Posted by: drracing | July 20, 2015

The Joy of Yaw Moment Diagrams

Hi everybody!

Today I will finally go back to more “vehicle dynamics” related topics, after the digression I had with my two latest articles.

Before to go into details about the object of today’s article, I am happy to share some good news about my upcoming driving simulation projects.
I have some close contacts with some teams, beside the LMP2 one I am supporting already. That means that, hopefully (and if time allows), I will model some new cars in a driving simulator environment for some professional teams/drivers, to be used mainly for driver training and for some basic setup investigations and, maybe, for some sensitivity study.
One of these teams is racing with Sportcars too, taking part to ELMS, but not in LMP2. The second one is running some cars in an OEM supported spec series here in Europe, using a very new product with very interesting performance, more or less between an GT3 and a DTM.
Two very interesting cars, anyway, that will hopefully give the same pleasure I had working on the LMP2 model.
More to come!

What about today’s topic?
Well, it is anyway partially connected to the above projects and to the LMP2 one, since I started looking into it when developing a tool for a preliminary vehicle modeling/performance validation, before to run the car model in a driving simulation environment. This follows more or less the same line I explained in the “LMP2 car modeling – first Steps – Tire data scaling” post, where I described a simple tool I developed to initially validated the tire models/data coming from the sources with a simple steady state cornering simulation.

What we will discuss today is the construction and the use of a Yaw Moment Diagram, with a procedure which is, somehow, an extension of what I used for the steady state cornering simulation.
Before going into details about it, I want to state clearly here that I am no long time expert about Yaw Moment Diagrams or Milliken Moments method in general. There are people out there who know about them much more and much deeper than me (beside, of course, Mr Milliken) and who used them to setup and developed race car already since long. It would be actually cool if some of these guys would want to comment what I am writing here. So if you know any of them, please share this link!
Yaw Moment Diagrams are anyway such an interesting topic (and, potentially, a useful tool) that I could not stop myself from trying to develop a usable Excel tool to plot them and extract some usable results/metrics, in order to describe car/model behavior and cornering performances.
Beside the chance to use this tool also for vehicle modeling, it was also a very interesting exercise: I learnt a lot both during the building process and finally also using the tool for some basic simulation.

Let’s proceed step by step.
First of all, what is a Yaw Moment Diagram and where do they come from?
First time I personally learnt about them was reading Milliken&Milliken “Race Car Vehicle Dynamics”, which was initially published in 1995, if I remember correctly (although I read it much later, of course!). The Yaw Moment Diagrams and, in general, the Milliken Moment Methods (also called Force-Moment Method) were used anyway already many years before, transferring to cars a typical aeronautical approach to stability and control studies. And, as we will see, stability and control are some of the features of a vehicle that can be investigated with this method and are, otherwise, difficult to describe and quantify.
A Yaw Moment Diagram is basically a plot which has lateral acceleration on the horizontal axis and Yaw Moment on the vertical one. To be precise, actually several versions exist with the Yaw Moment and/or lateral acceleration normalized on car mass and wheelbase, but the basic concept stays and that is what I will discuss here (following pictures have been taken from “Race Car Vehicle Dynamics”).

YMD Milliken
The original approach to build up such a diagram was based on a constrained vehicle test, with a car (or a car scale model) running on a flat belt and locked by two cables, one at the front of it (at a certain and known distance from the CG) and one approximately at the CG. The vehicle would be then put to several Body Slip Angles (Beta, β) while its front wheels would be steered at several Steering Angles (Delta, δ) through sweeps of one and the other, measuring each time (for example through load cells on the cables) the Lateral Force at the CG (Cornering Force) and the Yaw Moment (Force at the front cables times its distance from the CG). The following picture is, again, taken from “Race Car Vehicle Dynamics”:

YMD Milliken 2
Beside the possibility to build such a diagram through constrained testing, the Force Moment approach makes potentially very easy to investigate cars behavior through steady state simulations where, for each Beta value, a Delta one is picked up and Lateral Forces (and, if interested, self-aligning torques) produced by the tires are calculated.
This means, we can immediately estimate the lateral acceleration and the Yaw Moment acting on the vehicle.
As a consequence, since we investigate the vehicle status for several combinations of Beta and Delta, the Diagram itself gives a very clear picture of how the vehicle behaves more or less in its complete operational window (and inside the given boundaries, depending, for example, on speed and/or downforce).
The Yaw Moment is actually the main player here, since it enables us to study features of the vehicle and of its behavior that is normally not possible to study in steady state conditions (where, for steady state, we mean steady state cornering, so constant lateral acceleration and net Yaw Moment equal to zero).
What is extremely interesting about this method is that, with a steady state approach, we can take a look to some typical “transient” treats of car behavior: since we investigate mainly situations where the Yaw Moment is not equal to zero, we can really take a look at what the car can do in transients and how big is the available Yaw moment to make the car turn, stabilize it or control it.

