 The last class, we were looking at the lateral force development due to other reasons than just taking a turn on maneuvering. In other words, we found that even when the vehicle runs straight, there are lateral forces and these forces are generated due to three factors. We saw one is that what we called as Pleistier, okay. This is because of the fact that there are what we called as belts or composite material and we found that these composite structure which are due to the steel, for example in the case of a truck tyres which are called as steel in all steel radials, right or steel radials. If it is called then the steel is the material which is used as a reinforcement for the belt. It runs at an angle and we said that in the composite laminate Pleithiery, when you stretch this kind of composite material, there is a coupling that happens and that when I stretch it, there is going to be a moment or in other words, force and moment, there is a coupling term that is there. One of the questions that was asked by her is that she said what is this constitutive equation you suddenly said M versus kappa, right. What is this M versus kappa, okay. Yeah, I know you might not have had the background in composite material, okay and you might be wondering why this suddenly one kappa is coming and why is that it is just not sigma versus epsilon curves and so on. It is easier to write that way, you can solve any of these things, any of these structures using a general 3 dimensional concept but that is not the way it is done because for bending we have for example beam. So if you look at beam, you know how do I look at this kappa, if I look at beam all of you know M by i is equal to E by r of course is equal to sigma by y, very famously written, right. So I can for example rewrite this equation as M by, I will just remove that E i by r. So what exactly have we done? Note that I have say for example a beam say simply supported beam and say that I have some uniformly distributed load on it, okay. So what essentially have we done here? So there is a displacement, there is a displacement of course and which results in radius in other words change in curvature. So one by r is the curvature and so I can write that as E i into kappa, okay. So you see that even in your regular strength of materials you had used or you had learnt that there is a radius change. Of course many of you would not have used this equation. The equation that is usually used is M by i is equal to sigma by y is what you would have used. You might not have used E by r, I just wanted to point out that it is not a very new equation or new way of writing is exactly what you had written in your earlier classes, clear? Okay. So the other one if you remember, so in other words what did we do here? We changed the curvature, this is exactly what we said, we are changing the curvature. Imagine that I had a beam like this and I am applying a load and changing the curvature like that is essentially what you did when the tyre rolls. You change the curvature as it hits the ground, clear? The other one which we saw is called as conicity, okay. We said that conicity is basically because of the unsymmetric geometry that results when a tyre is inflated for example due to say a belt shift, okay and that produces a force and depending upon which side this belt actually shifts, the conicity also would vary whether what they call as the serial side where things are written or the other side. Depending upon that there would be a change, okay. Now there is as I said there is always a confusion whether the tyre is moved in the clockwise direction, anticlockwise direction, the forces change or they keep the same, there is a lot of confusion on that. The Pleist air, please note that the Pleist air for example if I have, let us say that is the top belt by which it is represented and if I have a force that is acting here imagine that you know even if you rotate it, it would look like this, right. This point say for example may take, go and take that point so it would look like this. So actually you do not rotate a tyre in the clockwise and the anticlockwise direction. So it is always clockwise direction, it always looks like this. So the Pleist air forces will be acting like that, right. The last one which we talked about is due to remember we talked about the Pleist air residual aligning torque and this is also due to the tread patterns, okay. We said that tread patterns are again, so let us say that I have tread pattern like that and just simple way of representing the tread pattern is much more complex than that and we said the tread also is has a coupling term, right. We saw that the day and that results also in a lateral force giving rise to a longitudinal force and vice versa and because of which there is a torque that is created and this is what we called as Pleist air PRAT, okay. There are standards for this, how much it should be. Some of them may say that PRAT should be, pratt should be 0 and so on and so the, what we call as the design of tread has a lot to do with many, many things. One of them being this, the other of course the most important thing is how water is carried and so on. We will have a brief discussion on that in one of the later classes before we close this tires, okay. So this has a number of effects, you know, this kind of the conicity in Pleist air has a number of effects. Yes, so the question is that all of them happen