The Engineer's Handbook to Racecar Tyre Inflation Pressure

In the high-stakes motorsport arena, tyre pressure is critical in determining a racecar's performance, directly impacting handling, grip, and overall tyre life on the track. Optimal tyre inflation pressure ensures the best possible contact patch, providing maximum traction and stability, while incorrect pressures can lead to excessive heat buildup, increased rolling resistance, or reduced grip. Additionally, tyre pressure is a dynamic variable influenced by track temperature, driving style, and track conditions, requiring continuous monitoring and adjustment throughout a race. This article explores the intricacies of racecar tyre pressures, examining the underlying science, methods for precise measurement and adjustment, and best practices for different racing conditions. It provides valuable insights for engineers and drivers striving for optimal performance.

How do tyres generate grip?

When a race car navigates a turn, two primary forces come into play: centripetal force and centrifugal force. Understanding these forces is crucial for grasping how a car maintains its path and how drivers and engineers optimize the performance of the car. ("For drag racers, these forces are irrelevant distractions—because turning corners is a challenge they’ll never face." )

Centripetal Force:

Centripetal force keeps a race car on its curved path by acting towards the turn's center, preventing the car from continuing straight due to inertia. In racing, this force is generated by tyre friction with the track, providing the lateral grip, also known as cornering force, essential for maintaining the car's intended trajectory.

Centrifugal Force:

Centrifugal force is an apparent force that feels like it pushes a car outward when it goes around a turn. This sensation is caused by the car's inertia, which is the tendency to continue moving in a straight line. Essentially, as the car turns, the inertia makes it feel as though there's a force pulling it away from the turn's center. In reality, there's no actual force pushing the car outward; it's just the car resisting the change in direction. This perceived force is equal in strength but opposite in direction to the centripetal force, which is the real force pulling the car towards the center of the turn to keep it moving along the curved path.

For perfect cornering, each tyre has to balance lateral and centrifugal force to have optimal lateral force at CG. If one of the axles, let's say the rear one has less lateral force than centrifugal force it can result in an oversteer condition, the same happens in the front axle (understeer condition). Now, the question is how does each tyre generate lateral force?

The simple answer is slip angle. It is defined as the angle between the direction in which a tyre is pointed and the actual direction in which the tyre is moving. When a tyre operates at a slip angle, both the tread and carcass experience distortion near the tyre contact patch. Essentially, the tread aligns with the direction of motion while the carcass adjusts its shape to accommodate this alignment.

The relationship between slip angle and lateral force is given below:

Fy=Cα*α (1)

where, F_y is the lateral force, Cα = Cornering Stiffness of the tyre and α = slip angle of the tyre

Higher cornering stiffness is required to increase lateral force but not the slip angle. As tyres project non-linear characteristics, the lateral force of a tyre doesn't increase in the higher slip angles phase. This can be well understood in Image 1.2 below. At higher slip angles, the adhesiveness of rubber starts degrading, resulting in the sliding of the contact patch's rear part, resulting in lower lateral force.

Influence of tyre pressure on the handling balance:

Tyres function similarly to anti-roll bars and springs in a vehicle by influencing the lateral load transfer distribution during cornering. Like springs, tyres have vertical stiffness, which determines how much they compress under load. This stiffness affects the vehicle's ride comfort and responsiveness to road irregularities. When it comes to lateral load transfer— the shifting of weight from the inner to the outer wheels during a turn—tyres also play a crucial role. By adjusting tyre pressures, which alters the tyre stiffness, engineers can fine-tune the vehicle's balance and handling characteristics, similar to how changes in spring rates or anti-roll bar settings would affect the vehicle dynamics.

Image 1.3 shows how tyre vertical stiffness (Tyre spring rate) is connected to main springs of the car. Both springs are connected in a series which helps us to calculate the overall spring rate of the whole oscillating system:

C_total = ( Cs * Ct ) / ( Cs + Ct ) (2)

where, Cs = spring rate of the construction spring, Ct = Tyre spring rate, Ms = sprung mass and Mus = unsprung mass

However, the spring rate of the tyre is higher than the construction spring rate but it will have a significant impact on the overall stiffness of the system.

