Physics Modeling: NTM V7 Info (plus more)

Physics Modeling: NTM V7 Info (plus more)

March 2nd, 2018 by
Dave Kaemmer 6 Comments

It’s been quite a while since I’ve written about the tire model, and I know some of you are eager to hear more about it. It has gone through several iterations since the new tire model (NTM) was first unveiled and I wrote the first incomprehensible blog about it. I am wrapping up the work on the seventh update to the model (V7). Because I haven’t written about it since about the first or second iteration, I’ll include info about version 6 as well. All the cars in the service are currently running on the V6 tire model. All versions prior to that are not worth remembering (except a modified V5 tread rubber model which is used in the V6 tires), so there will be very little here about those.

For those of you who haven’t been here since 2010, you can find some basic information about tire modeling and my approach here:

If you haven’t read that post, it’s worth reading before going on because it covers some of the terminology that I’ll be using later, as well as some basic information about what a tire model is.

As those of you who have read that post already know, the tire model I’ve been developing is a physically-based model as opposed to an empirical model. An empirical model doesn’t try to model exactly what’s going on in a tire, but simply tries to reproduce measured data and predict what will happen in some new situation by assuming it will be similar to what happened in the lab experiment. This involves a mathematical model, which has certain parameters (also known as magic numbers) that control its fit to measured data. In the tire industry, the most well known empirical model is Hans Pacejka’s Magic Formula. Google it! And it is pretty awesome at fitting tire curves measured in laboratories, especially when augmented with a couple decades worth of improvements.

But fitting tire curves as measured in a lab turns out not to be very useful when modeling (and therefore having to test) many different tires, and especially tires that are being pushed to the limit, i.e. racing tires. There are many parameters that need to be determined for each type of tire. And many of those parameters don’t have a real physical meaning, so without actually performing a tire test there is no way to predict what they might be, or how they would change with pressure and temperature changes or changes in speed. Further, while empirical models might fit part of a tire curve (usually they are fit to the initial part of the slip curve, before the limit), they tend to not reproduce what’s happening over the limit well. They also don’t do a great job with all the little transient behaviors that real tires exhibit. All of this is okay for most of the tire industry, where the focus is on passenger car and truck tires. Out on the highway, tires don’t undergo massive temperature swings, and they don’t often drive beyond the limit of the slip curve (except in the rain), so an empirical model works fine.

A physically-based model, on the other hand, tries to simulate the physics of a tire from first principles. That means it is based on the materials and the structure of the tire, and on the physics that apply to those materials and structures. In order for it to work, it has to model every real-world effect that determines how a tire behaves. Or to make it more do-able, it has to model at least all the effects that make a noticeable difference to that behavior. It becomes important to understand exactly how and why a tire generates the forces that it does. This is a hard problem, but in the end it is possible to predict tire performance knowing only how the tire is constructed. We can learn how a tire is constructed by cutting it up, and we can learn how all its bits behave by sending various bits off to a lab for analysis. Once we know its construction, i.e. what its tire cords are made of and how its cords are layed up, and we know what its carcass and tread rubber are made of, we can predict how it will behave on a racetrack under a great variety of conditions.

So what have I been doing on that front through the various versions of the model since the NTM was introduced? The short answer is two things: studying and improving the modeling of carcass constructions, and studying and improving the modeling of rubber. I tend to move back and forth between the two, as they are somewhat independent of each other. Usually when I have improved one, then I want to improve the other. In that spirit, the rest of this tome will go back and forth between carcass information and rubber information. It will keep getting more detailed until you don’t want to read anymore! Or until I’ve reached the limit of what I’m willing to reveal.

First I am going to describe how tread rubber is made, and how a real tire is constructed. Once we have a set of numbers that allows us to specify that construction for a particular tire, both rubber and carcass, then we are at the starting point. The goal from there is to calculate the tire forces and all the tire’s behavior using only those numbers (as well as knowing the position, orientation, linear and angular velocities of the wheel rim, and numbers describing the surface characteristics, of course).

Let’s start with some background on tire rubber. Tire rubber compounds consist of just a few basic things mixed together: raw rubber, oil, carbon black filler (soot particles), and sulfur. That’s an oversimplification—real rubber also includes stearic acid and zinc oxide (catalysts for curing), some accelerators for the cure process, as well as chemicals to protect the rubber against oxygen and ozone, and to make it tackier, if necessary. We don’t really care about those, but thank goodness someone does. In addition to or instead of carbon black filler, many tires are now made with silica filler, although that is less common in race tires. The first four ingredients listed above are the most important for determining the characteristics of the rubber that we need to know about. The raw rubber is usually SBR (styrene-butadiene rubber, invented during World War II), although natural rubber (or isoprene rubber) is also common. The oil mostly just dilutes the rubber, which makes it softer. I treat the oil simply as a diluent.

