 Let us start with the lateral dynamics. I thank Apurva who has done all these drawings on the board. She is my TA and thanks a lot for taking a time off to draw this. She has done it very beautifully. So, we will look at how lateral forces are developed now. After what we did, we are going to extend that and we are not going to derive again lateral force separately. We looked at in the last class just to summarize what we did. We looked at longitudinal force development and we did that using what is called as the brush model. We recognized that the contact patch is divided into two areas and we said that if this happens to be the contact patch then you have what is called as the sticking region and the sliding region. We developed a quantity called slip. It came naturally in our definitions and ultimately we saw that the force, the effects, the longitudinal force is a function of slip and it happens to be something like this. Now, we have to come back to this portion. We will do that a bit later. We got it straight mu into qz. The question is whether this is going to be a straight line or not but I do not want to waste time. She has drawn it again and let us go back to the or go to the lateral force development. Now, what do we mean by lateral force development, slip angle and restoring torque, pneumatic trail. These are the terms that we are going to look at in this lecture. Now, let us say that this is the picture of a single wheel. Let us say that the vehicle takes a turn. So you give a steering angle. When I say steering angle it is referred to the wheel. Let us say that delta is the steering angle is the steering angle. Of course, it is not the angle with which you steer the vehicle that would be a ratio between the two. So, I call that as the steering angle. Now, we are going to view it from this point of view. This is how I have steered it. So, that is what we are going to view. In other words, this is the plan view or you view it from the bottom of the tire and that is this direction. We are going to see what this direction is. In other words, this direction is. We are going to give a name to that slip angle. So, we will see what it is but before that get the global picture that when we talk about slip angle it is the angle from the steered angle, from the steered angle. So, what you are going to do is to sit here after the after you are given the steering input and then observing what is happening at that point of time. Have that coordinates clear. So, what is it that is required when you take to cornering. We have already seen this. I need a centripetal force in order to accommodate the centrifugal or centripetal acceleration whose d'Alembert equivalent is a centrifugal force and the centrifugal acceleration. So, I need one F y clear. Now, let us see how F y is developed. Very quite complex much more complex than what happens in the longitudinal case. Now, imagine that this whole tire is rotating. Now, what is that I need? I need a force here that is the force I need and that is the force which is pushing it out which is the what we call within quotes d'Alembert's force or centrifugal force. So, in other words this becomes the sticking region. The vehicle the tire sticks to the ground like before and that is the force which is now going to push the wheel out. This is the force which is going to push the wheel out. So, that is the d'Alembert equivalent force. So, both these forces now have to compensate. So, how is this force developed? Similar to what we did in the longitudinal case. Now, I am going to have this kind of sticking elements and as this wheel is pushed out pushed out in the y direction, these sticking elements are going to be stretched. They are going to be stretched and this stretch produces a tension which is going to pull the wheel inside. So, in other words this stretching in the y direction of these tread elements what we called as the tread elements or the bristles or the brush model that gives a push in the y direction and the result is that I get this centrifugal force. But then things are not just that all of them are going out. The things are not as simple as just getting a pull. So, as you roll now look at this. As you roll because this plane is now getting pushed out. So, as you roll you hit the ground note this carefully as you roll hit the ground at this place at one place. Now, that place is ahead of the sticking place. So, this is how you know it is rotating the corresponding point is here. Because of the fact that the sticking makes it not possible for that to roll in one plane, the contact patch gets a very skewed appearance, very skewed appearance. So, what happens is that this point comes in. So, this point later gets to this point that that point later gets it to that point and so on. Because it has to stick and when it comes out it has to go back to the plane to which this wheel has been pushed. So, it is not one straight line. So, it goes it goes like that and comes out like this. So, that is the shape of the contact patch. That is the shape of the contact patch when cornering takes place. That is the first thing I want you to understand. Any questions? Yes, exactly. So, it is a tread. So, what is this? This is nothing but if you if you sit in a tread with respect to the ground. First you will move you will come down like that. Then you will hit the ground at that point. So, then the next guy is going to because there is going to be a pull you know you are going to be here. The guy who is in front of you will go and hit that point and so on. The same fashion as the contact patch moved above you in that case here also the contact patch is going to move above you, but the contact patch is not a straight patch, but it is inclined because I have to cater to the needs of two planes. One plane sticking to the ground or one part sticking to the ground. The other is that the wheel is going out. So, it comes like this goes like that. Clear? That is how I can do it. Fine. Any questions? Yes, yes. So, so