 Okay, let us continue what we were doing in the last class. We were looking at rolling tires, we were looking at what we called as free rolling and we were also, that is what you are doing, free rolling and we were going in deep into the concept of how contact pressure influences various things that are happening at the contact patch. So, small slip of tongue in the last class, please note that when I drew the tire, I think that is how we drew the tire and that is the velocity. So, whether it is breaking or accelerating, the velocity, the direction is the same. So, I think that instead of acceleration being in this direction, I put velocity, please note that that is acceleration. So, the acceleration is in that direction. So, the ground we said will be, you can have that stationary and the ground moving in the opposite direction. We said that that is the relative velocity. So, please note that it is the acceleration. It is obvious, but now the less I just wanted to make it clear as to what we were talking about. There were some questions on tires like how tires are built and so on. That is a vast topic. In fact, tire mechanics can be a course. Unfortunately, there is only a part of this course. So, we will not be able to go into the details of tire mechanics completely. We will only talk about tire mechanics from the perspective of vehicle dynamics, how the tires interaction is going to affect the vehicle performance. That is what we are going to talk about. We are not going to talk about tire design and so on. That is a topic by itself. But now the less it is important that we talk about how or what is the construction of the tire and courtesy JK tires. I am giving you the slide, which talks about how tires can be classified. So, tires are classified based on construction into what we call as a radial tire or a bias tire. Today, radial tires have become extremely popular and all passenger car tires are radial tires. In this country, in India about 20, 25, 20, 25 percent of the tires in the truck or radial, but elsewhere in, for example, North America 90 to 95 percent of the tires in the truck or the bus segment or radial tires. Still, there are bias tires running in India, but in most of the developed countries, the tires are radial. There are a lot of advantages of radial tires. The foremost amongst them being the fuel efficiency. Radial data shows that the fuel efficiency or the difference in fuel consumption can be as I as 5 percent. So, radial tires are extremely popular in the truck market now and it is becoming popular rather in the truck market. The other tire is what is called as a bias tire. Let us understand what these radial tires mean. This is a cut section of a radial tire. So, you can see here that there are, look at that there are steel, what are called as steel belts and this is what is called as a ply. Let us get a cut section so that you understand this much more clearly. That is a truck radial tire. Look at that. It is extremely complex. It is complex from two points of view. One is that it has a number of materials and several reinforcements that make it stable, that make it efficient, that make it last longer, gives you a good ride and as far as possible reduces the fuel efficiency or reduces the fuel consumption and increases the fuel efficiency. Again, courtesy of J Guitars, we will look at what are the sections, I mean look at the different components in the sections and what the section is made up of. You would see that there is what is called as a body ply. You can see that the body ply which consists of a number of wires, a number of wires. So, they run like that from one bead to the other. So, you would see that it keeps running like this. So, they are radial in nature. That is why it is called radial tire. So, the body ply has steel reinforcement in a truck tire. We are talking about truck tire here. So, that is the body ply. Here you can see it may be here more carefully. You can see that that is the that is the reinforcement in the body ply and you see that the body ply. The body ply has rubber on either sides. The most important there are number of them. I am not going to explain each one of them. The most important which is towards the inner side is called as the inner liner. So, in other words it is inside. If this is the body ply and then what is inside this is what is called as the inner liner. And outside this, you can see that predominantly that is what is called as the side wall. In other words, the question is what is side wall. So, this is the side wall of the tire. So, in between the two, there are some components. Let us not worry about it. So, there is a reinforcement for us as far as we are concerned. There is a reinforcement steel reinforcement which is rather protected on either side by rubber and this side this is what is called as the side wall. This region of the tire is what you call as the buttress of the tire. Now, you can see that the steel reinforcements go all the way and then wraps around what is called as the bead or the bead core. You can see that here there is a bead and then the reinforcement, the steel reinforcement wraps around and comes out in the other direction. So, you can see that here very clearly that it goes like that and then comes around and then wraps around the bead. Now, around that portion of bead is solid, heavy. This is a very important guy. Last time we saw that he is the guy who is going to interact with