I will not go into the details about how the diagram itself is built, with the meaning of each line (above all the boundary ones), since it is explained pretty deeply and with much more competence in “Race Car Vehicle Dynamics”, chapter 8.
Looking to a Yaw Moment Diagram we can anyway immediately recognize two main groups of lines: constant Delta lines and constant Beta ones. The constant Beta lines are cutting the first and third quadrant and could be more or less straight. The constant Delta ones, are normally curved and, near the origin, they normally point downward for positive lateral accelerations. Of course, everything depends on the sign conventions in use and on the tire model formulation.
The sign convention I used, again, sticks pretty much to what is shown in “Race Car Vehicle Dynamics”. That means, in a right corner, lateral acceleration and cornering forces are positive, slip angles are negative. The self-aligning torques that arises in such a situation (and with the tires not yet at the limit) are anticlockwise oriented and negative. In general, Mz is positive when clockwise oriented (picture, again, taken from RCVD).

Now we should ask ourselves the (probably) most important question: why a Yaw Moment Diagram and the metrics we can derive from it could be so important/helpful to understand how a car behaves and what is happening in certain situations.
It is pretty easy to understand that, the first very basic info that such a diagram can give is the maximum lateral acceleration that the vehicle can achieve in its test conditions (see speed/downforce in place, for a car with significant downforce, among other things like any longitudinal force/weight transfer, for example; longitudinal forces are ignored here, for the time being), both in trimmed (net Yaw moment N equal to 0) and untrimmed condition (N not equal to zero).
How far away is the car from a trimmed condition when achieving its maximum lateral acceleration also tells us immediately what the car is doing at the limit in terms of over/understeering tendency (although this terminology could probably be not really telling much, here); this point can also help to understand why an ideally neutral car would produce the maximum cornering performance, at least in steady state: the maximum lateral acceleration would be then achieved in a trimmed condition (N = 0). Such a situation would anyway lead to other “issues”, as we will see.
Using the sign conventions above, if the maximum lateral accelerations (we talk about positive lateral accelerations, here, so right corner) is achieved with a net negative Yaw Moment in place, the car is stable or tends to push at the limit: the vertex at the far right of the diagram seats below the Ay axis and the distance from it is an indication about how “understeering” the car is (or, in better terms, how stable it is).
In such a situation, what is happening is that the front tires saturate before the rear ones, and cannot produce any (or nearly any) Yaw Moment anymore (or, anyway, the yaw moment they produce is smaller in magnitude than the stabilizing one produced by the rear tires). The net yaw moment is negative, meaning that the portion that the rear tires cornering forces are still able to produce overcomes the one coming from the front tires cornering forces, resulting in the car re-aligning itself: to use a proper language, we should say that the car is stable, more than understeering, since what is actually happening is that the Yaw Torque still available if the Ay further increase over the trimmed condition tends to steer the vehicle back, not to further rotate it.
The opposite is happening, if the maximum lateral acceleration is achieved with a net positive yaw moment in place. In this situation, the rear tires saturate before the front ones and the vehicle is unstable.
This explains why, ideally, a neutral car is able to corner faster in a nearly steady state condition: the right peak of the diagram would sit on the horizontal axis and the four tires would be saturated simultaneously, producing the maximum achievable grip exactly in steady state condition. On the other hand, anyway, such a behavior would leave the driver without any mean to further change vehicle’s path, since both the front and rear tires have nothing more to give.
All of the above is, of course, only an ideal situation, since the Yaw Moment Diagram itself is, in the real world, changing continuously its shape and dimensions depending on speed (downforce), tire conditions and temperatures, longitudinal acceleration, etc. This idealized approach is anyway useful to understand certain phenomena and to describe car behavior in certain situations.

