How a Diff Affects Roll

[johnc]

From Mark Ortiz's latest chassis newletter:

 

 

THRUST ROLL

 

I have always been curious about power on/off steer during cornering.  I am familiar with the usual suspects like weight transfer affecting slip angles etc.  But I have driven cars where I feel the amount is too great to be explained by this.

 

I have investigated this and I find that certain suspension configurations cause the body to roll when the suspension is unsymmetrical as in cornering.  The thrust loads fed into the body by the suspension attachment points can or cannot cause roll angle change depending on the geometry.

 

If the car has roll steer then of course the change in roll angle will cause steer.  But the really interesting thing occurred to me when I realized that when roll change is not caused by a side force the roll centers are taken out of the equation and only the elastic roll resistance applies.  Thus if a car has no roll steer but has unequal elastic roll resistance front and rear it will still steer during power on/off during roll due to tire loading.  Like perfectly balanced car ala elastic roll plus geometry roll resistance but unbalanced when power on/off applied.

 

I would like to know your views on this.  Maybe this has something to do with the difference in oversteer/understeer difference between tight corners and sweepers as the power requirement is different.

 

This is a real effect.  I have touched on it in some previous newsletters.  We might call it thrust roll: roll due to unequal anti-squat/pro-squat or anti-lift/pro-lift (in the case of powered front wheels) on a driven wheel pair.

 

As the questioner notes, only some cars exhibit this effect.  Therefore, thrust roll doesn’t explain why nearly all cars, unless they’re aero-tight, are looser in sweepers than in tight turns, but it does relate to differences between one car and another.

 

There are analogous effects in braking, as well, and even some due to drag forces on free-rolling wheels when the tires run at a slip angle in cornering.

 

Any difference in jacking forces between the right and left wheels of a front or rear pair creates a roll moment.  This is true whether the jacking forces result from longitudinal ground plane forces or lateral ones.

 

In the case of propulsion forces in rear-drive road or road racing cars with independent rear suspension, the existence of a thrust roll moment under power depends on curvature of the motion path described in side view by the wheel center as the suspension moves.  Usually, this trace is either straight, or concave forward.  It is possible to make it concave rearward, but this is uncommon.

 

If the trace is straight, but inclined identically on both sides of the car, the car has some rear anti-squat or pro-squat, but this doesn’t vary from side to side as the suspension displaces in roll.  If the trace is concave forward, anti-squat increases on the inside wheel and decreases on the outside wheel due to roll.

 

In that condition, equal forward thrust at the rear contact patches will roll the car out of the turn.  It will create a change in diagonal percentage: it will add load to the inside rear tire and the outside front, and take load off of the outside rear and inside front.  In other words, it will wedge the car.  It will make rear tire loading more equal, and make front tire loading less equal, which will add understeer.

 

If the traces are straight but inclined rearward at the top, there is anti-squat, but it stays the same on both sides as the car rolls.  There is then no thrust roll or thrust wedge.

 

If the traces are concave rearward, the effect is the reverse: thrust de-roll and de-wedge, adding oversteer.

 

Note that all of this assumes equal longitudinal ground plane forces at the rear wheels, as with an open differential and zero tire stagger.  That generally is a good approximation for road cars, and some race cars as well.  However, when we consider race cars with spools (locked axles), lockers, or limited-slip differentials, we often have thrust forces that are unequal, and in some cases even in opposite directions.

 

The biggest effect of unequal thrust forces is that they directly produce yaw moments that reduce or increase understeer.  However, in addition they affect thrust roll.

 

With systems other than open diff, the thrust forces not only are unequal, but vary as the car approaches the limit of adhesion.  As this happens, both yaw moments and torque roll are affected.

 

Taking the case of a spool, or locked axle, in a road racing car, when the car is cornering and the tires are well short of the limit of adhesion, the inside rear tire drives and the outside one drags.  The outside tire is not propelling the car at all.  Its x-axis force is rearward.  The inside wheel exerts a larger forward force.  The difference between these forces is the resultant that propels the car.

 

If both wheels have some identical amount of anti-squat geometry, the induced jacking forces will try to compress the outside suspension and extend the inside suspension, rolling the car out of the turn and adding wedge.

 

Now suppose we have a car in this state, and we slowly roll on more power until the rear wheels approach the limit of adhesion.  The percent slip on the inside wheel will increase.  The forward force at the inside wheel will increase.  The rearward force on the outer wheel will decrease, and its percent slip, which is negative, will decrease negative.

