 In the last class, we started discussing problem of grip in a tire. The fundamental aspect that gives grip to the tire is its interaction between with the road and that it develops what we called as the frictional resistance to its motion. Now, let us understand this from two perspectives. We have started this already. One is from the elastomer perspective. How does elastomer function when it contacts the road and the other from the road roughness perspective or the road perspective? So, there are these two things we have to understand because after all it is the interaction between these two which results in what we call as grip. Just to summarize what we did about the elastomers, we said that the elastomers are basically elastic viscoelastic material. In simple terms, they are viscoelastic materials. We said that there are two things that have an effect. One is the frequency and the other is the temperature. In fact, we saw that they have an opposite effect and we determined or we explained that the region where say for example, elastomer operates can be divided into three parts and that you have a region where it is say for example, this would be the frequency. We said the region where the frequency is low, the elastomer acts like a spring and in another region where the frequencies are very high, it becomes very rigid. So, it becomes something like a very rigid part essentially because it has to slide back to its original position and the sliding back is affected by the interaction of these long chain molecules and micro molecules with others which are present and hence it becomes rigid and between the two regions, we have viscoelastic part. We said that the graph for the modulus goes like that and because there is a viscoelastic part of the center, if I now plot the energy that is consumed due to deformations, then it would go something like that. This we said is the glass, corresponds to the glass transition temperature. So, if I now put the temperature here, we said that this whole graph goes in the opposite direction and that what happens at higher frequencies happens at lower temperature and so on. So, this is what we said yesterday. We said that there is a relationship between the two and it is given by Williams Landl and Ferry WLF equation. This E equation gives a shift say for a frequency when you shift the temperatures. When there is a shift in temperature, you would see that there is a shift in frequency as well. So, we will come to that in a minute. Just summarize what all we said yesterday, last class and we also said that there is a lag between the applied force in the deformation or other words applied stress and the strain that is developed. There is a lag between the two. So, if I now plot for example, the stress and then plot the strain, the strain develops something like this and that there is a face lag between the stress and the strain given by the equation delta or tan delta is an important quantity in the study of viscoelastic materials. In your classes and vibration, you would have studied about or in controls, you would have studied about face, face lag and so on and you would have expressed the quantities of interest in a complex plane. The complex plane is essentially used in order to bring out this kind of face information. Hence, here we have what is called as a complex modulus for a viscoelastic material, which consists of a real part which is called as the storage modulus plus an imaginary part which is called as the loss modulus. Note that there is nothing imaginary in this. The only thing is that we express the face information through a complex quantity and call that complex quantity into real and imaginary. You can say that this tan delta is an important quantity basically because that gives you the amount of energy that is dissipated. So, in other words tan delta which is from here you can find out what is delta. Tan delta can be written as energy dissipated. So, when there is a loading unloading, there is an energy that is dissipated and can be looked at as ratio of energy dissipated to the energy stored. So, tan delta becomes an important quantity which tells us about the dissipation of energy. So, if you now plot, so what does it essentially mean? If you now plot say for example, omega versus all these three quantities which is E prime, E double prime and tan delta, the graphs would look something like this. We have already seen how E prime would look. So, E prime would look like that, that would be the E prime value. E double prime would look something like this, that would be the E double prime value and tan delta would be something like that, that will be the tan delta value. This is at one temperature. So, this temperature is a constant here. If I want to determine this at any other temperature, it is not necessary that I have to keep on applying the same set of experiment or do the same set of experiments, that is not necessary. This whole graph gets shifted depending upon the temperature, this whole graph gets shifted. In other words, if I have data at one temperature here with respect to the frequency, I can find out the data with another temperature by shifting this frequency. In other words, I am shifting this graph. In other words, at t, what happens at omega and at the temperature t? If I now shift the temperature to t plus delta t, then what was happening at omega would happen at omega plus delta omega. This shift can be determined from the WLF equation, which is given something like this. Log of