Self Aligning Torque

Hi everybody.

This time, i will talk/think a bit about a topic that is often not taken into the appropriate consideration both in vehicle dynamics simulation packages and/or in real car testing and in vehicle dynamics literature: Self Aligning Torque.
I decided to write about this topic after some discussions I had with a friend about the way rFactor represents Self Aligning Torque and how far it is from a normal Pacejka MZ plot (which is, of course only another description of a physical phenomenon, but not reality!).

First of all, why is Self Aligning Torque important in a race tire (and for race car handling)?

As many of you probably already know, Self Aligning Torque is a moment that, in corner (so when slip angle is different than zero) tends to re-align tires, so it is a restoring moment that attempts to return the wheels to a zero slip angle state (straight running). It is the product of the Lateral Force and the Pneumatic Trail, the latter being the distance between the center of the contact patch and the application point of lateral force: these two points, in many situations, are not coincident and that is why a SAT actually exists.
The reasons why Fy application point is not in the same position as Contact Patch center relies in pressure distribution and slip condition on the contact patch itself. But I will not dig into that here.

What I am interested in is, what does SAT do that the driver can feel and that has an influence on car handling and performance?
To understand that, first of all we need to understand how SAT behaves with respect to the Slip Angle for a certain vertical load acting on the Tire (for the sack of simplicity, let’s ignore Camber for now). As we said, SAT can be seen as a product between Cornering Force and Pneumatic Trail. So, to actually understand how SAT evolves when SA increase from 0 degrees to the point where the Cornering Force is at its peak (or even up to higher values than this), we need to investigate how Pneumatic Trail magnitude change with respect to SA.

According to what is normally shown in all the most important Vehicle Dynamics books (including Millken “Race Car Vehicle Dynamics”) and assuming Fy greater than 0 when SA is greater than 0, SAT reaches a maximum for SA values of about half of the peak one (for a certain vertical load acting on the tire, which we assume to be constant for now), then going to be more or less equal to 0 at peak SA, thus deleting its effect on car handling at its very limit (parallel to that, we could say that PT is then at its maximum at 0° SA and then it gradually decreases, being equal to 0 more or less at peak SA. According to Pacejka and Milliken, it could even go negative after that, as also shown in the following picture).

What are then the effects of MZ that the driver can “feel”?
According to what we said above, we can think that, because of MZ effect, the driver should apply a bigger Torque on the steering wheel to negotiate a certain corner at a certain speed than in the ideal case of an MZ absence, at least if MZ is not gone negative up to that point. Moreover, it seems that MZ feedback to the driver could be misleading: since it is falling down rapidly as soon as we approach tire peak SA, then the driver will feel (at least in an ideal constant steer corner) a decreasing torque on the steering wheel as soon as car lateral acceleration (as well as Cornering Force) is approaching its maximum for the given conditions (or, anyway, as soon as front tires slip angle is going closer to the peak value).

And what are the effects of the four tires MZs on car overall handling?
Since SAT is trying to re-align the tire to their original direction, each tire MZ is actually counteracting the yawing torque produced by the front tires to steer the vehicle and it is helping the restoring yaw torque produced by the rear tires. In easy words, it has an understeering effect on the car that, because of MZ plot shape with respect to SA, can be felt more in transients conditions than at the handling limit. This can be easily understood if we think to bring the SAT torque action to the CG: the resulting torque will tend to realign the car to its initial direction.

According to the point above, we can already find a first take from this reasoning: to ignore MZ effects on handling would definitely lead to a wrong picture of what the car actually does, at least in transients situations (see corner entry, for example). How big the mistake would be depends, of course, on how big SAT is for a certain SA and a certain vertical load (or, easier, for a certain tire operating condition).

MZ influence on handling can be clearly felt when taking out the effects of self aligning torque in a driving simulation, like rFactor. The car is much less communicative in corner entry and it is much more difficult to trail brake into the corner, because of a lack of stability.
Anyway, we still see that when tire forces increase and get closer to the maximum value for the given condition, in an ideal cornering case where all the tires are working at their limit at the same time, its effect on handling would more or less totally disappear (or anyway it should become negligible).

That being true, we can then find another take: since MZ effects are much bigger in transients than at cornering limit, then they are very important in driving simulations (because of driver feeling and because of transients behavior) or in vehicle dynamics simulations where we depict somehow what is happening in transient (see a step steer simulation) but they could be not so important in situations where we simulate car behavior at the limit or where, anyway, transients are not really simulated, see in a quasi-steady-state lap time simulator; at least if all the tires are at (or close to) peak SA at the same time.

Anyway, what normally happens to a normal race car negotiating a corner and experiencing a high (limit?) lateral acceleration is that not all the four tires are at the limit at the same time, although this is exactly the condition that every Race or vehicle dynamics engineer dreams about. So we can assume that some of the tires could still produce a non-negligible SAT. Moreover, is the assumption that SAT is more or less equal to 0 at peak SA really what happens on real tires?

Looking to the pictures above and since this conclusion has been shown (basing on real data) by valid reference vehicle dynamics people like Milliken (his book is often referred as “The Bible” by people interested in vehicle dynamics and that is not a casa), I am prone to think that this is probably (and more or less) true in a most cases. So I am not questioning the complete story.

Anyway, I have direct experience of some situations where this is not actually the case. Sometimes, in fact, for some reason I don’t fully know (that I guess are connected to tire structure behavior), some tires are still producing some amount of SAT also when approaching Peak SA and they can be described by SAT vs SA plot that tends to be slightly different from the one above.