This brings up another very important point about the Yaw Moment Diagram, which is the concept itself of over and understeer. This discussion could become philosophical, I will do my best to avoid that, here.
Anyway, there is, there has always been and there will always be a very complex discussion about how to define understeer and oversteer and, even more complicated, how to measure, using the available (measured) data, how much a vehicle understeers or oversteers when driven on track by a real driver.
The truth is that, most of the theory behind the concepts of understeer and oversteer (covered again deeply in “Race Car Vehicle Dynamics”) has been developed under the hypothesis of the tires being in their linear range (where cornering forces are roughly proportional to slip angles) and using the bicycle model approach.
Under these assumptions (which are seldom verified with a race car cornering at the limit) one of the most known and used way to quantify if a car has understeer or oversteer (and this is advocated by several race car authors/engineers and also shown in RCVD) is to compare if the used steered angle is bigger or smaller than the so called Ackermann Steering Angle or Ideal Steering Angle (Wheelbase divided by corner radius). The truth is that, most of the times, you cannot do much else on the track when looking data, sometimes mainly because of lack of other information (it happens very often to me, to even have no yaw rate channel in my data).
Beside that, you always have to rely on driver feedback, since a good driver will always try to drive around a “problem” and find a way to use the car as good as possible, also when it doesn’t really handle properly; this means that, often, he will somehow “hide” the problem itself, making difficult to objectively evaluate it looking to the data: he will, anyway, inevitably be slower when the car is not giving him/her the right “feeling” or not exploring its full potential. And that’s why, often, good drivers prefers slightly understeering cars, although it could not be the best choice about performance: this behavior simply make them easier to push at the vehicle limit, even if this limit seats a step lower as where it could be.
Anyway, the above described approach to quantify how much understeer your vehicle has is formally incorrect, since a good driver will probably most of the times corner in a condition where the tires are not operating in their linear range anymore. All the linearity assumptions fall off and the approach itself becomes practically wrong (unless it could still be useful, when we don’t have anything better, and I have been happy to use it for several years).
It is clear, anyway, that, at least when developing a simulation tool, we should aim for something “better” (or formally more correct) and try to develop “weapons” that can tell us more about our car’s behavior, not only a comparison between Ackermann and real steering angle.
The Yaw Moment Diagram could well be one of these weapons, above all when backed with proper testing and calibrated with good and trustable driver feedback.
The idea should be to create some reference numbers/situations where we know the driver is happy and performance is good and identify which metrics are important and which values we should target for. This is what Claude Rouelle would call a “magic number”. As his other “magic numbers”, see for example Total Lateral Load Transfer Distribution, it should help us to identify when and why the car is performing as its best, to try to reproduce these conditions when needed.
Of course, the YMD should not be the only tool to be used, here: a race engineer has to look anyway to many other things when setting up a car, beside the pure cornering (see straights, for example!).
Anyway, when correctly developed, it could give a much better picture of how the car performs in corners than using only TLLTD, allowing for example to assess the influence of speed and downforce levels and distributions or a change of tires and to predict more realistically what to expect in terms of balance and stability.
Going back to the understeer/oversteer discussion, as I said, we actually should not speak about understeer and oversteer anymore, when working in the non-linear range of tires, at least not in the way we are used to think about it in their linear range (typical assumption for road cars investigations). This point has been analyzed already in RCVD, where the authors suggest to use the terminology “push” (for a car that tends to go straight in a corner) or, on the other side, “loose” (for a car where the rear axle tends to overcome the front one).
This is very much connected also to the concepts of stability and controllability, which are normally very difficult to define and equally difficult to quantify, at least for a running car.

The latest point that we need to discuss about the metrics we can extract from a YMD is the already mentioned stability and controllability.
Controllability is meant here as the ability of the driver to direct the vehicle as he/she desires or, in other terms, to have the power to control the direction of the vehicle through the steering wheel, creating a certain net Yaw Moment.  Consequently, to study if and in which extent a vehicle is controllable, we will look at how the Yaw Moment changes with respect to the steering inputs.
Stability normally refers to how the vehicle reacts to input and disturbances: a system in general is said to be unstable when, following an input or a disturbance, it doesn’t tend to recover its initial conditions but tends to diverge.
Since, as we saw, a racing car operates mainly as a non-linear system (at least at its limit), investigating stability and control-ability is pretty complex and, as I said, is very complicated to really quantify them objectively, above all when analyzing track data (since the driver has also a very profound influence on how the vehicle behaves, because his/her input directly dictates also the reactions that we can look at).
Being both stability and control-ability so strong related to the forces that the tires produce (which, as we saw, depend directly on Beta and Delta), we can understand why the Force Moment Approach (originally thought on the base of constraint testing) could give so many information about them, allowing to study the action that the vehicle can “offer” to the driver to control its path or to stabilize itself.
What is interesting to notice is that, as we mentioned, when the front tires have saturated and cannot produce any more force, the lateral acceleration could still increase (as a result of the forces produced by the rear tires) but the driver has lost its control on the vehicle path through the steering wheel. The vehicle would anyway be stable and, probably, easier to handle for many drivers.
It is then interesting to investigate how a certain vehicle/setup/tires combination performs in terms of stability and control-ability when the lateral acceleration is starting to increase (for example at the beginning of a corner) or when it has reached its maximum (or its steady state maximum value). In both cases, we will mainly look at how the Yaw Moment change with respect to the two main drivers for tire forces creations, Beta and Delta.
After this long (and hopefully not too boring) introduction, let’s now take a look at how the tool was practically built up.
As for the previous steady state cornering tool, Excel Solver and loop computing are doing here the whole computational work.
The vehicle model is a four wheel one; lateral load transfer is considered, although I am not looking into the effects of roll and heave on suspension geometry and wheel angles: although I am considering the effect of load traasnfer on final tires vertical loads, I am not looking at the effects of suspension movements.
Downforce is also considered, playing a key role in defining each wheel’s final vertical load. Static camber can also be considered (since I am using a Pacejka formulation to extract tire forces), but, for the time being, I am assuming it equal to zero, since I would anyway not consider its variation due to roll or heave. The wheels are supposed to always be perpendicular to the road for this simplified simulation.
I am also assuming a parallel steer setup or, more precisely, I am assuming that both the wheels on each axle are experiencing the same slip angles (so also no toe at the moment), using the bicycle model approach to calculate them based on Beta and Delta at each iteration.
Moreover, in a first iteration, I didn’t consider the tires self-aligning torque but then I inserted them in a second step. We will take a look at their effect on the diagram shape and on the results.
Summarizing, the assumptions we are considering are:

  • Equal left-to-right mass distribution
  • Fixed longitudinal CG location
  • Fixed downforce (ClA) and Downforce distribution values (no ride heights and rake effects simulated)
  • Fixed Lateral Load Transfer Distribution (TLLTD, which could be calculated in the future directly from an integrated excel module, where all the setup info are input and the suspension parameters are given)
  • No suspension kinematics effect
  • Same wheel angles left to right (corner radius calculated at the CG, using the bicycle model approach)
  • Tires produces Fy; cases with and without Mz, no longitudinal forces are considered

The simulation is performed assuming a certain speed and, as a consequence, the downforce acting on the car. For each Delta and Beta combination (which defines necessarily a front and rear slip angle, using the bicycle model), the tool iterates on the lateral acceleration using Excel Solver: lateral acceleration produces weight transfer, “deciding”, together with downforce, how much vertical load acts on each tire; this vertical load is fed into the tire model to output the cornering force and the Mz that each tire produces. The iterative process goes on (changing the Ay value) until the sum of the cornering forces produced by each of the four tires is not equal to the force required to balance the car at the picked Ay. During this process, being the speed fixed, also corner radius changes, coming into play in defining the final tire forces, through the definition of front and rear slip angles with respect to Beta and Delta.
When equilibrium is found, the values are stored (Lateral Acceleration, net Yaw Moment, slip angles, etc) and a new Beta and Delta combination is picked for the next step. For this study, Beta and Delta sweep between 12 and -12 degrees, with 1 degree steps.
The diagram is then built up drawing all the constant Beta and constant Delta lines. Moreover, I calculate some metrics of interest out of the results.
The scenario I´m presenting here is a very high speed corner (240 km/h). As a consequence, downforce and downforce distribution are key players on the final results.
The following pictures show the results for a high downforce, 1000kg heavy sport car cornering at very high speed and assuming a TLLTD equal to 0.5 (no tires Mz considered for now).

YMD-no Mz-TLLTD0.5

The main metrics I am extracting are:

  • Maximum Lateral acceleration
  • Maximum Lateral acceleration in trimmed condition (N ≈ 0)
  • Yaw Moment at Maximum Lateral Acceleration (tells how far we are from a neutral setup and gives indication about stability)
  • Slip Angles, Beta, Delta and Cornering Forces values at Maximum Lateral Acceleration
  • Maximum Yaw Moment
  • Slip Angles, Beta, Delta and Lateral Acceleration at maximum Yaw Moments
  • Variation of the Yaw Moment with respect to Delta when Beta is equal to 0: should give an idea about how controllable the car is in corner entry
  • Variation of the Yaw Moment with respect to Delta when Beta is equal to the Maximum Lateral Acceleration value: should help to understand how controllable the car is at the apex or, anyway, at its maximum cornering usage
  • Variation of the Yaw Moment with respect to Beta when Delta is equal to 0: should depict how stable the car is in corner entry
  • Variation of the Yaw Moment with respect to Beta when Delta is equal to the maximum Lateral Acceleration Value: should point out how stable the car is at the apex or, anyway, at its maximum cornering usage

YMD res-no Mz-TLLTD0.5

Let’s try to understand what their meaning is and, in a second step, to see how they change if we modify something on the car or test assumptions.
The first line (“Max Ay”, measurement unit being “g”) tells us the maximum lateral acceleration the car can achieve at the test speed or, in other terms, the minimum corner radius of the path the car can follow. It gives info about the maximum cornering performance envelope the car is capable of.

The second line (“N @ Max Ay”, measurement unit being Newton-meter) gives the Yaw Moment value at the maximum lateral acceleration. As we discussed, a positive value, as in the picture, means our vehicle is unstable at the limit, with the rear axle tires saturating before the rear ones. In our case, as we will see comparing this simulation/setup to the other I tested, N is very small, pointing to a nearly neutral car at the maximum lateral acceleration. This is further confirmed looking at another metrics, namely the maximum lateral acceleration in trimmed condition (“Max Ay @ N=0”): it is in fact very close to the overall maximum lateral acceleration in our case, confirming how the car is nearly neutral at the limit or, in other terms, how close is the steady state maximum acceleration (similar condition to what we could have in a very long corner more or less at constant speed, for example, although it is a very unusual condition in a race car environment) to the overall car cornering limit.
Since the only moment where we could have a close-to-trimmed-condition situation in a corner is probably the apex, having a higher steady state acceleration should mean having an higher minimum speed in the corner, so, potentially, better lap times. Although, assuming a close-to-steady state condition for corner apex could be sometimes not completely appropriate, above all in certain “highly dynamic” racing environments, like autocross.