From eq (1), it is clearly understood that to improve handling criteria of racecar we have to increase the cornering stiffness of the tyre. But the big question is "How to relate inflation pressure with the cornering stiffness of a tyre". This article describes the relationship between cornering stiffness and tyre vertical stiffness to predict the cold pressure of the tyre required in a race. Given that inflation pressure accounts for more than 80% of a tyre's overall stiffness, we can use this factor to improve the handling of our race car.

Let's do a simple simulation:

This article aims to provide a comprehensive and technical guide on setting the pressure of your race car's tyres while maintaining optimal grip balance. Let's start:

The data that I am going to show is replicated from a real-life scenario stint at Silverstone National Racetrack. The car used for this case is a poorly set Radical SR3 running on Hankook 200/580R15 (front) and 200/610R16 (rear). The starting cold pressure of the front tyres was set to 24 psi front and 23 psi rear. The track temperature and ambient air temperature were 27.8 C and 15.9 C respectively at the start of the stint and evolved up to 30.3 C and 17.7 C at the end of the stint.

From Images 1.4 and 1.5, it can be seen that after lap 7, the front tyre pressure was well over Hankook's recommended hot targets and the rear tyres were below their recommended hot targets. From this workbook, we can actually assess vehicle balance as the front axle tyre pressures are higher than the rear ones even if the car is a rear-wheel drive racecar. The front tyres display higher values as the vehicle is understeering and the front axle has a higher load transfer distribution.

Assessing the vehicle's grip balance:

To understand overall vehicle behaviour, whether the car is understeering or oversteering, we need to compare the yaw rate at COG of the car at each degrees of steering at the front tyres to that of vehicle speed.

Image 1.6 Ψ(dot) represents the yaw rate at the centre of gravity(COG) upon δsteer which rotation (steering) angle of the outer wheel. The understeer graph indicates that as the vehicle's speed rises, the steering angle of the front wheels gradually increases without resulting in additional yaw. Eventually, the steering angle becomes excessive, leading to increased resistance. Generally, cars are slightly biased towards the understeer side to make them stable. This graph gives a good understanding of vehicle balance in both low-speed and high-speed section. One can note that in the 20-25m/s region, it understeers heavily. The next aim is to find a relation between vehicle balance with driving style of the driver so that setup changes would be developed around him.

Jorge Segers in his book proposed a method to measure the understeer/oversteer effect of racecar by finding the difference between the outer wheel steered angle and Ackermann steering angle. If the outer wheel steered angle equals Ackermann steering, the vehicle is neutral steer condition. Similarly more positive values denote understeer conditions and negative values denote oversteer conditions.

Image 1.7 compares lap times with the average understeer angle to comprehend the driver's driving style. From the trend line plot, it is well understood that with higher levels of average understeer angle, the lap times are lowered. This means the driver is more confident with understeer setup. Of course, too much understeer will make him slow but giving him a slightly front-limited setup will make him faster and more confident.

The driving style and vehicle performance are well-understood now. To make the tyres perform the best, we need to make sure, they achieve the manufacturer's hot pressure targets quickly and then get stabilized at that temperature bracket. Tyre pressures increase linearly and then get stabilized at a range of pressure shown in Image 1.4, setting the cold pressures correctly will help us reach the plateau quickly and will allow the tyres to stay in the plateau region which will maximize the tyre's performance.

