you want. </div></BLOCKQUOTE>


Why do you say that?


Until the breakaway point on the inside, putting the throttle on coming out of a corner is going to make the outside (more heavily laden) wheel the faster turning one. The one that you want to send more drive torque to.

If you reach the point of breaking the inside wheel loose, it's still going to spin it, like with an open diff, it just won'd happen until you reach the tractive capacity split equal to the torque bias ratio.

At least that's how I thought about it.


***************************


The part I thought was interesting was that I always thought the Torsen and Quaife were identical on the inside, I didn't realize till just now the different arrangement of the element gears and the lack of the friction washer between the side gears.

On the FSAE section of the Torsen site it outlines a few ways to tune the TBR a little bit, namely making new friction washers out of different materials with different mu to affect how much locking effect there is between the side gears. The stock one is oil impregnated compacted powder bronze.

The Taylor diff article (http://www.taylor-race.com/pdf...ng_differentials.pdf) (nice one, BTW) goes so far as to say that the Quaife is insensitive to the lubricant you run in it, with is supposedly NOT the case with the Torsen. In fact, one way Torsen suggests to tune TBR a small amount is to run different weight or even type of lubricants.

They also specifically recommend not using grease to lubricate the diff, as opposed to gear oil.

We never had enough power out of the single to know whether it was a problem that we did or not, or what effect it had on lockup, but sure made keeping it from leaking easier.

Best,
Drew

[billywight]

<BLOCKQUOTE class="ip-ubbcode-quote"><div class="ip-ubbcode-quote-title">quote:</div><div class="ip-ubbcode-quote-content">Interesting. You wouldn't think you could get much "lock" from the wedging action without actual clutch plates. </div></BLOCKQUOTE>

That's why qauife and torsen have a max bias ratio of about 4, where the salisbury can be much higher.

It also isn't about the difference in wheel speeds, it's about the torque difference between the wheels. The only diff where wheelspeed is a factor is the viscous coupling.

Quaife and Torsen lock due to both drive torque and a torque imbalance between the drive wheels. Salisbuy locks due to drive torque and not the torque imbalance (for the most part).

I've explained in much more detail in previous posts both here and on the sports racer forums. Have a search, it's all there.

[Jersey Tom]

<BLOCKQUOTE class="ip-ubbcode-quote"><div class="ip-ubbcode-quote-title">quote:</div><div class="ip-ubbcode-quote-content">Originally posted by billywight:
<BLOCKQUOTE class="ip-ubbcode-quote"><div class="ip-ubbcode-quote-title">quote:</div><div class="ip-ubbcode-quote-content">Interesting. You wouldn't think you could get much "lock" from the wedging action without actual clutch plates. </div></BLOCKQUOTE>

That's why qauife and torsen have a max bias ratio of about 4, where the salisbury can be much higher.

It also isn't about the difference in wheel speeds, it's about the torque difference between the wheels. The only diff where wheelspeed is a factor is the viscous coupling.

Quaife and Torsen lock due to both drive torque and a torque imbalance between the drive wheels. Salisbuy locks due to drive torque and not the torque imbalance (for the most part).

I've explained in much more detail in previous posts both here and on the sports racer forums. Have a search, it's all there. </div></BLOCKQUOTE>

Yea but you don't know what that torque difference IS, in advance. That's what I want to solve for.

For example with the ATB.. let's say my car is making a left turn and I'm at absolutely neutral throttle (0 engine torque). LR wheel speed 1010 rpm, RR wheel speed 1040 rpm. At this instant, they're free-rolling.

I add +100 ft-lbf of drive torque from the engine. Not nearly enough to break the tires free. How does that get split to the LR and RR wheel? What if I then go to a different condition where all other things being equal I'm at 1015 rpm LR, 1035 rpm RR?

Taking it a step further... one way or another I can make the case very asymmetric. Let's say LR load is 1000 lbf, RR load is 2000 lbf. Mu = 1 for both.

Case 1: LR slip stiffness is 500 lbf/% slip, RR slip stiffness is 1000 lbf/% slip.

Case 2: LR slip stiffness is 1000 lbf/% slip, RR slip stiffness is 500 lbf/% slip.