the rim in order that the loads are properly transferred and so on. So, that portion is made up of a number of reinforcements. There is what is called as a steel chipper wrap around to protect the bead and then you have a chaffer and so on. The chaffer and chipper all of them help the bead to actually interact with the rim. So, there are other things there. Let us not worry about it. So, the first thing is that this body ply, important component, either side of it as rubber, side wall and inner liner wraps around the bead. Then when you come to the top, you have what are called as bells. These are what are called as the steel belts. You can see it here. They are in this case this is all steel radial. So, this is this ply is steel as well as these belts are made up of steel. They are steel belts. They give the stability for the tread and you can see that this is not only one belt, there are a number of belts. There is what is called as a transition belt and there are two working belts and then there is a protective belt. So, these belts have steel wires running. Say for example, if you look at it from the top, so you have wires running like that in one of the belts and then running like that in another belts and so on. So, the two belts working belt one and two are very important belts and as I said that gives that stiffness, the lateral stiffness and the longitudinal stiffness to the tread. The construction there is very complex. We will leave it for a minute because we are only looking at the mechanics of the tire and not into the mechanics of load carrying and stresses and so on into the tire. So, we are not going to look at the design. From design point of view, how you place that belt, what is the width of the belt, what is the angle and all these things, how many belts are used, all these things become important. There can be four belts, there can be three belts and so on. Then on top of it, there are some layers again of rubber and then ultimately you have the tread about which we were talking about. So, these are the, this is the tread. We will see that in a three dimensional picture and this is the tread. Tread is the one which is in contact with the ground and the mechanics has lot to do with the tread as well as the belt and how that interacts with the road. That becomes very important and that is what we have been talking about and we will continue to talk about that in this class. What is tread made up of this rubber? Tread is made up of elastomers, rubber. We will see, I am not going to the details of what is this composition, how it works and so on. This can be natural rubber, natural rubber, N R plus, natural rubber is usually called as N R, butadiene rubber, B R, sterine butadiene rubber which is S B R and so on. So, it is a combination of all these materials. So, we will not go into the details of these materials. Nevertheless, to say that, that is the passenger car, we will come to that in a minute. So, you can see that the passenger car tire for example, as well as the truck tire is made up of a number of different compounds of elastomers. In other words, different regions of the tires have different compounds of rubber. So, it is not one rubber. Rubber or elastomer is a very generic term. It is not that there is only one compound that covers the complete tire. In other words, tire is just not made up of only one compound, the number of compounds. As I said, that is not the intention in this course now. That is the passenger car tire. Essentially, the construction is similar but not exactly the same. The body plies are not necessarily made up of the same steel but it can be of different materials. Then, as I said, we will look at the biased truck tire. Unlike the radial tire where you had one body plier and then you had belts, here you have what are called as plies which run throughout. One, two, three plies as you can see which runs throughout and are arranged in a position, arranged in such a fashion as you can see that they are crossing each other and that gives the strength for the tire. Because of the way these are arranged, there is a performance difference between the bias and the radial tire but since the bias tire is almost given way to radial tires, we are not going to talk much about. There is no point in now comparing bias tire and radial tires. So, we will concentrate on radial tires and how that works. I just wanted to say that there is also what is called as a bias tire. Bias tires are still used in trucks. Now that is a typical passenger car tire. You can see the treads there. The treads are very clearly seen. There are various ways in which treads are designed. You can see that there are grooves there, there are grooves in the tread. These grooves, this is a passenger car tire and these grooves are there in order to carry water or minute sand as you go over say a sheet of water. In other words, they are channels for the water to escape as you go under wet conditions. The tires are basically classified into what are called as rip tires. This is a truck rip tire. You can see that there are rips. There are four rips in this tire and lug tires. These are what are called lug tires. The rip tires are usually used in the front wheel, not necessarily that it should only be used in the front because there are buses which uses this in the rear also. Basically, this is a steer axle tire. It performs very well when it is in the steer axle. So, this is a rip tire which