The oil actually affects the glass transition temperature of the compound. The glass transition temperature (Tg) of a polymer (and rubber is just a polymer) is the temperature at which the polymer starts to change from stiff, plastic behavior to soft, gooey, rubbery behavior. Any effect the oil has on the Tg of the rubber compound is taken into account by the fact that we simply specify the Tg of the whole compound, including rubber, oil, and black. That’s what we get back from lab measurements anyway. Typical Tg’s for tire rubber compounds range from -100 deg F (-75 C) for a snow tire to -60 F (-50 C) for a typical passenger car tire (or a high speed oval tire), up to -30 F (-35 C) for a typical racing slick, even as high as 14 F (-10 C) for an F1 tire. Given that rubbers typically only approach rubber-like behavior at about 90 F (50 C) above the Tg, you can see why F1 uses tire warmers. Without them the tires would start out like plastic water bottles.

The carbon black is an essential ingredient to make the rubber a worthwhile tread compound. Raw rubber by itself is too soft and weak to be an effective tread. Carbon black comes in the form of soot particles, and the size of the particles is important. Think of them as little spheres with diameters ranging from 15 nanometers to 250 nanometers. Most tire tread compounds use the smallest particles, up to about 30 nanometers. To the rubber polymer (and the oil), these little particles are very sticky. The polymer really likes to stick to the carbon black surface. A lot of things like to stick to the surface of carbon black, which is why carbon filters are so widely used and why it’s so hard to get the soot off your hands after touching your fireplace. In fact, even the carbon black really likes to stick to the carbon black. This requires careful mixing of the rubber, oil and black to make sure the particles are well dispersed, i.e. not in a bunch of clumps. Once you have black in your rubber compound, it has some startling properties. One, it’s much stronger. Two, it’s much stiffer. And three, it’s much more dynamically complicated, because it’s no longer possible to use standard polymer theories to determine how it behaves.

Sulfur is the last ingredient I need to talk about. It is used to crosslink the rubber. This consists of chemically attaching the many polymer strands to each other with sulfur bridges. Sulfur is mixed into the rubber compound along with the black and oil (and those other things we don’t care about), and waits, relatively inert, until the temperature of the rubber is raised in the tire mold after the tire is formed, to about 280 to 300 degrees Fahrenheit (140 – 150 C). At that point, the little rings of 8 sulfur atoms that make up the most stable form of sulfur at room temperature begin to break into little chains of 8 sulfur atoms. And those attach to the carbon-carbon double bonds in the rubber polymer, turning them into single bonds with an attached chain of sulfur. Eventually the other end of the sulfur will attach to some other bit of polymer and a crosslink is formed. On and on this process goes until the mold is cooled. The sulfur crosslinks also break during this curing phase, and then reattach to still more polymer, forming more crosslinks, until eventually there are crosslinks with anywhere from 2 to 8 sulfur atoms. At the end of the day, the only thing of importance is that the sulfur helps give us the number of crosslinks that the compound will have. Twice as much sulfur leads to twice as many crosslinks (roughly), and twice as many crosslinks means the rubber is twice as stiff. Crosslinking the rubber basically changes it from being a (very viscous) liquid into a solid that will hold its shape.

A rubber compound is specified with a recipe, which spells out how much of each substance is added to the compound. The raw rubber polymer is always specified as 100 parts by weight, and all the other additives are specified as phr, or parts per hundred rubber by weight. So for example, a compound recipe might be specified as: 100 phr SBR, 30 phr aromatic oil, 70 phr N330 black, 1.8 phr sulfur. And there would be other additives as well, as mentioned earlier, but we’re ignoring those because they won’t greatly affect the rubber’s dynamic behavior. N330 tells what grade of soot we’re using, from which we can determine the size of the particles. You might be asking yourself, “how on earth would you even find out this information?” Well, we can ship a piece of tread rubber off to a lab and get back a report which gives a pretty good guess as to the original rubber recipe. So it is possible to find out this information. Also, there are a number of standard rubber recipes published, although the tire companies guard their recipes more closely than Mrs. Field’s Cookies. While there are typically many different compounds in any given tire, the only ones that really concern us are the tread compound, and the carcass compound, which surrounds the tire cords. I assume that all tires use a similar carcass compound, and only specify the recipe for the tread compound. This is reasonable, because the requirements for the carcass are the same no matter the tire, really. And as far as the carcass itself, the tire cords are far more important than the carcass rubber.

So now you have an idea of how we specify the tread rubber, with a standard (but simplified) rubber recipe. Along with phr of rubber, oil, black, and sulfur, we also specify the Tg of the compound, and the mean diameter of the black particles in nanometers. We also specify a cure level from 0.0 to 1.0. Curing less than 100% gives an initially softer rubber that will continue to cure while at temperature on the race track. This is a simple dial that let’s us control for different curing times and temperatures, and different amounts of cure accelerators. The tread rubber model gives us most of the slip curve behavior (a bit is due to the carcass), as well as the ultimate grip and the feel over the limit, so obviously it is quite important to get the details right.

