 So, in this class we are going to talk about what are called tire models, very short introduction to tire models. We go over and look at what happens in the combined cornering and slip, a simple model and we will wind up maybe in the next class about tire models which are used for this combined slip. Now, what are, first of all, what are tire models? This is a very important word today in vehicle dynamics because all people who use or who are interested in vehicle dynamics, they use one softer package or the other, be it Adams or Carson or whatever it is. So, they are interested in the way the tire behaves. In other words, that becomes an input into the software. So, they have to give to the software how, for example, slip versus longitudinal force varies, how alpha to lateral force varies and alpha and moment varies and so on. These graphs have to be input. Either you can give it as a lookup table which some of the software has accepted as lookup tables or through formula, which is called as a model. So, a formula which links this slip, for example, if I write F, it can be Fx, Fy, Fz, whatever it is, as a function of, say let us say that it is longitudinal force, Fx as a function of kappa or sigma, my slip, then it can be normal force Fz which is acting. So, that camber angle, if it has an effect, for example, in Fy, it has an effect. So, if it is Fy, it will, camber angle will have an effect and so on. So, in other words, this tire model is nothing but a mathematical equation. There are a number of ways in which these mathematical equations are arrived at. The easiest of them, the easiest is to do a curve fitting, which we would call as empirical equations. For example, I can determine these curves using experiments, for example, it can be kappa versus Fx and then fit an equation to this curve. The foremost tire model used extensively by people is what is called as the magic formula tire model. You can use T i or T y or whatever you want, but the magic formula tire model is one of the most important models used today in vehicle dynamics, by vehicle dynamics community. There are other models, we will see that very short introduction to other tire models later. Let us look at magic formula model and let us look at how for longitudinal force, magic formula model is used. You will get a feel of it, then we will look at the combined slip. So, not that every single parameter, give me two minutes, let me explain this. First of all, what is magic formula? You may be wondering what is magic formula, that is the first question, right? A bit of a history, this is one of the greatest contribution of Prasip Pasekha, who is the foreigner in the whole of tire mechanics and he has a very interesting book, which I have already pointed it out. He with the old work company, Mr. Becker of old work company, they together in 1987, around that 1987, they put up this formula, initially called as Prasekha tire model or sometimes later it was called as magic formula tire model and so on. 87 to 89, this was, there were two or three papers in SAE, which actually formed the basis or laid the foundation. Later in 1993, this model was adopted by Michelin tire company and they came up with modified version of it and they called this as magic formula model or magic formula model using an empirical technique. After this, this tire model has undergone so many changes, there are so many versions, up to 2006, this is a 2006 version. We are not going to look at every version, you know it is a huge topic, I refer to the book of Prasekha on this. We will only go back to the original paper and just look at the philosophy of how this was done, that is what we are going to do. But before we go further, what is this magic formula, why is it called magic formula? The magic formula is because this formula, which I am going to write down now, can be used for all the three curves. In other words, the frame or the form of the equation is the same, which I can write down this as y is equal to d sin c r tan. Do not worry about how, you know why this is so complex that we will see it in a minute, how this came about bx minus e into bx minus e into bx minus r tan. This has undergone a lot of changes, let us first stick to this. Now, this formula, this is the fundamental form with some modifications, we will see that, is used for all the three cases. If I substitute now x for kappa minus lip in this form, okay, then I would get fx. If I now substitute this for alpha here, okay, slip, then I would get fy, right. The same way, I will get, if I substitute for x, the slip angle, then I would get m and so on, okay. So, these three cases are the ones, which are, which I would say, these are the three cases, which are important. There are other things that are there, like turn slip, that is now concentrated on these three. So, you will get, you will have the same form. It does not mean that you will have the same values, okay. It does not mean that you have the same values, okay. These values of d, c, b and e, what they are, we will see in a minute, how did they come here? Why is that so complex? We will see that. But these values, these are values, which vary with of course, fz, the normal force and they take different forms, right. So, in a very simple term, you can say that I put down a formula and I have three different curves. Say for example, sorry, I had a curve like that for moment and then, you know, this curve is anti-symmetric with