is used in the steer axle and that is a lug tire which is usually used in the drive axle. That is the rear. So, these are rear tires and the front is a rip tire. But in a passenger car, you cannot have two different types of tires. So, you have tires which are a combination or a hybrid of these two which would say all wheel tires. So, all wheel tires are a combination of this kind of lug pattern and a rip pattern. So, you would see that a passenger car tire is sort of a combination of these two. Even for truck, you have a combination all wheel tires where you can use which you can use the front as well as in the rear. So, this is sort of a very brief introduction on how the tires look like. Though we are not going to use that knowledge in this course, we are going to go more into how the tires interact. But nevertheless, I think it is good to know how the tires look like and so on. Now, coming back to our contact patch, coming back to our longitudinal forces and all that what we did in the last class, remember that we were looking at a tire which is under free rolling condition. Let us get back to what we were doing in the last class. We will switch and go back and look at some practical aspects of this contact patch as we go along. So, that was the tire which was rolling. We talked about the varying radiuses in this tire. We talked about R0 becoming RE and then becoming RH and so on. And under this we had drawn three figures. We talked about the contact patch. We just given a two-dimensional representation of the contact patch and let us now look at three-dimensional representation of this contact patch. Two-dimensional sense that it just cut at the center and looked at it. It is very theoretical, but let us look at it from a very practical point of view actually how the contact patch looks like. So, this is a finite element model. Again courtesy jacket as we have a contact patch, finite element model of a contact patch. Different colors here indicate the magnitude, red being the contact pressures are higher and blue being the lowest. So, in other words, you see that the contact patch is not uniform and as you go to say for example, a truck ruptire the contact patch is absolutely not uniform. Though we assumed that a shape in order to understand the fundamental interactions, the contact patch is not uniform. Let us understand a bit about contact patch because that is an important parameter for the vehicle-road interactions. If this tire is to be made up of a membrane whose bending stiffnesses are very low, say for example, you take a balloon, blow it and then just press it on the ground. The contact pressure will be uniform. Contact pressure has to be just equilibrate this, the inflation pressure and so the contact pressure will be uniform. Strict conditions that has to be the case or theoretically that has to be the case when there is no bending stiffnesses and other things that we are going to talk about are involved. Unfortunately, a tire has a lot more things than a balloon. The first thing is that the tire is not straight like that. It has, you can see that it has say grooves. That is what you see here that it has this kind of grooves. Contact pressure is actually, let us look at that. Let us say that that is the inside. I am just extrapolating or I mean exaggerating or blowing or zooming the area at the center. Now, here is the inflation pressure that is acting and here is the road surface. That is the road surface. In other words, there are grooves. The grooves become quite complex if you really look at that kind of tires. The grooves become complex and here again you have grooves. If you now, again we will do a thought experiment, if you go and sit here, this part is not supported. That is not supported. Because of which there is going to be a local bending, so there will be, this part of it will now start bending because it does not have a support. There is an inflation pressure that is acting that is going to sag. That is going to get deflected. In other words, there are lot of local deflection. Number one, when compared to a nice membrane or a balloon which you keep it at the surface, there are going to be local deflections. Yes, we are now, right now we are not talking about water or anything entering there. We are just acid rolls. This is exactly how it looks like in that picture, acid rolls. So, we are not, in other words, we are not looking at a soil road, I mean tire in a deformable soil. We are looking at tire which is running on asphalt road which does not deform. For all practical purposes, that is fine. If the soil, then the whole analysis is very different. How it, how the soil deforms, all those things have to be taken into account. That is why you have what is called off the road tires which is called as OTR or off the road tires. We are talking about truck tires which are running on asphalt or flat roads. So, the first thing is that the constant pressure is affected by these kind of local bending, right. So, the pressures in this region on either side in these regions, they are going to be affected. That is going to be affected. That is the first thing, right. So, that is one factor. The other is the side wall. So, that is the side wall. So, actually what happens is this. So, under, look at that. You know under pressure, the side wall will be like this. So, there is a deformation of the side wall and the shape of the side wall is different from what it, what it