 In the last class, we were looking at the tractor trailer and we were looking at braking. We also looked at how braking affects the performance as well as, in other words, the braking distances as well as we saw how when a tire is locked, whether the front gets locked or the rear gets locked or the trailer gets locked, how that is going to have an effect on the vehicle motion. So, obviously, we also gave a sequence on how it should be locked and obviously, the braking forces or the percentage of braking forces which are distributed would depend upon the W's that are acting as well as on this sequence which we said, you know that should take place. Without delving much into that tractor trailer, you have done some assignments, so that will give you an idea, we will now move over to a very important and an interesting topic on tires. So, in other words, we will understand what are tires and then we will go ahead and understand what is the mechanics of tires. Mechanics of tires is a very interesting topic as well as a difficult topic, though one would say that what is in a tire, I see this every day in every car, what is so important about tires. If you look at an automobile, there are two components which are extremely abused, abused to the greatest extent. One is the tire, the other is the piston rings, both of them are abused in sense that they are subjected to all sorts of loads, friction and so on in their lifetime. And what happens between the tire and the road is extremely important for the safety of the vehicle, whether you are accelerating or braking and whether you are taking a corner or maneuvering the vehicle for all these things, what happens between the tire and the road is important. So, in other words, we have to understand this carefully. In the last class, we said that we will dump all the effects, the effects that happen between the tire and the road into one constant and we call this as mu and we said that this is friction constant or friction coefficient. I put that within the inverted commas, friction coefficient because it is not as simple as what happens or what this equation says or what happens in just a block. So, friction coefficient is only a simple equation to depict the behavior of the forces or depict the relationship between the forces, but the phenomena is very deep. We will answer the question one by one as to why or how this friction coefficient or friction is developed and how it enhances the grip and so on. Now, before we go further, let us look at just the forces that are acting on the tire. Since all of you know, you have done a course on automotive systems, you know what is the tire and so on. I am not going to delve too deeply into the specs of the tire and so on, but we will maybe indicate this as we go along. We will not go into the construction of the tire and other aspects of materials, but we will suffice it to understand and whatever is required to understand, we will do that right now. So, first let us look at the forces that act on the tire. This is a longitudinal force, which is in the x direction, but that force is exactly not along, look at that, it is not along the wheel plane. Suppose, this is the wheel plane, then you would see that the tire direction of rolling or the travel is not exactly along that direction, but is at an angle, which makes this is lambda or alpha and that angle is what is called as the slip angle. Why is it so? We will see that in a couple of lectures. So, the first thing is that, it is very interesting to note that the tire just does not, as it travels, it does not go along the plane. Let us say that x direction is the direction perpendicular to the normal direction or normal to the wheel plane, so in other words, the tire does not travel along the wheel plane. So, it takes a direction and that is what I call as direction of wheel travel and that direction is at an angle of alpha, which I or lambda, which is called as the slip angle. That is the first thing, there are a number of forces that act as well as moment that act on the tire. Of course, there is a longitudinal force. We saw that the longitudinal force is along the x axis, but we modified it by saying that whenever a tire travels, it may not travel along the x axis. Now, the slip angle is necessary to generate what is called as a lateral force. If the lateral force is 0, slip angle becomes 0 and the vehicle or the tire travels along x direction. So, in other words, lateral force is developed because of the slip angle. So, we have, let us call this as longitudinal force, that is the Fx force, that is breaking acceleration and so on. Then we have the lateral force, which is Fy, which is the result of the slip angle. This you would see that this is the result of what is called as slip. We will define all these things carefully. The longitudinal force is the result of what is called slip. Very interesting, you know, when there is slip, a longitudinal force is produced. When I say that, it really sounds oxymoron rather, that when you slip, what is this, that there is a longitudinal force developed, how is that, that is what we are going to see. So, there is a relationship between a longitudinal force and what is called a slip. What is called a slip? We are going to define that slip carefully as we go along. So, we have a lateral force Fy and that is due to what is called as the slip angle. In other words, when the lateral force does not exist, the slip angle does not exist and the direction of travel is along the x