at, the shift is given by C1, this depends. So, we can write this as C1 into t1 minus t0 divided by C1 plus t minus t0, where C2 into t minus t0, where C1 is equal to minus 8.86. Usually, I mean, it varies and C2 is 101.5. t0 is the reference temperature, okay and which you say the glass transition temperature plus about 50 degrees, right. Now, glass transition temperature itself depends upon the frequency. So, when I shift the frequency, that temperature, glass transition temperature also shifts. Now, how do we use this? Suppose I have the data to be or data recorded at omega and t, okay and I want to determine what would be the, this data say at a temperature say tc, okay. Then, what I have to do is to find out the shift, say let me call that as q, which is log of at minus log of atc and shift the frequency and this is of course given in log. So, shift the frequency to that decade frequency, okay. So, each data point that is obtained in omega t, okay, that is this experiment, okay. Now, you say for example, you are plotting this graph of log omega versus, here if you have log omega versus the value say e prime or e double prime and so on, okay and you want to find out what would happen to the, in the new temperature, then you shift this a point say for the new temperature by this much amount, by that much amount. So, this is the shift factor. In other words, this shift factor is the relationship between the temperature and the frequency. Any questions? Yes, yes, yes. So, depending upon that is that is this relationship. So, what happens, what is the relationship between frequency and temperature? So, what happens at lower temperature happens at higher frequency, okay. So, that is understood, correct. So, this is just a mathematical expression for this shift, right. So, usually one decade, that is if there is a one order of magnitude in the frequency would result in a shift of say 6 to 7, 7 degrees is a shift, you know that is what would happen, okay. That is the first part. That is the shift. That is the, that is the shift. So, the shift is calculated by this formula. So, WLF gives you actually the relationship of shift or relationship between frequency and the temperature, okay. So, that is what I explain now with respect to Q. It is, in other words, what it simply means is that I can do all the experiment at a master, at a temperature, at a particular temperature and call this as a master curve and then I want it at any other temperature, okay. I can do that, I can determine that not by repeating an experiment, but by using this kind of shift factors. So, that is what WLF tells us, okay. That is a property which we will, T0 is the reference temperature which is usually, T1 is the temperature or T rather, T where the temperature at which I am determining the AT. Now, let us, that depends upon, see what is the value of C1, you know where do you apply, all these things depends upon the elastomer in question, okay. For usually, this is applicable for the range of interest, for what we have, right. So, 10 to the power of, say if suppose I have a 10 to the power of 1 in a frequency or 10 to the power of minus 1, I want 10 to the power of 5, it is possible to do that. Before we go further, let us look at the frequencies of interest. Then we will understand how we can apply this. So, let us understand what are the frequencies of interest. In order to do that, we have to now look at the road. So, this is the elastomer behavior. Now, shift to the road, right, okay. Because this friction is going to be very important and the friction is very different from what you see in metals. So, it is important that we understand how it works. Now, we shift to the road and we are going to look at the road from two perspectives. One in a macro perspective, okay. So, the road looks, say let us say that something like this. It is not that as big as this. Let us say that that is about 1 centimeter is this, 1, 2, in terms of centimeters. This is, what is this? This is those gravel stones which are sticking out as you see it in the road, okay. So, there is a binder between them, okay. The asphalt binder is what is, binds this kind of stones. So, these are the stones. This is at a macro level. If you now go and look at the stone more carefully, you have also small underlations or small micro roughness which you call as something like that, micro roughness, okay. So, these are the two things how actually a road is, can be looked at. A macro which can be in terms of a centimeter or so and micro in terms of a few microns which can at the maximum be 0.1 millimeters. That is this roughness is the order of say 0.1 maximum of 0.1 millimeter, much less 0.01 millimeter and so on. A few microns to maximum of 100 microns, right. Okay. This is how you look at the road. What does rubber do or what does the tire as it contacts the road, what does it do? It actually engulfs the whole road. So, you can see that this, that would how say a tread block would look like. So, it engulfs the road, right. So, because the deformations are quite large, it just goes and sits over this gravel or the stone, right. Now, what happens? Two forces are exerted. One is the force which is normal force and the other say let us say that traction, braking, whatever it is a tangential force, okay. These are the two things that act on this block. Now, again there are two things that happen. One is the micro-macro, the other is two forces and there are two phenomena that happens between the rubber, the block and the road. The