is now and when it is under pressure, then the side wall shape changes, right. And that is going to have an effect which is very important for this contact pressure distribution. In other words, you will see that if this is the undeformed side wall, then under load, this becomes the deformed side wall. So, under these conditions there are a number of forces that act on to this tire. There is going to be some tension or in other words, a membrane force that acts there and there is going to be an inflation pressure force that is going to act at that point. So, there is going to be a membrane, there is going to be an inflation pressure and there is going to be a, of course, the ground reaction and so on. So, because of this effect, if I call this as TX, because of this effect, we have a bending. This is actually, you can imagine that, first let us look at it physically, then we will look at the T equation. It has a tendency, so there will be actually, there will be, this will be the inflation pressures that will be acting and so there will be a force which will be acting in that direction. The sum of this inflation pressure forces, it may be not here, may be somewhere you can say here acting and so on. So, because of this, there is a bending because of these forces, these forces, these forces and all that, there is a bending that is bending moment that is created. The bending moment is the sum and substance of all the forces that are acting, one in the positive direction, the other in the negative direction and so on. So, that bending moment, what we will look at that bending moment closely now. So, that bending moment, let us get into this and let us just expand that. So, that is, that is the side wall. Now, assume that this whole thing is now bent and so we have a bending moment acting at this place as you had seen there. So, now what would happen because of that bending moment, there will be more reaction forces here, there will be a reaction force here, depending upon who wins, there are number of forces there, who wins, there will be a reaction force at this edge. In other words, this bending is going to alter the contact pressure distribution, this bending is going to alter the contact pressure distribution. Suppose the bending wins over or one of the forces here wins over and the bending is like this, what would happen to the contact pressure in this region? The contact pressure would increase. The contact pressure here would increase. So, in other words, the side wall geometry and configuration would result in a bending moment at the ends because of which the pressure here, the bending moment here will be compensated by the, that will be the reaction. This is the action and the reaction and because of which there will be contact pressure will be here or higher contact pressure would result at the edges. So, that is what you see here in this figure at the two edges, you would see that the contact pressures are higher. It is not necessary that it would be like this, it depends upon a number of forces that are acting. Number one is that there is what is called as the, the membrane force, this which we had drawn there as T. The membrane force depends upon two things. One is that there depends upon the radius of curvature. Please understand that there are two radiuses of curvature. One for example, here is a radius of curvature. The radius of curvature here is going to change and then there is a radius of curvature in the other direction. So, there are two radiuses of curvature in the tire and so, both the radiuses of curvature have an effect on this Tx. Both the radiuses of curvature have an effect on Tx. Of course, the other things that have an effect on Tx, on Tx are the inflation pressure and the force in the Ty direction. We will not go into the details of this equation. I just want to say that there is a force in the side wall which you could loosely call as a membrane force which acts in the side wall which has an effect of course, on the bending of at the contact patch edge. That is the first thing. The only thing I want to point out is that this force is affected by the shape of the or the profile of the side wall as well as the inflation pressure to which you subject the tire. These are the two things. When I say geometry, it is Rx and Ry are included. That is the first thing. The other things that affect the moment are the inflation pressures. The inflation pressures, there are two ways in which it acts. You can resolve it in two directions, vertical and the horizontal direction. That is what you see there, the inflation pressures acting in the horizontal and the vertical direction. They have a tendency to have a bending moment in this direction, one direction and this Tx has a tendency to have a bending moment in the opposite direction with the result that there is an addition and there is a subtraction. In other words, it is very simple to understand. If this is the W, what is that W? Let us get back. Say this is the W, this distance is the W. Hence, you will see that everything here depends upon the geometry. If that is the W, then you can see that there is a minus sign here which acts in one direction and then the other things is in the opposite direction due to inflation pressures in opposite direction and acts like this. Ultimately, what is this moment, bending moment that decides the variation or the difference in pressure? Just to summarize, just to summarize what it