Magic Numbers

Magic Numbers

Determining anti-roll bar stiffness using delta values

BY CLAUDE ROUELLE

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In the last few articles we have reviewed the art and science that help us to define our targets of spring stiffness and dampers force vs velocity curves.
The next logical step then is to explain how to determine anti-roll bars stiffness. But to do so we will have to first make a little detour in the world of magic numbers.
The best way to predict the future is to look at the past. Sometimes the best simulation, the best performance prediction, is provided by exploiting previous data collected on track or in lab tests.
When your car satisfies some objective and subjective performance criteria, there are in the analyzed test data and simulation outputs some key performance indicators (KPI), or ‘magic numbers’. Call them the best references if you want. Suppose you get lost during another test or race. In that case, these references will allow you to quickly and accurately get back to the level of performance you had when you had good lap time, lap time consistency, decent tire wear and fuel consumption, and ‘happy’ drivers. It works. It is magic.
These objective KPIs can be correlated with subjective qualification and quantification of car behavior by drivers’ notes written in organized debriefing sheets.
There are several magic numbers, but here are the four most important:

• Weight distribution
• Front and rear tyre cornering stiffness distribution
• Downforce distribution (often called aero balance)
• Anti-roll stiffness distribution

As you can guess, the latter is a reference that will help the race engineer decide front and rear anti-roll bar stiffness for the racecar.

Anti-roll stiffness distribution can be adjusted with springs, but most engineers try to avoid changing springs once they achieve a good ride as that affects the tire's mechanical grip and grip consistency.

And good ride height control, on high downforce, ground-effect cars affects downforce (therefore total grip) and downforce distribution (therefore balance).

If you can understand how these magic numbers, and their interconnection, influence a racecar’s performance, you will already have mastered 50 percent of the racecar engineering challenges.
Here are a few things to note:

1. These numbers are all expressed in percentages of the front value related to the total one. For example, the front weight distribution is 48.5 percent of the total weight.

2. Good racing drivers (and some good passenger car test drivers) will feel a difference of a fraction of a percent of any of these four magic numbers in a change of car set-up.

3. When cars and tires are all the same in a given race (which means the same weight distribution and tire cornering stiffness distribution), such as the Indianapolis 500, expect the other two magic numbers (anti-roll stiffness distribution and aero balance) to be with one percent for the whole field. That shows how tiny the window between the first and second positions could be.

4. During the car design process, good racecar designers will look carefully at both weight distribution and front and rear tire grip (quantified by cornering stiffness) distribution. In a passenger car, the front weight percentage is often bigger than the tire cornering stiff percentage. The majority of today’s passenger cars are front engine and front-wheel drive, with a weight distribution of about 55-60 percent front, if not more, but with the same front and rear tires. These cars will naturally understeer and that is what most nonracing drivers want. For a racecar, however, you want both numbers to be quite close. If during the season the designer changes the weight distribution but keeps the same tires, there will be a serious balance issue. The problems can be solved with different anti-roll stiffness distribution and/or different aero balances but, by introducing another imbalance to compensate for the first one, you can get back to the targeted balance but will pay for it in a loss of total grip.

5. The magic numbers can be accurately determined by simulation if, and only if, we have accurate inputs. Remember the old adage; garbage in, garbage out. If weight distribution can easily be measured with four scales, the knowledge of downforce distribution will depend on accurate aeromaps defined by CFD and wind tunnel tests, and then validated by on-track measurements.
The front vs total tire forces and moments distribution will require accurate and relevant tire models that also have to be validated on track. The anti-roll stiffness distribution might seem to be easily calculated with a spreadsheet and knowledge of springs and anti-roll bars stiffnesses, their motion ratios, the front and rear tire vertical stiffnesses, and the front and rear track width.
However, lab kinematics and compliance (K&C) tests will also take into account the compliance of the chassis, the stiffness of all suspension elements, their attachments on the chassis (what we sometimes call installation stiffness), and chassis torsion stiffness distribution. And that is without even considering the nonlinearity of these stiffnesses and motion ratios or the fact tire stiffness is speed, camber, pressure, temperature, vertical load, slip angle, and slip ratio sensitive.

