 So, we will continue with what we were doing in the last class. We were looking at a broad picture or perspective of vehicle dynamics. So, we were looking at how we are going to approach the subject of vehicle dynamics. We said that for us though there is a vehicle, it has its components and so on, when we are studying vehicle dynamics, we said the center of this whole thing is the mathematical model. So, mathematical model comes from our good old Newton, Euler-Newton equations and this has an input and an output. Remember that when we looked at the dynamics which is defined by using this mathematical equations, they are classified into what we called as longitudinal dynamics, lateral dynamics and vertical dynamics. We said that we classify the dynamics what we are going to study using this mathematical model into a longitudinal, lateral and vertical dynamics. We also said that for these of understanding its effect, we may most of the times delineate or decouple them and study them in isolation. Is that correct? Does it not have an effect or one has an effect on the other? Yes, there is, it is possible there is an effect but in order to understand the subject, most of the time we will be decoupling the effects of all these three. There are very important things that will happen like load transfer and all that. We will understand it as we go along. We also said that the input for this model is the driver's input through the steering or the acceleration and braking. In other words, how the driver interacts with the vehicle. That is what we are going to use as an input. We said that we cannot look at every scenario by the driver and we will have some test conditions in order to understand the behavior of the vehicle. Of course, we said that the vehicle itself which goes into this mathematical model will be defined by means of certain parameters, what we call as kinematic and compliance parameters, mass, moment of inertia, as compliances, stiffnesses and so on. The output from this model as we said yesterday are in terms of displacements, accelerations, velocities and so on. We said that what comes out as an output has an effect on the occupants and we are going to study that not to a great extent but at least as an introduction we are going to study how this is going to have an effect on the occupants. This is the broad, I would say, basis under which we are going to study the subject. We also said that we can again look at it from a different perspective and call this as driving dynamics, safety, right comfort. When we look at this from a different perspective, same problem, we are going to look at it from a different perspective. So, it is not organized like this. It is not that longitudinal dynamics is driving dynamics, lateral dynamics is safety, vertical dynamics is right comfort. It is to a certain extent it can be looked at it like that but they are not necessarily a clear demarcation like what we have here. So, we would look at the safety during driving both in the longitudinal as well as in the lateral dynamics, right. We will continue now with this short introduction and we are now going to look at this mathematical model and already we know or we had a very, very, very simple model in the last class. We are now going to extend this model, simple model into not a very complex model but just the same model, simple model making it more elaborate. It is not going to, we are not going to make it very complex. We are going to use the same equations f is equal to ma. So, in other words, we directly plunge into what is called as the longitudinal dynamics and look at a very simple mathematical model that we will be using in order to understand the longitudinal dynamics. Whenever we talk about dynamics, two things that comes to our mind, the very first thing are the forces that are going to act on this vehicle and the other is of course, the acceleration or deceleration and so on. And our good friend f is equal to ma from Newton is going to be of great help in writing down this mathematical model. So, the first things first. So, what are the forces that are acting on the vehicle? All of us have experienced this. The first one of the external, we are looking at the external forces. The external force that acts on the vehicle is what is called as the aerodynamic force which we would call as r a. The other force, next force which is important to us, which is acting on this body is the gravitational, pull or gravitational force and that and write it as m g or w and that can of course, be resolved into two directions like this and that if this is equal to, this is of course, is equal to theta s. So, if I call this as w, then we have, what is this w cos theta and w sin theta. You may have a trailer, you know, towing with this vehicle, which is called as a drawbar. So, if you have a vehicle or if you have another trailer with you, then there will be a drawbar pull which you would call as f t, clear? Apart from this, the vehicle itself is going to give us some force, some nice guys, some not so nice guys. So, if I am going to accelerate, we need what is called as the traction force. So, the traction force is now going to act in that direction. So, let us say, let us put the traction force in the front and the rear. Let us look at this tractive force and call this as FF. Of course, in every vehicle FF and FR will not act together or it may be a front wheel drive or it may be a rear wheel drive and so on. So, depending upon the front and the rear wheel drive, you will have either of the one or if it is a four wheel drive, then you will have all these things. Apart from this, one of the very important forces which