 So, in the last class we were looking at the lateral force generation. We said that the key factor in the lateral force generation was the distortion, which of course to the contact patch, which actually makes the tire assume a direction, which is at an angle to the direction in which the tire is supposed to be moving. So, if this is the straight running tire and if you now, let us say that that is the central part and then if you now give a steering input, okay, which you said is delta and then the actually there is a slip angle produced and so on. And we saw that the distortion causes that centripetal force that is required in order to sustain the centripetal acceleration. In other words, cancels out the D'Alembert's force called the centrifugal force. The end of the class we said that we get Fy and then we have what is called as the restoring moment. Restoring moment is due to the centrifugal sort of centripetal force not actually acting at the centre and that it acts at a distance which we called as pneumatic trail and then we had a curve which we plotted in the negative direction because it I think we plotted it together. Let me do the same thing here so that that is what we get and that is alpha. Remember that this is Fy and remember that that is the Mz which is the self-aligning torque, right. This is what we did. The question towards the end of the class why I just brought this to you notice is that why is this aligning torque, now why is this taking this shape, okay. In other words, why is the aligning torque becoming 0 when the force reaches that saturation point. Remember that the shear forces, okay the shear force which is responsible for that Fy, okay is something like this, right. So and there is a moment also sorry there is a normal pressure which is we will just remove this you know what is alpha now so which is now because of the fact that assuming that there is no slip because the fact that it is now shifted to the left in other words is an unsymmetric distribution obviously the resultant force will not be symmetric but is the resultant force because of all those shears or all those pulls of those bristles which were there they happen to be not symmetric but it is unsymmetric, right. Now imagine that slowly guys are not going to slip and it slips what happens it reaches a maximum, okay value which is mu into qz, right. So it sort of starts now pushing because all of them are going to assume the value of mu qz and this is this being almost symmetric, okay this one yes it is not exactly symmetric but it is assuming that it is almost symmetric this guy is now going to move towards the center. So in other words this situation at this situation where all of them have reached that mu qz, okay which is the maximum force that can be delivered by these bristles the resultant force position now shifts towards the center and T goes to 0. So if you assume for example a good parabolic distribution, okay the mu qz also will be parabolic and hence it will be exactly symmetric and the force sitting at the center, okay. So for a brush model with a parabolic pressure distribution the aligning torque goes to 0, right. Actually it does not go exactly to 0 because the fact that it is not very symmetric and hence it may actually be something like this and so on, okay. Now the question we asked in the last class was that is the lateral force development only due to cornering is there any other means of developing a lateral force that brings us to a very important and interesting topic which the tyre manufacturers call as Pleistia and Coenicity. Pleistia as people call it is something like a steer like behavior and Coenicity is something like a camber like behavior. Now let us now let us look at what is Pleistia, yes any questions, okay. Now let us look at what is Pleistia, let us say that that is the tyre which is moving this direction, rolling that direction, right. What is a tyre? Let us look at this say for example a track tyre similar is what you have for passenger car tyres and you have the tyre here, okay. Now it has what are called as belts, okay. In this case it is steel belts, okay and there are a number of belts it can be 3 belts, 4 belts and so on, right. If you look at these belts closely I will just zoom that here and draw it here. If this is one belt, if this is one belt I already told you this that there are steel cords which run like that, okay which run like that and that there may be another there will be another belt, okay which is where the steel cords run in different directions and so on. There is an angle to this let us not worry about that what is exactly the angle and so on, right. In other words this whole of this belt bundle, the bundle which is just above what we call as the body ply, okay behaves as a composite laminate and what is the importance of composite laminate? It is a very interesting property. Maybe you have studied this in mechanics of materials, okay let us do that here. So if I have now a sheet say for example made of steel and then pull it, we told this already in last class that it just gets pulled and that there is a Poisson's effect in the other direction, okay. Now let us see what happens when I pull this here in this case. Now that is a sheet, okay