Well that’s neat, but I’m sure you’re now dying to know how we specify the carcass construction! So let’s talk more about the carcass model. We’ll start with some background on how tires are built. You could not make a decent tire using only rubber. If you’ve ever inflated a bicycle inner tube with a hand pump you’ve seen that long before you get enough pressure in the tube to have it work as a tire, it will balloon out and pop somewhere around the tube. That’s because rubber isn’t really stiff or strong enough to hold its shape against the inflation pressure. So some clever chap came up with the idea of coating some cotton cloth with rubber, wrapping it into a tube, and curing it! The cotton fibers provide the strength to counteract the inflation pressure, and the rubber both holds in the air, and grips the ground. This worked great at alleviating the headaches of John Dunlop’s young son as he rode around on his tricycle, but as this pneumatic tire idea spread to automobiles, some problems arose. First of all, woven cloth has a tendency to chafe and wear through itself when subjected to the many thousands of deflections of a car tire as it rolls across the ground. Thus the cotton fibers break, and the tire pops. Flats were very common in the early days of motoring. Solution? Don’t weave the fibers over and under each other! Just lay a parallel set of fibers in a rubber sheet, then put another layer of parallel fibers in its own rubber sheet and lay that sheet on top of the first, rotated by 90 degrees. Now you have all the strength of woven cloth, but the fibers don’t chafe against each other. They are all separated by a little bit of rubber, which allows the fibers to move slightly relative to each other as the tire rolls, so there is a lot less chafing, and therefore less breaking of the cotton fibers.

Second problem is that cotton is not a great tire cord material. It’s not terrible, but the chemists of the twentieth century soon made much better ones. First rayon, then nylon, polyester, fiberglass, steel , and aramid (carbon fiber) cords were developed. Today all these different types of cords (depending on use) are found in cross-ply laminates of the type I described above, with parallel cords held in a rubber layer, and cords in a second layer laid on top at some angle (not necessarily 90 degrees) relative to the first. These cross-ply laminates are the building block of tires.

A single layer (ply) is not useful unless it either is wrapped around a tire completely circumferentially (i.e. the cords are parallel to the direction of travel when rolling), or radially (they are perpendicular to the direction of travel). That’s because if you take a single ply sheet of rubber/cord and stretch it along any angle other than parallel to the cords it will change shape in a way that is not helpful in keeping the tire stable when inflated. However, if you take two plies, and they are rotated so that one is plus some angle relative to the tire’s crown (the circumference around the center of the tread), and the other ply is rotated to minus the same angle, then this 2-ply laminate (as the two layers are called) is quite useful. Depending on the angle you choose for the laminate, it can have very different stiffnesses circumferentially and radially. In the early days of making tires, manufacturers tended to stick to 2-ply or 4-ply construction, and the angles were fairly large, plus and minus 35 to 45 degrees (approaching 90 degrees total as in a piece of cloth). These tires are called bias-ply tires, because the cords are laid along a bias angle relative to the crown (English dictionary definition of bias: n. 1. A line going diagonally across the grain of a fabric). If you take a handkerchief and stretch it with the grain of the fibers (pull at the middle of opposite sides of the square cloth) it is pretty stiff—cotton isn’t very stretchy. If you stretch it from opposite corners, though, then you are stretching it on the bias, and it is quite stretchy (using either pair of opposite corners). A 2-ply laminate is just like this, except that it is possible to lay the two plies at an angle other than 90 degrees as in typical cloth. If you could do that with the cotton fibers in a handkerchief you’d be able to make a handkerchief that is more stretchy when you pull one pair of opposite corners, and less stretchy when you pull the other pair of opposite corners.

It is important to have a material that is stretchy, because of how tires are built. A great video can be found here, thanks to Michelin:

In it, you can see that the tire casing is first wrapped around a drum that is the diameter of the wheel rim the tire will be mounted on. Once enough of the tire’s bits are added on this drum, the center of the drum inflates and pushes the tire out into a rough tire shape. In order to do that, the casing plies (or body plies) must be able to stretch from the rim diameter out to the tire diameter. They can do this only if the casing plies have their cords lying at plus and minus some significantly non-zero angle relative to the tire centerline. As the tire casing is expanded into the tire shape, the cord angles have to get shallower as they go from the bead up to the centerline (unless the cords are perpendicular to the centerline). So, for example, if a 2-ply bias-ply tire were being constructed, and the two plies start at plus and minus 45 degrees to the centerline on the drum, then when expanded the cords at the crown centerline would need to change to plus or minus some smaller angle, like 30 degrees. This is a fun thing to try to visualize—and even more fun is the math involved. So a bias ply tire is stiffer circumferentially than radially. This helps to flatten the tread area a bit relative to the sidewalls, but a bias-ply tire’s cross-section is still usually more rounded than a radial tire’s, and bias-ply tires grow more with pressure, generally.

