The physics of vehicle dynamics

Here Jeremy Spiering gave us an insight view about physics and dynamics of all vehicles and how they are handled. It is an old and extensive post but some way enjoyable and worth to read and contains a good bunch of basic and elemental terms to understand more advanced subjects.

I really enjoy vehicle dynamics and tire model discussions, and I know many others here do as well. I am writing this in effort to explain how a vehicle dynamics simulation works. This is going to be very general, but I think it will help many of us here to have a better understanding of vehicle dynamics and tire modeling and enable us to have more informed and inteligent discussions on such matters.

First off, I am a mechanical engineer who has been racing cars in some respect for 14 years now. I have designed and built suspension/braking/steering systems on formula cars for 5 years. I have also written several vehicle dynamic simulations involing 3D kinematics and steady-state and quasi-steady state vehicle handling. This post will not include all aspects of vehicle dynamics. It will not include anything related to engines, differentials, or transmissions. Just the basics of how a race car is simulated during acceleration, braking, cornering, or any combination of the three.

In many vehicle dynamic related discussions I see people misunderstanding one aspect of vehicle dynamics for another. It is my hope that this will help clarify and differentiate the different aspects. So instead of using the all encompassing “physics” we can break it down more clearly.

Also, please note, that I do NOT know how iRacing does it’s physics and vehicle dynamics computations, this is meant to be general, and is my understanding and experiences of how these things are done.

The three topics I want to discuss are:

Weight Transfer
Tire Model

Weight Transfer

Steady-state Weight Transfer (SSWT)
Steady state weight transfer (SSWT) is one of simpler things to compute with respect to vehicle dynamics and computer simulations. For the most part it requires very little computational power and in its simplest form does not even require iterations or loops in the code to achieve a result. SSWT is mathematical certainty given certain assumptions such as a rigid chassis. And the beauty of SSWT, is that the result is generally quite accurate even if a chassis isn’t completely rigid (and they never are). In its simplest form, to determine SSWT, you only need to know the corner weights of the car, the front and rear track width, the center of the gravity location, and a longitudinal and lateral acceleration of the vehicle that you want to compute. From this, you can determine how much load each tire should be carrying. Very simple calculations.

From here, you add in some more variables, such as springs and anti-roll bars. Now the calculations get a little more complicated, but still relatively simple. The springs and arb’s effect the the distribution of the weight under longitudinal and lateral accelerations. Also now you can determine the chassis roll angle now that we he have springs.

Taking it another step, you can determine the vehicles sprung (suspended) mass and unsprung (non-suspended) mass instead of just using the sum. The sprung weight is the portion of the vehicle weight that is compressing the springs. In generally that is weight of the chassis, the body work, and everything inside the cockpit. The unsprung weight is the tires, springs, and suspensions (although technically some suspension components are partially sprung and unsprung, but lets keep things simple). Now that we have a sprung and unpsprung weight, the SSWT calculations now have to include 2 center of gravities, one for sprung and one for unsprung weight. This gives us even more accurate results for our SSWT calculations.

Next we will incorporate kinematic roll centers into our SSWT calculations. I will not go into the details of what a kinematic roll center is, but it is geometric certainty that is inherent in the design of the suspension (generally the control arms). The kinematic roll center is actually constantly changing based on the dynamic position of the suspension, but for now, we will just pretend it is a constant value. The height of the kinematic roll center affects the distribution of the weight transfer caused by both the sprung and unpsrung mass of the vehicle. It also determines how much the shocks deflect (jacking forces) and therefore the roll angle of the chassis.

By using the above method of determining SSWT, you get 90% of the way there, and all by using very little computational power, and requiring no iterations (although you could use an iterative method, but the results would change by only a miniscule amount)
To really do accurate SSWT calculations, you would need to take into account chassis stiffness, tire spring rates, and you would need to use force based roll centers (as opposed to kinematic roll centers used above). Force based roll centers are based on the forces at the contact patch of the tire as well as the current kinematic positioning of the suspension. You would also want to use an iterative approach to your calculations instead of doing it all in a single pass, but that is more complex subject than I want to get into now. SSWT calculations can also take into account the track surface bank angle, pitch angle, cresting, etc…

SSWT calculations are the basics of vehicle dynamics, and can be as simple or complex as you want to make it. There are diminishing returns with SSWT transfer calculations though, so it is up to the programmer to determine where to stop, when for instance 24 hrs more of programming will only effect the accuracy of the model by a fraction of a percent (believe me, I know).