I know for damn sure that on a spool, those two cases will behave very differently under the limit. Is this not the case with an ATB?

The diff is just a dumb piece of metal. It doesn't know anything about the load on the tires, peak grip capacity, slip stiffness, or the like. All it knows is "Hey, the engine is giving me X amount of torque' and that it has two drive shafts attached to it.

Now maybe I'm just interpreting this wrong. If the ATB diff is continuously biasing the torque split all the time then there must be some relationship between the different shaft speeds, input torque, and output torque split.

Or, is it that with the ATB the torque split is always 50/50 while the tires are in their sub-limit longitudinal range, regardless of how fast the two different tires are spinning relative to each other? Ie with different slip stiffnesses, one wheel could be spinning much faster than the other and still receiving equal torque. Then only when traction is totally exceeded, there's a maximum about of torque difference that can be carried across the diff?

When it's written as:
<BLOCKQUOTE class="ip-ubbcode-quote"><div class="ip-ubbcode-quote-title">quote:</div><div class="ip-ubbcode-quote-content">When one wheel has better traction than the other, and that wheel begins to slip, the end thrusts of the helical gears become unbalanced, and the steep helix angle causes the planet gears to be forced towards the end of their cylindrical chamber. This causes a slight wedging action, biasing a greater proportion of the torque to the wheel with better grip. </div></BLOCKQUOTE>

It makes it sound, perhaps misleadingly, like that torque split is somehow a function of how much faster one shaft is spinning (and slipping) than the other.

[Chris Allbee]

Tom,

Its a bit early in the morning for me, but I'm going to take a crack at adding to this.

In order to accurately model the ATB effects of the Torsen (or Quaife) then you are unfortunately going to have to put together a gear train model to represent the differential. This doesn't have to run in tandem with your vehicle sim, but can be compiled before hand to generate a look-up table with the information you want. There really isn't an easy way to capture the information you want in a generic equation.

That part aside, the behavior of the Torsen is completely dependent on the torque that is being applied to the unit AND the wheel speed once excessive wheel slip begins to occur. In "steady state, non-limit traction" cases the Torque will be the main factor as input. As has been mentioned the torque difference across the diff is what generates a differing thrust load from the reactions between the spur gears and main gears, with the friction washers and bearings modifying the magnitude of those reactions in relation to each other.

<BLOCKQUOTE class="ip-ubbcode-quote"><div class="ip-ubbcode-quote-title">quote:</div><div class="ip-ubbcode-quote-content"> Or, is it that with the ATB the torque split is always 50/50 while the tires are in their sub-limit longitudinal range, regardless of how fast the two different tires are spinning relative to each other? Ie with different slip stiffnesses, one wheel could be spinning much faster than the other and still receiving equal torque. Then only when traction is totally exceeded, there's a maximum about of torque difference that can be carried across the diff? </div></BLOCKQUOTE>

I guess if the arrangement of washers and bearings were set-up correctly you might be able to approach this situation, but if the diff is symmetrically set-up this wouldn't really be able to happen. If the reaction torques were the same then there wouldn't be any advantage to either side, essentially at equal torques left and right the diff is "locked". Its also very important to realize that when traction is totally exceeded and the reaction torque of one wheel effectively approaches zero when compared to the other the Torsen will behave like an open diff. This was alluded to in an earlier post that described the gear arrangement as trying to counter rotate each other.

Its also worth noting that the maximum torque a diff will ever transmit is when you do have a perfect 50/50 split. Any biasing action at all will result in more mechanical losses, also the amount of torque that can be put down is a function of the wheel with the lowest reaction torque. Example: If the left wheel in a corner can only support 100 lb-ft of torque and your TBR is 3:1, then the diff will only be able to transmit 300 lb-ft of torque to the right wheel. After that any excess torque provided from the motor over 400 lb-ft will unbalance the dynamic friction in the diff and the excess torque will be delivered to the left wheel. Once it breaks loose it will spin up and its reaction torque will drop to say 10 lb-ft. You still have a TBR of 3:1 so you will only get 30 lb-ft to the right wheel.