direction. Then of course, we have what is called as the normal force. Normal force which is Fz is normal to the wheel plane, okay. That is the ground reaction as we called it in the last class. All the directions have an accompanying moment, okay. So, the, let us look at the Fx force. That Fx force is accompanied by what is called as the overturning moment. I would call that as Mx and that is what is called as the overturning moment. You have a moment which acts along the z direction which is called as a restoring torque or a restoring moment. So, here we have what is called as the restoring torque or moment and along the lateral direction there is a force as well as a moment and the moment is nothing but the rolling resistance moment. We said that the, that moment or rolling resistance torque is what can be converted into a force, okay, that acts in the x direction. So, we said that they are equivalent, you cannot put both of them. So, we said either there is a moment or there is a force. So, when we analyze vehicle dynamics, the whole of the vehicle, then we replace the rolling resistance moment by means of a force in the x direction. This is what we saw in the last class, right. So, these are the things that happen in a tire. The question is how is it developed? How is it developed? The mechanism of this development of these forces, okay, that is what we are going to see. One of you of course know how a tire is, say specified, okay. If you go to a shop to buy a tire, you would see that I want, say for example, you are buying a passenger car tire. So, you would see that I want a passenger car tire which is, say, 215, okay and you would specify one more, 55 or something like that, 55 or 65, 55, okay. Then you would specify r, okay and then you would specify another number, 15, okay. This is what usually you would specify, right. Sometimes you would go ahead and specify a number, okay and a letter. Each one of them have a meaning. P is the passenger car tire, 215 is the biggest section width of unloaded tire. Let us look at a section of the tire, okay. So that is a section of the tire. So this distance, okay, this is the section width, the maximum distance, okay or the maximum width is called as the section width. So you specify that section width, note that the section width is specified in terms of mm, millimeter. Then you have 55, 45, 65, whatever it is. That is what is called as the aspect ratio and it is in percentage. What is the ratio? When we say aspect ratio, what is the ratio? The ratio of the height here, this height which I would call, say, HT, okay, to the section width, right. So it is the height, this height to the section width. So if I have a tire like this, then that is the section width, then that is the height. The ratio of the height to section width multiplied by 100, okay. That is what is called as the aspect ratio in percentage. Lower the aspect ratio, okay, it is called as the low aspect ratio tires. In fact, colloquially called as a low aspect ratio tires, 35 is a very low aspect ratio tires. In fact, if you see today many of the high end cars, you would see that this height, you notice that they are very small, okay. These are low aspect ratio tires. These tires have a specific property or characteristics that it makes handling much safer. On the other hand, ride becomes quite wobbly or in other words, it has an effect on ride, a negative effect on ride and a positive effect on handling. So the low aspect ratio tires are compensated usually by a good suspension system, okay. R is what is called as the radial tire. Today almost every, why almost all the passenger car tires are radial tires. We still have tires which are not radial in the truck segment. That depends upon the countries. For example, in North America you have 90% or 95% of the tires or radial tires or even more. But whereas if you look at a country like India, maybe about 25% of the tires or radial tires and still what you call as radialization is going on where bias tires are converted into radial tires. We will see what is bias tire, what is radial tire. Again in the next class through some pictures, we will just note that two types of tires are there. One is a radial tire and another bias tires. We will just see quickly a section and see what is the radial tire but we will go from micro up from micro to we will go to the macro, okay. Now this is 15 is the rim diameter in inches. So that is the rim diameter in inches. Due to some peculiar history or legacy, you have mm combined with inches and specify a tire. Then you have a load rating, you have a table from which you can find out the load rating of the tire, okay and then you have what is called as the speed rating. So you have a load rating and a speed rating. A quick look at the section before we go further and explain the micros, that is a typical section, okay. That section of a truck tire courtesy JK tires, one of the leading manufacturers, we have a section of their tires. You would notice that there are a number of reinforcements, okay, the number of reinforcements. You would also notice that these reinforcements in this tire are made up of steel, okay and you would notice there are chunks of steel on either side, okay. Actually that is what is called as a B, okay. So the rim is here, the tire sits on this rim and it is inflated, right. So there are number of parts of these tires, we will explain that in the next class. We will go into the details of what is the material with which this tire is made of, how