Transient Weight Transfer (TWT)
Transient Weight Transfer (TWT) calculations as you might imagine are much more complex that SSWT calculations. SSWT calculations are fine for determining maximum lateral or longitudinal acclerations fairly accurately, but they simply will not do for a dynamic and real-time simulation such as iRacing. TWT calculations require a much more in-depth knowledge of the vehicle than SSWT. It requires full knowledge of suspension geometry to calculate real-time 3D kinematics. It requires knowledge of individual weights of many of the vehicle components to accurately determine inertias. It requires knowledge of dampers, inerters, tire damping rates and more. The more variables that you include in your calculations, the more accurate your results will be. But again, there are diminishing returns, so it is important to priortize the components of your TWT calculations.

TWT calculations are similar to SSWT calculations in the respect that they are for the most part a mathematical certainty given good input. There isn’t any guess work involved with TWT calculations. The equations of motions for multibody vehicle dynamics are well defined and have been for many years. Using methods of matrices and ordinary differential equation (ODE) solvers, it isn’t too difficult to accurately determine the transient weight transfer of a race car. This includes everything from lateral/longitudinal accelerations, to bumps, cracks or any other surface irregularities.

The computational power required for TWT calculations is on several orders of magnitude greater than for SSWT calculations. ODE requires iterations for each calculation, and the accuracy of your calculations can be controlled based on the amount of iterations you allow for the solver.

Once you you determine the TWT transfer of a vehicle, you know weight at the contact patch for each tire. You also know the angle of the chassis, the deflection in the springs, twist in the ARB’s, orientation of the suspension components based on the kinematics. You also know the velocity and acceleration of the different suspension components, the tire, etc…Now with this information (ignoring aero for the time being) ,you can start plugging some (or all of it, depending on the complexity of the tire model) into your tire model. But that is for the upcoming discussions.

That is all I have for now, in the next post, I will discuss aerodynamics, mostly from a single car standpoint, and to a lesser extent from a mutli-car (drafting) standpoint. I will discuss the different methods of computing aerodynamics and how they effect weight transfer as well. Finally, I will tie it all together with tire models both from an empirical and theoretical standpoint.


Aerodynamics (as defined by Wikipedia) is a branch of dynamics concerned with studying the motion of air, particularly when it interacts with a moving object. In our case, the moving object of concern is a race car. The two main components of aerodynamics that we will focus on are lift and drag. Our major concern with lift is how it changes the load on the tires contact patch. Our major concern with drag is how it slows the car and requires a force to overcome it.

Let us focus first on lift. In its simplest form, lift can be thought of as a vertical force acting at a center of pressure on the vehicle. This can be and upward force, or a downward force. For vehicle dynamics, we hope this force downward which increases the load the tires contact patch. Even for cars that do not have wings, these forces are present, although to a much lesser extent. Lift is caused by the tires, wings, bodywork, undertray, diffuser, and basically any other component of the vehicle that is in contact with moving air.

Race car teams generally design aerodynamics with the help of computational fluid dynamics (CFD) software. I will spare the details, but the bottom line is that all of the individual lift forces from the different components of the car can be summed and translated to a center of pressure which, similar to a center of gravity, is an imaginary point where if a vertical force is applied, using weight transfer calculations (such as the ones in part one of this discussion) , you can determine the effect it has on the tires load at the contact patch, the deflection in the springs, etc… This is the basis of why aerodynamics is important for vehicle dynamics. They allow an engineer to increase the load at the tires, without increasing the weight of the car. Thus higher lateral and longitudinal accelerations can be reached since lateral and longitudinal forces of the tires increase, without increasing the centripetal forces.