respect to this and then I have another curve, which may be like that for fy. So, this can be fy, suppose I kappa or alpha, okay. This can be the fx curve and this can be the m curve and so on. The reference, which I, which I am going to follow in this class is from the 1987, very first paper I said. Yeah, here it is 87, 0, 4 to 1. SAE paper 87, 0, 4 to 1 is the paper, which I am going to follow for this work and of course, there are, after that there are number of papers, which have come about. But this paper, this is a very fundamental paper and this actually gives the basis for obtaining the shape, right. So, that is what we are going to follow. Fundamentally, we have four of them, four factors, okay. C, let me put that as B, C, D, D, C, B and E. These are the four things that are there. I would say constants, constants, they depend upon, these are the four things. Now, it started, the whole thing started with a very simple equation. Suppose, now I want to represent this in an equation form, okay. Equation form is very easy or important if I have to do a multi-body dynamics analysis, right. So, I will talk about that a bit later, but just have that in mind. In other words, equations are important in order that I combine the dynamics of the vehicle with the tire road interaction. If I have to combine them, then equations are important. Remember, when we talked about launch student, and we are going to talk about lateral dynamics a bit later, talked about launch student dynamics, we were interested in the traction force, you look, okay. Fundamentally, F is equal to M, A we wrote. Now, we just wrote that as F, okay. Now, if I now have to go to the next step, then that force F has to be replaced by this curve, right. Where did that force come from? From the tire. So, how did that force develop? We saw all the mechanics. If I now want to go back to that equation, okay, then I have to put down this. So, in other words, in other words, the force develop would now be a function of my kappa, right. So, in that equation, go back to the equation, in that equation you have a traction force and that traction force will be replaced by an equation which combines kappa and Fx, okay. That is why mathematics, mathematical form of this equation becomes important. I can find out, in fact, if you want me to get this kind of acceleration, okay, whatever you will see, you will have seen a lot of advertisements. So, who, where they would claim that 0 is to 60 in 6 seconds or 0 to 18, 7 seconds or whatever it is, okay. Then I need that kind of traction force and whether I can develop that force depends upon this equation. I have a maximum force that can be developed. Say, for example, this would of course, depend upon the road and so on, Fz and so on, okay. Of course, depends upon the friction characteristics. So, this becomes very important. Where am I going to develop this? In other words, it also tells me whether I am going to have the wheel spinning, locking and all those things. So, in other words, that equation has more meaning, okay, and I now combine that with the interaction with the road, clear, okay. You would definitely know. Suppose, just to make the point very easy to understand, let me remove all this. There have been lot of questions before as to what is this formula, why am I using it and so on. Yes, a very experienced user after sometime, this would be nothing. For a new person who gets in it, this is going to be difficult. So, suppose I have a curve like this. So, what is the maximum force? I know this is the maximum force and that is the kappa, afx versus kappa. Kappa is our slip. We have defined two slips. Remember, theoretical slip called sigma, the practical slip kappa. So, that is what I am plotting here, kappa. Yes, I had plotted before sigma, right. So, kappa is the slip, percentage slip if you want to call it, clear. We had already defined that. Remember, V minus omega r and so on. Okay, go back and look at that if you have questions. Now, the point is this. If I now want to develop a force, suppose you are saying that I have a vehicle, it has this kind of rolling resistance, this kind of in the tire and if this is the kind of aerodynamic forces that are acting and this is the mass of the vehicle and all of the things and then now you say that I want to have this kind of acceleration. I want to have 0 to 60 in 6 seconds. So, you get an acceleration. Okay, first of all that acceleration would result in a requirement for a force and that has to be realizable. Whether it is realizable or not, this graph would say. Suppose I require a force like that, obviously that force would not be realized. If I write, so that would be a problem, right. Yes. So, when you are plotting this curve, all the other parameters on which fx will be depending on will be constant. Yes, that is a good question. So, in other words, this parameter is this curve a constant. What are the other parameters? That is what I wrote first. Suppose I change fz, okay. This curve would change, okay. This becomes very important. For example, when I break or when I remember we had that redistribution, when I break or when I accelerate, then fz changes. So, that becomes an important question. So, can I have only one curve? Does this curve not affected by fz? Even if you say that mu is independent of pressure, we know that this is f and that would depend upon mu and so it should depend upon fz and so on, right, precisely. So, this