that consists of a number of all these what we call as reinforcements, these belts. Let us say that they are all together and now I am going to pull this sheet, okay. If it were steel I know what you would expect it would just go like that and this will be in one plane. There would not be out of plane, out of this plane there would not be any deformations, right. Now let us see what happens here. Yeah you pull it, oh wow I did not apply any force in the other direction. I just pulled it and what happened? The whole thing twisted, okay. So in other words there is a coupling between this longitudinal force or one force to the other moment and so on, right. Clear. So this is one, running in one direction. If it were to run in another direction then it would go the other way, okay. So these are called as coupling stiffnesses, okay. There is in other words a coupling happens between one plane what happens in one plane to another plane. In order to explain this let us put down, let us go into some details on the laminate theory. We will not go too much details into laminate theory but we have to understand this so we will go into some details of the laminate theory and what does this laminate theory say? Let us look at what we call as the constitutive relation for a laminate plate or laminate plate or what is called as the laminate theory, right. So what are the forces and moments that actually what is the constitutive equation? I have the in this case forces moment and have some terms here, constitutive terms and this is what I would call as strains and so on, right. So let us say that I have nx force ny, nxy, mx, my, mxy, right, okay. What is this? If I have laminate like this just give a small thickness to it so that the thickness is say 2h and let us say that there is a v direction and that is the x direction and the force that is happening here is this nx force here in this plane is sorry ny, force that is happening here or force that is pulling here is nx and nxy, nxy is this and nxy, nyx and so on or in other words if you want to write that as ny, okay, y, x is the plane, y is the direction so if you want to write it like that this becomes nyx, plane, direction and nx. We know that nxy should be equal and so on. Now what is the right hand side? The right hand side of the strains, epsilon x, epsilon y, epsilon xy and then it is a laminate theory so you will have what we call as curvatures. So you have curvatures x, curvature y, xy. Now these two are related by this constitutive equations. Let us say that we will divide it by 3 by 3, 3 by 3, 3 by 3, 3 by 3, 3 by 3 so we will call that as A, we will call that as B, we will call that as B and that we will call it as D, okay. Now let us see what A is. So there are terms here. Let us not worry about how the terms are calculated, any composite books will, book will give you the terms how it is calculated. So we will call that as A11, A120, B11, B12, B16, okay. So these are terms which depends upon the new values EX, E values and so on, okay. So let us not worry what it is, that is not our intention. So you can go on like that, B12, B22 and A so this is a way of, I mean the numbers are important so we will write this as B226, okay and this would be A, we will say A12, let us be correct to a book, A2200, A66, B16, B26, B66 and that B gets repeated there again, okay. Same way you will have D11, D12 and, yeah they are values, okay which are coefficients. What are they? Very similar to what you know for example in plane stress and plane strain, okay. Say for example you have sigma is equal to E epsilon, right. So you have a constitutive equation so which relates stress and strain, okay. So this, what do they depend for example in isotropic material, what enters into this, into this? E and nu. So you require two quantities for an isotropic material E and nu, okay. So here this is not, this is a laminate and so this is not an isotropic material, okay and hence you have, okay, a matrix which is similar to this. That is why I said, let us not worry, let us not go into the details, it is not to, not to explain laminate theory but to explain why it happens and how that causes what, what is called as the Pleist here, yes. Mxy, okay, good. So Mx, My and Mz are the result of, they are the moments, okay, sigma x, sigma that is the sigma x multiplied by z, okay, dz minus h by 2 to plus h by 2. Similarly, My minus h by 2 to plus h by 2, sigma y into x into dz, right and what is Mxy? Sigma xy multiplied by z, so Mxy is, okay. So the moments are the result of stresses that are acting in this plane. Remember we had a simple beam, remember that we had a beam bending, that is the beam say for example in your earlier classes on mechanics of materials, so when the beam bends you would have, okay, the, how do you have this? So you have for example, things like this, okay, a moment that is created and that would be the stresses that are acting, okay, sigma xx is a neutral axis and that these stresses cause or equilibrate the moment, so that is what you get, right, okay. It is exactly like that, it is a plate. What is a plate? You can assume that a beam is extruded in the other direction, so you have a plate and the plate is extruded in the third direction, it becomes