Lift forces are generally the desired result of aerodynamics. However, nothing comes without a price, and this price is called drag. Drag is generally the undesired part of aerodynamics. There is no way to increase lift forces without also increasing drag forces which slow the car down. Engineers can do their best to minimize the effects of drag however, and even use drag in some ways to their advantage, such as creating stabilizing effects. The more drag a vehicle produces, the more power required from the engine to achieve the same speeds.

Single Car Aerodynamics
Now how do racing games/sims simulate aerodynamics? That is a much more difficult question to answer then the weight transfer discussion above. As I mentioned in the previous post, weight transfer is mostly a mathematical certainty given the proper inputs. Not so much with aerodynamics. CFD calculations, similar to weight transfer calculations, can be as simple or complex as you want them to be. In the simplest form, you take a wings surface area, a coefficient of lift, air pressure, an air velocity and you can calculate the lift forces it will generate. You can do this for a front wing, and a rear wing, and then calculate a center of pressure, and apply the sum of the forces there and get a rudimentary value for the aerodynamic load as a result. You can also calculate forces from the undertray, diffuser, etc… Then recalculate the center of pressure and apply the forces there, and your rudimentary analysis, becomes more accurate, but still very rudimentary. The problem is, the components of the car do not separately generate lift forces, but the car as a whole does, and each component interacts with the others.

To properly determine aerodynamic forces (without a wind tunnel) you need to use a CFD package. CFD calculations can be very quick and run real time, or they can be extremely complex and take hours to calculate even an instant of time. It depends on what you need the data for and how accurate you want it to be.

Again, back to the question of how do racing games/sim simulate aerodynamics, and to be honest, I can’t really answer that question, and I suspect different simulations and possibly even different cars within a simulation use different methods of calculating aerodynamic forces. I would venture to say that some games/sims simply use a coefficient of lift and drag that varies with speed and apply those forces to a center of pressure of the car. Some games might use best guess methods coupled with partial data and even vary the coefficient of lift and drag with speed, or move the center of pressure with speed, or vehicle attitude with respect to motion. Some games might use a more complex (but still relatively simple) form of CFD that uses the actual vehicle geometry of the wings, bodywork, etc…However, as I mentioned before, CFD is very complex, and I am by no means an expert to be able to tell you which of those methods would give more accurate results.

Another method that can be used for aerodynamics is called an aero-map. An aero-map is similar to an empirical tire model (which we will get into later) in that you input certain values and get a result back from a black box basically. That black box comes from either extensive CFD calculations, or hopefully, from actual wind tunnel testing. Basically an entire vehicle (full size or scale model) is simulated (CFD or wind tunnel) at a range of speeds, slip angles, roll angles, pitch angles, ride heights, and other factors that affect aerodynamics. From this, a giant lookup table(aero-map) is made which contains the desired lift forces, center of pressure, etc… Now in a racing simulation, with each step of the physics, the orientation and velocity of the car is sent to the aero-map. The aero-map then interpolates the output based on the closest inputs.

The beauty of the aero-map, is that is requires virtually no computational power, and is pretty darn accurate, as long as the input is good. The difficult thing, is how do you obtain an aero-map? First off, not many cars ever have an aero-map created for them. NASCAR, F1, LeMan Prototypes and other top high level racing cars are generally the extent. The amount of time and money involved in renting a wind-tunnel to create an aero-map for a company that makes racing simulations is pretty much out of the question. (I could be wrong, but I am pretty darn sure about this). Creating a CFD aero-map is still extremely time consuming, and significantly less accurate than a wind-tunnel aero-map. It also requires software that costs in the 10’s of thousands of dollars to license for just a year. Plus you have to have an engineer who knows race car vehicle/aerodynamics EXTREMELY well, and knows how to run the CFD software and create an aero map. Those kinds of engineers are generally working for NASCAR and F1 teams and not working in game development (except Eric Hudec of course). Luckily many companies involved in racing games/sims have licenses with the cars they are simulating, and hopefully those teams/manufacturers provide that data to be used in the simulation if they want their car to be simulated accurately.