consumes our fuel is what is called as the rolling resistance of the tire. The rolling resistance of the tire acts opposite to, we will understand this rolling resistance in a minute, it acts opposite to this tractive force and in fact, it is something like a breaking force that acts on the vehicle. Now, what is, let us first understand, see up to this it is not very difficult to understand all the forces that are going to act. What I am going to do is, what I am going to do is very simple. I am going to find out the reactions of the front and the rear. Let us say that we are accelerating force, we are applying a tractive force. So, we can say that I have a D'Alembert's force. Actually, it is not a force, it is a pseudo force which can be written as W by G into A. It is not a good practice actually to put this D'Alembert's force. It is nice to write F is equal to M A, but then when I take some moments, then it becomes easier for me to have a force there and that is the reason why I have a force and call as the D'Alembert's force. Now, all these forces are familiar to you. In order to take the moment, of course, you need some dimensions. So, let us call the dimension something like this. Let us say that, that length is equal to L 1, that length is equal to L 2. L 1 is the distance from the front wheel to the CG location. L 2 is the distance from the CG location to the rear axle, the rear wheel and let the total length of the vehicle L 1 plus L 2 let it be called as L. The other thing that is important to us is the heights. So, let us call this height as HA, let us call the CG location height is equal to H and let me call that height to be HT. You know how to determine the two W's or the reactions at the wheels W f and W r, if I want to find out W f, I take a moment about W r with proper signs, I can determine W f. In other words, W f into L is equal to whatever the moments that are due to the other things. But before we go further, there are two comments that are important to us. One is the system that we are going to use in this course. The x, y and the z direction that we are going to use in this course. This comes out of an ISO standard and we call the direction which is along the direction of travel as x perpendicular like that as y and the other direction normal to the ground as z. So, longitudinal lateral and vertical directions. Of course, you know that there are motion, the angular motion along these directions. For example, the angular motion in the direction of x, in other words that angular motion along the direction of x, let me think that has to be a correct one. Let us say positive, that is positive is called as, what is that angular motion called as? Roll. So, this is the roll and the angular motion here, which is in the y direction, that is that angular motion is called as the pitch and that is what we call as yaw. So, when you say colloquially pitching, that is moving in that direction. So, that is the first thing. The second is, let us go into the details of what is called as rolling resistance. Rolling resistance is today very, very important for fuel consumption, especially in trucks can imagine that the rolling resistance whose origin are the tires consumes nearly 30 percent of the fuel of the vehicle. We are going to do quite a bit of tire dynamics in this course, but let us understand what is rolling resistance and how do we get it. Quickly, we will go into details later just to understand, because I am putting a force there. So, you would understand what this rolling resistance is. Now, there is a misnomer. Many students assume that the rolling resistance is just the frictional resistance of the tire, absolutely not. It is not the frictional resistance. Rolling resistance comes from the property of the elastomer or rubber, which is the material of the tire. Elastomer or rubber as it is called, that is what goes into the manufacture of the tire. Elastomers have a property called viscoelasticity, usually depicted by a dashpot in order to understand, in order to understand the effects. Now, what is this viscoelasticity and how does that going to have an effect? We will see that in a minute. As I said, we will elaborate it later. Any material can be looked at, small digression here. Any material can be looked at from three simple models, one a spring, other dashpot and third one is what I would call as a friction. Suppose, I say that a material is purely elastic, then you can say that the material can be represented by means of a spring. This is not a very correct representation. We are not going into too much of details. We can say that spring, a linear spring especially, is good enough to model say, a linear elastic material. It is something like an understanding of the material behavior. A linear spring, where the force is proportional to the displacement with the stiffness k, can be looked as if it is the material and k is something like E. When I leave the force, the spring comes back to its original position and that is what we loosely call as elastic. Now to this, we can add other material behaviors. For example, if you look at elastomers, elastomers are elastic, of course elastic and then viscous behavior. So, in other words, spring comes because I can model elastomer or I can understand elastomer as if this is made up of a spring and a dashpot. This dashpot can either be attached in parallel to look at it or we can understand the behavior by attaching it like this and so on. There are names to these models, Kelvin and the Maxwell models, but we are not going into details of these models. We are putting this in order to understand the behavior. For example, if you have a metal which