 In the last class, we defined a very important parameter which we called as understeer gradient. We looked at it from a state space form, the equations were written in a state space form and we found out that this understeer gradient plays a very important role in the stability of the week. In fact, if you go and look at that equation again, we see that b c alpha r minus a c alpha f is the key term and the whole concept of understeer gradient is based on that particular expression. So, that is what is going to make it positive, negative and so on. Then we moved over, we have not yet given a very physical meaning to it. We have not given yet. We said that that is definitely a parameter which has a problem or which is going to create a problem on the stability of the week. We also made it very clear that it is not that when it is negative, the vehicle immediately becomes unstable. We also said that there is a speed, a critical speed at which this vehicle becomes unstable. That is what we said. In other words, we had two terms and we have to make sure or we have to get that some of those two terms to be negative in order that it should be unstable. This is where we left and then we started with another small digression on steady state cornering. In order to, why are we doing this? In order to understand what physically this understeer gradient is, we will go back to the state space. We are going to spend some more time in state space. I know you have done this in controls but we will go and refresh your memory on the state space as well. Before that, let us look at this. So, since I said steady state, we do not have those terms, i is at r dot, r dot does not exist because it is a steady state expression. So, we have an expression of this form. These are the two equations. In other words, the governing equation or differential equation or reduces to these two equations because this r dot and we said that v dot goes to 0. One of the key things that we found when we started writing down this equation is that the two forces that is force acting in the front and the other acting at the rear. Remember that we were looking at the bicycle model and remember that we have one wheel or one tire where the left and right tire actually is collapsed. There is some confusion I said front and rear in the sense that front and rear has one tire where the left and the right tires are collapsed into one. So, the stiffness of this tire is the sum of the stiffness of the left tire and the right tire in the same way it is for the front. So, if you are given say in a problem the stiffness of a tire, then you have to take it as 2 times that of the stiffness of the tire, right. So, that is just a small input. Remember that one of the key factors or key conclusions that we made was that the front and the rear centripetal forces are distributed in the same fashion as that of the weight of the vehicle, right. So, we said Wf is equal to W into b by a plus b and so on and it is distributed in the same fashion. So, Fyr can be written like this. This is what we wrote clear and Fyr is written in this fashion. So, in other words Mu square r that is the term that is like our W a by a plus b and that is equal to b by a plus b, okay. So, you can write that as Wr into u squared substituting it and you can write like that. That is a very straightforward expression, okay. Why are we doing this? We will know that in a minute. Now this is also things which we wrote in the beginning of this derivation that the forces that are acting can be expressed in terms of the stiffness, okay. So, in other words we are using a linear model for the tyre, right. Now let us substitute these expressions, okay. Here all these, this one, let me call that as 1, 2 and 3, okay. Let me substitute that so that now I will write this as Fyr by C alpha f minus Fyr by C alpha r, okay. That would be the first two terms. Note that one minus cancels out and so on, okay. Even if you had written alpha f to be positive you would get the same expression, okay plus l by r. And note that l by r is in radiance because we are replacing the angles so if you want to put it in degrees correspondingly you will make the changes, okay. Now let me substitute that expression, this is what we have here from 1, okay, on to the new expression which we would call as 4, right. Substitute it and write it down. You will get a very interesting factor there. You will get m into, no, is it clear? I am substituting these two into that expression, okay. In other words, I am going to take here first these two expressions and then substituting it there, okay. Let us see what you get. Expand this. You will get Bc alpha r minus Ac alpha f, wow. So that is what is the key factor here, right. Bc alpha f minus e, this is Bc alpha r, sorry, Bc alpha r minus Ac alpha f is a key factor here, that will be divided by, we have A plus B, so A plus B which is l, A plus B is nothing but the length of the vehicle, l into C alpha f into C alpha r, okay, multiplied by that u squared by r is there, u squared by r, that is the term here, plus L by r. Let me call that expression phi and spend a few minutes here. Look at this term here. Let me call that term as K1, okay. He is a very close relative to K3. Why? It is just removed from K3 by that factor l. Go back and look at that. We had put l squared there, so factor l, okay. So many people call this as an understeer gradient, right, understeer gradient. Now look at that expression, Bc alpha r minus Ac alpha f is here, right. And what is your conclusion from this? That when the understeer gradient is positive, is high, okay, is high, then for the same u, very simple, first expression, first things, for the same u, okay, you are going to apply more steering input, right. That's the first right, the beginning, right. We will also note that it is possible to express this in terms of w as well. We will come to that in a minute, okay. Now let's go into the details of this. Let's say that I am plotting u squared by r, do not be surprised if they say that understeer gradient, okay, is expressed in terms of per, you know, g value. In other words, you can also substitute for instead of m, you can put a w, you can put a gr, okay. It's also possible that we will call this as k2, let's not worry about that right now. I just want to point out that it's possible to write that as well, okay. Now let us