Multi-car aerodynamics

Unfortunately this section will be brief. I know enough about multi-car aerodynamics to realize that I know and understand very little. The simple calculations above (using coefficients, area, and velocity) for single-car aerodynamics simple do not work when other cars are involved. Even when a car runs next to a wall, those simple calculations won’t factor in the changes to lift and drag, which are very real and have a significant impact. For multi-car vehicle dynamics, it is essential that you use CFD and/or wind-tunnel testing with multiple cars to get accurate results. Accurate real-time CFD calculations with multiple cars is highly unlikely to be a realistic option (my opinion only, I could be wrong). Luckily aero-maps can be used to some extent for multi-car aerodynamics. The distance between the vehicles and how many vehicles becomes important. With CFD, you can even calculate the relative velocity of the vehicles and how that effects the aero. Unfortunately, when it comes to aero-map wind-tunnel data, the cars are not moving relative to each other. Also, it is difficult to get several vehicles inside a wind tunnel (very few wind tunnels exist in the world that can even hold a single full size vehicle)

How do racing games/sims handle multi-car aerodynamics? I really don’t know. I suspect most of them take the single car data they have and tweak the values based on proximity to other cars, # of cars, relative position to other cars, walls, etc…I suspect they tweak until the cars behave believably. Again, I really don’t know though, that is beyond my area of knowledge, but I do find it fascinating to think about and discuss.

That is really all I have to say on the subject of aerodynamics. Remember, aerodynamics effects weight transfer, and power required to overcome drag. This has a great effect on vehicle velocity and tire contact patch loads which (wait for it, its coming) leads into the next segment….Tire models. This will probably have to wait till tomorrow though.

Tire Models

Ok, finally onto part three of this vehicle dynamics discussion…Tire models. By far, on several orders of magnitude, a proper tire model is more complex than the previous two discussions. Before we discuss tire models and the different types of tire models, it is important to note that there is no single definition of tire model that I would agree with. The term “tire model” is extremely dependant on context and the particular conversation it is involved in. What is a tire model? To determine this definition, in a way that can be applied to all levels of vehicle dynamics discussion is not possible. It simply isn’t, don’t even try. A tire model could be the way a tire feels in a sim. It could be the correct levels of lateral and longitudinal force produced for given inputs of a car. It could be the levels of aligning torque produced to determine accurate force feedback on our racing wheels. It could be accurate levels of overturning moment. It could mean the temperatures of the tire react realistically with the way the tires are being used. It could mean the tires degrade realistically with respect to wear, graining etc…It could mean the tires have realistic spring rates and damping rates based on pressure and temperature. It can mean any number of other factors and variables that make a tire behave realistically to a given simulation, which in this case we are speaking of a tire model for a racing simulation. Ultimately, a good tire model should incorporate all of the above mentioned factors and more. The point I am trying to make, is that a tire model in its most pure form, can be any number of things to any number of people. When somebody says, the tire model is crap because the level of grip is unrealistic because the temperatures are too hot or too cold for that level of grip, they may be right about that certain aspect of a tire model, but to say the entire tire model is crap based on that one aspect, is just wrong. There are so many levels to a good tire model for a racing simulation, that to write off a tire model based on one aspect, is a crying shame. The tire model might be 95% accurate, but to certain people, who expect a certain behavior, they might write off the entire model based on one aspect that they consider important. That isn’t to say this feedback isn’t important, because it certainly is, but is important to understand the other aspects of the model so that one might appreciate it, and provide feedback for the parts of it that they consider wrong.

Before diving right into the two major types of tire models, let us just discuss tires from a non-simulation point of view and make sure we are all the same page and up to speed with how a tire works and the terminology involved.

Tires are a composite structure made of multiple synthetic rubber compounds combined with belts or radials made from nylon and or steel (very simplified definition). The engineering that goes into a racing tire is nothing short of incredible. Tire technologies are changing constantly and even the best tire engineer will tell you they have so much yet to learn that they do not understand.

Tires produce forces (grip) from several different mechanisms at the same time at both a macroscopic and microscopic level. Paul Haney wrote a great book about these mechanisms called “The Racing & High Performance Tire”. His book is written in such a way to be useful for anybody interested in tires, and isn’t simply reading for engineers.