it works and then we will look at the different parts of the tires. The way the reinforcement is done, it is not, first of all, it is not a pure rubber, it has reinforcements, number one. The way reinforcements are done, the tires are classified either as a radial tire or what is called as the, either as a radial tire or what is called as a bias tire, radial or a bias tire, okay, though you see that they are all steel radials, it is not necessary that all the reinforcements are made up of steel, okay. Some of the reinforcements you see here, which is called as apply, the reinforcements which come here, okay, section of which we will see in the next class is made up of polyester materials and the belt as it is called, these are the belts, okay, the reinforcements are made up of steel. So in other words, you can have a combination of materials which can reinforce this. So the reinforcements are basically made up of steel, nylon, polyester, rayon and so on, okay. Before we delve deeply into tire per se, we will go from the material of the tire then to the construction of the tire, just to give you a background. So let us now look at what are the materials and what or how does it behave, right. All of them, all the tires which are used in the automobile basically consists of what are called as elastomers. Of course tires are classified into pneumatic tires which we use all the time or solid tires, you know non-pneumatic tires which has very specific applications but now we are in this course we are going to only see the pneumatic tires. So the first thing is that we should understand elastomers and how it helps the tires to develop the forces that are required for it to accelerate, break, take a corner or maneuver and so on, okay. So we will go into small bits of information which we will, you know collate together in order to understand how a tire interacts with the road, in other words we will understand the elastomer, we will understand the road then we will understand how these two guys talk to each other in order to develop the forces that are required, right, okay. Now what are these elastomers? Elastomers are basically long chain molecules, okay, they are long chain molecules. Now these long chain molecules are vulcanized through what are called as the sulphur bonds. In other words these long chain molecules do not exist, it is not that there is one long chain molecule like this, another long chain molecule like that, okay they are independent long chain molecules that exist, no. These long chain molecules are bonded by what are called as sulphur bonds and that is what we call as vulcanization. Now if you leave this long chain molecule, okay, just leave it, it now what happens it becomes something like a bundle of twine or wool or something like that, okay. It is very interesting to note how they behave, how they behave independently and how that is different in their behavior or what is the difference in their behavior when it comes to these molecules, okay, these molecular environments have other molecules as well. In other words, it is interesting to note the difference between how one molecule behaves and how a number of molecules or molecular chains, okay, they behave, right, okay. Now if you look at one molecule, one molecule then when I, the molecule can be like that, let us take that, let us say that I take these two points and they stretch it and then I stretch it, okay. When I stretch it, it increases, the length increases, okay. What is length? That is the original length, we call that as the original length. Now when I stretch it, that is the final length. The first point you have to notice is that this stretching is different from the stretching that you would notice in a metal. For example, if you look at steel, steel does not have molecules like this obviously and that the elastic behavior of steel is due to what is called as lattice deformations. In other words, they have depending upon the type of steel, it is or the type of the structure of the steel. Say that it is FCC and BCC if it is an austenite, if it is FCC and so on, okay. Now when you have the steel and subject to a force, then the lattice gets deformed and that is what we call as lattice deformation and when we leave the forces, okay, the lattice goes back to its original position. So lattice distortions or deformations are responsible for the elastic forces that are developed in steel. You also know that the plastic deformations are due to a phenomena called slip and slip is due to the presence of dislocations and so on. Look at this. So you have a wire, say for example, if you take the mouse, okay, let us say that this is the length of the from my left hand to right hand, this is the length of the of this long chain molecule as you see it outside. So now let us see what happens, say when I stretch it, okay, look at how much I am able to stretch. Look at how much my hand moves, okay. Actually I am not disturbing the bonds like what I did in the previous case, okay but I what I did was to straighten this out, okay, look at that difference. So the phenomena of deformation of an elastomer is different from that of steel and the phenomena is controlled by our good old entropy, by entropy. Now you can imagine that the entropy which you normally understand it to be higher the entropy, higher is you would imagine that very simple way of understanding entropy is that higher entropy means that more, there is more chaos, more disorder and so on and so forth, okay. This is what