phenomena can be classified into what is called as hysteresis, molecular adhesion or adhesion. You can call this as molecular adhesion, okay. So, molecular adhesion again happens at a very small length scales and hysteresis happens at a much larger length scales, alright. Okay. So, let us quickly look at this because then we will go there and understand this. What is meant by, what do we mean by this adhesion? Actually, these long chain molecules which we represented by means of a simple model of spring and dash pod, these long chain molecules get attached to the surface, assuming right now that the surface is dry. It gets attached to the surface through what Van der Waals forces or Van der Waals bond is what is formed between these long chain molecules and the surface. So, they form this bond. Let us say that this is a rubber piece, okay. They form a bond with the surface molecules or the surfaces that exist, right, the surface and then they are subjected to a force. These bonds that are present here, it breaks. So, it sticks, breaks, slips, then again sticks, breaks, slips and so on. So, at the adhesion, at the molecular adhesion level, there is a sticking, slipping, sticking, slipping and so on. So, it is the Van der Waals bond that forms, breaks, forms, breaks and so on, right. But what happens here? What is this hysteresis and what happens here? That is a very interesting phenomena and we already know that when a rubber piece is subjected to loading and loading, it loses energy through this phenomena of hysteresis, okay. Now, let us say that for a moment forget about that. We have normal forces. Obviously, the normal force would give rise to a contact pressure which would act something like this, which would act something like that. Now, when there is a lateral force that is, that this block is subjected to, then obviously, this kind of symmetric distribution, okay, which equilibrates the normal force is no more maintained and this pressure distribution becomes unsymmetric because it has to now equilibrate that lateral force as well. So, what was nice like this, which is equilibrating this, would also have a force which has to equilibrate this and hence the vectorial sum of these two forces would be something like that and with the result that the pressure distribution now shifts. Superpose this on to this, this two phenomena. What really happens at this point is that there is a slip, okay, there is a slip and this whole block can move out of this hill, okay and slide down. In other words, the loaded region, do not look at it as this big. Now, we are talking about this whole distance itself is 1 centimeter, like maximum of 1 centimeter, right. So, keep that in mind. So, what happens is that this slips, it goes over the hill and slips. So, when it goes like that, what was loaded becomes unloaded, becomes unloaded, okay. So, when it slips, it may again get reloaded and so on. So, the frequency of loading, unloading because of these macro, this kind of macro pieces, okay, depends upon the velocity of sliding and is given by simple v by d. The distance between the two, this I would call this as d, then v by d is the, is that frequency with which this slips. The frequency with that stick slip happens is much less. So, if this is around 10 to the power of 4, this would be actually 10 to the power of 8. So, that is the frequency at which it slips. So, in other words, in other words, there are two things that contribute to what we call as friction. There are two things that contribute to what we call as friction. One is the hysteresis loss and the other is this kind of micro-molecular interactions. So, that energy is what gets converted into a frictional resistance in rubber and can be written in a very simple form as the friction forces or the friction coefficient, if you want, can be divided into two categories, one due to addition and the other due to hysteresis, right. So, we can call this, we can, or we can quantify this by means of a simple equation. This is from by Moore in his very good book on friction on, of rubber. There is a book on pneumatic tires, very early book, very good book. So, I am taking this from there. So, that is the reference. So, that is given by addition and let me, let me write down that carefully K 1 into S into E prime by P power R into tan delta and mu hysteresis is equal to T to K 2 into P divided by E prime N into tan delta. S being the effective shear strength, S is the effective shear strength of the material. P is the pressure that is acting, the normal pressure that is acting. The exponent R is usually about 0.2 and N is greater than or equal to 1. So, essentially friction is the result of these two. In order to put this in proper perspective, we will go a bit back and forth, so that we get a macro picture of the tire whole tire. We are going to micro, we are going to molecular level. We look at the link scales again when we summarize, but what I want you to understand is that, is that this is happening in a rolling tire, that is phenomenal. So, a rolling tire is subjected to this kind of phenomenon. So, as it rolls, let us say that these are the tread blocks, that is the carcass on which we have what are called tread blocks, we saw that yesterday. So, these tread blocks go through that kind of cycle, that kind of you know, engulfs the stones that are there, gravel that are there, they develop a bond between the surfaces, all these things happen when this whole tire rolls. So, that is very interesting. You can look