Tires generate force (grip) from slip angle and slip ratio. Or, slip angle/ratio is caused by tire forces. It is a chicken and the egg scenario. Depending on how you look at the situation or for what purposes your discussion is based upon, either can be correct. For the sake of this conversation, it really does not matter. Todd Wasson and I had a good discussion about this very topic a few months in another thread if anybody is curious enough to look it up. Let us define slip angle and slip ratio.

Slip Angle – The angle between the way the tire is facing (or the wheel rim) and the direction in which it is moving. Does this mean that slip angle implies that a tire is sliding? Not necessarily. At large slip angles it probably is, but at low slip angles, the footprint (contact patch) of the tire is being stretched, so the tire is “crab-walking” but not necessarily sliding.

Slip-Ratio – The difference between the speed a tire is spinning and the forward motion of the tire. So if a car is not moving but the tires spinning, your slap ratio would be 1 (or 100%). If the tire is completely locked while the car is still sliding, the slip ratio would be -1. If the tire is rolling freely at the same speed the car is traveling, the slip ratio would be 0. In general, a driven wheel while accelerating or braking is always somewhere between -1,0 and 1.

Now let us define some of the forces and moments the tires produce at varying slip angles and ratios. (We will not discuss the ones not important to this discussion though)

Lateral Force – Lateral force is generated primarily by slip angle. It is a force that resists the slip angle imposed upon a tire. As slip angle is imposed upon a tire, the contact patch of the tire is stretched, and the carcass of the tire is subjected to all kinds of twisting and pushing and pulling forces. The lateral force generated by the tire is partially the tire trying to pull itself back into its normal shape. It is also the rubber of the tire filling the macro and microscopic road imperfections and being stretched or even broken off (can you say skidmark). Lateral forces oppose the centripetal forces of the vehicle trying to make the vehicle move away from the instantaneous center of the radius it is attempting to traverse. If the lateral force is equal to the centripetal forces, then the tire can travel on this instantaneous radius. If the centripetal force is greater than the lateral force, the tire begins to slide, and we all know what that means.

Longitudinal Force – Longitudinal force is generated primarily by slip ratio. As a tire generates slip ratio, similar to lateral force, it causes the tire footprint to stretch and flexes the carcass. Part of the longitudinal forces generated are due to the tire trying to restore its original shape. It is also caused by the rubber filling micro and macroscopic road imperfections and the resulting stretching or even breaking off rubber. Longitudinal forces oppose the tires resistance to motion. If the longitudinal force is the same as the resistance force than the tire will generate very little slip ratio. If the resistance to motion is greater than the longitudinal force, large slip ratios can occur causing wheel lockup, or spinning tires under acceleration.
These slip angles/ratios and lateral/longitudinal forces are all occurring simultaneously when a vehicle is in motion and they interact with each. These interactions mean that a tire producing a given lateral load at a given slip angle will not be the same at different levels of longitudinal force and slip ratios.

Aligning torque – Aligning torque is what we are interested in specifically when talking about steering wheel forces or force-feedback in simulations. Aligning torque is not a force, but a “moment”, which in engineering terms basically means torque. It is a force with a distance applied to it, so you can describe forces related to rotational resistance, rather than linear resistance. Aligning torque is the “moment” that makes a tire want to move in a straight line rather than being turned. This moment is transferred through the steering linkages and is eventually felt in your steering wheel or sim wheel. The mechanisms behind aligning torque are pneumatic trail and mechanical trail. Or just trail, the sum of the two. Mechanical trail is caused by the design of the suspension geometry. It is similar to a shopping cart caster which has the center of wheel located behind the pivot point. As a result the wheel always follows tangentially the path of the front of the cart. Pneumatic trail is caused by the characteristics of the tire itself. The center of the contact patch of the tire is rarely directly vertical to the center of the tire due to forces on the footprint. Since lateral and longitudinal forces act at the contact patch (although this is very complex subject in itself, we won’t go into details) and since the contact patch isn’t in line with the vertical centerline of the tire, a moment or torque is created which similar to mechanical trail is transferred through the steering linkages to your wheel.