Today bias-ply tires are rare, except for off-road vehicles and motorcycles. And some types of racing, notably dirt. A bias ply has a basic trade-off if you are trying to make wider treads. In order to make the crown ply angles small enough to control the cross-sectional curvature (i.e. to keep the wide tread area flat), you need to use shallower angles overall, which makes the sidewall less stiff, so less able to hold pressure. And, of course, wider tires give better grip, so racers want wide. The radial tire solves this issue, allowing for both a flat tread area and stiff sidewalls that won’t balloon under pressure. It does that by introducing another set of plies, the belt, which is wrapped around the casing plies after the tire is expanded into shape. A little secret: most radial tires don’t actually have purely radial cords, i.e. the body ply angle isn’t 90 degrees from centerline. In order to get enough longitudinal stiffness, the cords are often placed at a slightly smaller angle, say 85 degrees. Radial race tires more often use 65 to 80 degrees. This is key to determining the correct value of the longitudinal stiffness kx. More about kx later.

One more ply is often added to a belted tire, the cap or overlay ply. This is usually a single ply with the cords lying parallel to the crown. It is used primarily to keep the belt attached to the tire at high speeds, although it can also add longitudinal stiffness. So finally we are at a point where I can describe how we specify a tire carcass construction. First, there are some basic dimensions that need to be specified: A reference pressure at which these dimensions were measured, the rim width upon which these dimensions were measured, the centerline circumference (with new tread), the section width, tread width, tread radius at the center of the tread, tread radii at the edges of the tread (tread radii are the radii of curvature in cross-section), the percent of tread width that has the center tread radius, and a bead apex percentage (what percentage of the sidewall length consists of the bead apex, which is a relatively stiff section of the sidewall near the bead that acts for our purposes somewhat like additional rim radius).

Second, we have to describe the body (casing), belt, and cap plies. Each of these consists of ply depth plus cord information: cord modulus (how stiff), cord density, the cord loss tangent from a carefully specified dynamic experiment (used for rolling drag and vertical damping), the crown ply angle of the cords in the as-built tire, and the volume fraction of the cords in the ply (the remainder is carcass rubber). Finally we specify the tread depth and the dimensions of tread blocks (if there are any). All of this information can be determined simply by looking, and measuring, and cutting, and by taking a deep dive into the textile world of fibers, yarns and cords, and how twist levels affect cord modulus in a quantitative sense.

While that seems daunting, it’s really a fairly small number of parameters, each of which can theoretically be determined. Combined with the rubber recipe for the tread described earlier, we have the complete description of our model tire. Well, one more thing: we need to describe the rim on which it will be mounted. That’s very basic: a rim diameter, rim width, flange height, mass, and material properties—density, heat capacity and thermal conductivity. So how do we go about turning all these measurable numbers into tire forces?

Short answer: with mathematics. Lots of it. For a couple reasons, I’m not going to talk too much mathematics here. One, much of it I’m not eager to reveal. It took a long time to derive all of it, and we consider it a key asset of iRacing’s. Two, I’d quickly lose most remaining readers out of the few of you still reading. I will cover some of the theory that underpins the model, but know that I am just scratching the surface. A complete description would be an entire book, mostly filled with equations.

Since we’ve been talking carcass construction for quite a while, let’s return to rubber. The paramount thing we’d like to be able to compute for the rubber is what’s called its shear relaxation modulus. This is known as G(t) in the literature—since G is a universal mechanical engineer’s term for a solid’s shear modulus, and it is a function of time. Fortunately for us, the basic experiment we’re conducting over and over again is how the tread rubber shears as it passes through the contact patch. Shearing a rubber block (especially since the amount of shear isn’t too large) doesn’t require us to use the full mathematics of solid mechanics. And that is fortunate, because solid mechanics makes organic chemistry look like kindergarten class. Instead we can use the (simplified) notion that the shear stress exerted on the tread by the road is equal to G times the shear strain. Ok, the undefined terms are piling up. Here’s an illustration:
___________                     ____________
|__________|                  /___________/

(a) rubber block           (b) sheared rubber block

Worth a thousand words, right? Shear strain is just the distance the bottom (or top) surface moves divided by the thickness of the block, essentially an angle in radians. Shearing a rubber block requires that a shear stress be exerted, which is basically a force parallel to the moving surface, stress being a force per unit area. Another simplifying thing about our experiment is that the shear rate (just the speed at which the shear strain is changing, in radians per second) is pretty much constant. Shear strain is usually represented by the Greek letter gamma, γ, and the shear stress by the Greek letter sigma, σ, both of which are used in the following integral equation giving us the stress as a function of time spent straining the rubber at the constant strain rate dγ/dt:

σ(t) = dγ/dt ʃ G(t-u) du

This is a well-known equation from viscoelastic mechanics. It actually contains a lot of information in its relatively simple form. Knowing only G(t) and the strain rate we can determine how much of the exerted force after a time t is due to elastic deformation, and how much has been lost to viscous friction. The sad thing is that reality is nowhere near this simple. The strain rate in reality isn’t actually constant, there is a lot more of importance going on in the contact patch than just shearing the tread. But this at least serves as an example of the usefulness of knowing G(t) for the rubber compound.