simply means is that the inflation pressure, the inflation pressure that is acting would have been the same as that of the contact pressure, but for the presence of local bending which we saw here as well as a global bending, I would call it as we had seen here. Because of this bending and of course because of the membrane forces and so on, the contact pressure is not the same as that of the inflation pressure. That is the first important point. Now, there are number of design parameters which affect the contact pressure. Let us agree that they are not, it is not the same. Actually, we were looking at the rolling tire and right now our explanations are more to do with the stationary tire. Rolling tire has other factors which are very interesting. Before we go to the rolling tire, say free rolling tire, then we will go to a tire which is breaking, how the contact pressure changes or accelerating. And lastly, the most difficult of these cases, what happens when the vehicle takes a turn or cornering. So, in this class, in today's class, we will concentrate only on rolling tire, free rolling tire and we will just get into what we call as breaking or traction. Next class, we will expand that and so on. Clear? Any questions? In the y direction, right? In the y direction. Yes, in the y direction. x and y, x I had given here, y is in the other direction. F y is in the other direction. P i is the inflation pressure. P i is the inflation pressure. Let me write down, if you have doubts, let me write down. T x and T y are already explained. T x is the side wall tension in x direction and T y is the side wall tension in the y direction. P i is the inflation pressure. Yes, one minute. Inflation pressure. This is the free body diagram from which we are determining the... What I have done is essentially, I have cut this and then put the equivalent forces here. Yes, see what we are... Good question. Actually, in most of these analysis, we have left out centrifugal force, so as it rotates. In some of these conditions, if you look at literature, centrifugal force becomes important, but under many conditions, we neglect the centrifugal force because of rotation. You have to be careful. Usually, in a truck tire, we are looking at the truck tire. The centrifugal forces are not that very important because of the speeds, but in a passenger car tires, centrifugal forces become important. Right now, we are not considering that, but then there will be another force, other forces and there will be some changes here and so on. This is only to explain more physically the bending, what happens. I am not going to rely on these equations. Today, most of us do not rely on analytical expressions. We go to finite elements. All these things, all the results that you get, all the niceties are beautifully brought out in finite element analysis. But, nevertheless, this becomes important in order to understand what really happens or why the results are what you see in finite element. That is the reason why we explain what is bending, how does it bend, what are the forces which cause bending. We cannot do hand waving. That is the reason why we write down these equations and see, look, there is a moment and the moment consists of... The moment is due to various forces and forces act because of inflation pressure, because of the force of the membrane or the side wall forces. These participate in this moment and so on. All these effects are accurately brought out by the finite element analysis and hence we look at the results ultimately in the finite element, you know, contact pressure distribution. Clear? In which case, you can also add the centrifugal force and so on. So, we will look at now rolling time. There are other things that become important. Why coming in x direction? This is the way actually the... This is a membrane theory. We are not going into the equations. So, we would write now not going to the derivations of these equations y r y r x r x r y and all that. We will not... Because then the topic becomes very difficult and so this is not F. This is not F. This is F v. I thought you have to ask me r y v. This is F v and F v is this force that is acting here in this direction. The force which is acting at the center here and that is F v. That is not F y that is F v. But r x and r y come into picture and that is F v. Actually, it is not the total vertical reaction force that is acting. This is not F y. But r x and r y come into picture because of this radius is on either x and y directions. The m is the bending moment that acts at the edge. So, there what we do is simply we take bending moment due to distributed loads due to inflation which is divided into two parts. That is what we do this and this. As if there is a distributed load like that here. This is the result of the bending moment due to the distributed load and that acts in the opposite direction. Ultimately, you have a bending at the edge period. Yes, yes. See what is this? What do we mean by distribution of this forces? If you want to understand this, it is again very simple. If you have a doubt, let us say that I have a cantilever beam. Now, I have it is a distributed load that is acting on the cantilever beam. So, there is an action here and there is a reaction at this point. So, we are talking about the same action and reaction here. So, this is something like a cantilever beam when there is an action and a reaction. So, the reaction is in the opposite