Centre Diff Lock dynamics

Posted: 
Sunday, April 2, 2017 - 20:00

I've trying to get my head around how the programming differs between early SLABS and the later CDL aware versions.

I've been toying with the idea of building a test rig to see how differently the two behave. As part of that thought process I've been work out how the CDL changes the way is distributed power to the wheels, and what influence that might have on the ETC programming.

Diff Dynamics

One of the problems (for me at least) is that without an understanding of the physics of diffs it’s almost impossible to make sense of what is going on.

One of the most BS free and straight forward pages covering the diff behaviour is Your differential and μ . The μ is the symbol of the traction coefficient and the article looks at how much force can be applied through each wheel. It's well worth a read, in fact I'd suggest reading the article before continuing on.

After doing a bit internet reading the summary of open diff behaviour seemed to be:
- an open diff distributes input torque equally between outputs (Torque bias ratio is 50:50)
- average output speed is equal to the input speed
- output with lowest tractive force sets the maximum torque applied to all outputs.

Anyway, lets look at this from the perspective of a Discovery with open diffs, unlocked centre diff and no ETC. The plots are created using a basic model of the D2 drive train in Simscape Driveline. The main idea is look at the general behaviour that specific values.

Driving on tarmac there is no problem all wheels have high friction connection through the tyres to the road surface so torque and power splits 25% to each wheel. You can see this behaviour in the plots below where the torque and wheel speed is close to identical for all four wheels. Note that the gear reduction of the transfer case and front and rear diffs multiplies torque and divides rpm by the overall gear ratio.

Wheel speed with unlocked CDL and full traction on all wheels
Torque with unlocked CDL

Torque distribution with unlocked CDL and full traction on all wheels Torque with unlocked CDL

One wheel with minimal traction
Next the tarmac conditions are modified so that one rear wheel behaves as if it is on a surface with less resistance than ice.
I was running into problems with engine rpm going over 6000rpm in less than 2 seconds with a wheel in the sim, so this is an almost no traction alternative.

Notice that the speed of the wheel with minimal traction spikes to over 500rpm, while the wheels with traction are below 25rpm.

Open CDL Wheel Speed with one wheel slipping

The torque output per wheel is significantly lower and all wheels have the same torque output which confirms the observation that the wheel with lowest traction sets the maximum torque output for all wheels.

Open CDLTorque with one wheel slipping

Two wheels on same axle with minimal traction - Open Diffs
This is an interesting case - two wheels on the rear axle have essentially no traction. Both wheels have the same surface parameters so both slip at the same time and spin to the same speed. Torque is low - about the same as with a single wheel with no traction.

For ETC to be effective in this situation both wheels on the same axle need to be braked simultaneously to redirect power to the wheels with traction.

Open CDL Wheel Speed with two wheels slipping

So what happens when the CDL is locked?
The result is slightly surprising - despite the lack of traction the rear wheels are moving at the same speed as the front wheels.
At first glance this seems a bit odd, but consider that the locked CDL means the input to the rear diff must rotate at the same speed as the input of the front diff. Further, the average speed of the left and right outputs is the same as the input speed. If the level of (non)traction is same on both rear wheels they will rotate at the same speed - which means the speed of the front wheels.

Locked CDL Wheel Speed with two wheels low traction surface

The big difference is in how torque is distributed.
Open diffs always split torque 50:50, while a locked diff can distribute torque 0:100.

So in the case of the locked centre diff and open axle diffs if one wheel on one axle has no traction the other axle can use all available torque up to the limits of traction.
And this is exactly what the simulation shows occurring.

Locked CDL Torque with two wheels low traction surface

randomness