Beside the “only-registering” metrics, like the slip angles, Beta and Delta the car experiences at maximum lateral acceleration (you can see more or less the same data also immediately below the “Max Ay @ N=0” metric), another useful information is given by the maximum Yaw Moment achieved during the simulation (“Max N”), which gives an idea about how big is the torque at our disposal to rotate the car, for example in corner entry.
The following line (“Ay @ Max N”) indicates which lateral acceleration the car is experiencing when it is producing the biggest Yaw Moment possible.
The latest metrics we will discuss here are probably the most complex ones, being them mainly related to stability and control.
The first of them shows the variation of the Yaw Moment with respect to Delta, when Beta is equal to zero (“dN/dδ @ β=0”). Since we are looking at a snapshot of how the car could perform in the region where Beta is close to zero, we can assume that the result should be representative of how the car behaves in corner entry in terms of control (what the driver could obtain acting on the steering wheel). We are here moving on a constant Beta line, in particular on the one “β=0”: this graphically explains the positive value of this metric, basically stating that, if Beta is equal to 0 and we start acting on Delta (steering) to steer the car toward right, we produce a positive variation of the Yaw Moment, cause we initially only have a cornering force on the front tires (the rear slip angle, being Beta = 0, is at the beginning also equal to zero). It will be interesting to compare the absolute value of this metric with different setups, to see if and when it is increasing or decreasing, meaning a more or less controllable car.

The second one shows the variation of the Yaw Moment with respect to Delta, when Beta is close to its value at maximum lateral acceleration (“dN/dδ @ β Aymax”). This time, we look at a region close to the maximum lateral acceleration; thus, we could assume that this metric depicts what the car does when at the apex or, anyway, to its maximum usage in terms of lateral forces. Again, we are moving on a constant Beta line but, depending where we are in terms of tire (or axle) saturation, now the variation could even become negative, as in our case: that means that if we would try to further increase Delta’s magnitude by another degree (still keeping Beta constant), we would see the Yaw Moment decreasing. In such a situation, the front tires have lost the power to generate any Yaw Moment and, actually, increasing Delta any further (so, as a consequence, increasing the front slip angle), the tire forces generated at the front axle would decrease.
In our particular case, considering a right corner, lateral acceleration and lateral forces are positive and the Yaw Moment is positive if clockwise oriented. Delta is also positive, so what we are analyzing here is what we get when increasing it by one degree (with its variation being positive). Since the net overall Yaw Moment we get at Delta = 6 degrees is smaller than the one we have if Delta is equal to 5 degree, its variation is negative. That means that, at its cornering limit, the car is loosing the ability to be controlled through the steering wheel.
The third metric of this section is the variation of the Yaw Moment with respect to Beta, when Delta is equal to zero (“dN/dβ @ δ=0”). We are looking here at stability now and moving on a constant Delta line. Being again in the region close to the origin (Beta and Delta both close to zero), we can assume this metric shows how the car behaves in corner entry. More precisely, here we are analyzing the stabilizing effects produced mainly by the rear axle, when trying to rotate the car. Since having a positive Beta here produces a positive rear slip angle (being Delta equal to zero), the rear tires see a negative force and that produces a positive Yaw Moment and Yaw Moment variation (see sign convention picture and assumptions for reference). From a pure sign perspective, a positive Yaw Moment would create an angular acceleration “bringing” the car toward the corner, actually “destabilizing” the car or, in other words, making it turns, not realign: as I said, this is mainly the result of the sign of the rear Slip Angle and, being the behavior symmetrical, we can imagine a similar phenomenon in magnitude also in the other direction.

The fourth metric is the variation of the Yaw Moment with respect to Beta, when Delta is close to its value at maximum lateral acceleration (“dN/dβ @ δ Aymax”). As for the previous one, this parameter gives us a feeling about car stability but in proximity of maximum lateral acceleration (so, again, here we look at the corner’s apex or where the maximum usage of lateral forces takes place). We move here of course on a constant Delta line, so we are only looking at the effects of Beta’s variation on Yaw Moment.
As we may see in the picture above, this metric’s value is, in our case, negative. We have to be careful about signs, here. We look at what does it bring to increase the Beta magnitude by one degree (from 5 degrees, value where we find the maximum lateral acceleration, to 6 degrees in magnitude, in this case). In particular, let’s consider now what it happens in a right turn, with positive lateral acceleration: tire forces are positive and so is the Yaw Moment if clockwise oriented, so if “turning” the car into the corner. On the other side, in a right corner and with the car close to its maximum lateral acceleration, Beta is negative and its value is here equal to -5 degrees. What happens if we increase its magnitude by another degree, bring it to -6 degrees?