what happens here is look at this diagram. Look at this diagram. Initially when it when it was rolling straight or if it were to roll straight this whole plane would have been like this this whole plane would have been like that. So, now because of this force the plane gets like this. So, that is what is manifested as this twist in more simple terms with this twist. When you get back you have to get back to that plane. You have to when you are rolling you are in one plane. You are sticking you are in another plane. So, you have to come from here to here, from here to here and then get back and go to this plane. That is what is seen here. So, that is why this skewed appearance. What is the result? The result is that the tire just does not roll like that. The velocity there are two velocities. Now because there is a lateral velocity and there is a longitudinal velocity with the result that the resultant velocity is not like this, but is like this which is equal to an angle which would call as alpha. So, the wheel now moves actually in the resultant velocity direction which is alpha to the plane which I would love the tire to move. So, I am going to move and I do not want it to move because if I if it moves like that all my pulling does not exist. So, I am not going to get the lateral force. What is our we will come to that? What is t? We will come to that in a minute. Now go to the next step. This creates a obviously a shear force which has to which has to be compensated by mu into N. In other words, this this guy who is sticking is pulling these are the the bristles which are pulling. If that friction there the same old guy which we saw in the longitudinal direction if that friction is overcome what will happen? It will slide in. So, it will slide in right. Same concept nothing new right. Now let us see what happens what happens let us let us just put that and what happens during the sliding get the normal normal view. Let us say that that is those are the deformations. You know now by now you know that there is a race between mu into N the normal force as well as the the frictional force right you know that right. So, let us say that at a particular x which we call as x sliding this is overcome. So, there is a normal force you know that there is a normal force and there is a tangential force. Obviously as as I move look at that as I move towards the rear of the contact patch the pull is going to increase shear force is going to increase. So, I am going to go ok. Then later I have to go back to that original plane. So, my pull is going to decrease and that. So, I have something like that right ok. Now let us say that up to this point up to that point the tangential force is able to hold. So, it will be sticking after that point it is going to now slide at that point there will be sliding right ok. So, in other words so after x s let us say that x s then it is going to slide. In other words the the development of f y is not symmetric. How do I get f y? Integrate this right and in other words it is the sum total of these pulling and obviously this pulling is not symmetric about the center. The pulling is less in the front and the pulling is more as I come come down right. So, this f y if this were to be the contact patch if this is the center of the contact patch ok. Where does the f y act it would not obviously can it act here will it act here unless it is symmetric it is not going to act at that point right. It is going to be shifted right. So, where will it be shifted to the rear and that is what we have here that the because of this this unsymmetric distribution I have an f y which is not which is not symmetric which is at the center. In other words in this diagram I cannot draw it exactly at the center if this were to be the center. So, that f y will be displaced clear. So, that is what happens here and that distance what is the effect of that distance exactly. So, it would create a moment. What does that moment do? What does that moment do? That moment rotates the tire and aligns it back to the straight running position and aligns it back to the straight running position. Hence this is called as the self aligning torque or moment. So, we call that as self aligning torque or aligning torque or aligning moment whatever you want to call it. After the straight running has kind of intersect with the ground there will be any contact with the ground. No it is it is it is it is say it is not that it is please note that it is not that it is not in contact with the ground because this contact patches this. It is it is in contact even if it slips yeah if it slips or slides rather that is correct word as I said. Even if it slides it is not that it is away it slides and still is in contact with the ground. Yes of course okay that is the maximum force which is mu into q z correct. So it is it is though the figure looks like that it is it is a three dimensional figure and it is not that it has lost contact right this whole thing is the ground clear okay. Now now please note that it is both these guys are now you know reducing towards the end and there is a race between two reducing quantities. So it starts slipping as it starts slipping at one point of time okay it may so happen that the slipping may engulf and and that that would be the force distribution of all these bristles and so f y will be uniform when when when q z when sliding completely sliding takes place okay of every every bristle that is there then every bristle will have that q into mu into q z and then it will be completely symmetric right okay. So now if I plot this graph that will be my f y and what do I plot it against what is x x it should be slip angle which I would call as sigma y. I am not going to derive this this is exactly the same the deformations are exactly same derive that the so it just replace it I am I am going to derive a combined slip in which case you will understand what is going to happen okay. So that is what happens now if I plot plot now m y it is a usual