For polymers, it turns out that there is a lot of existing theory for finding G(t). Except when you add carbon black to it, sadly. G(t) represents how stiff the rubber seems to be at different time scales. At very short times scales (on the order of a picosecond, a trillionth of a second), rubber will seem to be like a hard plastic. At typical temperatures, this holds true up to times as long as nanoseconds (billionths of a second). After a nanosecond, but before 10 microseconds (millionths) has gone by, the rubber softens rapidly. It continues to soften, but more slowly, up to quite long times, as long as several seconds to minutes. At colder temperatures, all these times are lengthened considerably. At hotter temperatures, they are shortened considerably. In the field of polymer mechanics, this is known as time-temperature superposition. All polymers (long chain-like molecules) behave this way. Changing temperature is like speeding up or slowing down time for a polymer. This is simply because at cold temperatures the polymer molecules are moving more slowly. At high temperatures they are moving more quickly. In most research in this area this shifting of times with temperature is described by the Williams-Landel-Ferry transformation, or WLF. The WLF transform is useful for temperatures from a little above Tg up to Tg plus 100C or so. That’s laughably inadequate for the purposes of the temperatures we see in racing tires. However, after quite a bit of work and research I have found a good way to model this shift over a greater range of temperatures, and knowing this transform allows us to collapse this complicated function G(t) into a single curve that can be used at all temperatures, as long as we change the time properly. Another secret: adding carbon black to the rubber throws that out the window—there is no longer a single curve. But after a lot more work and more research, that has also been addressed, although now our function is G(T, t), where T is the temperature and t the time. But let’s call it G(t), for old times’ sake. Lest you think this is all a waste of time, it turns out that G(t) plays the predominant role in how the tire feels over the limit, as well as to grip levels at different temperatures. Some of the improvements in V7 come from that work.

Another set of improvements in V7 come from better carcass modeling, but first I’ll cover a bit of V6 carcass information, because that is a building block for V7. Fundamentally, the carcass model gives us the tire’s foundational stiffnesses as a function of load, pressure, temperature and speed, and also determines where and how large our contact patch is. The foundational stiffnesses are basically how strong of a spring the tire appears to be when the contact patch is moved in any direction relative to the wheel rim—longitudinally, laterally, and vertically. Just as suspension springs are labeled in pounds per inch of travel (or Newtons per meter for the metric crowd), so can the contact patch be considered to be mounted on springs that have different stiffnesses in the x (longitudinal), y (lateral), and z (vertical) directions. Those foundational stiffnesses can be measured, but we’ve found that the measured numbers for kx, ky, and kz (as they are referred to) differ quite a lot for different tires, and they differ in how they change with inflation pressure. Kx is usually quite stiff, but doesn’t grow rapidly (or linearly) with pressure. Ky and kz have a more linear response to inflation pressure, but they differ in magnitude as well. Ky is usually the smallest (though not always), and kx is usually the largest. As an example, a typical tire might have a kx of 2,000 pounds per inch, a ky of 800 pounds per inch, and a kz of 1,300 pounds per inch. A hot NASCAR right-side tire can have a kz over 4,500 pounds per inch! They can be pretty stiff. Just in case this still makes sense to you there is actually a fourth stiffness that is important, the torsional stiffness. That is how many inch-pounds of torque per degree of rotation the contact patch exerts as it is rotated relative to the wheel rim (metric is Newton-meters per radian). The torsional stiffness varies strongly with load and pressure changes.

We have seen quite a lot of data for kz, a fair amount for ky and kx, and very little (but some) for ktorsion. The data doesn’t generally have any kind of predictability to it. Different tires can have very different numbers, even if they are similar in size and shape. A better understanding of where these stiffnesses come from is required. That was the bulk of the work for the V6 model. We have covered cross-ply laminates and how we describe the tire carcass by specifying those along with some basic tire dimensions. As a point of interest, the carcass code actually proceeds in an analogous way to how the tire is built on the drum in Michelin’s aforementioned video. We construct the body plies, change the body ply cord angles appropriately due to inflating the drum out to the tire shape, then we add the belt, then the cap/overlay ply, and finally the tread rubber. As the plies are added to the carcass (virtually, using C++, not a tire drum), some matrix math along with some formulas that approach solid mechanics a little too closely (and so probably burned off some of my hair) give us a way to describe the various stiffnesses in different directions of the carcass tread belt and the sidewalls. Armed with those stiffnesses, and a bunch more mathematics that I toiled on for a long time we can find our way to the foundational stiffnesses. Those are important because the contact patch doesn’t stay in a fixed spot with respect to the wheel rim when the tire is cornering, accelerating, or braking. Neither does it point in exactly the same direction as the wheel rim. We need to know how the tread belt (and therefore the contact patch) moves and steers relative to the wheel rim to know exactly how the road is moving past the rubber in the contact patch. Other goodies from all this work: we have a better way to determine the contact patch dimensions and pressure in the contact patch, and the stiffnesses all change properly as the tire heats up and the pressure builds. Most of all of that is already in V6, and you’ve been driving on it already, albeit with our older V5 tread rubber model.