<BLOCKQUOTE class="ip-ubbcode-quote"><div class="ip-ubbcode-quote-title">quote:</div><div class="ip-ubbcode-quote-content"> It makes it sound, perhaps misleadingly, like that torque split is somehow a function of how much faster one shaft is spinning (and slipping) than the other. </div></BLOCKQUOTE>

It is a little misleading, because the initial slip is caused by the torque imbalance at the wheels (i.e. torque). If there were no reaction torque and you spun one wheel, then the other would spin backwards at the same speed, just like an open diff.

Now because of the gear arrangement and its similarity to an open diff, there is a maximum speed difference that the two shafts can see while still moving in the same direction, but this speed differential is NOT the same as the Torque Bias Ratio.

Bottom line, if you want something useful for your sim then construct a diff model and play with it for a bit, you'll come away with a better understanding than you are likely to get from reading all the papers on the subject.

Good luck!

[Jersey Tom]

Good input, Chris. I appreciate it.

Doing a physical model of a diff.. probably a good idea. Just sounds like a bitch since I've never done geartrain modeling.

I'd still like to have some "simple" models.. which I think I can do for Open, Spool, Cam&Pawl, and Salisbury.

With the Salisbury I'd think I could get something simple and quick if I assume the clutches work on Coulomb friction.

Am I right in thinking.. the Salisbury basically makes a certain amount of "preload" as a function of applied driving or braking torque from the engine? In which case you have to overcome that preload to have a difference in wheel speeds?

If that's the case, I'd suspect you'd get something like this?

http://i936.photobucket.com/albums/ad202/tomszelag/salisbury1.png

And if THAT's the case, then how does that change when you add "real" preload to the thing? Do you get this...

http://i936.photobucket.com/albums/ad202/tomszelag/salisbury2.png

Or..

http://i936.photobucket.com/albums/ad202/tomszelag/salisbury3.png

?

[Gareth]

I have found it useful to define the governing equations that dictate the behaviour of the diff:

Input Torque = Output Torque Left + Output Torque Right
(Ignore losses)

Locking Torque = (Input Torque * Ramp Angle Factor - Preload) * Clutch Pack Factor
(assuming Coulumb friction)

Locking Torque = abs(Torque Output Left - Torque Output Right)

Based on these equations you can build a simple simulation that will predict how the diff will behave. The two factors can be estimated using some simple theory (ramps generate thrust based on their pressure angle, clutches packs are complicated, but you can guess enough to get decent inputs). There's a discontinuity when both wheel speeds are equal, which causes issues with numerical simulations. You can determine locking torque direction based on relative wheel speeds.

The Torsen diff can be modeling in a similar way, as its function is not that different. The Torsen's helical sun gears thrust against some friction surface (washer, bearings, etc) that behave like the clutch packs in a Salisbury. The pressure angle of the gears is like the pressure angle on the Salisbury's ramps. The advantage of the Salisbury is that the clutches and ramps are more easily varied.

Without preload springs, both diffs are just torque multipliers and, as noted by others, cannot transfer unlimited torque to the wheel with grip. If you spin up the inside wheel on corner exit, you'll need to slow down to 'hook up' again before the diff will transfer appreciable torque to the outside wheel. Torque transfer is generated by binding clutches, which is proportional to the input torque. If you can't keep the clutches engaged, then the locking disappears and the transfer drops. When a wheel spins the diff sees a drop in total output torque, which equals in input torque and the clutches lose their thrust force.

Preload springs are useful because they cause a flat spot in Tom's bias curves. This is useful because it means there will be less or no locking at partial throttle. This is good for balance on an aero car where mid-corner the driver may still be using appreciable engine torque. This may still be useful for FSAE cars? Regardless, it allows you to run higher TBRs without the undesired understeer at part-throttle.

I haven't considered dynamic effects in the above discussion - the wheel/axle inertia plays some role in the above, but in general I believe the above is a reasonable start. I hope it helps!

[Flight909]

Sorry, for this old thread.

I trying to model LSD differentials, to understand a setup will work best for our car.

So I try to correlate it to tire data, because I can achieve the same overall "torque" within the limits of the TBR with very different slip ratios (during low lateral weight transfer).

Different SR will of course affect driving force at the wheel, but also the combined lateral performance of the tire (more SR will give less lateral force).

Has anyone made any progress on this topic since Jersey Tom?