 

At some point, the slip on the outside wheel will reach zero, then pass zero and become positive.  At this point the car is driving with both rear wheels.  As the tires approach saturation, the propulsion force from the outer one will generally exceed that from the inner one, since both are now grip-limited and the outside one has more normal force.  The inside tire may also be at a percent slip that is past its peak for longitudinal force, at the lateral force it’s generating.

 

Interestingly, if there were a linear relationship between percent slip and longitudinal force, and if the hub motion paths are also linear, then as this process progressed the overall upward jacking force from the anti-squat would increase, but there would be no change in thrust roll.  However, the relationship between percent slip and longitudinal force is highly non-linear.  It is also different when the tire is making lateral force at the same time than when we are using the tire’s full capability longitudinally.

 

The rate of change of longitudinal force with respect to slip is greatest near zero slip, and diminishes with increasing slip, eventually reaching zero, then becoming negative.  That zero point is the point of tire saturation: the point where the tire’s longitudinal force peaks, and it is at impending wheelspin breakaway.  When we are not cornering, on dry pavement, a typical percent slip at peak longitudinal force might be around 10%.  If we are cornering, this number decreases.

 

Because the relationship between slip and longitudinal force is non-linear, and because there is more normal force at the outside tire, in the case we are examining we get a decrease in thrust roll and thrust wedge as we add power.  This will intensify the transition from power understeer to power oversteer.  If the car has pro-squat geometry instead, and that geometry produces side-view hub motion paths that are linear and slope forward at the top, we get the opposite effect: the jacking force at the outside wheel again increases faster than at the inside wheel, but this time it’s downward, or pro-compression.  This makes thrust roll and thrust wedge increase as we add power, making the transition from power understeer to power oversteer less in magnitude and abruptness.

 

Now let’s consider the same situation as above, but with a locker.  A locker drives whichever wheel rotates slower, and lets the faster one (the outside one, in a road racing car) overrun.  To release and overrun, the faster wheel has to exert enough torque to overcome a spring load in the mechanism.  Consequently, there is a slight drag from the outside wheel, but it’s much smaller than with a spool.  If neither wheel tries to overrun sufficiently to unlock, the device drives both wheels at identical rpm.

 

At moderate power application, the inside rear wheel drives and the outside one coasts, with just a little drag.  If the suspension has symmetrical anti-squat, this produces some thrust roll and wedge.

 

If we roll on more power, the inside rear will produce more thrust, while the outside rear continues to coast.  Thrust roll likewise increases.  Inside rear slip percentage will also increase.

 

As we continue to roll on more power, inside rear slip will increase enough to equalize shaft speeds.  Then the locker will lock, and drive both rear wheels at identical rpm.  At this point there is an abrupt increase in thrust at the outside rear, which adds oversteer and produces the characteristic “locker twitch”.  With symmetrical anti-squat, there is also an abrupt decrease in thrust roll and wedge.  This will tend to intensify the twitch.

 

As with the spool, symmetrical pro-squat will have an opposite effect, and moderate the transition.

 

Gear-type and clutch pack limited slip differentials have behavior somewhere between spools and open diffs, depending on the amount of locking torque.  The amount of locking torque depends on the amount of power applied, the preload in the unit, the size and number of frictional elements, the ramp or gear tooth angles, and the type and condition of the lubricant.  This makes modeling the unit’s behavior, or generalizing about it, rather difficult.  We can say, however, that the closer the unit is to being locked, the more it acts like a spool, and the less the locking torque is, the more it acts like an open diff, with respect to thrust roll as well as otherwise.

 

The questioner notes that the front/rear distribution of elastic roll resistance affects the amount of wedge or diagonal percentage change that we get from thrust roll, and that geometric roll resistance doesn’t matter for this.  That is quite true.

 

However, it is not true that there will be no diagonal percentage change if the front and rear elastic roll resistances are identical.  Rather, the rear suspension has to have all the elastic roll resistance for this to be so.  In other words, either the front elastic roll resistance has to be zero, or the rear suspension has to be completely rigid in roll.  Short of this point, having less front elastic roll resistance, and more rear, will reduce the magnitude of wheel load changes from thrust roll.  This is also true for diagonal percentage changes from driveshaft torque roll in live axle rear suspensions.