For V7, in addition to the tread rubber improvements I referred to before, there is also a sophisticated model for handling the dynamics of the carcass motion. Every tire has a critical speed beyond which the tread belt takes up a great wavy shape and much heat is generated. Generally, tire manufacturers want you to stay far below this critical speed, because any prolonged operation near or above it will quickly cause the tire to fail. Only exception here is drag racing. At the end of a Top Fuel run you can see how the rear tires become pentagons due to all the inertia in the tread belt, and how the belt bounces back and forth around the rim between each impact on the ground. Fortunately, the tire only has to do this for a few seconds. In all other forms of racing approaching the critical speed is a no-no. It will be a no-no at iRacing, too, for the same reasons. The belt dynamics equations provide a lot of information about the shape of the tire belt as it is spinning around, and that shape changes as you approach the critical speed.

From the shape and the speed of the tire, the rolling drag (the energy dissipated by the tire while it’s rolling) is now also computed from first principles, using nothing but the tread, carcass, and wheel rim descriptions outlined earlier. The vertical damping of the tire is also computed from first principles, and also matches what real world data we have well. In addition, the amount that the tread inertia of the belt “lifts” the tire as speed increases is also calculated. That is important for changes in grip with speed. These effects change with inflation pressure—lowering pressure lowers the critical speed, which affects how the tire is going to behave at a given high speed. A tire may have higher grip in a 40 mph corner when pressure is lowered, but that will make the handling in higher speed corners a bit sketchier. So there will be trade-offs, and those should be similar to trade-offs you’d see at a real race track with real tires.

If you’ve made it to here, congratulations, you now know a lot more about the V7 tire model that will be making an appearance on a few cars (we hope) next season (June, that is)!


With V7 the variation in laptimes from cold to hot temperatures is much better, there is no longer a “Golden Out Lap” syndrome, and the grip loss with temperature and feel over the limit has been improved. Plus a whole lot of other good stuff!



UPDATE: Dave answers some FAQ!

There have been a bunch of questions raised in this thread and I’d like to answer as many as I can. I picked questions that came up multiple times, and questions or comments that were particularly on point. So here we go, somewhat in order of most asked:

1) Given what I know now, that this NTM effort was going to consume 10+ years of my life, do I wish I had pursued an empirical approach instead?

No. Absolutely not. I actually did pursue an empirical approach for 20 years, while at Papyrus and during the early years of iRacing. That taught me something important: an empirical approach gets slower and slower the more tires you try to model and the more cars you try to model. There are just too many variables, and to adjust them requires a lot of testing and feedback between changes. Inevitably someone will bring up an issue where the empirical model deviates from real tires, and that involves adding some new bit to the model, with more parameters, which now need to be added to all the tires and cars… more testing, feedback, ad nauseum.

With the physically based model, there has been a lot of work required to get to where we are now, but the development is getting faster, not slower. That’s because the more we know, the easier it gets. There have been several times that the model produced a behavior that was surprising, but upon looking more closely at real data we realized that behavior was correct. Very often the numbers and concepts that we already know allow us to think more clearly about what we don’t know. It’s absolutely the right approach–I am stoked about the V7 model. Lots of things are right about it. Will it have some issues? Probably. It’ll take a while to transition over to it as well, but it’s a big step in the right direction.

As an example apart from tires, think about Ptolemy’s epicycles as a way to describe the movement of the planets, versus Kepler’s elliptical orbits around the sun. Ptolemy’s model was definitely empirical. No way to figure out how many circles upon circles any particular planet needed, or how large those circles should be, or anything. It’s just a massive curve fit against a model that has no basis in a theory of why. Kepler came up with a much better, simpler curve fit, though he also didn’t know why. Sadly, he died thinking his 25+ years of work had been a complete failure because he hadn’t been able to explain the divine order of the heavens. Isaac Newton, though, was aware of Kepler’s work (amazingly, if you’ve ever read Kepler’s work, which is… way out there) and it was one of the things that helped lead to his theory of gravitation. Look how much more progress resulted from Kepler and Newton’s theory than from Ptolemy’s. A good predictive theory beats big data any day of the week.