a beam extruded in third direction, it becomes a plate, okay, right. Any questions? Yes, yes, this is, right, this is the way you express, I did not say it is the same, I said it is similar, okay. So this is the way you express for a laminated plate, okay. So express it in terms of force, express it in terms of strains as well as curvature, right, okay. And those forces are of course related to the stress through those kind of relationships, exactly not the same. That is exactly what I am trying to say that they are, that is the force that is acting, nx is the force that is acting and why? Is that sigma x or sigma x? When I define mx, I define it as sigma x or sigma xx if you want to call it. That is not nx, the sigma xx. No, no, no. That is sigma xx, okay. Is it sigma xx into what into z? Z, right, that is what creates the moment, okay. You have nx, ny and nz, right, that is the force that is acting in the x, y and the z direction, okay. Any questions? Right? Stress is in the moment acting. The right hand side, the extreme matrix that presents the deformation in x, y. Yeah. Please note nx is due to sigma x, which means that what is nx? How do you define nx? Just integrate minus h by 2 to from the same thing. If mx is like this, nx is, that is it, clear? Okay. So, note the difference the way a plate is written and a regular constitutive equation. Can you call this as constitutive equation? Yes, you can keep debating that because constitutive equation is stress and strain for a material. Now, you are talking about plate, of course, a grid, right. Any questions? So, this is the equation which you look at. Now, what do you mean by, okay, having understood that, what do we mean by coupling? What do you mean by coupling? When I apply epsilon x or epsilon xx, right, when I apply epsilon xx, I would expect a force nxx or nx. That is all I would expect, okay. Now, there are in the same fashion x, epsilon y and so on, okay. But these coupling terms here give a different picture. So, for example, here I have b12, b22, b16, right. What is the picture it gives? What is that you get from this? Okay, I give you a minute to understand that. So, what is it that, that it gives? Correct, that is it. So, in other words, there is a coupling between the two in the sense that elongation also gives a moment, okay, or shear gives a moment, okay, mx. Yeah, of course, because that is how it is populated, right. That is how it is populated, isn't it? Right. So, in other words, if I change a curvature, if I change a curvature term, say for example, kx term if I change, then it is, it is not that if I change this term, it is not that I get only a moment for changing the curvature. You would, you would imagine that I have a plate like this, you know, I change the curvature, the curvature is changed only by giving a moment. This is what usually you think. But it not only does that happen. In other words, not only is the moment, of course, moment has an effect, but you would see that there are terms which would also give rise to your force, okay. So, in other words, that is why what happened here is that when I stretched, twisted and so on, okay. How do I apply this? So, what is that we are talking about? Why are we talking about this? Changing curvature gives rise to a force, fine. So, why? So, what happens? So, why is that I am worried about it? Why is that I am worried about it? Because the deformation in one direction, there may be a reaction in all three directions as well as... Suppose that is the belt, okay. Forget about direction, forget, let us understand the physics. Suppose this is the belt, okay. Of course, you know that it is just circumferentially going on. I am looking at a cross section, okay. That is the tire, that is the tire and that is the belt. So, when this tire hits the ground or when it is sitting in the ground, in effect, what am I doing? I am changing the curvature, okay. I am making this guy flat, okay. In other words, I am sort of I am giving a moment to make it flat. When I give a moment to make it flat, okay, what am I attacking? I am attacking this. I am attacking that. But when I attack that or in other words, when I change kappa x, if I call that as kappa x, the curvature will change it, then there is one term which is sitting here, which I will multiply by kappa x in order to give n x y, right. I change this, I give n x y term. It is a force term. I change this, here is one guy who is going to give me a force n y term and so on, right. So when I change kappa x, it is not just moment. So you would think that, fine, I have a curvature like that. I am making it flat. How do I make it flat? Maybe give a moment like that and make it flat, right. But that is not just what happens and you have forces. So in other words, I am producing a force here, right and I am producing a force in this direction. That is why I said that let us not, why is that zero? Okay. There is no coupling between those elements. Of course, why? Because there is no coupling term between the shear and the longitudinal force. Poisson's effect is only x to y direction. So gamma x y is g into, sorry, tau x y is equal to g into gamma x y, okay. Go back to your earlier class on mechanics of materials, all right. Okay. So there