practice to plot as minus m y for the simple reason that if this is positive this is positive okay x y z should be inside the ground yes yes what is what is the physical difference between a slipping bristle and sliding sliding sliding bristle and a and a sticking bristle what is no it is sticking that is exactly what we said it is only deforming it is not moving this is not motion this is only sticking here and it is not moving okay it is it is it is just the head is here this is exactly how we developed in the last class the longitudinal force okay. So this is the guy who is sitting in the carcass and moving and this guy is here is sticking okay so that is what we call I mean the difference between that is what cost the force clear okay go back and look at last class we did exactly that and that is what it is so we will come back here and usually it is the minus sign that becomes I mean that is what is used because z is in this direction but actually the m is is in that direction so this being positive that being negative so usually it is plotted as like this so it should be should be something like this so where should it come to 0 and I think it is better that we plot here where should it come to 0 full sliding so when you reach this point that should be 0 right so that is the region where it is sticking few sliding complete sliding okay fine we will we will now look at a small a simple mathematical model to understand the derivation for for F y m z as well as F x in other words let us look at a situation where just a second where we are going to have both a longitudinal force as well as the lateral force in other words you are breaking while cornering a common thing that would happen breaking and cornering yes no we are not right now we are when I say sliding a part of it is sliding okay yes that is in other words in this region itself there is some parts which are sliding okay here is where it is full sliding okay now what is loss of control that is the question okay we will we will get back to this question in a you know what do you mean by loss of control in other words what happens in the in what we are going to call as the unstable region okay in order to explain that it is we have to start with F x I will just make a passing remark now but we will come back to this later in the course maybe I mean after a couple of days okay we will come come back to this and let us just go back to F x I thought I will do that later because I do not want to you to live with this doubt so let us do that now so if I now plot actually sigma F x which we had done okay using that brush model remember that we got a graph nice straight line like that in practice in actuality in reality this is not a straight line like that you know it is not a flat line rather maybe straight line was not a flat line because of the temperatures because of the fact that mu is not a constant as it slides when and when it slides the temperatures are very high flash temperatures as it is called as the temperatures can reach hundreds of degrees some of the reason papers say that it can reach up to 200 250 300 degrees okay so when there is a temperature increase this mu is no more a constant you just said that that is mu into F is it mu into F is it is this okay that is no more a constant because temperatures play a very major role right remember our first graph with viscoelastic material behavior we had three regions okay remember that there is a region with higher temperatures we plotted that at temperatures at higher temperatures where the viscoelastic properties drop or hysteresis drops and so the friction coefficient which is a function of the hysteresis would again be affected when you are in that region and that is what happens I move towards that region that is what happens in here after I reach the peak with the result that my graph is not going to be a nice graph like that that would go like this and start coming down so here what happens the the the wheel has not you know come to a halt but what happens when it is what do you mean by when it come when it starts coming down so in other words this is an unstable region the larger you you break you know this the force is going to come down and that region where increase in sigma results in loss in FY okay is the unstable region and warrants the application of ABS okay anti log breaking system right so if I have to break I would like to break here that region rather than go into this region so the best way is to get back to to this curve to get back to this curve okay so that my force breaking force will increase that is the breaking force effects okay rather than stay in a region where breaking force is going down fast clear okay that is what happens in FY as well okay some physics before we take this take on to this derivation as I told you before that there is going to be a race between these two quantities now we are going to remove this so this is the slipping this is what this is where you know there is a sliding region that is the sliding and these are the deformations and that is the sliding region it is very nicely brought out in this picture thanks to more friction of pneumatic tires we have a very good picture of what really happens okay now let us start with combined lateral or combined breaking and cornering let me call that as combined breaking and cornering before we proceed to this derivation there were some questions on how we can find out that a remember that we had a a what is that a to a was the contact watch the questions on how do I get a how do I get qz these were some of the questions at the end of the class very simple let us say that that is the tire let us say that that is the ground in other words that is the tire which has been deformed so in other words if that is the center of the tire that is the R I think I used R right R hope I used it R let me call that as A B okay call that as C and call this point as D so let me call that DC as the deformation in the Z direction or perpendicular