The Magic Formula is like Ptolemy’s epicycles. It fits data well (sort of) and can predict with fair accuracy what needs to be predicted in most common situations. Our physically-based model is like Newton’s theory, in that it is able to explain what’s going on and even predict things we didn’t know. Well, in fact it’s way more sophisticated and interesting and way cooler than Newton’s model of gravitation, but we’ve been able to stand on the shoulders of a lot more giants than Newton could, so he still wins the smartest guy award by, like, a long way. For example, Newton did all his important work before Euler was even born!! ? Plus we use Newton’s theory of gravitation in the sim, and it works really well, so we have to give him that. Also, because calculus.

2) How do you go about validating all this work?

This is a great question, and unfortunately it has a much more complicated and unsatisfying answer than all of us would like. Ideal answer is: “For every element of this model we’ve devised experiments which have verified our theory, and we have published many papers in peer-reviewed journals, and our results have been independently corroborated in other laboratories around the world.” Sorry, not that way. It’s more like a mundane, everyday part of the process.

Fortunately, a great many scientists around the world have published books and papers in peer-reviewed journals which provide results that we can use to build our model. The most important thing to remember is that once we see a paper with some data, and we use that data to come up with a bit of theory, we can NOT use that same data to validate the theory. We have to find new data, which we haven’t seen before. We often get data from auto manufacturers, or race teams, or even (rarely) tire companies, in addition to finding more books and more papers. Sometimes, we use data to theorize, sometimes we use it to validate. But not both. Data is often very dependent on the exact experiment conducted. Good papers will explain in great detail the conditions of the experiment reported and the materials used. Using that detail, I can reproduce that experiment virtually, sometimes with code, sometimes in a spreadsheet, and see what my model outputs. I have read a lot of papers reporting on dynamic experiments on rubber. Most of them provide the rubber recipe; if not, the data is of more limited value.

In creating any complicated simulation, there will always be a lot of unknown numbers. But the more we learn, the more those numbers become constrained to have a range of reasonable values. It’s a bit like putting together a jigsaw puzzle. At first, it seems impossible, but once you begin to fit a few bits together, and the outline becomes clearer, the easier it gets. The process is always one of double, triple, quadruple checking against new data.

As a general principle, I’m always trying to eliminate any “magic” (i.e. inexplicable) numbers. Every number we use should have a basis in reality, either it was measured (and we know under what conditions), or it is derivable from a largely accepted physical theory. Some numbers are quite easy to derive, like the moment of inertia of the right front wheel about its spin axis. Some numbers much less so, like how hot will the right front carcass be after 20 laps at Charlotte? But many numbers are related to each other–the right front carcass temperature is related to the pressure build after 20 laps, and so if we know one of those numbers (we might learn pressure build from a race team), we can deduce at least a small reasonable range for the other. For a long time, the rolling drag for a tire was computed with an empirical model, because we had no way to really calculate what it was from first principles. And in order to make all the cars and tires end up with reasonable temperature and pressure builds we’d have to turn a magic knob, the Rolling Heat Multiplier, in order to get numbers that were in the right ballpark.

Now, we are able to compute rolling drag from the carcass model, and the material parameters in the carcass, which are also measurable. We’ll retain the Rolling Heat Multiplier for now in case we miss the mark from time to time, but it had better turn out to be 1.0 (i.e. no change required to the model rolling drag) for almost every tire. If not, then that will be good information for us. Maybe it will be just about 1.2 for all the tires, in which case there is something that isn’t being accounted for in the model, or maybe it will be some very different number for every tire, in which case I will start trying to figure out what I’m doing wrong. There aren’t very many magic numbers like this left in the tire model. No magic numbers and all right answers = good model. In that sense, validation is a continuous part of the work.

Some validation certainly comes from “feel,” and the feedback from real racers who should know how it feels. We do a lot of that sort of testing, too. But feel can be illusory, so we have to be careful. There is always lag in force feedback wheels, and a lack of seat-of-the-pants forces, which make controlling a slide more difficult than in real life. I always find that I’m over-driving the car in-sim and being too aggressive with the car (which I believe is fairly common among all of us). In the past, every time I’d do a race weekend, I’d be much faster back in the sim afterwards. Why? Because the real car would force me to be smoother and not over-drive it as much. Using that same approach in the sim is better. If it “feels right” in the sim, sometimes that means it’s too forgiving of your (probable) over-driving. Once you get used to a more realistic, less forgiving feel, it’s just as easy to control. You just have to be smoother and keep your eyes up so you can respond faster to over-rotation, or any other bad situation.

3) When will flat spots be modeled?

95% of the behavior of a lockup is modeled already. Nobody is being denied a championship because somebody else is locking up in every corner and paying no price for it. Modeling the remaining 5% is on the to-do list, but currently it comes after rain, which is another feature everybody wants–until they get it.