you are taking into the plastic region, then I can model this metal using the spring and the friction element. So, you can join together in parallel also. So, you can model series and so on. These elements you can join them and then write a mathematical equation which can form the basis of the constitutive equation or stress strain behavior of the material. Now, we are not going into this as I told you into the characteristics and I am not going to write down equations here. We will understand only the elastomer part in this case. May be pass a comment afterwards about this friction and why friction is used to model what we call as plasticity. Now, what is the difference between an elastic material, the viscoelastic material? Sometimes people call this as hyperelastic, viscoelastic material and so on. First of all, let us understand that elastic material is not necessarily linear. It can be non-linear elastic as well. So, if I now have a load deflection of the stress strain curve, when I load it, I load a material which means that I am applying forces. I keep increasing the force because of which the stresses increase and there is increase in strain and so on. So, when I load the material, let us say that the path taken by the stress strain curve is something like that, goes like this. When I unload a material, an elastic material, it would actually, all of you know that it would follow the same path. On the other hand, a viscoelastic material does not follow the path when it is unloaded and would now follow a different path and that amount of energy is lost and usually called as the hysteresis loss. So, there is an amount of energy that is lost. It is the same as plastic. There is a subtle difference, a good difference that though at the end of loading, there is a residual strain here. The strain will come back to zero with time. So, time is an important factor in viscoelasticity. Time and frequency are an important factor in viscoelasticity. So, in other words, what I mean by time and frequency are important is that the material behaviour is affected by the rate at which you load it or in other words, the frequency at which you load it and so on. So, time and frequency are important factors. So, the first thing is that to conclude whatever we have been saying that there is a loss of energy when the material is loaded and unloaded. How is it going to affect us? Why is it that the tire should develop a rolling resistance and that is what we are coming now. Now, let us say that obviously all of you know it, but I am just reiterating what is well known. Let us say that I have a tire. We have what are called as treads. Let us say that that is a tread and that is the ground. So, this tread material as it approaches is going to get, let us say that it gets compressed and then again gets released. Why tread? The material inside the tire which we are going to see what they are also gets compressed and released or in other words, there is a loading unloading cycle as the tire rolls, a loading unloading cycle as the tire rolls similar to what you see in this stress strain curve. So, in other words, if I go and sit here in this tread and go through the cycle of rolling, I will go through a compression and then whole compression. As I come here, I get completely compressed. So, go out, the load on me gets released. Because of this cycle, I lose energy or there is an hysteresis loss. So, this is now who is going to compensate for this hysteresis loss because your vehicle has this tire and tire is losing energy. So, who is going to compensate? The vehicle has to compensate. The vehicle has to compensate. So, the first thing is that because of the material of the tire, there is a lot of advantages. Why then rubber? Let us not talk about that because a lot of advantages, we will see that. So, because of the material with which this tire is made off, we have a hysteresis loss and the loss has to be compensated by the engine ultimately and so this opposes the motion. Now let us understand how did I get this force? I said that there is a rolling resistance force, which let us call this as FR, rolling resistance force. It can be the front and the rear. So, how did I get this as a force? So, in order to understand this, we have to look at what is called as the contact patch of the tire. What is a contact patch? A patch that is formed obviously by contact of the tire with the road. In other words, more precisely it is the pressure distribution at the contact. It is a pressure distribution at the contact. So, what is the pressure distribution? We will see the three-dimensional pressure distribution later or rather two-dimensional pressure distribution later. Now, let us understand a section of this pressure distribution. So, let us say that I come into contact at that point and leave contact at that point. In other words, that is where my contact is. It is not necessary that the contact pressure exists only when the tire rolls. When the vehicle is stationary also, you have contact pressure. Let us for a moment stop the vehicle and look at this contact patch. So, the contact patch now is not that of a one tread, but there are number of treads. So, the contact patch would look something like this. In other words, rubber is symmetrically compressed about the center. This is a vehicle that is standing symmetrically compressed about the center. So, whatever is the force that is compressed by this and it has to come out in the other side. Whatever is compressed has to come out the other side. Now, let us understand one or two more things about tires before we go into the details. The first is that the tires that we use are what is