Ok, I think we are finally ready to begin our discussion on tire models.

The two types of tire models that I would like to discuss are as follows:

The empirical tire model
The theoretical tire model

Empirical Tire Model

An empirical tire model is a tire model based on measurements. Measurements of forces, moments, slip angles, slip ratio’s, and any number of other factors that one wants to implement in his empirical tire model. Some of the more popular empirical tire models are the MRA(Milliken Research Associates) Non-dimensional tire model, the Pacejka “Magic Formula” tire model, the SWIFT model, and the Brush model. The line between empirical and theoretical tire models is actually a lot more gray than you might expect, and some of these tire models could even be considered semi-empirical with bits of theoretical thrown in, depending on how they are utilized.

Let’s talk about how an empirical tire model is created. It all starts on a tire testing machine, such as the one at Calspan.
A tire testing machine constrains the tire while a rolling belt is rotated under it. The tire is subjected to various loads, slip angles, slip ratios, inclination angle (camber with different sign convention). Sometimes multiple tires are tested at varying pressures and other factors. It is a very expensive and time consuming process (a racing team can easily spend over $20,000/day on tire testing). The forces, moments, angles, ratios, pressure, etc…are all recorded.

From this data, you now have a basic tire model. Using the MRA Non-Dimensional approach to a tire model, you can now non-dimensionalize the data (I won’t go into specifics) into a table where you can plug in given inputs and it will give an output. The input might be a certain slip angle, inclination angle and load, and the output would be force. Or you could input lateral force, inclination angle and load, and your output would be slip angle (remember what I said before about the chicken and egg regarding slip angle and force) . Using this data one could code a simple lateral steady state simulation (skidpad) and determine the maximum speed, g forces, lateral forces, etc…for a given race car using the steady state weight transfer calculations I mentioned in the first post. Basically, you determine a radius of corner and you use an iterative approach at a low speed, or g-force (they are really derived from each other so it doesn’t matter what you use) and you increase the speed with each iteration. At each iteration you determine the lateral acceleration of the car. You determine the weight transfer that occurs at each tire contact patch, and you calculate the centripetal forces of the vehicle (the forces trying to push the car away from the apex, or instantaneous radius). You then plug your values into your basic tire model and you find the lateral forces the tires are producing. If the tires are producing values higher than the centripetal force values, then your car is continuing to traverse the radius as planned. You keep iterating this loop until the centripetal forces exceed the value of the forces of the tires. At this point the car can no longer maintain the radius of the corner. If the front tires give up the ghost first, then the car understeers out of the radius, and the car must slow down to maintain the radius. If the rear tires can’t keep up with the centripetal forces first, the rear end slides out and the car spins, unless you reduce the speed.
Similar simulations could be written for maximum braking, and maximum acceleration. These are the very basics of how a steady state simulation is coded. Keep in mind, these simulations are not real-time, but are simply determining the highest levels of speed, g-forces, acceleration, or braking, etc…If coded properly these simulations can run in a split second, even calculating the kinematic(changes to camber, caster, toe, etc.) changes to the suspension based on the weight transfer calculations from the first post.

You might be surprised at how many race teams rely on steady simulations as described above to make setup changes to the car, and quickly be able to see how it will affect the max accelerations, steering forces, and balance of a race car. A car that is setup well in steady may not be fastest car at a track, but it can certainly get you most of the way there and allow you arrive at a given track with a setup very close to what will be used in the race. Steady state simulations can be further expanded to quasi-steady state using yaw moment diagrams or MMM(Milliken Moment Method) which were originally used for aerospace simulations and developed by Milliken Research Associates for automotive purposes. A yaw moment diagram takes a lateral steady state simulation to the next level and allows an engineer to determine accelerations that can be achieved which are higher than what steady state simulations can show you. This is because in transient conditions it is possible to have combinations of slip angles and forces which are higher than what are a capable in steady state conditions. It also can show handling characteristics of a vehicle such as stability and control, or resistance to yaw (rotation) and steering response. This can be very powerful information that allows a high-end racing team to be very competitive, and all this without a true transient or real-time simulation.