4) How much rubber curing is happening on track?

Not much at all, unless your carcass temps are greater than 265 Fahrenheit (130 C). This pretty much limits it to high speed oval racing, and even there it’s just one effect out of several that reduce grip.

5) Tell us more about tire failure from excessive heat.

I don’t want to over-hype this, it’ll be some kind of rapid deflation event (i.e. inflation pressure goes to zero, just like current tire failures) based on exceeding some temperature for too long a time. The main thing is to provide a reasonable downside to running very low tire pressures. But even before failure, running pressures that are too low will in many cases not be fastest. Also, for now the behavior approaching, at, and beyond the critical speed can be a little weird due to some simplifications in the model. That is something I’m aware of and will try to address.

6) How reliable are lab results when reverse engineering rubber compounds?

These more provide a range of values, and constrain what’s possible. We get a pretty good idea of how much of the compound is carbon black, less accurate is the determination of polymer vs. oil content. The lab tests give some raw data sweeps that are very useful, as well. Another double check is testing Shore A hardness of the tire’s tread. But that depends on temperature, how long a reading (one second, three seconds, ten seconds?), and on the thickness of the tread, so care must be taken. Telemetry data is useful for grip determination (especially since our track geometry is close to exact), but aero forces need to be known precisely.

7) Why not update all the cars at once with the new tires?

Going to the V7 model is like having a new tire manufacturer come into a race series. Cars are often designed around the tires. Changing tire constructions and/or rubber compounds can change a car’s handling quite a lot, even in the real world. So a lot of testing and setup work needs to be done. It’s better to get feedback from a more limited number of cars at first. If there is something in the model that needs to be changed, we can do that without messing up setups for all the cars. Also, more work is necessary in the V7 code to properly do dirt. I think in the long run it will be better, but we’ll probably stick with V6 on dirt for a while. Dirt masks the over-the-limit breakaway that is one of the biggest issues on pavement.

8 ) …let’s should be lets.

Oh jeez, I have to admit to being an apostrophe Nazi myself, so it’s quite embarrassing to have let this slip through. Fixed! Thanks, Todd! Won’t happen again…

9) How does this compare to the tire manufacturers’ models, and would they be interested in what you’re doing?

I don’t really know the answer to this, but I can speculate. A large tire manufacturer has very different goals than a sim-racing developer. One, they don’t need a model that runs in real-time. So they can use finite element analysis (FEM) extensively. For us, that’s a little like ray-traced graphics from the movie industry. We may be able to get there some day, but probably there will always be a faster way to draw things. Tire manufacturers probably care the most about being able to sell more tires to a public that has little interest in driving at the limit. We are interested in selling more memberships to the limit-driving cognoscenti. The public buys tires based on the three things that are mandated to be measured by the government: grip rating (essentially how good the tire is at stopping while locked up at 40 mph on a wet road), mileage (miles per gallon of fuel burned, that is, essentially rolling drag on the highway), and wear rating (how many miles it will go before I have to buy a new one). Also, the public don’t want their tires to pop when running over a beer can. On this, we can agree–racers don’t want that, either. The public want tires that are not too noisy or harsh on the highway.

So much of the FEM work that tire companies do (but not all, I’m sure) has to do with: where and how does heat build up in the tire? What are the stresses at key points and how long can the tire materials last with that stress and heat? How can we better remove water from the tread area? How do we make the tread blocks quieter? If Joe Public doesn’t pay any attention to his inflation pressures and is driving on the highway with 5 psi in a rear tire, how do we protect him from himself? And us from liability? How do we simultaneously increase wet grip, decrease rolling drag, and make the tire last longer? Can we do all this with cheaper materials? And of course, can we make a really cool video to run during the Super Bowl? (preferably with a little baby being saved by our FEM model graphics)

We care about the first (how and where of heat build up), but much less about the others. My guess is the tire companies are already able to get better numbers for their purposes than we are. But we are better at getting numbers for our purposes than they are.

10) Have you seen Nicki Thiim’s video?

Yes. And I have to say, Nicki’s videos are as much fun to watch as it is to listen to Jens Voigt announcing a bike race. In other words, awesome! And to be honest, he is right about the V5 slip and grip over the limit model. Although, I think it is not as bad as he makes it sound. Ok, it is if it’s hot out. But he also says that he loves the “curb-surfing” (a great term) and that has much to do with the V6 carcass stiffnesses, which I think are very good. And he liked everything else about iRacing, including the graphics and track geometry realism. I think he’s right about that, too.

11) What is the sampling rate of your physics model?

360 Hz, is the short answer. But we calculate forces twice per update step, so we do player tire force calculations at least 4x2x360 = 2880 times per second. More if some of the tires are contacting multiple surfaces (i.e. curbs). We use IEEE floats.

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