called as pneumatic tires. In other words, we inflate the tire to a particular pressure. So, many of you may have driven a car even now when you go to a gas station to fill your or inflate your tire, still you talk in pounds per square inch units, 32 psi. If you are driving a huge vehicle truck, it is 120 psi and so on. So, let us go into some details and look at this section from this angle. So, let us say that the tire, that is how it is deformed. Let us say that the tire is deformed like this. So, when you look at it from this section or whatever be the section, that is how the tire is deformed. For a moment, I am taking out the tread and I am saying that the tire has a thickness something like that. That is what is the inflation pressure which we have used in order to inflate the tire. When we inflate the tire, we get what is called as the inflation pressure. So, now we know very well that we know very well equilibrium equations. We know very well that whatever infinitesimal element you take should be under equilibrium between the forces that are acting on those infinitesimal elements. So, obviously, when I take an infinitesimal element here, I said contact pressure is what is acting in that region. So, if I want this to be under equilibrium or if I want it to be under equilibrium, then the pressure that is acting, the contact pressure that is acting should equilibrate the inflation pressure when it is full contact, when it is full contact, when it is in full contact. So, the contact pressure should be equal to the inflation pressure. Contact pressure should be equal to the inflation pressure. So, strictly speaking, the contact pressure should have been uniform. But, contact pressures are never uniform. We will see more about it a bit later because of the inflation pressure, the local bending because of the bending of the side walls, these are called side walls and all that. So, the contact pressure is never uniform. It has a particular shape. We will study this after two or three classes. So, now let me come back. So, in other words, there is a lot of theory as to how contact pressure develops, how contact pressure is distributed, whether it is uniform, whether it is not uniform and all those things. Now, here when I talk about this, I am only talking about the pressure because of the tread as it travels along the or around the circumstances, I mean, circumference. So, here I am looking at the pressure on the tread. So, the pressure on the tread compresses, goes to a maximum and then gets released. So, let us not right now confuse between this and this. We will come to that later. So, the contact pressure, what we are talking about is because of the tread getting compressed. When the tire is stationary, then we have a contact pressure something like this because there are a number of treads. There is one tread that is getting compressed, another tread getting compressed a bit more, another tread much more, another tread slightly less and so on. So, number of treads are involved at various compressive positions and hence we have a contact pressure like that of the treads that are formed. On the other hand, let us now roll the tire. Now, when I roll the tire, let me follow a tread. So, that is for a static. So, I am just removing that. Let us now roll the tire. When I roll the tire, one tread or one block, that block is what we are going to follow, that block gets compressed, goes to a maximum compression. Then gets released. So, one block here, that block as I rotate the tire as the tire revolves goes into this position. So, same block goes into this position, maximum compression goes out and gets completely released. So, in other words, a block gets loaded like that and then gets unloaded. Now, how is that? It is going to be unloaded. It is going to be unloaded like this. So, unloaded like this. So, as the blocks gets loaded and unloaded, blocks gets loaded and unloaded, blocks gets loaded and unloaded, these blocks lose energy or hysteresis develops in these blocks. It is not only the blocks that gets compressed or loaded and unloaded, but the sides of the tire, they also go through the same thing. So, in other words, the sides of the tire gets also loaded and unloaded and so on. So, in other words, this loading and unloading cycle gives rise to this energy loss and that has to be accounted for by the wake. I am just repeating that so that you understand it and that is quite clear. Now, how does this loading and unloading cycle affects that contact patch in a very simple setting as we had seen? How does that gets affected? Because the loading cycle or loading path is different from the unloading path. How does that get affected? So, it was symmetric when it was stationary, that is fine, but when it gets loaded and unloaded, look at this carefully. For the same strain, in the unloading path, the stress is less. So, now we are talking about the pressure that is acting on the treads. Since, for loading and unloading, for loading and unloading, they are different, this curve cannot be symmetric because they are, both of them are not the same. So, they cannot, this guy is due to loading, this is due to unloading. So, they cannot be symmetric because I am following the same tread which is going through the cycle, so it cannot be the same. So, how it should be? This has to be a different curve, this has to be a different curve. So, the curve actually shifts and becomes like this. The curve actually shifts and becomes something like this because it is, the loading curves are different from unloading curve. The curve, you know, the symmetry is lost, becomes something like this. If I now say that the reaction force