you roughly understand from your, let us forget about the mathematics. Here we say that it is the configuration which decides the entropy, when it becomes straight the number of configurations that this molecule, macromolecule can take is limited, okay and so the entropy actually drops. So when I leave it the entropy actually increases and so it goes back to this position. So there is in other words there is an entropy change that is responsible for the deformation of these macromolecules, okay. So at one level you can imagine that these molecules are like a spring, okay. So you can say that I will fix it here, I will pull it, when I pull it, okay it becomes like this and then when I leave it it goes back to this position. You can say that the same thing you can say or imagine the same situation when you have these springs which are bonded by sulphur. So if I have a spring bonded by a sulphur here, spring bonded by a sulphur here, this springs bonded by sulphur, this spring bonded by sulphur and so on, okay this keeps increasing and so you apply a force, okay similar thing happens, fine. So this gives that kind of spring effect. So on one hand this whole, this can be modeled as a spring. Don't forget that there are on long chain molecules though they are bonded, they are bonded, okay they are not arranged like happily like that, right. So when they are pulled, these long chain molecules start interacting with one another or with itself and so on, okay. In other words, the long chain molecules have let us say some sort of a friction, it is actually not a friction, you can say let us say that it is some sort of a friction between the other molecules, right. So when I pull it, they also interact with other ones, okay and that gives a very interesting effect which is the viscous effect or the dashpot effect, the viscous effect or the dashpot effect. You can for a moment imagine that these molecules are in a tube, this is called as a tube model and that elastomer consists of a number of tubes, okay number of tubes into which these molecules are placed and these tubes now start interacting and you can say that there is one more, okay the chain as it goes there is a tube like that and they start interacting with one another, right. So because of this interaction, I said there is a viscous part. Two factors become important in their interaction. One is what is called as the frequency and the other is what we call as temperature. These two become important and interestingly you will see that the mechanisms are similar and hence there is a relationship between the two. What do I mean by frequency? frequency tells me how fast I am going to pull, say let us say that I am pulling it and releasing it, how fast I am going to do this, okay. I can do that slowly, okay in say 1 cycle per second, 10 cycles per second or I can do it in a very fast fashion or I can just keep doing it at 10 to the power of 8, 10 to the power of 10 cycles per second. Now what is the effect of this frequency? On the mechanism just we said what we indicated right now that there are interactions at very low frequency since we had put a viscous effect which means that velocity has a role to play at very low frequency, okay. The force that you, when you apply the force, okay, the chain has the time to react to that force and hence they would react to the force almost in the same fashion as you applied it and since you are going to release the force again in a very slow fashion, okay, it has time to react and come back to its original position. So at low frequencies the effect of time is, I mean the time available is quite large and because of that the molecules have the ability to expand and then come back. So in other words at low frequencies it behaves something like a spring, okay and so all the forces are dumped into the spring. So the force here is say for example a very simple model, the force here is kx, x is the displacement and for a dashpot it is always cx dot, okay. So the forces that act are able to get back or in other words the whole rubber behaves like a spring. What happens when I increase, let us look at the other spectrum, the other end of the spectrum is that I am applying the force at a very high frequency. So when I apply the force at a very high frequency then the guy does not have time to get back to its original shape, okay because by the time it gets back you have applied the force again, right, it is a viscous effect we said. So what happens, it would behave like a very stiff material because when you want to release it it does not come back, when you want to again apply it, it would not again go back so it would become very very stiff. So at very high frequency they behave as a very stiff material, okay. So in between the two at very low frequency it is a nice spring because the guy has enough time to recover and at very high frequency he does not have time at all to recover and hence it becomes very stiff and in between the two it has a spring and a dashpot effect. So if I now plot say for example this frequency versus what we call as the stiffness in other words modulus how would it be, it would be like this, like this, here is the spring, here both the effects are there, the spring and the dashpot and here it is quite rigid, okay. Now clear, so these are the three effects, so to summarize at low frequency the time required is good enough for the rubber molecules to come back and so they are springs and at higher frequencies they do not have time and hence they become