It is possible to combine any of the basic six channels (read more) in order to create other channels which usually is not logged. For instance, a math channel that takes all the lateral acceleration data and plot the absolute value of them. The same can be done for the longitudinal G. Then it is possible to build math channels to compute the combined G. These are just examples to describe that math channels are used to expand amount of information regarding the vehicle. It is also a way to work-out some limitation of the data acquisition systems due to regulation restriction or just due to the team budget. The objective of this first article is to introduce the filtering and sampling rate procedure and discuss about the basic math channels.

Sampling rate and filters

When math channels are being created it is common the necessity to define a sampling rate. Most of the data acquisition softwares ask for that. For instance, usually the lateral and the longitudinal accelerations are logged at 50 and 20 Hz, respectively. Hence, a combined G channel can not be logged at 50 Hz if one of the signals was captured at 20 Hz. The rule of thumb is that, when creating a math channel that combines two or more signals, the sampling rate will be lowest frequency between them. The filter should be used considering if this signal will suffer a derivative. For instance, the steering signal and the steering rate. Usually, the filters considered usually are the following:

• Moving average;
• Low pass.

Another important detail regarding the derivative of signals is if this should be performed before or after filtering. For instance, considering the spring deflection channel logged at 50 Hz. The objective is to create a channel which is the derivative of this signal, that is the damper speed. The correct approach is to firstly take the derivative and then filter it. In this way there is no significative loss on the information. Sometimes the filters are toggled of just to perform the derivative. For example, the steering signal, logged at 50 Hz, usually has a 5 Hz filter just for the graphics. If the steering rate information is necessary, the derivative of the steering signal should be done on the raw data. Or rather, it will be a great loss of information.

Combined G (G_comb)

This is the easiest math channel that can be created. It is given by the square root of the sum of the lateral and longitudinal accelerations at power of 2. Figure 1 illustrates these parameters for a lap around the Portland race track. The black and the gray lines are respective to the lateral and the longitudinal accelerations, respectively. The other curve is the one from the combined G channel. It is possible that, when the car is under pure cornering, the combined G equals the lateral G. Conversely, when the car is in pure braking, the combined G is basically the longitudinal acceleration. In Figure 1 it is possible to see the corresponding peaks for turns 1 and 2. At the first, the car is in fully cornering, the same occurs in turn 2, but there is a small valley between them. This means that the driver is not exploiting the grip of the tire fully. Therefore, the combined G exhibits the general grip level that is extracted from all tires combined into one. Hence, it is a sort of lumped mass, a general indication of the grip. The sum of the four grip forces divided by the vehicle mass, is the combined G. In Figure 1, at the turn 6, the combined G is low, is not reaching the peak as in the previous corners. The reason is that the corner 6 is not a grip limited one, instead it is a speed limited corner.

There are another forms to plot the combined G information, one of them is using an x-y graph. This is a good one to evaluate the performance envelope of the car. The lateral, longitudinal and combined G can be plotted together. In addition, it is possible to build ratios between them. Figure 2 illustrates a plot of multiple data related to the combined G.

As can be seen the are the ratio between the modulus of G_y and G_comb and the same for G_x and G_comb. These ratios have a range that goes from 0 to 1. Hence, if one these become 1, this means that the lateral or the longitudinal G is equal to the combined G. Then, the tires are operating in fully cornering or traction/braking, respectively. The fact that the lateral and the longitudinal G are in modulus do not discriminate if the car is turning to the left or the right. It just tells the amount of performance that is being requested. In terms of the ratio between the longitudinal and the combined G, it is possible to notice that this can be 1 on the straigths and on braking. However, in the first case G_comb will be rather low since G_x in this condition is lower is a bit than 1. Hence, starting from two channels, the longitudinal and the lateral accelerations, it is possible to built three math channels, the combined G and those two ratios.

Considering the x-y graph, this reproduce the combined G as a vector, while x and y are the two acceleration components. Figure 5 illustrates another example, which these parameters are used in a similar form wit respect to the previous one.

In this case the three accelerations are plotted together in order to allow the evaluation of the performance envelope. The graph illustrates all three envelopes together. These allow to see the maximum performance of the in longitudinal, cornering and combined maneuvers.

Corner radius

The corner radius, assuming a circular and uniform movement, steady-state and no transients, has this definition for each section of the corner. Hence, the corner is defined as a sequence of steady-state circular movements at different radius. The corner radius is given by the following formulae:

R = V²/G_y

Where V and G_y are the vehicle speed and the lateral G. Since this acceleration can be to the left or right, the sign changes, thus the radius can be positive or negative. Figure £ illustrates the graph with the radius, the modulus of radius and the other related parameters as underster-oversteer, lateral G and speed. In addition Figure £ is also about the same Portland example seen in the previous article (read more). As can be seen, the radius channel alone is useful to recognize the left and the right corners, while the radius modulus is usually adopted when the analyst is familiarized with the circuit. However, the use of supplementary parameters is interesting since the different corners are recognized because the lateral G goes to zero. There are some cases, as the Portland first corner, that it is possible to recognize that this is a continuous radius one. In addition, previous to the second corner there is a small reduction of the lateral acceleration. A good tip to recognize a corner through the data is to keep in mind that if it has a lateral acceleration, an apex and a radius, it is a corner.

Figure 7 illustrates the approach to the chicane A and B, which comprehends the corners 3 and 4. There are three lines in this graph, red, blue and green. These represent the same outing, but with three different filters. The green curve is filtered at 1 Hz, the red one is filtered in 20 Hz and the blue one is the raw data. This is an interesting example of math channels and their filter selection. When it is being programed, it is possible to choose a filter for it. The problem is when the signal is excessively filtered. The spikes seen in the blue channel are lost. This miss could be a small steering correction. When this occurs, the car reacts with the lateral G. This parameters is used to calculate the corner radius, thus a small variation on the steering angle results in small variations on the lateral G, which is translated to instantaneous variations in the corner radius. Actually, those spikes in the corner do not have a physical meaning, but they are an indication of the driver smoothness. Hence, it is important to choose the right sampling rate of the math channel in order to not miss those details. For instance, in the chicane A, it is possible to spot that there is no significative variation between those lines. However, in the chicane B, there is a significative variation of the green line (1 Hz).

Conclusion

This article described the main procedures for filtering and the first basic math channels. The filtering is important considering the influence of the parameter, but also if this will be derived. The combined G is a parameter that indicates the maximum grip that the car is able to reach. In addition, it can be used to verify if the driver is being able to race at the peak of grip. The turn radius indicates from which side the car is cornering, but it a math channel that have to be used together if other channels.

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.