Predicting cold pressures of the tyres using the ideal gas equation:

Let's assume our tyres are filled up with nitrogen or dry air which behaves like an ideal gas, by using ideal gas equation we can calculate the cold pressure targets of the tyre. For this article, I would show only for front tires to keep the article short. This can be applied to all four tyres of the car. The ideal gas equation gives us a fixed relation between a particular gas's temperature, volume and pressure.

where, p1 = desired cold pressure targets, p2 = optimum performing pressures (manufacturers hot pressure targets), t1 = temperature measured inside the pits/ ambient air temperature and t2 = optimum performing temperature. All units are in bar and Celsius, 1 here is atmosphere pressure in bar

Except for t2 which is the optimum performing temperature and p1 which desired cold pressure target, every other value is known to us or can be measured in the pits. To find t2, we generally look at the data from the tyre temperature Infrared sensors projected towards the tread of the tyre. Plotting GSUM values against front and rear axle tyre temperature will give us an idea of optimum performing temperatures.

GSUM is the combined acceleration of the racecar, higher values of GSUM show higher cornering potential of a racecar. Image 1.8 gives peak GSUM values at 66°C at the front axle and 56°C at the rear axle. ("tyre temperatures are quite low because the damping ratio is way lower than 1.0. I will make a different article about dampers"). As t2 is known from this plot, we can calculate the cold pressure for front tyres using MATLAB code.

clc;
clear all;
P2 = 28; % Example value for P2 in PSI
T1 = 17.6; % Example value for T1 in Celsius
T2 = 66; % Example value for T2 in Celsius


function P1 = calculate_P1(P2, T1, T2)
    % Add 273.15 to the temperatures T1 and T2 to convert them to Kelvin
    % and pressure to bar
    T1_K = T1 + 273.15;
    T2_K = T2 + 273.15;
    P2_B = P2/14.505;

    % Calculate P1 using the given formula
    P1 = ((P2_B + 1) * (T1_K / T2_K)) - 1;
    P1 = P1*14.505
end

P1 = calculate_P1(P2, T1, T2);
disp(['P1 = ', num2str(P1)]);        

From the code, the cold pressure of the front tyre is 21.93 psi ≈ 22 psi. And rear tyre cold pressure is 22.83 ≈ 23 psi

"Are we done? Not yet !!"

From Eq (1), it is clear the higher cornering stiffness increase the lateral force generation capability of a tyre, hence improves handling. As tyre pressure affects vertical stiffness which in turn affects the cornering stiffness of the tyre, in the next section, a relationship will be made between the three different factors to maximize tyre performance.

Cornering Stiffness vs Vertical Stiffness vs Inflation Pressure:

The direct relationship between tyre pressure and cornering stiffness was first given the Magic Formula tyre model introduced by Pacejka in 1997.

where cornering stiffness was calculated using tyre load, nominal tyre pressure, some magic formula and fitting parameters.

"We are not using this equation because lot of parameters are unknown!!" I will give a workaround to relate cornering stiffness with inflation pressure.

First of all, let's find the cornering stiffness of a tyre, for that, we need two things: Lateral forces acting on a tyre and the slip angle of a tyre. Lateral force can be calculated using chassis accelerometers and suspension displacement. There's a whole lot of literature available on this topic. The second challenge is calculating slip angle if you don't have optical sensors to find carcass deflection in tyres.

Calculation of tyre slip angle:

We will calculate the tyre slip angle using the geometric method:

where, αfl = Slip angle of the front left tyre, αfr = Slip angle of the front right tyre, αrr = Slip angle of the rear right tyre, αrl = Slip angle of the rear left tyre δf= Steering angle at the front wheels, vy= Lateral velocity of the vehicle, vx: Longitudinal velocity of the vehicle, γ= Yaw rate of the vehicle, lf= Distance from the centre of gravity to the front axle, d=Track width of the vehicle.

Using the equations mentioned above, respective math channels can be created to determine the slip angle of front tyres. One thing to note, these are calculated values, accuracy may be low but gives a good understanding of vehicle performance.

(-('Steered wheel angle FL' [deg])+atan(('Lat Patch Vel FL' [m/s]+'Yaw angle' [deg]*1.57)/('Long Patch Vel FL' [m/s]-'Yaw angle' [deg]*(1.515/2))))        

The data can be viewed when plotted against the lateral force of each tyre.

