In this second part article about the math channels created for race car data analysis it will be described two important channels, understeer and gear ones. From these it is possible to build more math channel in order to expand the magnitude of the analysis. The first one is a sort of stability index, while the gear is usually a complementary channel.

Understeer-oversteer

An important math channel is understeer, this is also an engineering parameter. Understeer is different from understeer gradient. This is the difference between the front and the rear slip angles divided by the lateral acceleration. Hence, understeer is just the difference between the front and the rear slip angles. The main point about the understeering is that this is a stability index. Stability is a characteristic of the equilibrium, this can be unstable or stable one.

US = α_f – α_r

UṠ = (α_f – α_r)/a_lat

Therefore, oversteering is an unstable equilibrium and understeer is a stable equilibrium. Understeering occurs when the front slip angle is higher than the rear one. Actually, understeer describes the quality of the quasi steady-state equilibrium. In the vehicle dynamics there are just two parameters with this description. One is the understeer and the other is the J-turn. Since understeer is a parameter that is amplified, it has units, which is degrees (°). This unit can also be given in radians. The understeer formulae can be expanded by the following formulae.

US = α_f – α_r = δ – L/R

US = α_f – α_r = δ(°) – (L∙180)/(R∙π)

Where L is the wheelbase and R is the turn radius. To come to this conclusion, the term R should be understood.

R = V²/a_y ; V = ψ’∙R ; ay = ψ’²∙R = Ω²∙R

Hence, it is possible to notice how the understeer can be given in degrees or in radians. The sign of the understeer can be positive or negative. This is obtained by multiplying that formulae by the sign of the lateral acceleration. The reason is due to the convention adopted, that the defines for left corners a negative slip angles and a negative radius. Consequently, for the right corners the sign for these parameters is positive. Hence, if it is negative, the understeer has its sign changed, while if it is positive, the understeer has its sign retained.

Meaningful events

The additional parameters used is the wheelbase L and the steering ratio δ. An important detail about that, is the position of the sensor that capture the value of δ. If it is at the steering column, the δ measured regards the steering wheel. Since the steering ratio is channel, it is possible to calculate the average of the meaningful events. This suggests that there are events which do not count. If it is performed the average of all values, the understeer value obtained will be very low, because it is being added many events there are zero. Hence, it is important to perform the average only where there are countable events. For this reason it is performed the event counting previous the average.

Analysis example

Figure 2 illustrates an example of an analysis of the understeer together with other parameters. It is usually included in the graph the radius, the understeer/oversteer, the lateral acceleration, the throttle activation and the speed. It is possible to notice some marks regarding to what understeer signal means. At the beginning it can be observed a mid corner understeer followed an exit corner oversteer. The same situation is also observed at the end of the outing. Both exhibited the same pattern, a mid corner understeer followed by an exit oversteer. However, the oversteer occurs just at the moment which the car is at full throttle. This is called, power oversteer. Around the mid section of the track, it is possible to observe a big mid corner understeer, a huge exit corner oversteer and an entry oversteer. With those quantities, it is possible to compare the understeer trace between setups, drivers and tires.

Understeer is a stability index

There is no value that characterizes too much understeer and oversteer. Actually, understeer is an stability index that depends on the car. The faster the car is, more pronounced will be the oversteer. Hence, a good strategy is to set reference values of understeer respective to speed or to radius. For instance, it is possible to calculate the understeer according to a second order polynomial equation. Then, plotting this one against the value measured in order to have a reference of how far the car is from the target. The understeer channel is more synthetic than the steering angle itself. This is not a measure of the understeer, it is required to subtract the kinematic steering, which is the wheelbase (L) divided by the radius (R).

Understeer times throttle signal

Figure 3 illustrates a which one of the parameters plotted is the steering angle. Hence, it is possible to understand that the steering angle itself is not enough to determine the car behavior regarding the understeer. Another channel that can be create from this one, is the understeer combined with the throttle signal.