due to road-tire interaction. So what happens because of these properties or because of these forces generated? That is the question, right. This is exactly what it is. So when you say for example, if I have a tire. So what really, in other words, I will just post this question slightly differently. So what happens to this force? What are the result of this? Do these forces really happen when the road-tire interaction is there? And because of which, what would happen to the vehicle, right? This is the question, good question, very good question. And let us see what happens to the vehicle. So in other words, first thing is that when I roll the tire on the road, remember that Pleist air is due to the change in that curvature. So when I roll the tire on the vehicle, automatically a force is generated, okay, a lateral force is generated, right. So lateral force is generated in the front and the rear. Let us say that that is my rear tire. Let us say that is the vehicle, okay. Now I have a number of forces that are acting. Let me do that. Let us say that that is the say a force due to Pleist air. Let us give some values, okay. Taking from original paper, say let us say that I have some values. Then I have other forces, right, cone-city force. For example, for the time being, this talk, we will just leave it for time being. And we will look at cone-city forces, okay. Cone-city forces depend upon, as you said, because of this belt offset can be in any direction. Let us say that it acts in this direction, all right. And it can be anything, it is not a constant because it is due to the manufacturing and so on. It is 25, 20 or 15, 10 or whatever it is, okay. Then the rear tire, let us assume that they are the same as 70 because the tire design may not be the same but let us just assume that 70 or 60 or whatever it is, okay. Let us say that the rear cone-city factors is say 10 in this direction and 20 in this direction, all right. Now, so in other words, definitely there is an imbalance in this vehicle, okay. The force distribution now, remember that it is going straight, clear. Now this is the distribution as it goes straight. So what is the effect of this kind of distribution? Obviously, the vehicle is not going to go straight, okay. So there will be a tendency for the vehicle to actually travel like that, okay, right. There will be a tendency to pull, in other words, there is a steering pull. So you have to constantly steer the vehicle in order that it goes straight when these force distributions are not uniform. So one of the things that you can do is to actually shift the tires so that this kind of distribution is not bad, okay. Eliminate it completely difficult but at least you can make it small but what is the effect of this? The effect of this is obviously that I am going to constantly steer the vehicle which means that this force, opposite force have to be generated which means that there will be a slip angle and hence there will be wear, okay. This kind of forces would cause wear and that is the reason why, you know, wear will depend upon whether it is a front or the rear and that is the reason why there is always a rotation. What the tire companies call as a rotation plan so that the forces that are generated does not harm the wear of the or does not extenuate the wear of the tires, right. So this is what is the effect. There are the other effects which is called as dog trailing when the rear gets shifted and the vehicle and the front is like this and that is what is called as dog trailing that happens. Yes. Yeah, no, I am just doing that arbitrarily, okay. You can do, you can have 15 that is why I said this can be 15, 10, 15, 5, 0, yeah, that is exactly. In the direction, so this is what we plotted if you remember that if I take, if I go to a factory and if I take 100 tires, okay and then plot the lateral forces say for example, that is the lateral force, forget for a moment, pleist air, I know what is the pleist air value, okay so I can find out what is the cone of city value then the distribution would something be like this. But in other words there will be positive, this is what is this number of tires, the lateral force distribution will be something like this because I said that this is due to belt offset and belt offset can be in either direction, in either direction. So because of which you will have a minus lateral force and a plus lateral force, okay whether the cone is going to become like that or the cone is going to become like this. Why is there a cone? Remember that when there is a belt, okay and when there is a belt in the tire and the belt gets shifted, one side becomes stiff, the other side is not as stiff as this. So in other words, just go to the only the crown of the tire, let us say that that is the center, okay and that is the buttress region and so on, let us forget about that for a minute and so I have a distribution of the belt and if the belt now gets offset, is say for example, instead of being symmetric about the center, if the belt happens to be something like this, okay offset in one direction, we saw those belts the other day and when the belt is shifted then due to inflation pressure, okay so now the belt is like this. So this side due to inflation pressure would go up more, okay than this side which is stiffer because of that belt offset. So it would not get a nice symmetric shape and the shape of the tire after inflation would look something