curve cannot be one curve, so it has to depend upon fz, okay. Now, then there can be a series of curves, right. It is a function of fz. For example, if you look at cornering, you go to see later, okay, maybe from the next topic that there is my cornering forces f5 becomes important, okay and its effect on the vehicle dynamics, we will see it. Now, here again there is going to be a load transfer due to the roll, right and again there is going to be a change of this forces with respect to f5 and so on, right. So, when I now calculate it, for example, when I do a calculation I have to take into account this kind of transfer of loads. So, in other words, if I have to calculate fx or fy, then I should have an equation, okay, which would now give me a new fy or a new fx because of the load transfer. I have to shift from one curve to the other curve when you have fz. So, in other words, in other words, this equation dcbe and so on should be a function of fz, should be a function of fz and that is what we are going to see. Hold on to your questions. Let me finish this, okay, then we will look at your questions. I know that that is why I am going bit slow, okay. People who know this, experienced users bear with me. I will develop this slowly, okay. Now let us now get into this equation. How did you get this equation? It looks very complex. Some r, tan, sine, this, that, b, c, d, e, okay. That is the genius of these people who worked on it, right. So, what they did initially was to see whether this whole thing can be given by a very simple formula, okay. Now I am going to modify this. Please wait for me for a minute. So, they wanted to know whether they can just put sine of bx. See, they designed bx. They first wanted to do this. This did not work. So, actually, because let us get back to my, all the three curves, you will see why it is not going to be easy to work, okay. One curve looks like this. Yeah, it looks very much like a sine picture. The other curve looks like this and the third curve, say, let us say that it looks like that. Accordingly, you have kappa and alpha that you know already, fx, fy, yeah. So, this was not working. The sine curve is not working, okay. It does not work. So, I had to now accommodate that what is there in the sine, okay. In another words, I have to adjust what is there inside the brackets in the sine, okay. So, if you now look at this curve, for example, this looks as if it is an elongated sine, an elongated sine, okay, right. So, how do I adjust that? And I want some function which is elongated in the x direction, right, okay. So, what is a function which is elongated or tan is a function which is elongated? Or in other words, if you now plot an arc tan curve extensively used in so many applications in mathematics, x versus y, the curve looks something like this, sorry, I mean, okay, right. It asymptotes asymptotically converges to a value. What is that value? Pi by 2, right. So, if I now that is the first thing. So, this asymptotically converges to a value of pi by 2. The same thing here is minus pi by 2 and so on. So, that is an elongated curve, right. So, I will now replace, if I replace what is there in sort of x, I would replace that value, okay. So, that I would now put this as y is equal to d sine, okay. So, arc tan bx, okay, okay. That is a, let me introduce one more term here, okay, arc tan bx, then I would elongate it. But the amount of elongation I have to now control. Here it is not very much elongated. Here it is elongated more and then here it is elongated less and so on. This looks like a sinusoid with a lesser elongation, larger elongation, okay, very short elongation. So, I have to now adjust this elongation. How do I adjust that elongation by multiplying it with a constant called c, right. So, what am I doing? I am looking at this graph and looking at mathematical expressions for which can actually model this graph and adjusting this equation. That is all I am doing, right. Obviously, right now you can immediately tell me what should be the maximum value of y. This is purely a graph. You know, this is, I am just fitting that. So, what is the maximum value of y is equal to d, obviously. So, this should be the value of d, okay. So, c in fact gives me that kind of the shape, okay, how elongated it is, how sharp it is and so on. Because it sort of has an ability to shrink the frequency if I can call it of, do not get confused with that word frequency of this sign, right, okay. Now I am not still happy with it because this has some, look at these two graphs. This has some peculiarity as to how actually it rises up. Apart from the slope, it is there it has a curve, okay. I now introduce into this equation or change the form of the equation in order to introduce that variation, okay. Let me call that as the factor e, okay. So that I will now write down this expression to be, okay, slightly variate and write down this equation to be y is equal to d sin, I am going to keep all that constant r tan, okay. B instead of x, I am going to introduce that phi, a quantity called phi, okay and phi is equal to 1 minus e into x plus e by b into r tan. This e, you know, in other words what I have done is I have introduced this e which actually changes the curvature at this place. Now without confusing c and d, let me call this c as a shape factor because it is gives the shape and e as a curvature factor, okay. Now, substitute that into this expression for