at how fast that is going to happen. So, in order to get again a macro picture from a tire perspective, let us see what happens and then again we will come back here. So, let us see what happens in a tire. Let us now consider a tire again, you know in isolation and then tie up after we finish that, we will tie up the concepts that we have explained now with the tire. The first very first thing that we notice in a tire, pneumatic tire is that tire which is like this, say let us say that it is an inflated condition, kits deformed or is deformed when I apply a load and put it on the road and it when it rolls, this deformation again takes place obviously. So, in other words, in other words, there are different radiuses that are present in a tire. One is this free radius. Now, because it gets deformed here, the free radius is not maintained and hence you have at this point, we will call this as loaded radius. So, I have a free radius R, let us say that that is a loaded radius and in between the two, what you do not see here is what is called as a effective radius R e. What is effective radius? Very simple, if this guy is not to be deformed, is not deformed and it just rotates with a angular velocity omega, then the tangential velocity V is given by omega into right. So, in other words, as the wheel rotates with an angular velocity omega and the wheel travels with a longitudinal velocity V, then V is given by omega into R e. So, R e is defined with the same formula that you are all familiar with from fundamental dynamics. Before we go further, let us understand this contact patch. From a plan view, let us say that the contact patch looks like this, lot of niceties in this, we will come to that a bit later. Let us say that that is the contact patch. First things first, when the tire is stationary like that, it is not, omega is not there. So, when this I am saying that this is stationary, then you have a contact patch of course, the region where the tire is in contact with the road. There are different points on the tire which sit at different point on the road. So, they all these guys sit in different points of the road. It is simple to understand, nothing very difficult. On the other hand, let us see what happens to the contact patch as the tire rolls, as the tire rolls. In order to understand that, let us do a small thought experiment. The thought experiment is necessary to dispel some of the ideas which you may have, which I have over the years students asked. So, we are going to do this thought experiment. Let us say that these are the treads or the bristles that are sticking out. So, let us say that you are going to sit there. Do not worry, this is only a thought experiment, you are not going to be hurt. Just sit there. Ask your friend to go and sit at this place and another friend who sits here. Now, the tire rolls. So, as the tire rolls what happens, you would go and hit the ground. You would go and hit the ground. So, the new situation would be like this. So, you would have hit the ground here in this region. Let me just extend it. So, it is not that you are going to become zero. So, you will get compressed. So, you will be at this position. You are just going to hit the ground. The first thing is that you are not going to travel in space like this. You are not going to travel in space like this. Then what happens? So, you hit here. Let us say that you are not slipping. You are just your forces are such that addition is such that you are going to stick there. Then what would happen? Your friend who is sitting here, that guy would hit the ground before you. Then you are the other friend. He will hit the ground before that guy. So, in other words, in other words, you can imagine that the contact patch is actually traveling over you. It is not that you travel over the contact patch, but the contact patch travels over you because as new guys keep hitting, relatively, you would be moving back in the contact patch. So, in other words, what happens is I will hit here. I will stand here. There is another guy who goes there, someone else and so on. So, with respect to the ground, with respect to the contact pressure, actually I will go, I will be going behind. So, ultimately, I will come to a situation where because of this rolling, I will come to a situation where I am at the end of the contact patch, I will rise up and go out. Now, let us see what happens when you stick. You are subjected to a pressure. So, you are subjected to a pressure. So, let us draw that situation. Let us just take that out. You are there. You are subjected to a normal pressure and let us say some force is applied and you are also subjected to a tangential force, which becomes a shear stress and so on, traction. Under these conditions, under these conditions, what actually happens? I will go stick to the ground. As long as possible, I will be at the place. I will be pulled. So, I will be stretched and so on. Depending upon there is a braking or there is acceleration, I will be shifted in the two directions, different directions. But I will stick to my ground because I have already established my relationship with the ground and the nice guy is holding me. And at one point of time, I would not be able to stand there because of all these forces. I have to slip a bit. I will just move. In other words, there is a micro slip. When I, from here, I will just move like that, some micro slip. Then again, I will be subjected to a contact force, which will not be the same as what it was before. We