is, in the words obviously there is a shear term that does not have any effect on the force term, okay. Why is your, sir? In other words, there is no coupling. What it simply means is that I would eliminate all these guys. These two guys will be done with zero, no coupling and I will have only these two terms. These terms are your familiar terms. For a minute, forget about all this. So you would see that just force and, well known, okay. a11 into epsilon x, a12 into epsilon y, 0 into epsilon xy or gamma xy or how you call it, okay, right. That is all. So the shear force is equal to, or the shear stress, which is the result here, is nothing but this multiplied by the shear strain, right. Okay. Does that clarify your doubt that there is a shear term that is involved? That has nothing to do with the normal force terms. Okay. Let us come back. There is a lot more to look at it. So kappa x I am going to change now because I am going to make it flat. When I make it flat, obviously I am introducing forces, okay. I am introducing forces because of this coupling term. And those forces introduce the lateral force, okay, the lateral force and a moment and a moment. So simple fact that as the tire rolls on the ground and a part of the tire becomes flat results in a lateral force called Pleistia. Okay. So if I have this, these are the two tires which are running and it is, right. So this produces, has a part of its belt flat and hence there will be a force that will be acting. Yes, very good. So the question here is that does it get cancelled? It is a very important question. Can it get cancelled? See let us say that I have this belt structure. Okay. Now I have this belt structure. In other words, what happens if I rotate the tire, okay, if I rotate the tire and keep it, okay. You have to be very careful in this. There is a lot of confusion for people in Pleistia in that. Suppose I have this tire, okay. How do I cancel it? If I have this force to act in the opposite direction, can I ever make it happen like that, happen that way? Can you do that? You rotate it. Whatever you want to do, you do. You rotate it, okay. You rotate it, the structure will exactly be the same. Structure will exactly be the same. You rotate it like that, it will be like this. Put a point here, okay. That point will go here, this point will come here. So it is not, you cannot cancel that, okay. But if you rotate in the opposite direction, the force will be in the other direction, okay. You can change that in other words by the structure of the ply in the two directions. So there is always a confusion that Pleistia changes direction whether you rotate it in clockwise or counterclockwise. True, but you do not rotate the tire, okay. You do not run the vehicle in the counterclockwise. You run it only in one direction. So you call this as the right tire, for example, and the left tire, okay. And now superimpose on the, say for example, I plot alpha versus both the aligning, this torque which we called, which we called as aligning torque, torque that happens due to the NXY term and lateral force, the NX term, okay. So that happens to be like this and it so happens that because of the directions that you will get, if this is the lateral force, not left force, lateral force, if you want to call it like that, lateral force, okay, that is the aligning torque. And these are, in other words, when they are 0, not when alpha is equal to 0, in other words, when alpha is equal to 0, okay, what is alpha? Slip angle. We said slip angle alpha is required in order to produce a lateral force. So when it is 0, you would see that there is a lateral force and a corresponding clear. Hold this for a minute. We are going to come back to this graph. You could explain it again. Is this clear? So when you, when it is free rolling, okay, on a ground, nothing to do with the ground, you will get a lateral force, okay. Now there is also what is called as conicity. What is conicity? Name indicates that, conicity. As you had seen that there are what are called as belts, we saw that just now, okay. So assume for a minute that the belts are not aligned exactly at the center. In other words, suppose I say that I cut this tie, okay, let us forget this part of the tie, okay. We will concentrate on only this part of the tie, top of the tie, right. That is where the belts are. The belts are supposed to be nicely placed. That is the center line. The belts are nicely placed like this, okay. They are nicely placed with symmetric about the center, right. I inflate it. I inflate it. There is a stiffness that is given. I inflate the tire. There is a stiffness that is given to the tire because of these belts, okay. So it assumes a uniform, say radius or in other words, it, it are the, the deformations on either side is the same, right. Let us say that that is the deformation due to inflation, okay, of the belt, inflation with the presence of symmetric belts. Let us do a thought experiment. Let us say that the belts are not right symmetric about the center. They are not symmetric. They are not right. They are not symmetric about the center. They are not correctly placed. So I will have, say, let us say that there is a belt, what we call as belt offset. So this guy, okay, left and