Interestingly enough, the above pictures come both from raw data. In both cases we can notice how the align torque is never equal to 0. Ok, we can argue that the first picture is probably not showing a peak Fy condition up to the the highest measured slip angle (looking to the Fy plot it was actually the case, since it seemed Fy could still grow) and that the second one shows a final MZ value (after peak SA) which is, anyway pretty small compared to the maximum one (more or less 1/3 than that). But still, they are not 0 or negative amounts and it both cases curves tend to stabilize to a certain non-0 value at higher and higher slip angles.

In a situation where MZ, even in a limit handling cornering condition, is not negligible, then its understeering effect could still be felt also at high Fy Forces and to ignore it would lead to a mistake when evaluating both car balance and cornering final potential. As we have seen, in fact, MZ is somehow reducing the effectiveness of the front tires in creating a Yawing torque to steer the car into the corner.
Ok, its magnitude is normally smaller than the one of the yawing torque produced by the front tires cornering forces, but in some cases (above all when the rear tires are already producing a significant amount of cornering force as well, it is not negligible and its effect can be felt, above all in driving simulation environment)

Another thing that personally makes me think that to reduce at “negligible” SAT torques effect could be misleading is also the feedback I had from some drivers. It honestly never happened to me to find a driver that could really feel a decrease in the torque to be applied to the steering wheel before the car was actually understeering.
Now, this could be not the definitive proof, since i guess no real track driving situation could be similar to something where SA little by little increases till you finally reach Fy peak on one or more tires.
Moreover, when your car front suspension has a not negligible longitudinal mechanical trail (for example because of rather big Caster), then pneumatic trail reductions effect could (at least slightly) be mitigated.

Finally, a race driver handling his steering wheel, will somehow feel the forces going through the steering rack, which are a direct consequence of the forces passing through the tie-rod, not directly the actions exchanged by ground and tires.  So, actually, what a driver feels is strongly related to suspension geometry and kinematics, not only to what happens at tire contact patch (although of course this is what is driving the complete process).

To have some numeric data about this, I did a static loadcase simulation with a famous Multibody software where I built up a F3 car model.
I applied to the front tires contact patches the Fy and Fz that I could expect in a cornering simulation where the car was travelling at 150km/h and 2 g of lateral acceleration. The numbers I used are shown below:

Mass = 550kg
Front Mass Distr = 0.42
CGH = 300mm
Front Track = 1585mm
Rear Track = 1535 mm
TLLTD = 0.5
ClxA = 2.5
ClxA Front = 0.42

Lat Acc = 2g
Speed = 150 km/h

Doing the math, you find that vertical loads would then be:

Left Tires       Right Tires

653.7 N

2728.9 N

And, using tire Data I came to the following tire Forces (it’s of course not 100% precise, but I just needed some numbers to input into the multibody simulation):

Left Tires       Right Tires

1032.9N

3499.3N

I did the simulation in the following three cases:

1)      SAT = 0

2)      Left Tire SAT = 17Nm; Right Tire SAT= 75Nm (again coming from some tire data I had)

3)      SAT torques = 0.5 of case 2

and always assuming a steering angle slightly above of the Ackermann steering angle (so I assumed a slight understeer).

Results clearly show that SAT should still make a pretty big difference on driver feeling and on its effort to turn the steering wheel (below you find the output steering torques in the three cases):

1)      12.5 Nm

2)      24 Nm

3)      18.3 Nm

Ok, this simple simulation doesn’t take into account some side effects: increasing SAT would probably also require to increase steering angle, to still produce the right amount of Cornering Force (or it would require to reduce speed if we were already at tires limit). Changing steering angle would also change mechanical trail…
But it should still give an idea of the magnitude of the forces playing in our picture.
The conclusion to me is that, in case of SAT absence, the steering torque reduce drastically. So the difference should be felt.

Is this the proof that (at least in certain situations) MZ is not really equal to 0 in limit cornering situations? No, because there is nothing similar to a steady state cornering on a race track and not only speed but also loads, camber, temperatures etc are always changing.
But the results should still show that MZ could not be lightly ignored.
Moreover, this shows that a significant steering torque still need to be applied also in case where Mz=0. So, although sensibly smaller, the driver still have a feedback.

As I said at the beginning, all of this has originated from looking to the way how rFactor represents Self Aligning Torque vs Slip Angle.

While in “normal” vehicle dynamics package you can specify Pacejka coefficients for both Self Align Torque and Cornering Force, in rF you simply input a look up table defining the main shape of Fy vs SA curve (I found some similarities between rF model and Milliken Non-Dimensional Tire model, although I need to explore more their common points yet) and a Pneumatic Trail value, which can potentially also be 0. What rF then does is to define how the Pneumatic Trail evolves with SA, starting from the input (maximum) value at 0 SA and gradually decreasing in a way that depends on Fy curve shape (or, put in another rF way, on tire slip).
This is on one side a very smart and efficient approach, that allow for a very quick and easy calculation of SAT. On the other hand, since the Pneumatic trail will never be equal to 0 or smaller than 0, you will never get a pacejka-like representation of MZ, since it will always stay positive and it will be bigger or smaller depending on your Fy curve shape.

Some say this is a very poor representation of tire Self Aligning Torque, because it doesn’t allow to depict how it somehow deletes (or should delete) its effect at peak Slip Angle. But is it really so wrong? As shown, for some tires it certainly is not. And the representations of the effect on vehicle handling seemed overall pretty realistic to me, till now.

Ok, it is probably true that sometimes you could still have a too high value of MZ at peak SA, also comparing the above pictures to the ones showing raw tire data. But still, it should represent car cornering behavior in a much more realistic way than when no MZ is actually simulated. Moreover, the shape of the above plot (SAT vs SA) could still be influenced by changing the look up table of Fy vs SA, although this could have also other consequences.