Differential Effects On Vehicle Dynamics

Prerequisite: Critical Specifications For Balance

A lot of the following logic path is overkill for young teams.
But this may get you thinking about smaller and smaller slices of vehicle dynamics.
Differential or spool selection and tuning is fractal the whole performance envelope.
The process will parallel:

a. Determining the overall traction envelope of the car.
b. Breaking that down into the traction envelope at each wheel.
c. Using that information to find areas for improvement in each Dynamic event at competition.

List of things that are usually more important than differential yaw effects (though the differential plays a role):

• Minimum apex speed.

• Traction on corner exit.

• Stability under braking.

• Speed through slalom (only example where vehicle yaw responses dominate).

• Consistency in all vehicle responses.

Section 1: Definitions

1.1 Only longitudinal torques are discussed below.
Any slip created by differential locking may lead to additional changes in yaw moment due to reduced lateral force at the slipping tire, which is not covered in this article.
Don't bother with second order tire effects in any simulation until it's working well with just point mass.

1.2 Effects of a single brake on the differential carrier vary with brake pressure.
Except for spools, any single rear brake will struggle to pass the brake test without sufficient breakaway torque.
Regeneration can be as powerful and have the same effects as a single brake on the differential carrier.
Engine braking effects function in the same way, and vary with throttle position and gear selection.
Engine braking and the resulting yaw moments are generally much lower than a single rear brake.
All of the negative braking torques above are referred to as "braking on the differential carrier".

1.3 Four wheel braking torques are not considered part of the differential yaw moment.
They must be added to any differential torques described below to get total wheel braking torque.

Note: Deceleration from braking may be quicker than the drop in engine speed in neutral.
Then the brakes decelerate the engine, and the engine inertia puts a positive torque on the differential during braking.

1.5 Spider (Open)
Averaging the axle speeds gives the speed of the differential carrier for any spider or Torsen differential.
Axle torques through a spider are always equal, and limited by the wheel with the lowest traction.
The breakaway force is zero.
Therefore, no yaw moment is generated by an open differential, even if one wheel is slipping.
Note: You will not pass the FSAE IN.12 brake test with a single rear brake on an open differential.
The wheels will rotate in opposite directions around the locked single rear brake.

1.6 No differential (Locked, Spool)
Wheels are locked together at a constant speed.
The torque at each wheel is limited by its traction.
At equal traction, the torque is assumed to be equal.
Tires may slip in opposite directions at the same time.
Torque at the wheel with lower slip is torque applied minus the torque to slip the wheel with higher slip.
A yaw moment is created by the difference and direction of wheel torques.

Section 2: Limited Slip Differential Types

2.1 Spider With Viscous Coupling
Breakaway is determined by the viscous coupling, usually close to zero.
The shear fluid in the coupling will increase locking torque based on differentiation speed.

2.2 Spider With Clutch Pack (Positraction)
Breakaway is determined by the static torque of the clutch pack.
May have a pump to control clamp depending on carrier speed or differentiation speed.
Electronically controlled active differentials usually involve controlling the pressure on the clutch pack or packs.

2.3 Spider With Ramp Clutches (Salisbury, Drexler)
Torque applied to the differential loads internal ramps that clamp clutch packs, controlling both breakaway and differentiating torque.
May or may not include additional preload.
Over a series of maneuvers, the differential may be in any of the following states:

• completely open and no yaw moment

• partially locked with yaw moments proportional to torque applied and clutch pack torque

• completely locked and a maximum yaw moment caused by differences in tire slip

The amount of lock at any time depends on the amount of positive driving torque or negative braking torque on the differential carrier, the ramp angles used, the number of opposing clutch faces, and preload.

2.4 Torsen (Helical, Quaife, Taylor Race, Truetrac)
All types have the same principles, and function.
May have different bias ratios for positive driving torque and negative braking torque.
May or may not have additional preload or clutch elements.
At equal traction, the torque is assumed to be equal.
Below the bias ratio, torque at the wheel with higher traction is torque applied minus the torque to slip the wheel with lower traction.
Differentiation will not occur until the difference in axle torques reaches the bias ratio (plus any additional elements.)
Differentiation occurs at very low and very high loads, as long as the bias ratio is reached.

Section 3: Description of Limited Slip Differential Behavior

3.1 All Limited Slip Differentials are locked below breakaway.
Or below the bias ratio in the case of a Torsen.

3.2 Limited Slip Differentials do not optimize torque distribution.
Limit slip differentials only transfer torque in the following way:
Locking torque is generated during differentiation by internal differential friction.
Outside the carrier, whether under positive driving torque or negative braking torque on the differential carrier:
Half of the locking torque is always negative on the faster axle, and the other half is always positive on the slower axle.

3.3 In practice:
Differentiation under positive driving torque: slower axle torque = faster axle torque + internal friction
Differentiation under negative braking torque on the carrier: faster axle torque = slower axle torque + internal friction
The differential creates a yaw moment proportional to the difference in axle torque and the track width.
For a given setting, the direction of the yaw moment is reversed by changing from positive driving torque to negative braking torque on the differential carrier.
Torsen convention multiplies by a bias ratio instead of adding internal friction.
A Torsen can technically can transfer torque without differentiation.
The result is the same.

3.4 Note on Electronically Controlled Active Differentials:
In a spider differential with a clutch pack on either side, measured at the axles, locking one side or the other just changes the total amount of locking.
The axles and tires can't tell which clutch pack is being adjusted, just more locking or less.