like that in fact, okay. So the crown of the tire is not going to be symmetric but would incline. So when this tire is rolled, you produce what is called as the, okay. Now unfortunately these are not the only things which produce lateral forces. The lateral forces are also produced by what is called as the camber, this is actually a motorcycle camber exaggerated, just enlisted a point that there is a camber and the camber you know about camber and the camber also produces a thrust, yes. Why do we have the offset in belt? We do not introduce that, the offset, why is that there are offset in belts, this is a manufacturing process, this is due to the manufacturing process. As the tire is built, the belt is placed, okay on a drum so and then it is taken over to a curing press, okay and it goes through what is called as a curing process. So when this tire is manufactured and they are placing the belt and then cured and so on, the belts do not stay in place. Remember that these materials are viscoelastic materials and the viscoelastic, it is when it is in a stage where it is in a pre-cured stage when the curing is does not, has not yet taken place, okay, it is in a pre-cured state, okay. This viscoelastic property, it becomes highly viscoelastic, in other words it starts flowing very short time. So due to so many reasons, these belts are not necessarily symmetric. So this belt gets you know is offset, whatever be the belt with it is going to shift, okay. There are other things you know into design is a good question but then we are going into tire design, there are very good question, why is that we have, we just have a long belt and be done with it. One of the cardinal principles in tire design is that your belts edges or singularity points, okay and as far as possible you do not get the singularity point into regions where the stresses are high, okay, you do not get that singularity point. This is what is called in our mechanical engineering design a stress reser. So the tip of the, of the tire of the belt, okay is a stress reser. So it is good not to get that into a region where the stresses are already high. You know that when you have a very sharp corner for example to singularity point then you will have stress concentration and so when there is a stress concentration the stresses increase, okay. So that is the reason why you cannot have as long a belt or as big a width of the belt as you would like to have, okay because you would now enter into regions where the stresses are high so the belt edges would become very vulnerable, right. Would not be able to go further on than that because of lack of time but tire design has so much of engineering principles. So it is such an exciting field because there are so many things that act. So in fact if you, if you look at tire design you would learn the mechanics of material at the highest level, okay because you are dealing with a composite material. You are dealing with a material which is not linear. You are dealing with a system where not only failures you have to avoid which is very, very important, okay but also the durability, the wear characteristics have to be very nice, very good. You have to design in such a fashion that handling characteristics are not affected, the noise of the tire is low, the wife does not produce vibration, oh you have covered the whole of engineering, right. So that is why tire design becomes very exciting because the whole of engineering, whatever you have learnt, tell me a course and you have an application here. You studied for example long chain molecule, macromolecules, right, the behaviour of macromolecules in the first year that was your chemistry course. So macromolecular chains, how they act completely forms the basis for the behaviour of the tires, okay. How does the macromolecule, this is a macromolecular structure, this is an elastomer. So whatever you studied in chemistry is applicable here. So tell me a course and I will tell you where it is applied in tire. So that is why it is very exciting, right, okay. Come back and we will look at, does that answer your questions or there are forces. But unfortunately the forces do not stop there and we will come to the camber part of it, okay. There are two things that are important, one is the camber of the vehicle, okay. It can be 0 camber in a car, it can be 0.5, it can be 1.5 in a truck and so on, okay. So there is a camber of the tire. The camber of the tire is superimposed on the camber of the road, right. So there are two cambers and they act together. For example, the camber of the road can be like this and the vehicle can be going here like that if it happens to be a two-lane traffic and in some of the, what you would call as kacha roads or village roads where it may not be like this, you may be going in this way, whether it be one and so on. So these kind of cambers are given for the water to be drained out and so that is also going to have an effect and what is the effect? The effect of camber is to give what is called as the camber thrust or camber force. From the point of view of a vehicle stability, this becomes very important in a motorcycle tire. From the point of view of wear, especially in truck tires, the camber thrust becomes important. Let us see just what happens as is our practice, let us go and sit down in the tire and see what really happens and how this force is generated. So if assume that there is no road, if