The Yaw Moment (which is positive at maximum lateral acceleration, as we saw when looking to the “N @ Max Ay” metric) increases by about 55 Nm, while the Beta variation is negative. Hence, the negative value of this metric. The Yaw Moment’s increase is driven by the rate at which front and rear cornering forces change: they both decrease, switching Beta from -5 to -6 degrees, but the rear cornering forces change is bigger in magnitude than the front one, determining the net Yaw Moment’s increase.
This confirms the car tendency to instability at the limit and gives a “number” to quantify this behavior: this number is what we can use to somehow quantify if and how car behavior changes with a different setup or different assumptions.
If we consider this latest metric result with the one we got for the “dN/dδ @ β Aymax”, we can see a very interesting point. The car (or we should better say, its simplified model) is loosing its control-ability and it is also, in this particular situation (we have to remember we are, for example, ignoring the Self Aligning Torque effect at this stage), unstable. A pretty funny situation, isn’t it?
As we said, this model reaches its maximum trimmed lateral acceleration in a situation where the overall yaw moment is very close to 0; as a consequence, the maximum lateral acceleration in trimmed condition is very close to the overall maximum lateral acceleration. The vehicle is nearly neutral at the limit or, more correct, our vehicle model is, under the simplified assumptions we are considering, pretty neutral at the limit.
With such a vehicle behavior, the driver has not many means at its disposal to really control the car, although it is probably the configuration that would theoretically produce the maximum cornering performance. Anyway, the question we should ask ourselves is: can a real driver use it comfortably, fully exploring the best performance of the “car-driver” package?

It is anyway worth to remember that, what we have just analyzed is the result of the simulation where we ignored the self-aligning torques produced by the tires.

Now that we have taken a look at the meaning of the main metrics I am extracting, we can finally take a look to what happens to each of them and to the Yaw Moment Diagram itself when changing any of simulation parameters.
To follow up what I just said about the Self Aligning Torques, let’s take now a look at how the model’s behavior changes if we take them too into account.
Before to look at the bare results, just a word of warn: what I present here refers to this vehicle and, most importantly, only to the tire model I used for this study (and, here in particular, to how the Pacejka set I used depicts the tire self-aligning torques produced by the tires). Any generalization could lead to an error, since how a vehicle behaves at the limit depends of course very strongly on how its tires perform.
The picture below shows how the diagram itself looks like, if also the SAT is considered.


And here below the results table referring to the same simulation:

YMD res-Mz-TLLTD0.5

The first thing to notice is that, although the Maximum Lateral acceleration stays the same (at least up to the second decimal), the maximum lateral acceleration in trimmed conditions drops slightly in comparison to what we have seen in the previous simulation.
This aspect is strictly connected to another very interesting metric, namely the “N @ Max Ay”, which became now negative, meaning the car is now stable at the limit. That is also confirmed, of course, by the shape of the diagram itself, with the extreme right peak now seating below the Ay axis, although only slightly.
As we already saw in other entries here, the Self Aligning Torque produces a stabilizing effect and, in this particular case, this actually means switching from a potentially unstable vehicle to a stable one.

Beside this, which is probably the most important point here, it is interesting to notice how the maximum net Yaw Moment slightly increases: if you look to the Yaw Moment signs, it is easy to understand why: in a right corner, the greatest Yaw Moment value in magnitude is actually negative and is achieved with a set of negative Beta and Delta. This is not a realistic cornering situation, as far as I know! This metric should anyway give a feeling about how big the overall Yaw Moment can be and it is nonetheless interesting to see how this value increases when adding the self-aligning torque to the game.
While the “dN/dδ @ β=0” (Control-ability in corner entry) decreases a bit (confirming how the Self Aligning Torques produce a stabilizing effect in this phase, here reducing the overall available net Yaw Moment to steer the car into the corner), it’s intriguing to see how the “dN/dδ @ β Aymax” (Control-ability at corner apex) switches from a negative to a positive value. Interesting enough, this change is strictly connected to the sign’s change of the net Yaw Moment at maximum lateral acceleration, which is now negative. That drives, practically, a negative net Yaw Moment both at the maximum lateral acceleration Delta step and at the following one, under the influence of the Self Aligning Torques produced at the corresponding slip angles by this particular tire model.
It is anyway interesting to notice how the net Yaw Moment (now being negative, as we said), is, in magnitude, still bigger at the maximum lateral acceleration Delta value as it is at the next computational step. Here, this phenomenon is driven by the SAT itself, reducing its effect and magnitude as the front slip angles increase.
So, again, the tires have actually already saturated (exactly as it was in the “no SAT” case), as we can also understand looking at the cornering forces produced at the front and rear axle, which stays exactly the same as in the previous case; what we see here is only the tires MZ effect. It is still true that, at the limit, the car is loosing the ability to be controlled through the steering wheel, as in the previous case, but here the self-aligning torque is coming over the Yaw Moments produced by the front and rear tires (which were indeed very small). The car is, as we see, effectively stable at the limit now, but still, increasing the steering angle would not produce bigger yaw moments magnitude.
Nonetheless, the sign of this metric has changed, suggesting how, in this situation, we have to be a bit more careful about our analysis and always keep in mind what the effect of each “player” could be.