phi, okay. Rewrite this expression, rewrite this expression. What do you get? You get an expression which is written there, okay. So in other words, I what I do is to use the property of arc tan, use the property of sin and use the property of combination, okay and then get a shape which can be now adjusted. I have now 4 values which can be adjusted, okay. Let me give some names to it. Let me call b as a stiffness factor. B as a stiffness factor. We will see that stiffness factor is not actually the stiffness, okay. Stiffness factor. C as the shape factor, right and d is the peak force that we had already seen. d sin of something, sin, the maximum sin value is equal to 1. So peak force d and e is the curvature factor. Why is it called curvature factor? Because if I now vary the value of e, okay, suppose this is the curve which is e is equal to 0, then depending upon the value with if, if whether e is between 0 to 1 or e is equal to minus, there is the curvature here would vary. The structure of the formula remains the same. So our real job would be in relating all the coefficients as a function of your slips, right, say kappa, gamma. Yes. Yes. So I am first establishing this formula which by tweaking these values, I can get whatever be the shape, period, right. Now I have some, I have to introduce some modifications to this, further modifications. Is this clear? Any questions? Yes. Now PCB are going to take care of all the other factors that contribute to Fx rate, including Fynfz. Yes, I am coming to that. I am coming to that. How is Fz introduced? How is, there are a lot of things that are introduced. In other words, in other words, yeah, I, I understand the question. First Fy, already I have, I have told you that Fy is affected by Pleistiae. I have told you that Fy is affected by Conicity. I have told you that Fy is affected by Camber. Now the question is where are, what happened to all that, right? Okay. Now what, what do I do? What can I do? Can I, should I look at this carefully, look at the question carefully. Now what is the question that I have to introduce a force when slip angle is equal to 0? Okay. So in other words, I have to introduce a force. In other words, it simply means that, let us take Fy. It simply means that this curve should not pass through 0. Origin. Fantastic. So it has to pass through something like that. Okay. It is an exaggerated, no it would not be so high but just, so in other words I have to shift Y. Conicity the other way. So I have to shift X and so on. Okay. So I am going to introduce a horizontal and a vertical shift. Okay. In order to take care of, in order to take care of the Conicity and Pleistiae. Okay. I am going to shift that, that curve. So, let me call that as Sv and Sz. Yes, yes. This is only for F. Yeah, this values would, yes, of course, of course. When we talk about Conicity and Pleistiae, we are, we are looking at the values of, of this. This formula is unique. So you can adjust this Sv and Sh, of course, right. You can adjust this formula for Sv and Sh. Yes, when I said Fy, obviously we are talking about Conicity and Pleistiae. So let me redraw that graph very cleanly. Let me call that as Y and let me call that as X. Just introduce this small x and small y and let me call that as Sv. Let me call that as Sh. Okay. And let me say that this equation is actually, that is the, that is the curve and the whole curve as well as x axis, I am shifting. So that I would write X to be X plus Sh and Y to be Y of X plus Sv. So I have, I have introduced two factors, right. Okay. Camper, we will, we will wait for a minute with our friend here is in a hurry. What happens to Fz? Right. So what I am going to do is, I am going to introduce all these factors as a function of Fz. Okay. Before that, there is one question. What is actually the slope? The slope of this curve is very important, the initial slope of the curve. So all safe drivers, okay, drive in this, this linear range. Okay. So what is the slope of this curve? Look at that. What is the slope dy by dx? Differentiate it? Yeah, yeah. dy by dx, b cos, okay, whatever is inside, then put octane is 1 by 1 plus x squared, right, and so on. Okay. And then put x is equal to 0. So you will get that, that is equal to bcd. So dy by dx at x is equal to 0 would now become bcd. So bcd is the initial slope of the curve. Okay. Again, do not get confused. I am not getting this curve from these equations. Okay. I have got already this curve. For example, I can get this from an analytical formula. I can get this from finite element. And more importantly, I can get this curve from experiments. So I have got the curve with me. I am only fitting an equation, right. So the next, my next job is to find out how I am going to express these factors bcd and so on. I want to express bcd as a function of f set, but f set itself depends on this. No, f set is a normal force. It depends upon, see it varies depending upon how, what is the way you do a maneuver or how severe is your cornering. Okay. Absolutely. Absolutely. So that is why I am now expressing this. Okay. In other words, as I told you, I have a number of graphs here and I want to express all these graphs in terms of one equation. No, no, no. We are not like, that is a good question. So we are right now looking at, okay, independent cornering. This equation now, that is what I said right in the beginning. We are now looking at cornering and or sorry, cornering separately and braking