there is no road, actually this would be the tire profile. That would be the tire profile. Imagine as if it is penetrating the road and now when you roll it, the tire would take an elliptic curve goes like that. Now that is not the case. Road has an important effect and the road actually pushes a point, say for example, which is here which is here onto this position. So let me call that as P and P1 and P2, P1 a point, if you go and sit at a point P1, you will not be following P1 but you will be shifted to P2. Actually, you will come here, is that clear? So this is actually the shape that is taken by the tire is that, what is seen in the white line. So in other words, there is a shift in the z direction, if I call it as the z direction and there is a shift in the y direction. This is our y direction. So this is shift. Of course, this shift in the z direction produces from the road that reaction which we called as FZ. Remember that this is what we did all the time before. But the road also pushes this point. So it goes like that and then goes like this. So just exaggerate that. So from here to here when a point is pushed, which we call as P1 and the other one is P2. So it takes the route from here to here and here to here. So in other words, there is a force which actually pushes the point in the lateral direction so that the P1 reaches P2. If you look at the plan view then for example, the points which are lying here are now completely shifted in the plan view to points here. So I will get a number of points which are here. So these points now go to those places. So because of this pushing of these points, I develop a force. So that force is the camber thrust and it acts like this. So there is again a lateral force that is generated. I did not do anything. I did not even corner. Having a lateral force is generated due to the camber that is present either in the vehicle or on the road and the result is that that is to be added to the forces that I already have. So the camber thrust produces a lateral force. Yeah, camber, yes this is a positive camber and a negative camber. You know, we are looking at camber just in one of the, say for example, this is a camber I have provided depending upon you can have a camber like this or you can have a camber like that. If the camber is in the opposite direction, the force will be in the other direction. So it does not make a difference how the camber is. For example, you can look at it as a motorcycle which is taking a turn where you are just leaning and that is what happens. Clear? So the thrust of course depends upon what the camber you give and also please note that the camber is quite complex, is made quite complex because of the road, okay. Let us say that I have a camber which is given like this, I am just exaggerating, okay. It is very important for trucks and it is very important that the truck manufacturers actually look at what is their camber, you know, they give a big range but that would have a lot of effect on tire wear, okay. It also goes with the road, road camber. Now this is straight road, okay. Suppose you are going like this in the road then what would happen to the road? The road is inclined, road is given a camber, okay. What do I mean by road camber? That is the question. The road is always not straight, it is not a flat region like that, right. A road is given a camber and the road camber is, say for example, is something like this. Why do I give this? Because when there is, all of them are exaggerated, will be so, you know, like a mountain, no, no, no. We are talking about 1 degree. This angle is, in India usually it is about a degree, okay. Now this may vary from place to place and so on but let us say that it is about 1 degree, okay. Why are we giving it? Because it is a beautiful drainage that has to be brought. Go and look at any road, you would not have noticed it, stand here and look at it, you would see that camber given automatically, right. So it would be not so sharp, okay, something like that, right. So now if this is the flat road, let us say that this is the camber and now I am going to change the camber of the road, okay and make it say like this. I am giving a camber, right. So what happens now? What was sitting, this is the angle, now it becomes that is the angle, okay and what was this angle now becomes that angle, right. It is something like, it is like this, I changed the camber of the road, okay or in other words I put one more angle. Let me draw that clearly. Let us say that that is the road, right. I have an initial angles, right. So now I am giving a camber to the road. So the road camber, I am putting it like this, okay. Now the tire is going to now sit like this. So there are two angles now. This is how the tire is now going to sit, right. So in other words what was straight here, you have in for this side, you have rotated it in this direction which means that you have made it more towards zero camber or reduce the camber, okay. Now what happens this side? Now this side actually it is actually going like that, sorry. Actually it is going like this. So in other words this angle now is now reduced. Yes, of course gravity is always acting in that direction, gravity we are not, of course. No, no, no, this is geometry. This is geometry. Please note that I cannot resolve it, you know it is not, camber is not resolved. Note that carefully. Say for example, let us forget about the road for a