Regarding the two following metrics in the list, what we see is that the “dN/dβ @ δ=0” (Stability at corner entry) increases, under the effect of the Self Aligning Torque (as we could already expect), confirming the car is now more stable compared to the previous case of study, while the “dN/dβ @ δ Aymax” (Stability at corner apex) further decreases (or, in other terms, increases its magnitude still being negative): again, here we need to be careful with the signs and with the effects produced by the Self Aligning Torque.
If we still consider a right corner, as we did in the above explanation about this metric, we already saw how the net Yaw Moment produced by the tire forces is positive and increases its magnitude switching from the maximum lateral acceleration Beta’s step (-5 degrees in our case) to the following one (-6 degrees), determining, because of the negative variation of Beta, a negative value for this metric.
In particular, we saw how both front and rear tire forces decrease, when switching Beta from -5 to -6 degrees, but the rear tire forces ones do it at a higher rate, leading the Yaw Moment to increase.

Now, we have to consider inside this picture also the effect of the self-aligning torque: it makes the net Yaw Moment at both Beta’s steps negative, with a bigger magnitude at -5 degrees as what we have at -6.
Hence, we see again the effect of tire saturation, now also on the self-aligning torque. Since the net Yaw Moment is negative at both the two steps, we have a positive variation, which leads again to the negative value of this metric.
An interesting point coming out of this analysis is that it seems to be possible, with and without considering the self-aligning torques, to understand what is going on with tire saturation by looking to the sign of this “variation” metrics.
Nonetheless, I would not jump to conclusion too quickly, when interpreting the diagram. I would not exclude that engineers with more experience about this topic could maybe extract their takings quicker and easier than me.
In any case, the interpretation process itself can teach a lot about car behavior in general and about this particular simulation in particular.

We can now take a look at what happens to the Diagram and the relative metrics as a consequence of a setup change. We can analyze, for example, how the vehicle behavior changes if we increase the TLLTD value from 50% to 60% (that could be, for example, the result of a stiffer front antiroll bar or softer rear one, or a suspension geometry change that moved the Lateral Load transfer distribution toward the front, for example reducing the Roll center height at the rear or increasing it at the front).
The setup change we are looking at here is pretty substantial and it is not something you would want to do so lightly on the track. This should anyway help to better see how and how much such a modification could change our model’s behavior and balance and the Yaw Moment diagram’s shape and its metrics.
From now on, the Self Aligning Torque of each tire will always be considered.

So, without hesitating any further, here below is how the YMD itself looks like after the above described change:

And below you can see the results table referring to the same case:

YMD res-Mz-TLLTD0.6

The first thing we see is a very slight decrease of maximum lateral Acceleration: although slightly, the setup change effectively reduce car’s cornering potential (actually both in trimmed and untrimmed conditions); moreover, the maximum lateral acceleration is now achieved at a bigger Delta value, as we could expect from a setup change that should increase the understeering tendency.
This later point is also confirmed by the diagram’s extreme right vertex position (or, looking at the table, from the “N @Max Ay” metric), which now seats lower compared to the previous case (the “N @Max Ay” has now again a negative value, as for the previous simulation, but a bigger magnitude, confirming how the vehicle has now become more stable).
While the overall maximum Yaw Moment doesn’t change significantly, we see a pretty sensible reduction of the maximum lateral acceleration achievable in trimmed conditions: this is directly connected to how the diagram itself moved toward a more stable behavior, consequently making the extreme right peak to slide downward and making its interception with horizontal axis to move more on the left.

What is happening to the “Variation” metrics?

First thing we see is that the “dN/dδ @ β=0” (Control-ability in corner entry) is further reducing, confirming how the vehicle became more stable (here, this means less available net Yaw Moment to steer the car). The “dN/dδ @ β Aymax” metric (Control-ability at corner’s apex) is, on the other hand, further increasing compared to the case with 50% of Lateral Load Transfer distribution at the front and Self Aligning Torque considered. Again, beside the pure number, it is interesting to notice how, again, the net Yaw Moment shows negative values at both the maximum lateral acceleration Delta and at the following step, with the latter showing a slightly smaller magnitude. Beside this, the Yaw Moment magnitude is anyway significantly greater than in the 50%TLLTD case (-300Nm vs -1600Nm c.a).