separately. I have not come yet to combine cornering and braking. Okay. We will develop first a simple mathematical model in order to say that, okay, what it is or not to enumerate what are the things that act when I have combined cornering and braking and then we will indicate how it is done. But even today, most people do only, they do not do combined cornering and braking and all these formulas that are used even today are only for a decoupled longitudinal force and a decoupled cornering forces, right. So yeah. Why are we interested only in Kappa and slip angle? Is it because they are the only parameters? Yes, of course. So these are the parameters ultimately I am finding out. Why are we interested ultimately we are finding out what is Kappa, what is a slip and so on, right. Now, so what am I doing? I am as being repeated questions on FZ. So I am going to write this as quadratic equation A1 FZ squared plus A2 FZ. No, no, no, no. BCD is the, when I said B into C into D, that is the initial slope, okay. That is all I am saying. It is the initial slope. dy by dx at x is equal to 0 is BCD. They do not depend upon x, y, z, okay. That is the initial slope. Now, these are the parameters let me reiterate. These are the parameters which gives the slope of the curve, right. Now, this curve that is point number one, most important point. This curve depends upon FZ. I have repeatedly said this and I am going to write that, right. So D is equal to A1 FZ squared plus A2 FZ and E is equal to A6 FZ squared plus A7 FZ plus A8. BCD, the initial curve is A3 FZ squared plus A4 FZ divided by E power A5 FZ and C, okay. These are initial curves. Then later models have changed the C. Let us stick to this first model. C is equal to 1.3 for the side force and is equal to 1.65 for braking and acceleration being symmetric and is equal to 2.4 for self aligning time, okay. So now what essentially I have done is I have shifted the owners of this curve from BCD and so on to A1, A2, A3, A4, A5, A6, A7, A8 and so on. So in other words, I have now a set of parameters A1, A2, A3, okay which would capture these curves, okay. And all these points or all these A1 to A8 are determined from experiments. They are all determined from experiments. So these are the coefficients, okay. If you want the type of coefficients or how it would be, so you can say that for example Fy A1 would be minus 22.1. Please note that it is not necessarily positive, positive or negative. A2 will be 1011, okay. This is in terms of kilo Newton, okay. This is a tendency for many people in the entire industry even today to use pound force, okay. Then these coefficients would be accordingly adjusted. For A9, we will have A9. We will see how this comes. Then we have properties for A9, A10 and so on. So A9 and A10 and other things which we are going to see now in a minute. Give the values. Take care of the camber aspects for Fy, okay. We said that the camber gives you a camber thrust. In other words, that gives you an Fy, okay. So that would again be a shift factor, okay. And that is the shift factor is given by, due to camber is given by horizontal shift factor is for the camber is given by A9, gamma and gamma is the camber angle and the delta B change in stiffness, okay. That is also obtained as delta B, okay with change in delta B, gamma, okay. Again there are number of parameters. Let us not also, I will confuse you. This is, I can again put down 9, 10, 9, 10, 11, 12. There is another 13, okay, which again a factor for E and so on. Let us not worry about it. So I don't confuse you. Let me summarize what all we said, okay. So in other words, what we said is that we have an experimental curve and the experimental curve has to be input into my, say for example, into the mathematical model of the whole vehicle, okay, because I am interested there to determine the forces, okay. I want to know whether my tire would develop that forces. If it develops the force, what would be the slip angle or what would be the slip at which this forces will be developed, where do I sit actually in that curve and so on, okay. And that I have to meaningfully take into account what is the change in the FZ values or the normal forces that are acting due to the dynamics, okay. So in order to take into account all that, I have an experimental curve, okay, and then this experimental curve is fitted by means of an equation, okay, which has so many values a1 to a13, right. So these are determined, this is not sacrosanct. This is for one tire. If I have another tire, these values will be different. In fact, I do not have time, but if you are interested, we will discuss it later as to how to fit this. It is very important that I get some unique parameters for this formula. So fitting this becomes very important. I had a student who worked on this and so there is a way of fitting this, right. So what do you get? I get a curve. Then how do I get from this? I get a curve, not one curve. I have to change FZ. I will get a series of curves. I have to change gamma. I will get a series of curves. So with all these curves, now I have to fit I have to fit a1 to a13. There are special softwares available for it and there are quite an amount of research told. It is an optimization problem that can be used in order to fit a1 to a13. Yes. Definition of D, V, C, D changes for reference. No, no, no. Definitions do not change. Definitions do not change. They are all, the form of this is the