minute, okay. So suppose I have, this is very important for the point which you said. Suppose I have a camber like that, see here let us take this, okay. So let us say that, let us forget about a road for a minute that makes it complex. Let us say that I have a flat road where I have a camber and how does the load act, what you call as gravity load, say it is a g load, it acts like this and camber is how, how does it act perpendicular to it. So how can you resolve a perpendicular force into, you know vertical force into a horizontal force and resolve. So camber is not resolving the force, okay, along a horizontal direction. This is a very important point. Camber is because of deformation of rubber. The camber thrust is because of the deformation of the rubber, okay. And that is what we saw here, moving from P1 to P2, right. So you do not resolve it. Effect of camber thrust is of course a force that is acting on the tire which is now, the tire is equilibrated by an equal force on the axle and there is a two forces that are acting on the tire. If you suppose somehow over, if you are able to generate a body force in horizontal direction, is it equivalent? There is no body force involved in this. What is body force? Body force is a force that acts throughout the volume of the body which is typically your gravitational force, okay, or magnetic force and so on. So this is not a body force. It is a force that is acting at the tire-road interaction, I mean due to tire-road interaction and is acting at the interface between the road and the tire, clear. So it is not a body force. Body force means it would be acting throughout the volume of the body. So I know this confusion. I understand this confusion that many people think that camber is due to an inclination and resolve this in this direction that would not work because this force, this force you cannot resolve it to get this force, right, okay. So in other words, if you look at road camber, you just look at it as if there is a change in the camber angle, okay. So that is what is called as a camber thrust or a camber force and that is added to all these forces that are happening there, right. So the situation is not that straightforward, you know, that is why this is so exciting. So the tire manufacturers recommend that you rotate the tire because so many forces are acting. You do not know the forces and these forces have an effect on wear and you want a uniform wear. The next thing to do is to have a rotation plan, okay, right. Let us get back into the, so one more derivation which I have to do and we will stop here for a minute or maybe for 5-10 minutes with this class because I know that you have a quiz coming up and if there are questions you can ask them so that we will clarify some of the time again now. No, no, no, no, no, no, no, no. Now camber is from the vehicle, okay, cone city is from the tire. The tire becomes a cone. So the reason is not the same, okay. The deformations may be similar but the reason is not the same. When we say camber thrust, okay, it is coming from the vehicle where I give a particular camber, right. So hereafter, when you look at a tire, go and see, see how, what is the camber, how is the wear of the tire? The deformation in the tire. Absolutely, I agree with you. So there are, this is due to deformation of the tire and so the origin is tire. The ultimately the force origin is due to an interaction between the road and the tire. No doubt about it but, but why is that interaction taking place in camber, the interaction is taking place because there is a voluntary camber that is given, okay and that camber is what is causing this kind of minute deformations that are taking place at the interface, right. On the other hand, the shape of the tire when it is put in the road is changed by the road, by the, not by the road, okay. It is first changed by the manufacturing defect or whatever it is, okay. Then there is a deformation in the road which makes it flat like what we have said and then it causes. So in that sense, both camber thrust as well as conicity is produced because of deformation of the tire which is running on the road, no doubt about it. Any other questions? What is PRAT? Unfortunately we do not have these slides but I hope you remember the slides, right. Remember that when the question is, do not understand what is, how do you create PRAT? direction and what is PRAT, right. So this is a very simple thing which I did but let us understand what it is, okay. The fundamental reason for this PRAT is what is called as unsymmetric behavior, okay called as say unsymmetric bending and so on, right. So it is the unsymmetric bending, you are going to face this quite often in your next course on automotive structures. You are going to look at unsymmetric bending very closely with lots and lots of equations. So equations can wait for some more time but let us look at what is unsymmetric bending. All of us had looked at symmetric bending, you know where there is a symmetric, say for example it can be a rectangular beam or a square beam where there is a, very clearly draw it, there is a symmetric, this axis is symmetric about, I mean the geometry is symmetric about the two axes, right. Now when this, say for example I have a cantilever beam and when there is, when it is symmetric, okay and you bend it, when there is bending then it would not get, suppose I displace it, okay I displace this and then the