okay so that is the ground how do I calculate that a very simple expression delta Z is equal to the force fz or fz this acting divided by the stiffness okay let us say that the stiffness in the Z direction C z right okay then what is a so what is what is this now R-delta z so a is equal to root of R squared that is R R squared minus R minus okay which you can expand it leave out delta z squared term and you can say a is equal to root of 2 R delta z this is very I would say an approximation okay nice approximation and that is what you can use if you want to use a on the other thing is how did we get Q z right that was another question what is this Q z okay we got a very nice expression for it how did we get this very simple again we assume let us say that that is the pressure distribution parabolic pressure distribution so write down an expression write down an expression for the pressure distribution K into right some K that is what I want to find out how do I find that out Q z into dx integrate this from minus a to plus a should result in fz substitute that here do the integration and you will get the expression 3 by 4 fz 3 fz by 4a into 1 minus x by a whole square now this is what we got yeah I fully understand I agree with you that when rolling tyre you you just said that you this is not okay yeah very symmetric distribution yes we make an assumption here in other words rolling resistance is not taking into account in this okay precisely for this reason precisely for this reason we have we have moved slightly away from all these brush models so I will just since you ask this question we have to we have to talk about models I will come back to this combined braking and cornering in a minute but we have to talk about what are called as tyre models these brush models I mean they have been extended they have been beaten to death with some very smart analysis but still it is a long way to go okay to get the correct what happens exactly in the tyre it is not very easy because of the complexities that are involved you are going to see more and more complex I am going to spend some time on tyres because this is one of the most important topics in vehicle dynamics and it is slightly difficult topic to understand the reason why we are going to spend some more time the rest of them before we go to lateral dynamics the rest of them are not as difficult as this topic because here you have to get the physics and get a hang of this mathematics the reason why we are spending some more time okay tyre models are extremely popular I know some of you would have used packages like carsome or maybe even Adams that is a which has a number of tyre models tyre model is nothing but an equation a mathematical equation I will devote a class or two for different tyre models that are available okay they are mathematical equations so model means of a mathematical equation which gives us fx fy mz which we just now saw okay as a function of say sigma and other quantities okay which we would see there will be some stiffnesses and other things cp and so on right so it is just an equation which gives fy these equations can be derived by various fashions can be experimentally determined as well what we saw in brush model like very nicely he pointed out that what happened to all those things that you said you know suddenly you say it is a parabolic distribution okay so the brush model is a simplified model but it brings out all the physics you know if you want to understand it it is a very good model because you now know friction coefficient how it affects you know this there is sliding there is sticking and all that so it brings out all the physics but it is a very fundamental model okay it agrees to a certain extent does not agree there have been a lot of things that is happen okay so that is the fundamental model so if I say this is the model that is the brush model but people are not happy with with that model so they go to what is called as an empirical or semi empirical models in other words they express this as an equation we will see that typically what is called as magic formula models we will see that in the next class we will see that by first put forward by Prasekhar it is called sometimes Prasekhar's tire model magic formula model is the next level where you write down an equation and simply do a nice curve fitting of experiments that are done so it is a semi empirical model in other words you do an experiment I have a curve and do a curve fitting and then express this as an equation for FX, FY and MZ and so on so happens that it is called magic formula because there is one equation where the structure of the equation is the same whether it is FX or FY or FZ so that is the next level formulas. So then these models have again some limitations so as that tire speed increases okay and you want to look at ride okay the roads which are not very nice and smooth and you have all the undulations on the road okay and so on then I have to move to a different level of models I would like to now look at the dynamic behavior of the belt I want to know actually how the belt vibrates that would give rise to noise that would give rise to vibrations and so on so I would go to a next level of models which also would take into account road the road profile the envelopment characteristics of the tire and so on maybe up to a certain frequency so you have in that category so this is the second category and that is the third category I mean more sophisticated as we go up our models are becoming more and more sophisticated okay again we are capturing this with an equation. So you have at the next level you have what is called as short wavelength intermediate frequency tire models or swift models a lot of work done by Professor Prasheka at Delft University you know we are having we have a number of models like this of course we will talk about other models birds eye view of other models there are other models as well okay. Now lastly we have models which are based on on say finite