rigid. Our region of interest in this whole thing is this, that is our region of interest. If I now plot a stress strain curve for that region of interest which I would call as the viscoelastic region of interest, okay how would it look like? Let us say that I apply stress like this, okay then the strains do not follow the stress in other words if it were to be elastic material like steel then so let us say that that is how the strains would be ratio being the Young's modulus. On the other hand here in this material because of the viscous effect there is a time delay, there is a time delay between the strain or stress in the strain or in other words there is a phase lag, okay between the stress and the strain, right. So there is a phase lag between the stress and the strain and that is a very important that phase lag is what gives you what is called as the hysteresis and is important property of rubber and which gives advantages and disadvantages in the case of functioning of the tire, clear. So in other words what does it mean? It means that when I apply the stress the strain is not immediately developed, okay it takes time to develop and I release it, okay he does not immediately get back to its original position, it takes time for him to get back to its original position, okay. So this is a very important property which we will see more and more. We will call this phase lag as delta, okay this is the phase lag we would call this as delta between the stress and the strain, right. Now let us look at the effect of temperatures and then get back to this again. The temperature has just opposite effect, when the temperature is high, the temperature aids these, so let us say I am fixing the frequency, I am taking the temperature to be high. What would happen when the temperature is high then it aids in the recovery of these molecules to its original shape and hence it would behave as if you are at a low frequency. So if I now plot the temperature at high temperatures since it aids the molecules to recover the modulus would be something like this. At very low temperatures what would happen at very low temperatures just opposite, okay. The molecules are under difficulty to get back to its original position. So the modulus is high, intermediate temperatures the modulus is between the two. So if you compare this graph with this you would notice that the temperature effect is the inverse of the frequency effect, okay that is the inverse of the frequency effect. We will see that there is a relationship between the two which is called as the WLF relationships, okay. Now if I now plot for that frequency effect let us say that I plot for the frequency effect, the energy loss versus the frequency then the curve would look something like that. There is a region which is that central region at which the energy loss is high that is that region, okay and that is the region at which we are going to use our tires. So same case temperatures would also look something like this, right, clear, okay. So the temperature also would have the same type of behavior. There is one particular temperature of interest which is called as the glass transition temperature popularly denoted by Tg. So Tg is the glass transition temperature below which the material is vitreous and above which the that is that is this region, above which the material becomes very soft, okay. So the glass transition temperature would obviously depend upon the frequency because both of them have an effect, right, both of them have an effect. In other words if I do a test at one frequency at a temperature and I now do with the same frequency at another temperature the effects would be different vice versa. If I do a test at one temperature with two frequencies again the effect would be different. So there is an interaction between these two, in other words there is an equivalence between the frequency and the temperature. The equivalence is inverse. So it is usually said that when the frequency increases by a factor of 10, increases as a factor of 10, there would be a change of temperature to 7 to 8 degrees. What is this change when the temperature, temperature would it increase or decrease? When I increase the frequency, when I increase the frequency would the temperature be? Increase or decrease, decrease, okay. So you go this side in frequency, you would come the other side in the case of temperatures. There are equations for this and these equations are given by William Landl and Farid's WLF equations. We will not go into the details but we will understand this more physically, okay. Now why are we talking about this? How is that the behaviour of elastomer helps in the development of say grip? That is the first question which we are going to answer. In other words, let us define what is meant by grip. So we are going to now go into the micro level. So in other words what we are trying to do is we are now, that is the section and say let us say that this is the tyre, we are now going to go and sit at the interface between the rubber and the road and understand how this concept is going to be applicable at that circumstances. One of the things we know about the road is that the road has what are called undulations. The undulations can be looked at from a macro view point or it can be looked at from a micro view point, from a micro view point, right, okay. How does this rubber interact with the road is our next question given that we have these two kind of roughnesses, okay. That we will see in the next class.