Special Cases:

3.5 Ratcheting (Detroit Locker)
Breakaway is defined by spring preload and tooth angles, then the faster wheel ratchets, the slower wheel stays locked.
Under power, carrier speed is equal to the slower wheel.
Differentiation creates a yaw moment proportional to the torque applied minus the ratcheting torque.
If there is insufficient preload and sufficient clearance, braking on the differential carrier potentially has no effect aside from ratcheting torque.

3.6 Torque vectoring
Best done with individual wheel motors.
Also possible by using individual wheel CVTs, with or without a connecting differential.
Possible with individual wheel clutches with no connecting differential, or additional gear paths around the diff.
See 3.4, Note of Electronically Controlled Active Differentials.
Often, this claim means individual wheel braking under power, separate from the differential, which must be added to any differential torque to get total wheel torque.

Section 4: Team Responsibilities

4.1 Know the maximum positive driving torque provided by the differential.
And the maximum negative braking torque if braking on the differential carrier.
And the relative advantages or disadvantages to other differential options.
Including 4-wheel braking if braking on the differential carrier.

4.2 Know what causes differentiation to start or stop.

4.3 Know the yaw moment direction of your differential (oversteer or understeer) through the stages of a corner:
As the car brakes, turns in, hits the apex, gets back on the throttle, and straightens out.
And ideally, have an estimate of the differential yaw moment's magnitude (relative to the total yaw moment.)
Remember that the yaw moment will increase or decrease along with power or braking on the differential carrier.

4.4 If your differential is adjustable:
Make an estimate of the best setting to start testing.
Have a plan for changes depending on data and feedback.
See Race Car Vehicle Dynamics, Milliken, Sections 12.2, 12.3, and 20.2, 14th printing.

Section 5: Defining Requirements

5.1 You have a huge amount of data from competition.
Even if all of your sensors recorded nothing, you have:

• The layout of each course.

• Individual laptimes for every course.

• Everyone else's laptimes for each course.

• Drivers to mercilessly debrief for every scrap of information.

  • What gear, how much throttle, how much steering, braking points, what line?

  • Is it always brake-turnin-midcorner-power-exit?

  • Are there different situations for different events?

• Brain dump. Make everyone write reports for reference and then talk and then write more reports.

And on top of that, you have anything you did record or measure.
Obviously, you and I are hoping there is longitudinal acceleration, lateral acceleration, steering angle, throttle position, brake pressure, and individual wheelspeeds.

5.2 You'll have to calculate/approximate/assume+test+check any of those that are missing.
Try to figure out what the car is doing speed vs. lateral G, lateral G vs longitudinal G, radius vs lateral G.
Ask questions involving the differential, and consider the characteristics of differential types.

• Is wheelspin under power one or both wheels?

• Is there instability under straightline braking?

• Is it difficult to increase or decrease yaw?

• Can you narrow down the situations where the car was unpredictable?

Section 6: Calculating and Validating Models

See Prerequisite: Critical Specifications For Balance

6.1 You can make a diff-behavior table that functions a lot like a traction envelope.
Your mechanical understanding needs to be accompanied by a mathematical understanding.
You don't need every coefficient, start with: 'If this slows down or speeds up, this force increases or decreases, and this other force increases or decreases, etc."
You are plenty smart enough to start adding numbers and qualifiers on each axis.
Random guessing can get your models started, but you ALWAYS go back and fill these in with actual calculated/measured numbers:

6.2 Straightline is really easy.
Load transfer from CoG height and acceleration/deceleration, plus drag reaction.
Simplified linear traction model vs engine/braking torque available.

6.3 Start simple with cornering.
There is load transfer as soon as there is lateral acceleration.
Split the load transfer front/rear based on longitudinal acceleration.
Start by assuming the same % split left/right at each axle based on lateral acceleration.
From the simplified linear traction model, determine the driving/braking torque available each wheel.
Determine the total traction and lateral capability based on the desired wheel torque and any limited slip effects.
Determine the total potential yaw moment from differential effects. (Probably small compared to overall.)

Once you've done this, your differential effects are ready for a lap-sim with straightline acceleration, straightline braking, and steady state steady radius cornering.

6.4 As always, validate by simulating and testing straightline acceleration, straightline braking, and steady state steady radius cornering.
Remember the stopwatch is a huge data tool.
Rain testing can be done with a hose.

If there is any preload on your current diff, measure the breakaway torque to differentiate.
If you can find a way to measure dynamic differentiation torque, all the better.

Section 7: Minutiae

7.1 By the time you understand differentiation torque in combined longitudinal and lateral acceleration, you have a powerful tool to check for tuning opportunities.
You can see how this will tie in to other team projects.
Just like one finals team told me their clutch sim accidentally turned into/turned out to be an acceleration sim.
The most valuable of your time may be to adapt the tool for other projects.
Reap the benefits, and avoid duplicating work.

7.2 Potentially big effects:

• Second order tire saturation: empirical, Pajecka, whatever the rest of the team has.

• Front and rear roll stiffness sweeps.

• Impulse loading on each downshift.

• Vehicle yaw velocity and acceleration

7.3 Potentially interesting case studies:

• Differential effects through a slalom.

• The relationship between yaw acceleration forces and lateral acceleration forces depending on speed and radius.

7.4 Likely small effects:

• Load transfer due to CoG movement relative to tire contact patches.

• Velocity forces from dampers.

• Oscillation/vibration due to spring/damper/tire/drivetrain interaction.

We've gotten pretty far away from the differential at this point.