elements okay which brings out all the physics okay carefully done carefully done brings out all the physics that exists in turn mechanics so the finite element of tires is actually I would say is a very sophisticated finite element analysis it has all the shall we say difficulties or challenges because the material that you are going to use is geometrically non-linear materially non-linear in sense that they are hyper elastic visco elastic means at least if you want to look at hyper elastic city it is a hyper elastic material and you have what is called as geometric non-linearity because the strains are extremely high okay you have contact you have reinforcements we saw that there are bells and radial plies. So the model becomes very complex but because of the fact that even contact is taken care of well and even the friction coefficient need not be a constant it will change it with respect to pressure as well as the sliding velocities and so on okay the model becomes as sophisticated as you would like it to be of course you got to do lot more work for it you can predict temperatures and if you want to do look at mu as a function of temperatures that is also possible okay. So these are the models that are available in this sophistication okay there are other interesting things okay let me before we get in two more comments before we get into this we started this and I would like to start it in a new class because derivation is going to be involved in a lot more things I have to say maybe we will start a new class or whatever I want to say it was end of the class I am the next class I am going to say it now right okay. Now this is this is fine you know I have an alpha I have that alpha and at alpha gives me deformation and so there is an Fy that is generated right. The question you may ask is I am sure at the end of the class you are going to ask I am taking a turn okay this you say when I take a turn say a constant radius okay I am I am I am driving a constant radius steering pad as it is called I take a constant radius okay around this pad I agree with you whatever is happening. On the other hand I am entering I am entering a curve or maneuvering you know so what happens in other words Fy does not exist you are talking about after the development Fy. So there is a region where from 0 to Fy I require Fy but Fy is not immediately realized I turn then there is a time at which because after all this is also a viscoblastic material so I I have that time lag before this Fy is developed okay in other words there is a transient region before which I get into this steady state Fy okay. So that transient region has its own mechanics where okay you develop theories on on the force development over time and it is usually said that it takes about half a revolution or more slightly more in order that the force is developed and is characterized by what is called as the relaxation length right time permits we will talk more about relaxation length the other very interesting phenomena is the development of this Fy now Fy development as we had seen is when the car takes a turn okay strictly speaking then if I have a car let us say that is my car I do not want to take much time in drawing it so it is going straight okay let us say I have a beautiful road to go okay and I leave the steering would the car go straight or the car go like that forget about even an actual experiment I do not want to do this experiment okay do not do this experiment with your cars do this experiment in in one of the numerical codes say for example Adams you would see that the car does not go straight in other words even at when there is no camber okay and no steering given there would be a lateral force that is generated okay this is what is called as we will talk about two things place here in cone city but we will first talk about place here this is a very involved concept so let us talk about first place here okay so in other words even in straight run here we had we had just D linked it I have an FY because I am taking a turn I have an FX because I am breaking okay and both are as I said are modeled using what are written as a numeric as a mathematical equation using tire models fine but is that what is that all what happens is it that can I get an FY by just going straight you will and that is exactly what is called as place here will I get oh yeah I know you are all surprised then and always tire mechanics has lot of surprises that is why this whole topic is extremely interesting this is this is engineering by itself okay anything in engineering you say there is a small role here in tires why does this happen that is because of an effect on of composite laminates deformation which is very different from that of deformation of steel for example if this belts what we are talking about you know this belts these guys who these are the belts we saw that the guys who run around okay I said that if I look at it from the top it look like this I said there are steel cords that are running like this there are more than one belt and then there is another belt where I place it on top and there will be steel cords running like that and so on okay so in other words that makes tire a composite now let us forget for a moment that it is a composite material and let us say that that is made up of complete steel okay now this is this rubber embedded I am sorry steel embedded in rubber let us say that it is steel just steel sheet obviously when you pull this just pull it what happens there is no twist there is no twist okay just pull there is thinning which is called as the Poisson suffer straight forward what if this material is a composite laminate then the behavior is not as simple as this we will see that in the next class what is the behavior and because of which what happens to the force development and why the car is not going to go straight and so on okay we will finish that and then we will come back to the combined braking and cornering television okay we will meet in the next class.