will see that. So, again I may stand here. So, in other words, if at all I move in the contact patch, I move over very short distances, micro distance, micro slip, very short distances. I do not slide or go from here to here. The contact patch, as I said, goes above me and at the maximum I am going to just move like that. So, the first thing that is important to understand is the motion of this contact. It is you are not moving in the contact patch. Contact patch is moving over you. That is the first lesson. Let us get back to the tire again. Let us see how actually the change in radius has an effect on the velocity. So, that is my tire. Since I am going to hit the ground, that is my ground. I cannot penetrate the ground obviously. So, I will be like this. That is the free radius. What did we call as R0? What is that around? I think we should maintain, let us call that as R0. Let me call that as original radius so that, let me call that as R0. If I change a symbol, please let me know so that I do not confuse you with various symbols. Let us say that the original or the undeformed radius R0. So, that is R0. Let us say that the RE factor. What is RE? This is the equal, like we used RE. Equal and radius. That is smaller than this R0. So, let us say that that is RE and the loaded radius RL. Let us now look at what happens at the contact patch. We look at the physics and then we will look at the actual values. Now, again a thought experiment. Go and sit here. My tangential velocity V omega R0 where omega is the angular velocity with which I am rotating. So, there are three conditions that exist in the contact patch. What are these conditions? Three conditions are that there are situations where the radius is more than, so that would be the radius, would be more than RE or equal to R0 more than RE but less than R0 and lastly it will be less than RE or in other words, in other words this is the actual case. In other words, R0 changes to a loaded radius RL and RE sits between the two. So, there is a region. Let us just take that line, this line. When I just hit it, I am R0. Somewhere at this point, my radius is equal to RE, that is the point and somewhere at this point or the center I am RL. Again as I go back, I have RE, go out, I am at R0, just go out. So, there is a continual, continual variation of, continuous variation of radius from R0 back to R0. Clear? Now, let us now look at the velocities. The velocity when the tangent velocity here is omega R0. We tangent, let me be very clear so that you do not get confused. Let us say that, that is the velocity with which the vehicle is moving. Either you can look at the velocity of travel V and a stationary ground or you can say that the wheel is stationary and the ground is moving. It does not matter. It is a relative velocity. Ground is moving in the opposite direction. It does not matter. However, you want to imagine, you can do that and it is not going to make a difference. Now, obviously this velocity, tangential velocity will be more than the velocity of travel which is equal to omega RE. So, in other words, here is a region where the tangential velocities are more than the tangential velocity. Velocity of travel V, we call that as more. Here exactly the same. From here to here, it is less. Here to here, it is more. Get up and go. You are back to or not. Oh, that is a complex situation. Imagine that this is happening as the tire rolls. The guy is really confused. Initially, he is happily coming, hitting the ground. Suddenly, you know, he is slipping in one direction. Then he meets RE and again goes in the other direction. Then again comes back to a situation where he has to go the opposite direction. So, mechanics is very complex and that is what makes tire mechanics very interesting. So, if you look at the direction, you will see that they are in the opposite direction in one direction up to this. From here to here, they will be in the other direction. From here to here, they will be in this direction and it goes up. What are they? They are likely slip. An interesting point is that we just now saw that without that slip, there is no friction. So, your friction concept has to now change because friction is now contributed due to hysteresis. It is that energy loss as well as due to addition. So, without slip, it is an oxymoron, but it is actually like that. Without slip, there is no friction here. So, the tread is subjected to a very complex phenomena at this position. When it breaks, the situation is very different. Added to this are some subtleties. What are the subtleties? We will just mention the subtleties and stop. So, imagine that you are now you are so tired standing there, getting hit all the time in the ground. Let us say that you are going to lie down there happily in a bed like this. Now lying in that bed, you are going to approach the ground. There is already a signal that ground is approaching. How? Because of this radius change, when you are here, the circumference gets reduced or is reduced and you are going to feel that as you go near, you are going to shrink actually. Your bed is going to be compressed. So, that is going to be compressed. So, as you go in, you are going to get actually compressed and as you go out, you are going to get elongated and go out. So, in a circumferential sense, the radius change results in a circumferential change. So, there is a compression, more compression, tension, in the sense release of compression. So, what is that effect? We will see that in the next class.