in order to support, there is not a very correct picture. That is why I introduced this inflation pressure. Keep that in mind. We will come back to this topic again. Not a very correct picture. We are going to see very interesting things, how inflation pressure is going to act and how actually the tire carries a load. We are going to get to details there. So, we will come to that a bit later, but let us now understand this from a different angle and give an explanation only to the rolling resistance. We will refine it as we go along. So, if this is the load that is acting on the tire, then the load is now equilibrated from the ground or in other words, that is the load that is going to act, which opposes the weight. Now, since this symmetric distribution is affected, what will be my resultant force due to this contact with the ground? The resultant force which is developed due to this compression which opposes the load that is on the tire would now get displaced and hence actually instead of acting right at the center, the load now acts away from the center and that is how the load acts. When it acts away from the center, then if I now look at that load, with respect to this center, not only I am going to equilibrate this load with this force, but I am also creating an additional effect correct. So, what is that additional effect? That will be a torque, that will be acting or a moment that will be acting like that right. Watch carefully that the moment is now going to oppose the motion of the tire. So, there is an opposing force or opposing moment that is acting. Now, I do not want to put that moment here. I know that the moment opposes the motion. So, I just want to replace this moment by means of a force that is acting here because that will oppose the motion of the vehicle. So, I replace this moment which in reality exists because of this co-elasticity by a force here and call this as rolling resistance force and say that this force, rolling resistance force creates the same moment which opposes in other words, F r into r is equal to this into this. We will give names to that in a minute. So, first let us understand the philosophy of development of a rolling resistance force. So, the philosophy of this rolling resistance force to summarize is the viscoelastic behavior of the elastomer which means that there is a loss of energy, which means that the symmetric contact pressure distribution when it is stationary gets affected or in other words it becomes skewed and the skewed distribution produces a force, normal force which not only opposes or not only supports the vehicle or the tire, but also creates a moment which opposes the motion and the opposing moment or motion or torque is now also depicted as a force which opposes the force. Motion of the vehicle or the tire and we call that as the rolling resistance force. So, that is why we have a rolling resistance force. The rolling resistance force, of course you can see this very clearly rolling resistance force since it comes out of a moment which supports the weight W. So, this force has to be proportional to W. So, we usually write the rolling resistance force to be a rolling resistance coefficient multiplied by W, multiplied by W. Obviously, the rolling resistance opposes the vehicle motion and hence is not within quotes a good force. It is not eating us to travel. Actually it is opposing you. Since it is opposing you or opposing the motion of the vehicle we consume energy because I have to overcome that like you have the aerodynamic forces, we have rolling resistance forces which opposes the motion. Interestingly note that when the vehicle breaks, this rolling resistance force would act in the same direction as that of the braking force which is now going to flip and act from the other direction. So, rolling resistance force aids in braking and opposes traction. So, the first thing you would tell that why not I completely reduce rolling resistance, go to zero is it possible? How low you can go? There are lot of issues. We will come to that later when we talk about time mechanics. So, first things first. So, that is the rolling resistance force which is written in terms of a rolling resistance coefficient and W. Our next step is to find out WF and WR. And WF determined by taking a moment about the point A and WR determined by taking a moment about the point B. So, on one hand we have WF into L is equal to on the other hand you are going to write down the moment due to the forces. So, you know this very well. So, WF into L is the clockwise direction. So, accordingly put the forces in the moment rather the moment due to the forces put the signs properly and we will see how we end up with this equation in the next class. We are going to make some assumptions with respect to this heights. We would see that usually in a passenger car these heights are almost the same and we will make an assumption that HA is equal to H is equal to HD. That makes our life simple. One of the things which is obvious which all of us experience which you would immediately notice is that WF and WR is going to get affected when a vehicle is accelerating or in other words that is what is called as a load transfer. You would have noticed this when you go in a vehicle in a car obviously all of us know it very simple mechanics that when you accelerate you tend to fall back and when you break you tend to fall forward. In other words there is a load transfer to the axles as well. It has a very interesting effect. So, we will write down this equation. We will find out WF and WR then we will look at traction and braking and so on. We will stop here and we will continue in the next class.