Its negative value in a right corner indicates an aligning torque. Interesting enough, here it is clear that the front tires have saturated, while this is not the case for the rear ones: between the two steps, we can clearly see a reduction in front lateral forces, while the rear are still (slightly) increasing. The car is clearly understeering and, to gain control-ability, the only way would be to reduce the cornering speed: any further increase of the steering angle produces only a further reduction of the front tire forces.

The following two metrics also confirm the same tendency. The “dN/dβ @ δ=0” (Stability at corner entry) one, has an higher value than in the previous case, confirming how the tire forces that the rear tires produce in corner entry can be bigger and help to further keep the vehicle stable in this phase.

The “dN/dβ @ δ Aymax” (Stability at corner apex) line, on the other hand, has again a negative value, which increased in magnitude in comparison with the previous simulation. As in the previous case, here the overall net Yaw Moment is negative and its magnitude decreases if we increase Beta by one degree (from its maximum lateral acceleration value to the bigger one). Being N negative in both steps, what happens here means, again, a reduction of the overall aligning moment when increasing Beta. In any case, still being the vehicle stable, this seems to indicate that, increasing Beta, we loose some of the self-aligning effect and, in other terms stability. This makes physically sense, since our Beta’s change practically produces a growth of both front and rear slip angles (lateral acceleration is close to its maximum, here) and, while the front axle’s cornering force stays nearly the same because of front SA change, the rear axle one decreases more significantly, reducing its stabilizing effect.

This closes our analysis about what happens and how the diagram changes for a certain setup change we put in place on our car (in our case, we are pretending this has produced a switch in Total Lateral Load Transfer Distribution from 50% to 60% at the front axle).
As we have seen, if, on one hand, looking to the diagram and reading the metrics gives an immediate understanding of some of the reactions that we could expect when changing any parameter on the car, a more detailed analysis of the simulation results could anyway show important information that should not be taken for granted.
In any case, the Yaw Moment diagram could be implemented as a more completed method to classify and predict what to expect in terms of balance, control-ability and stability when changing a certain setup parameter.
Of course, as for each and every simulation we could perform, the better and more detailed is the data we can access, the more reliable are the results we can expect. What seem to be anyway necessary, at least at the stage I personally am with my knowledge about this topic, is a kind of “validation period” during which the engineer can cross check driver feedback to the results shown by the diagram, at least to identify which is the “right” window where the car should be for the package (driver-vehicle) to work at its best.

One interesting point about this is the possibility to understand and study in more detail car reactions in both low and high speed corners: here below you can see two pictures of the resulting diagrams with the vehicle in the 50% TLLTD configuration cornering at very low and medium-low speed (I purposely didn’t change the diagram scale to let you see how much smaller it gets, compared to the previous ones).

YMD-Mz-TLLTD0.5-Low Speed

YMD res-Mz-TLLTD0.5-Low med Speed

The insight we could gain from this approach would be at its best if, as I said, we could access detailed and reliable information about the car and a well validated tire model.
Some of the data/assumptions that we ignored for this article but could very beneficial to have and implement are:

  • Proper suspension geometry model, above all with regards with roll and ride height variation
  • Complete aeromap, including effects of front and rear ride heights
  • Any non-linear suspension behavior, like the effects of bump stops of any kind

In any case, also if detailed data are not available, I still think “something is better than nothing”. Such a tool could still be useful, even if we don’t have access to detailed info or we have to guesstimate some of the values we need for our models.
Beside the availability of useful results, the building process of such a tool can still teach a lot about vehicle dynamics in general and about your car in particular.



  1. sir, what kind of software you’re using to make the YMD

    • hi Ali,

      this is something i built up in excel using some VBA coding.

      It is no commercial software, but a purpose built tool that i did.

  2. sir, i have generated a matlab code to make the CN-AY code and your post helped me a lot understanding the parameters, i am a senior mechanical engineering student and i am participating in formula SAE this year as my graduation project, so if it doesn’t bother you i want to ask you some questions regarding vehicle dynamics design and simulation.
    kind regards,

    • As i mentioned in the article, i am honestly no real expert about YMD, but it could be interesting to discuss, nonetheless.

      You can post your questions here or leave your email address and i will do my best to answer, as far as i have an answer at all!

  3. Hi,

    Great discussion, thanks for putting this together. Just one thing, I think there may be an error in you discussion of stability at corner apex for the 60% forward lateral load transfer example. You have dN/ddelta @ B Aymax. Should this be dN/dB @delta Aymax?

    Thanks very much, looking forward to reading more.

    • Tim,

      thanks a lot for your comment.
      You are absolutely right! I corrected it now and should be correct.

      Good to know that somebody didn’t fall asleep reading and even find errors!

      Thanks again!

  4. my email is
    thank you sir.

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