Cornering Stiffness:

Now, we can easily calculate the dynamic cornering stiffness of the tyre by dividing lateral force by slip angle. Modifying eq (1) as a math channel:

Fy/α = Cα

Point to note that this cornering stiffness is a dynamic one not actual Tyre data. It does not have a fixed value but when plotted against a tyre's vertical stiffness gives the best operating bracket of the tyre. It is beneficial to make sure that tyres perform in these optimum brackets.

To find tyre vertical stiffness we divide the tyre load (from strain gauges) by tyre vertical deflection.

'Tyre Load FL' [N]/'Vertical Tyre Deflection FL' [mm]        

Image 2.1 gives the best vertical stiffness bracket of the tyre. From these data density regions onwards, any increase or decrease in the tyre's vertical stiffness decreases the tyre's cornering stiffness. The highest cornering stiffness occurs around 254 N/mm for FL, 257 N/mm for FR, 249 N/mm for RL, and 250 N/mm for RR of vertical stiffness.

Vertical Stiffness vs Pressure:

To find the influence of inflation pressure on vertical stiffness, we can do FEA testing using Ansys Workbench. By applying varying vertical loads during braking, cornering and accelerating on quarter-car model with different tyre pressures, we can find a tyre's vertical stiffness.

From the optimal range of the tyre's vertical stiffness, new hot pressures can be predicted shown in image 2.2. Therefore, the new hot pressure targets are around 24.65 PSI on FL, 25 PSI for FR, and 23.5 PSI for both RL and RR. Now again using Method 1("ideal gas equation"), the cold pressures of the tyres can be estimated.

The new cold pressure targets are 19 PSI on FL and FR and 18 PSI on RL and RR. ("I have rounded off the decimal places")

Note: Always look out for the minimum cold pressure set by the tyre manufacturer.

Track evolution:

"Track evolution" describes how a racetrack's state and performance characteristics evolve during a race weekend. The way that cars perform as well as the strategies that drivers and teams use can be greatly impacted by this phenomenon. Track evolution occurs due to several factors like track temperature changes, more rubber on the track, weather conditions etc. In this article, influence of track temperature on tyre pressure would be discussed. I will go slightly technical into the thermal modelling of the tyre but this will give you a idea about why are we concerned so much about track temperature. ("We are engineers, we have to deal with equations and numbers !!")

Danny Nowlan explained how temperature gets build up at the tyre thread surface from a formula used in the thermal modelling of a tyre. ("Although the equation looks terrifying!!")

where,

ρ = Density of the tyre material, cp = Specific heat capacity of the tyre material, dT/dt = Rate of change of temperature of the tyre.

H.F. = Heat generation factor, a coefficient that translates mechanical work into heat, Fy = Lateral force acting on the tyre, α = Slip angle of the tyre, VT: Tyre velocity, Fx = Longitudinal force acting on the tyre, SR = Slip ratio

κ = Heat dissipation coefficient to the ambient air, Tt = Temperature of the tyre, Tamb = Ambient temperature,

κtrack = Heat dissipation coefficient to the track surface, Ttrack = Temperature of the track surface

if we compare the parts of the equation to a race car tyre illustration:

1: Heat generation at contact patch due to lateral and longitudinal forces

2: Heat transfer between tyre and air

3: Heat transfer between tyre and track

Thus, temperature evolution of a track has lot of importance in managing tyre temperature and pressures throughout a stint or a race.

The Ideal Gas Law (PV=nRT) ("I have explained it earlier in predicting cold pressure section") is often used to understand the relationship between pressure (P), volume (V), and temperature (T) of gases. However, in the context of setting tyre pressure in motorsport, relying solely on the Ideal Gas Law can be problematic due to track evolution and other factors like complex thermal conductivity between rubber and track surface. While the Ideal Gas Law provides a basic understanding of the relationship between pressure and temperature, it is insufficient for accurately predicting tyre pressure in the dynamic environment of a race.