direction which means it should be in this direction. So, you will have a load distribution here such that it will oppose this moment that is acting here in the same fashion as you have in the cantilever beam. So, the only thing difference between these two is that here we have a cantilever beam within quotes, cantilever beam in sense that it is curved beam and with forces acting. Of course, there are effects of local bending. You can see that there are green in some places there are green and there are as you near the grooves and then you have again red and so on. So, there is a sort of relief here that is that is happening. In other words, in other words what happens is that when it really bends then there is a small lift and so on there is a redistribution and that is why you will see that near the grooves they are not red there is a change. So, there is an inflation pressure that is why the two components or what we give that. No, there is no this p i only one pressure one pressure that is acting this direction. That is the inflation pressure the force the result of the inflation pressure can be divided into two parts or two in the x and y direction. So, what we have drawn here is the inflation pressure. Let us come back here and so we are talking about rolling. So, this is the what we call as the static pressure rolling has a very interesting things that are going to happen when you when it is rolling with respect to contact pressure. Let us see what happens and then we will explain. There is a contact pressure of a rip tire look at that when it this is sudden change. You will see that there is a change between the two contact pressure when it is stationary and you can see the whole thing gets twisted. You see further when it is the same thing with the different limits you would see when it is breaking look at the way the contact pressure now changes when it is traction look at the way the contact pressure changes. So, in other words you can see that let us let us get back to this. So, you can see that there is a there is a change in contact pressure. There is an adjustment of contact pressure why does that happen? That is our next question. We are looking at contact pressure in this is the contact pressure. Now, as I travel as I travel from one end of the contact pressure to the other in other words just look at that contact sorry contact patch this is the contact patch. As I travel from one end to the other there is going to be a longitudinal force we saw that in the last class. There is going to be a longitudinal force goes up and comes down. Remember last time we saw that something like this. We also saw that this longitudinal force reaches some maximum R e remember that we talked about R variation of R in A to B, B to C, C to D, D to E and so on in the last class and this longitudinal force has a very interesting effect. Now, just see how that is infinite element let us yeah that is the that is a longitudinal force. I hope you are able to see one is in the positive and the other is in the blue is in the negative the red is in the positive. So, it goes up goes to 0 in the middle comes up and this is exactly what you see here is what you see in the in this case. But unfortunately since there is no unfortunate here, but anyway the rip all the ribs as you can see here do not have the same longitudinal force due to the construction of the tire as we explain. They do not have the same longitudinal forces they are they are different in different ribs you can see that. Now, the tire so that is the let us yeah. So, the first point is that the longitudinal force as we had seen there is not a constant in all the ribs they are different at different ribs because of which the tire now rotates a bit about the vertical direction. So, that the total strain energy total strain energy becomes minimum. So, in other words this distribution of or uneven distribution of forces causes the tire to adjust adjust or in other words the contact pressure now is such that becomes a minimum that the total strain energy is minimum as a system to like to go to a minimum strain energy and so there is a twist. So, that you can very well see that here look at that. So, it it sort of twist in such a fashion that it becomes slightly more uniform. So, in other words the longitudinal forces that are acting has an effect on the contact pressure distribution itself. And for example, the longitudinal forces which we are going to see in the next class due to braking has further look at that has further effect on the contact. Any questions? Just to summarize what we what we said we will continue this that the contact patch is an important or influences the interaction between the tire and the road and that the contact patch though one would like it to be very uniform is not uniform and that there is a there are some some local effects and there are some global effects which has an effect on the contact patch distribution and that the contact patch is also affected by the uneven distribution of longitudinal forces between the various ribs and the ribs tend to or the contact patch tends to adjust itself. So, that the strain energy is brought to a minimum or there is a redistribution such that the strain energy levels as a system is brought to a minimum. So, there is a small rotation that is as far as the free rolling is concerned. We will look at braking and then go to the contact patch in the next class right we will stop here.