same. Definitions do not change. The values of a1 to a12 would change. But shifting. Yes. But for camber, what exactly? Camber, the same thing. This is shift and I call this as delta SH. This is shift. The same way I am going to shift. That is the shifts here. Yes, coefficient friction automatically comes in because I am not interested in mu. If I now, that is exactly what I said right in the beginning of the class. So if I now replace FX for example by a normalized FX by FZ curve, I would, people tend to call this as a mu curve. Here I am plotting FX directly. So it is also important. That brings out an important topic as to what is this, what is the, what is the role of friction. How do people determine this graph, this curve? There are two ways in which they determine this curve. One is by doing an experiment in the road by a tractor trailer. For example, TNO in DELF in Netherlands, they have a facility to determine this by means of, I mean by taking the tire to the road and finding out this value. But many, many people what they do is to do an indoor test. They do an indoor test and find out they have a machine on which they have a surface on which this tire is mounted and then they give, they have facility to measure these forces and the slip, slip angle and so on and they determine this in the laboratory. Okay. There your question of surface becomes important because the question is asking is how do you, how do you characterize mu? Okay. So the surface becomes important. So people have various surfaces if they have to do this in the laboratory. Okay. These surfaces are, I mean there is a big topic and these surfaces are actually or do have a roughness factor which would simulate or hold good even if you have to go or these equations would hold good even if you are to do it in the road. Okay. So they have, they mount the tire may be next, next class we will see this. So they mount the tire okay on a spindle and they have a flat surface okay which is the surface may be moving right and then they measure forces slip and so on in the laboratory scale. There are many tire companies who have this kind of facilities in order to determine Fx, Fy and other factors. No because it would, it would depend on F, see first of all yeah to a certain extent I would say that this is a good question that that is why people use in those days they used to do what they used to do is to just normalize this curve and then plot Fx by Fz. Okay. And then, and then have yeah the kappa value okay and then have this curve and they say that this is a normalize curve is fine. But then this brings out a very important topic on, on friction itself, the friction coefficient itself. Can it be, can this curve be like this? So there have been lot of work as I told you before as to how actually friction works between the road and the, and the tire, it is a big topic by itself. So this depends upon as I said on the, not only the pressure, contact pressure but what is called as the V you know slip and so on. If you want to model a very, in a very detailed fashion the tire behavior then you have to go into that friction models. Absolutely that is exactly what he was asking what happens when there is a friction coefficient is different. So you have to be careful yes the question is if I do it indoor okay, won't I get a different curve than if I do it outdoor in the road you know because the road characteristics are different. Absolutely you know there will be a difference between the two okay whether you do it indoor or whether you do it outdoor in this. In fact, there is a paper for remember right entire science and technology by continental tires which actually bring out and tell you what are all the differences okay on friction coefficients and so on. So there is, there will be but the theory is such that the differences are not very high it also depends upon the enveloping characteristics of the tire okay. So we would say that there would be a difference but for all practical purposes people use the data which is generated on this kind of rough paper or sand paper or whatever you want to call it okay that is the type of thing that is used in order to get this data okay. Yes any other questions? Yes I am not looking at the tire design here, tire characteristics I am looking at the final result. In other words the question is how does the tire characteristics affect this? For example if I change the compound or if I change the side wall profile, if I change anything in the belt angles where is it reflected here? That is the question it won't it is not reflected here this is for a tire, a given tire you are doing an experiment. Absolutely go and look at the paper which we published it is a very interesting question in fact I and my student we published a paper on how to link it is a very good question I am very happy because it gives importance to the paper which we had written. So how A1 for example is affected by design okay. For the next class I will give you the reference of the paper where we have linked it is a very important question because ultimately the tire manufacturer wants to know okay how he has to change the design in order that he would vary A1 and so on. So there is a link between this and the characters in fact you will be interested to know that these are all affected by the contact sorry by inflation pressure as a role to play as well okay and lot of design parameters. We will postpone that till the next class.