Anyway, back to our discussion on empirical tire models. We left off with a huge amount of data being recorded from a tire machine. Using this data we were able to create the simulations described above. The problem with this data is that it is very noisy. It is important to look at this data and filter it. Remove stray points on your plots that were clearly caused by vibration, hysteresis, etc… Still, the data is very noisy. Steady state simulations might stop at points they shouldn’t because an output from one of the iterations might be lower than the other outputs in the same range. To get around this, algorithms such as the various forms of the Magic Formula (Dr. Hans Pacejka) were created to generate curves of best fit to match this noisy data. Remember your pre-algebra class back in high school or Jr. High with all of those f(x) functions and you had to fit very simple curves (parabolas, etc..) Well the magic formula is similar, only it takes easily over 15(depending on what version you are using) variables instead of just 2 or 3. It does a reasonable job of matching the raw data, and gives you a smooth curve instead of just points on a plot. It gets rid of the stray points, and fills in the blanks. So now we have a new tire model, although really it’s not all that different (and I would argue it isn’t any better than the MRA model from above). The biggest difference is how your code is written and the functions you use to input the weight transfer and kinematic data to get your output. I personally don’t think either is more accurate than the other and that it is completely subjective and determined by the programmer’s method of utilizing the model properly. I have personally used both methods in simulations I have written, and both have worked equally well.

Ok, so now we have an empirical tire model, any of the above models, for the sake of this discussion, it does not matter. How do we use this model for a real-time simulation such as iRacing? Let us pretend a sim car is going 75mph in a straight line. And suddenly, instaneously, the front wheels are turned 5 degrees. So now we have 5 degrees of slip angle, and our code checks our tire model based on the current load on the tires, camber, etc…and determines the level of lateral force that each tire should now produce. The car now turns, and weight transfer happens, and several times a second we keep checking the tire model and updating the forces and updating the weight transfer calculations. But something feels wrong. The car feels like it is either on rails, or completely out of control, very little in between. This is because our tire model is completely steady state. In reality, if you were going 75mph and suddenly turned your wheels 5 degrees, you would not have instantaneous forces that match your tire model. This is because the tire carcass has to flex first, and the contact patch has to stretch, and slip angle has to build up progressively, not instantly. Based on our tire model, we have no way to account for this. It takes time for forces and to build up in a tire when a slip angle is imposed (or vice versa). This concept is called “relaxation length”. It describes the delay that occurs between slip angle and force. In the earlier days of simulation, when transient simulations were coded, these relaxation lengths were fudged numbers that were tweaked until the tire “felt” right. Nowadays however, a clever engineer can actually calculate relaxation lengths from tire data that is recorded in a certain manner.

Ok, so now we have a transient empirical tire model, but it is still empirical. The simulation is starting to feel more real. But still based on our empirical model, we aren’t dealing with temperature, heat cycles, tire pressure (or maybe we are, some empirical tire models do account for tire pressure). Temperature is very difficult to account for. We can measure temperature on the testing rig, but temperature is increasing and decreasing so quickly that it can’t be controlled to get all of the values we would need to fill our data tables. Plus there is more than one important temperature in the tire. There is the tire surface temperature, not to mention the different components in the carcass that are operating at different temperatures. They each affect the tire differently and can drastically change the steady state results that we obtain from the tire testing machine. There isn’t an easy way around this. Even the tire companies that make the tires don’t know all the factors involved in the different temperatures. And they certainly won’t give out the detailed information that they have, even to race teams. As a result, developers have to use data from engineers on race teams, drivers subjective input, common sense, and some good old fashioned trial and error method until the tire feels right. Functions are written to alter the steady state data in a manner that seems likely and realistic at different temperatures (and this probably even is way more in depth than the methods many sims out there use). Similar methods are used to alter the steady state data based on wear, on other factors.

I think this is about as far as you can take an empirical tire model. As I said before, the line between an empirical model and theoretical is grayer than you might think. There are certainly factors that set them apart from each other, but an empirical model taken to a very high level, will definitely have many aspects of a theoretical model in it.

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