displacement would be only in this direction and would not get displaced in the other direction which I would call that as x, okay. Now if my loading is not in the same direction as this or in other words if the loading is the same but if the structure does not have symmetry like this, say for example I have a structure which is like that, okay and then I have my axis is defined like this and I have a force that is acting in the y direction and that being the y then the displacement is also in the x direction, okay. So a y force also produces a displacement in the other direction. When would it not produce? There exists what are called principal axes for this kind of structure, okay which may run like this, let us call that as psi and eta which we are going to see in the next course that the, if you apply forces along these principal axes then this kind of coupling would not take place. So there exists these principal axes about which say for example if I apply force along the psi direction or eta direction there would not be any displacement psi direction and so on. But as far as, as long as there is an unsymmetry in this kind of loading then it would produce a force in the other direction, okay clear. So now this is the fundamental. Now I have lateral forces and longitudinal forces even in free rolling, okay. Let us say that I am doing a free rolling of lateral forces and longitudinal. You have already seen that there are forces say let us say that I have, this is my tread profile at a particular region at a particular point, okay. So this is the, that is the say contact patch, okay. I have tread profiles like that and let me say that I have at the edges tread profiles like this, okay. Now we have seen for example that the lateral or the, the longitudinal forces, yeah any questions? There are longitudinal forces that are acting, okay. At the, this is the leading edge in other words that is how the vehicle moves. There are longitudinal forces at the either end, okay. And remember that is a positive and then comes back negative, you know the longitudinal force, right. And there are also lateral forces in other directions, okay which is spreading out, which is spreading out. So there are lateral forces at the edges at the shoulder of this contact patch. Yes, yes. There is a side wall which there is a thrust and because of which there is a lateral force that is up, okay. That is what we saw on the last slide, right. Remember that we said in contact patch there is a local effect, there is a global effect and all those things. Let us forget it for a minute, okay. Let us say that I have in a tread which is something like this. So now these guys are going to now produce additional forces which are perpendicular to the forces, these forces which are actually thrust on them because of the contact patch. So there will be a force like this, there will be a force like this, there will be a force like this and there will be a force like this, right. These are due to the coupling term, right. So these coupled forces now create a torque, okay, around it. Now the question is, are the treads as simple as this? No. We are not saying that the treads are as simple as this. But you can use for example finite element analysis in order to understand what is PRAT, how it is produced and what are actually the shape of these treads and all those things, blah, blah, blah. You can do that, okay. So actually the next question is can I treat the tread like simple cantilever beam? No. These are stubs, you know because why are we producing, I mean what are the other things that are happening because there are shear stresses become important without what are called shear deformations and so on. All of them are brought out very cleanly by a finite element analysis and the theory of which as I said we will learn later, okay. Is that clear? So this kind of cross linking due to unsymmetric bending is the reason for what we call as PRAT. PRAT is because of the stubs, yes, this is the shape. That is the shape which is not symmetric. It is not necessary that it should be a shape like this rhombus or something. As you have seen in any tire you would see that the shapes are very complex, okay. The shape of the tread also depends upon each ability or its requirement that it carries water, okay. So that is what it is. Maybe after some time we will see about what is wet traction. I mean we will go ahead, we will do lateral and then we will come back and look at wet traction and so on because we are spending a lot of time on tires. Understand it is an important topic but we have not yet gone to lateral dynamics which is exciting in its own sense and how the tire is going to have a huge impact on lateral dynamics is what I am also interested to project and we have already seen, we have already entered lateral dynamics in the sense that we now know how lateral forces are developed by the tires. Yeah, any other question? Yes. So when we have uneven contact pressures actually all the points on the ground are actually the axis of rotation rate means if we are writing tau equals to i omega we can write about all the points. Can you show me what is the question so that I can repeat it to others? So we have done like we have found f equivalent equals to dr into dr equals to f into rd about this thing. So actually all the points are equivalent in this case so we can actually, can we apply about the point where the equivalent