To understand track evolution, we need to find a relationship between track temperature and tyre pressure.

Performing a linear regression equation for all four tyres we can represent the best-fit lines for each tyre's pressure as a function of the track temperature. We will use these equations to predict tyre pressures at different track temperatures. Note: Tyre pressure curves are not completely linear it is a slightly logarithmic curves, but if we isolate the linear region, we can use the y=mx+c equation.

The equations are:

• Tyre?Pressure?(FL)=1.6943×Track?Temperature?20.8074
• Tyre?Pressure?(FR)=1.3368×Track?Temperature?11.3200
• Tyre?Pressure?(RL)=1.4062×Track?Temperature?14.9203
• Tyre?Pressure?(RR)=1.2905×Track?Temperature?11.7724
• Creating a tyre pressure prediction chart based on linear regression equations for track evolution, we can predict next session/day pressures (Image 2.5)
• Final cold pressures are decided by comparing the prediction values against Hankook's recommended pressure (Image 1.5 and image 2.6).

The final cold tyre pressures for the next session is:

Evaluation of all the aforementioned methods:

In this section, different methods of predicting cold tyre pressures mentioned above are used to assess the performance of the car. The best way is to look at the stopwatch. Any improvements in lap time will give us an idea of choosing the best setup from the proposed methods.

All proposed setups were utilized to evaluate race car performance. Image 2.7 shows that the setup using tyre pressures determined by examining cornering stiffness against vertical stiffness and inflation pressure (method 2) resulted in the fastest lap times, approximately 2 seconds quicker. The other two methods were each about a second faster than the baseline.

After analyzing the tyre wear rates of various setups, it was discovered that the quickest setup, which used tyre pressures determined by evaluating cornering stiffness and vertical stiffness, exhibited the highest tyre wear rate. The other two setups experienced significantly less wear. The setup that considered track evolution had the lowest wear rate. This happens because the tyre sidewalls flex more at lower pressures. The tyre structure experiences greater heat and mechanical stress as a result of the increased flexing, which could accelerate wear and degradation.

This is very crucial when deciding what tyre pressures to run in a qualifying session or in a race. From images 2.6 and 2.8, the setups that optimise grip and lap times at lower pressures should be given priority in qualifying session. Since tyre wear is less important in a single-lap sprint, the cornering stiffness approach may be chosen despite its higher wear rate. However, during a race, a setup that balances performance with tyre longevity is chosen. The ideal equation approach with the lowest wear rate would be the best choice to maintain competitive lap times while minimising pit stops.

An essential component of racing, tyre energy management affects grip, wear, consistency, safety, strategy, and condition adaptation. Teams spend a lot of money studying and improving tyre performance because it has a direct impact on the speed, handling, and final result of the race. ("This article has covered a lot of ground. I will be discussing tyre energy in a separate article.").

Conclusion:

To balance grip, handling, and tyre longevity while optimising?performance, proper tyre pressure is essential. The theory and techniques of exact tyre pressure management were examined in this article, which also showed how various setups impact lap times and tyre wear. Lower pressures maximise speed and grip during qualifying, but a balance is needed for race conditions to last. Strategic advantage, reliability, and competitive performance are all ensured by efficient tyre pressure control. Further insights into tyre energy will be discussed in a separate article.

I hope you like this article. Follow me for upcoming articles on trackside engineering. Happy Racing!!

References:

• Soltani, A., Goodarzi, A., Shojaeefard, M. H., & Saeedi, K. . Optimizing tire vertical stiffness based on ride, handling, performance, and fuel consumption criteria.
• Nam, Application of Novel Lateral Tire Force Sensors to Vehicle Parameter Estimation of Electric Vehicles
• Nowlan, Deriving a Tyre Model from Nothing. ChassisSim Technologies.
• Segers, J. (2008). Analysis Techniques for Racecar Data Acquisition. SAE International.
• Ralph and Ton, Race Car Handling Optimization_ Magic Numbers to Better Understand a Race Car