6. You do not need to be perfect to be efficient. There is no reason to be discouraged if you do not have access to all these accurate inputs and/or ultra-performant simulation software. The wise engineers will tell you there are too many inputs and too many inaccuracies in those inputs to be able to work in precise and absolute value. You progress thanks to relative comparisons, experience, and successive approaches. You work in Delta Values.
Example (simplified, I insist): when your driver was happy at a previous race or test, balance was quantified as neutral in a subjective appreciation on a debriefing sheet by a zero on a scale of -10 (big understeer) to +10 (big oversteer). The anti-roll stiffness you calculated with your simplified spreadsheet in this instance was 47.8 percent. That is a reference, make a note of it. Maybe the ‘crystal ball’ perfect simulation, with perfect inputs will tell you the real antiroll stiffness distribution was 46.9 percent, but you do not know that, and so you decide to ignore that possibility.

So, 47.8 percent of anti-roll stiffness distribution is now one of the four magic numbers. Two weeks later, on another track, your driver complains about oversteering, quantifying the balance this time as three on the same scale. The new track is smoother than the previous one, and your driver likes a responsive car with stiff anti-roll bars, so you increased the total roll stiffness. But, by doing so, you changed the anti-roll stiffness distribution, and your magic number is now 47.1 percent.

Remember, you goal is to get back to your magic number of 47.8 per cent that made your car, your tyres and your driver happy. By playing with your spreadsheet, you will know by how much you need to increase your front anti-roll bar to get back to that same front anti-roll stiffness distribution.
It is not always going to be accurate, and you will not always reach the desired balance at the first set-up adjustment, but it will not take too many tests for you to master the correlations between the driver's subjective balance description (on the -10 to +10 scale) and an objective measurement of the anti-roll stiffness distribution target.

By accepting you do not have full and accurate knowledge of all car characteristics (in the example here, an absence of tyre models and aeromaps), you have decided to work in relative, not absolute value. It works, good enough!

7. On the same circuit, with the same car, and the same tyres, magic number targets can be different from one driver to another, but only by a few tenths of one per cent. Some drivers like cars with a bit of overseer, some with a bit of understeer, but the difference between magic numbers should be quite small.
In fact, experience has shown that in a team of four Formula 3 or Formula cars, if the magic numbers are more than one per cent apart, there is a high chance the cars are not the same. There could be a difference in chassis torsion stiffness, or stiffness distribution due to a crash repair, or because the repair and mounting of underwings on the cars have not been accurately described and policed. There may have been a slight change in the shape of the underwing or diffuser and the aerobalance is not what you think it is.
A different aero balance will require a different anti-roll stiffness distribution.

8. A racecar aero balance is front and rear ride height, yaw, steering, and roll angles sensitive (usually in this order of importance). It changes a few percent during a lap. As a reference, take the aero balance in a straight line at an average speed that corresponds to the most critical corner speeds. It is a number. Remember, you work in Delta.

9. Ideal magic numbers could be slightly different from one circuit to another. Usually, street circuits require a slightly higher anti-roll stiffness distribution than circuits with high-speed corners.

10. Ideal magic numbers could have to be changed significantly, by as much as three percent or more, if the front and rear tires are changed. That usually happens when a team, or a whole racing series, switches from one race tire manufacturer to another. The teams that have access to comparative tire models will cope with such changes in an easier and faster way.

11. Weight distribution is nearly fixed during the car design process, but can change with the amount of fuel onboard. A GT car with a central fuel tank will not get any significant weight distribution difference from a full to an empty tank, while that number can change as much as 1.6 percent on a NASCAR that has a rear fuel tank. All other things being equal, that will give the engineers and the driver a challenge in terms of balance variation from the beginning to the end of a stint. And all other things won’t be, or stay, equal. Consider the unequal loss of grip through front and rear tire wear as an example.

12. Your anti-roll stiffness distribution target could also change with tire wear, especially if you have different front and rear wear and/or if the same front and rear tire wear corresponds to a significant tire grip variation. That is why adjustable front and rear anti-roll bars from the cockpit help a driver cope with balance change during a race.

13. Weight distribution and aero balance are interconnected. Usually, the aero balance is one to three percent smaller than the weight distribution. An aerodynamic centre of pressure behind the c of g helps with aerodynamic stability. On slow circuits, the difference between the weight distribution and aero balance percentages is typically smaller than on fast circuits. In some specific cases, like Formula Student, an aero balance bigger than the weight distribution could be beneficial because, with good drivers on the typical low-speed corners of Formula Student competitions, stability is a lesser problem. But again, that consideration cannot be made without paying attention to the front and rear tire grip distribution and anti-roll distribution.

There are other magic numbers such as heave, roll, and pitch damping stiffness and damping ratio, percentage of time spent by the dampers at low and high speed, to name but a few. We will further develop those in future articles, but for the next one, we will return to our suspension stiffness definition, when talking about anti-roll bars.