is? Yeah, see in other words what the question is that how, so suppose that is the, you know let me understand, let me see what I understand from your question. Suppose the, that is the contact pressure distribution which we said okay which is unsymmetric because of the viscoelastic forces right and there is a contact force that is act, I mean the whole of the reaction that is happening like this okay and that is going to rotate the tire in the opposite sense in the sense that there is a torque okay. The question is can I just replace all these forces that are acting by an equivalent force there? Of course, of course okay and that causes the moment, just one second right. So in other words, oh God I removed it maybe what Appulva drawn, I drawn it beautifully I am going to replace that. In other words, in other words when there is camber for example look at a motorcycle okay there is a lot more to it we are not there is what is called as turn slip and so on. We are not entering into a lot more tire mechanics and tire dynamics but please note that the complete force doesn't act right at the center of the contact patch. It acts here in this case like this if you look at it from the top it acts actually to the side which gives a overturning moment. So in actual case okay that gets deformed and so the force is acting like that okay. So in other words completely resolve forces do not act symmetrically at the center because of which there are so many moments that are created clear. So can I replace the complete contact pressure by means of an equivalent force that is the question of course you will do it and of course that is what we did for all the longitudinal forces and that is exactly what we are going to do for the lateral force. So for the lateral force generation we are going to look at the equivalent force as well as a moment okay which we called as the aligning. What are the activities? Yeah no no absolutely there are moments in the other direction okay which is an overturning moment. There can be moments which tries to overturn okay which is say for example you have the three directions okay say for example this is the travelling direction of the vehicle say let us say V and that is the lateral direction of the vehicle Y okay and this is the normal to the ground. Yeah any questions? Yeah we talked about viscoelasticity remember that is what that is what it is. So this is with respect to that but the other moments are created due to other reasons. Remember that we had what is called as an aligning torque right. Remember that we had okay if it is a plan view a vehicle which is going straight like that had to take a turn right and so we gave a steering angle and that produced what is called as the alpha in order that we get the centripetal force and the centripetal force again does not act right at the centre okay and acts if this is the line that is the centre it acts away and because of which there is a moment that is generated and that moment actually makes the vehicle or makes this to go straight again okay so it is called as the aligning moment. So that aligning in other words that is the moment aligning moment right and the rolling resistance because of this produces a moment about Y right this is the moment is produced like that there is a rolling resistance moment. The other third moment in the X direction is because of the fact that there is a camber and the contact point is not exactly along the X direction and can be shifted away from X. So the normal force acts not exactly here but away from it okay look at this this is the centre point but actually it acts away from it. So because of which okay so that is the force that is actually acting if this is the contact patch so because of which there is another moment that acts along the X which is the overturning moment. So there are three forces the normal force the longitudinal force which gives the traction or braking and there is a lateral force which is the centripetal force along with it there are three moments okay these forces produce moment in other directions that is what we saw now clear. So it is not symmetric the contact patch is not symmetric that is the most important point yes the though you may say we assumed a parabolic nice parabolic distribution we are going to do that again in the next class for the combined cornering and braking but that is an assumption which we made okay. Sir will that unloading and loading be constant throughout the free rolling like suppose the contact patch will be yeah but but then we are looking at the the graph which I drew was for a point see is the the the question is how do we view this loading unloading view the loading unloading from the point of view go and sit in air tread and look at it okay actually the tread is at different positions in the contact patch. So that will maybe we will do that in the next class and then you see that each one of this point will get loaded and unloaded and that is the graph which we saw. But then sir how can we assume it to be a constant brake track like yeah because because there is a steady state okay and every tread undergoes this is what we call as a steady state and so every tread undergoes a loading unloading characteristics okay. So at in you can look at if you would something like this freeze the position and then view it that is what you get right okay. So we will stop here and we will continue in the next class on Wednesday.