This channel allows to identify situation which the understeer is motivated by the engine power. This channel is not a logical value, 0 or 1, it is a proportional one. For instance, when the throttle signal is 50%, the understeer channel is multiplied by 0.5. The reason is that the understeer behavior is different when at 50% and 100% throttle. It is just a simple multiplication of the with a 0.1 logic with the understeer channel. Hence, the focus of this channel is to analyze the understeer angle while throttle is being applied. Otherwise, the two parameters must be compared, but the product of these channels highlights the most severe oversteer. However, it hides the mid corner understeer. These details make the understeer parameter more synthetic. As previously described, it is possible to perform the understeer analysis only when throttle is applied. The same process can be performed for braking. In this case, the channel that is combined whit the understeer one is the longitudinal acceleration. For instance, the understeer channel is multiplied by the longitudinal acceleration and divided by G_y at the peak. The importance of this parameter is to evaluate the corner entrance. Hence, it is a useful tool when the driver is complaining of understeering at the entrance.

Gear signal

The gear channel is derived from the formulae seen at Figure 5, it is equal to the speed divided by RPM. These two parameters are measured by common sensors from any tool kit. Hence, the gear can be easily calculated. Each level corresponds to a gear, this is the ratio between the engine and the wheel revolutions. Therefore, it is possible to calculate which gear the driver is running.

Lateral acceleration times throttle

Hence, similar to the understeer channel, it is possible to combine the lateral accelerations time throttle. The reason is to spot the lateral acceleration at full throttle. Figure 6 illustrates this channel plotted together with gear, G_y, RPM and throttle. Actually, the choice of the channels to be plotted together with gear is completely arbitrarily. The gear channel is meaningful when there is a direct connection between the engine and wheels. For instance, when gear is being shifted. An interesting point is that any parameter can be combined with the throttle signal. Even gear channels are useful when combined with throttle. In this case, the objective is to introduce an emphasis of where to change gear.

Gear channel analysis

Throttle channel can be a 0-1 channel. For instance, below 10% means no throttle, while above 90% means 1. Based on the speed levels it is possible to superimpose the gear channel, that allows to observe the gear at a determined speed (Figure 7). As can be seen, the first corner is ran at second gear. If this corner was ran at the first gear, the vehicle speed would be too high for this gear. However, if the car was running at this corner at the third gear, the speed would no be improved. Hence, the difference on the gear that is being used has no influence on the vehicle speed, only on the engine rev. The gear information is useful to have a wide combination of channels. This actually give a lot of information about the vehicle movements and behavior. This is all about data interpretation, for instance, mid corner understeer and power oversteer are information contained on the data.

Multiple laps

The multiple laps plots are useful to evaluate consistency. Figure 8 illustrates a graph with combined parameters, these are gear, corner radius modulus, throttle, speed and steering for 4 laps. Each color represent a different lap, thus it is an evaluation of the driver consistency. In the gear channel it is possible to notice that these are quite identical. However, in the blue lap, at the first corner, it is possible to spot that the driver waited a bit more to shift-up the second. The same situation can be noticed at the last part of the lap. Apart from these two points, the gear shifts are quite consistent. To complement this information, there is the throttle signal. This is also quite consistent, but exhibits some variations in the same points described previously on the gear channel. Hence, it is possible to notice that in the blue lap at the last section, the reason behind the extended period at the second gear was a hesitant throttle activation. In addition it is possible to see that this results on a lowest speed at that track section when compared to the other laps. The main point in this example is that, those events can occur owing to many reasons. Usually, a senior data engineer can spot and understand the reasons very fast. However, in some cases, those reasons can be just driver decisions. The gray, blue, red and green lap times were 59.62, 60.08, 59.94 and 59.68 seconds, respectively. The variation on these could be a tire wear, a different line or a mistake.

Conclusion

In this article was described two important parameters, understeer and gear. In addition, a final analysis example was discussed. First, these two parameters can improve the amount of data, thus it is easy to get lost. Hence, it is important to understand how to combine these with throttle, because this one is what really gives stability to the car. Second, the multiple lap analysis is not definitive. The reason is, the context (read more) is important. Even though what was discussed is important and real, it is not known if these events occurred due to a mistake, a different setup or car.

References

• This article was based in the lecture notes written by the author during the Applied Vehicle Dynamics lectures attended in Dallara Academy;
• Segers. J. Analisys Tequiniques for Racecar Data Acquisition, 1° Edição. Warrendale, PA. SAE International. 2008.