the right placement of the belt are not symmetric. So what happens, what is that you conclude looking at this left hand side and the right hand side? What is that you conclude looking at the left hand side and the right hand side? Fantastic. So radius in the left when I inflate it will be higher because this is going to be stiffer. So this, these guys sitting here, these, these material elements are not going to move to the same extent as to the left and hence, okay, my tire is going to become something like that. In other words, my tire is going to look like a truncated cone, right. So now you have a truncated cone, let us say, I mean just exaggerate that and roll a cone, roll a cone. What happens? So there will be a force generator. Yeah, yeah, bottom, yes, bottom surface will be flat but still I am going to, okay, like, it will be like this. There will be a conical shape. In other words, the pressure distributions on either side is not going to be the same. So with the result there is a cone, it is as if there is a cone sitting and then roll. The surface, correct, agree with you, surface is flat but the pressure distributions are not going to be the same, okay. That is exactly what happens in a cone. The cone is also when it is placed, okay, it assumes the surface and one side becomes flat. It is not that it is like this, it becomes flat. So when you rotate it, why is that it is not going straight, it goes like this, right, exactly the same thing, right, okay. So the cone you are placing it on the ground which means that, please note that one side of the cone is sitting like this. So the cone will be sitting like this, okay, fine. So this is what is called as the cone city effect. That is giving rise to a force, right. Now this is one thing which is not necessarily is one side or the other side because it depends upon which side it is shifted. So if you now take a number of ties, you go to one of the manufacturing plants, entire manufacturing plants, you take hundreds of ties. Now and then, okay, say that I want, I measure hundreds of ties, I measure the lateral force. Let us say that that is the lateral force. Let us say that is positive and that is negative and let us say that this is the number of ties. I take 1000 ties. What would happen is that, okay, you will get, suppose I get a peak like this, a very pointed peak, okay and say I get a peak, I get something like this. I have ties which are, so if I get a peak like this, suppose I get a peak like this in the ties, if I get a peak like that in the ties which I have tested, which this means, one minute, this means that this peak, peak is what? All the ties, whatever I have tested, okay, let us say that it falls within this gap. All the ties which falls within this gap, small gap, almost the same, then that lateral force is Pleistier because Pleistier is due to the design, okay. Here, if it so happens that the tire, the force distribution is something like that, then there is, that is not Pleistier. That is not Pleistier basically because it is distributed on either side and so the belts are offset this side or offset the other side, okay. We will come to the effect, this diagram in a minute. Let us forget, let us go and look at, are there any other things that cause a peculiar behavior of the tire, right. So in other words, straight running tire, is it going to be a nice guy who is going to run straight or he is going to do other magic, other things. Yes, yes, it is most of the, most, in most instances, it is this belt offset that is going to cause cornicity, okay, that is a defect, right. Now that is not the end of the story. There are other things that are happening which are peculiar and which is not covered by all your courses in mechanics of materials. So what are the other things that are happening, okay. Let us look at that now. Now let us look at the contact pressure distribution, okay and the lateral forces that take place or that is present when I roll the tire, okay. Now you see two things there. So that is, that is the lateral force distribution, right. There is a lateral force distribution that is happening. What is the importance of this lateral force distribution, okay. Now why is this important and what is that we are going to look at. Actually that is the previous picture is just to zoom in and show you the values at the inner ones. Now that is in the longitudinal direction. In order to understand that, in order to understand that picture we will come back again to the lateral force and longitudinal force. Now let us assume that the tread is a block, okay. Now if you look, if you look at these blocks, they are not straight, you know, squares like this. That is a plan view, this is a plan view and so they are not nice squares like this, okay. They are say a rhombus type of figures, okay. All those blocks are rhombus type of figures, right. We will explain this in a minute. We will come back to that. So in other words, they are not what is there at the top, okay. They are more a rhombus type of figures. Now what happens when I apply, see let us say that let us now model this blocks. This is a plan view, okay. So now let me model this block, okay and apply a force, say a square block, apply a force in this direction. I am applying a force in this direction, okay for both these things, right. Now let us see what happens. So I can do two things. One is that, one is that both of them, the bottom and the top are fixed not to move in the z direction, okay. Both of them are, the guy is stuck between two platens. This is what happens to a block when it is in the middle of the contact patch. At the edges of the contact patch, one guy is free, okay. So I can free him, I can free him, okay. We will apply this load. Let us say that I have a square cross section, I apply this load, okay. What do you expect? You expect it to move the direction or bend in the direction in which I am applying the force here for a square block. I change this block to rhombus. You would see that there is a difference, okay. Now that is the top view. That is the top view. See what has happened? Of course it is out. See what has happened? Let us concentrate on the left side. So left side clearly shows that this is a finite element picture. It is just an isotropic rubber block. So it shows that when I apply a force F, the displacement Fx, let me call that as Fx. The displacement is only in the x direction. For that rhombus, okay, which is very similar to that there in the block, in the tread block, if I apply a force Fx, that is Fx, there is also a displacement. Look at that. There is a displacement in the other direction, y. You can see that in the bottom picture. Civil engineers long ago recognized this and called this as unsymmetric bending. In fact, when you take this next course on automotive structures, you are going to talk a lot about unsymmetric bending, okay. They realize that there is a principle axis directions, okay, along which there is no coupling. And again there is a coupling factor that is involved because I am not applying the force along the principle axis direction, which happens to be the symmetric axis, okay, that you see, for example, in the first picture. So if I now have a block whose plan is like that and if I have symmetric axis, okay, it is symmetric bending. It is a simple cantilever beam. It bends. But of course, it is not a cantilever beam as you know or you have studied it because it is a short, fat fellow, okay. Hence, shear deformations become very important. There are other things that are happening. And a simple beam theory which you have learned, a cantilever beam theory cannot be applied, okay. It is one thing. Now if this plan view happens to be not so symmetric, say now it is something like this and I am applying a load here in that direction that is not a symmetric axis and hence a coupling happens in the other direction, okay. And hence there is a push, okay. Now that in other words, that results in, if I stop it, that results in a force. If I stop this, that results in a force, okay. So if now go back to my first picture, we will explain that in the next class again. Now I have, I have shears that are happening, okay. Let us say that I have, I have this kind of rhombus blocks sitting, sitting like this, okay. And there are going to be, even if it is, when it is rolling, remember that there are going to be lateral shear forces that are going to happen, okay. That would be the lateral shear forces that would happen or that would act on the stretch and that lateral shear forces results in forces that are acting in that direction, okay. And hence would result in a torque which is called as the Spratt PRAT, Pleistia residual aligning torque. I have to be careful in this but anyway shape is, you know there will be, this is a very simple method, simple thing but there will be a, because of this, this shear forces that are acting, okay, as it goes straight, the shear forces that are acting, okay. Remember that the way the contact pressure shear distribution is there. So because of the shear forces that are acting, there are forces, because of this coupling, there are forces perpendicular to those shear forces, okay. At the, at the ends there are shear forces like that and this is the shoulder region where there are shear forces which are happening like this and at the end of the contact patch you have shear forces that are happening like that and because of which, okay, you have coupled forces which come along with these shear forces and the coupled forces which are acting perpendicular to these shear forces. So if there is a shear force like this, this is the, that is the contact patch. Say that is the end of the contact patch and so the contact patch you have, okay, shear happening like this, here it is happening like this and these two they are happening like that and so there is always a perpendicular force that is happening, okay, that is, that is the result of, this is the result of this coupling. So when you have perpendicular forces that are present and that results in torque, okay, towards the centre of the tyre which is, this is the centre of the tyre, okay. In other words you see that, that is the, so that is the, that is the centre of the tyre and so there is a moment that is acting. So this is another reason why there is an aligning torque that is happening when you are going straight, okay. We will come back to this and we will continue with this and the other derivations in the next class.