now look at this graph of, so this is for a steady state cornering, you know, that's the result which we are looking at. Let's consider the simplest case, understeer gradient is equal to 0. That means that bc alpha r is equal to ac alpha f. What happens? If I now plot this u squared r, this is equal to a constant which is l by r, okay, the kinematic angle, that is a constant. In other words, when I take a turn, whatever be the speed, I will apply the same steering input, clear? Okay. Now let us look at a condition where k is positive, k is positive. So this is, yeah, bc alpha r is greater than ac alpha f, right, k is positive. Obviously that's a straight line because I have plotted this as u squared by r, okay. So what happens now? That straight line would keep increasing, that's the l by r. That straight line keeps increasing. So what does this mean? That means that if I have to take a cornering or in other words, maneuvering with the same r, with the same r, for the same r, if I now increase the speed of the vehicle, then I have to increase my steering angle. So in other words, the steering input that I give depends upon the speed of the vehicle and it increases with an increasing speed of the vehicle, okay. So that's why we say that it is understeered, okay. So I have to give more and more steering if I now increase the speed, okay. Comes out very clearly from this expression. What happens if it is an oversteered vehicle? In oversteered vehicle, this quantity is negative. So when I increase u, since it is multiplied by a negative quantity, what happens to my delta decreases because I have obviously look at it mathematically, it's just a straight line with the slope is equal to negative. So that guy goes now like this, right. What happens? There is one value of this u squared by r at which delta that is required is equal to 0. In other words, you don't give any input to the steering, delta is equal to 0 and the vehicle takes a turn, okay. So it takes a turn. In fact, you cross that and come down. What happens? Delta becomes negative. What does it mean? It means that if you are going to take a turn to the right, okay, your cornering, you give an input to the left, okay. You give an input to the left, oh, right. It comes out very clearly mathematically that that's what happens. I know, you know, puzzle phases I see but it is very simple mathematically, right. No, you are not convinced, okay. Whenever mathematics gives a result like this, you look puzzled, you know, you really want to know whether it is true or not, right, okay. So you want to look at physics, what really happens, right. First let this get into our mind. Is this clear? Okay. Why is this happening? Let's go to this mind, front tire. Let me put that front tire, okay. Now, actually it is this alpha f and alpha r which is playing the role of cornering. Those two guys are very important, right. That's why tire is very, very important. Look at that. You know, tire plays such an important role in this whole handling of characteristics. Very important role, right. Just for a moment, look at this more carefully. Please note that the b into, first my second expression there, here, a into f y f minus b into f y r, okay, is equal to 0, right, because that's a steady state, right, okay. Now coming back to this, remember my figure that was delta, right, that was delta and then I said that there has to be an alpha f that has to be created in order that or that should exist in order that I develop what is called as the front force, clear, okay. So when alpha f now increases, which means that my expression is going to become more and more positive, okay. What happens? It is going now up. Note that this is the direction of travel, okay. Now if I now keep going up, then I have to give more and more steering, more and more steering in order that I maneuver my vehicle, okay, to take a turn, to take a turn, right. So when alpha f is small, when alpha f is small, then already the guy is aligned towards that turn and so delta needs to be small, right. At one point of time what really happens is that this alpha f goes to the other direction, okay. If it is like this, then we get into lot more trouble of instability, right, because I have to now bring it back again, so heading becomes a problem. So that is what happens and in the words that is this alpha f, role played by alpha f becomes extremely important, clear. So the key factor here is this b c alpha r minus a c alpha f, that is a key factor here. Now what is a and b? Remember that they are the CG locations. Imagine that or why imagine, you know you have so many front wheel driven cars, most of the cars today are front wheel driven. So in a front wheel driven car, what is, what do you think will be the distribution of a and b, will a be small or b be small, a will be small. So all front wheel driven cars are understeer cars. So the first factor, right, they will be more towards the understeer vehicle. The first factor that the vehicle manufacturer has to look at is what is this understeer gradient, okay. Towards the end of this part, we will look at what are the tests that are used in order to determine this, let us not right now worry about it, say. But we need to know that understeer gradient, okay, is an important part of the or important characteristics of vehicle for handling. Do the manufacturers produce an understeer vehicle or an oversteer vehicle? They do not produce an oversteer vehicle, okay. Basically because the worry is it can become unstable, you know, very difficult to handle it, sometimes it can become counterintuitive. So all vehicles are understeer vehicle, they are not oversteer vehicles, okay. Please note they are not oversteered, they are all understeer vehicles, right. Now the question is how much understeer it can be, okay, that is the important question. If it is, I want to be very safe and I want it to be highly understeered, then tomorrow the newspapers or the magazines which does a review with your vehicle will say that this car is a damn squib, it does not just turn, okay. You need to apply so much of steering for this vehicle to turn, response is very bad and blah, blah, blah, blah, you are in trouble. So I cannot have so much of understeer that the guy who is going to drive the vehicle, he keeps on steering, right, so that is a problem, it cannot be an oversteer. It is just above the neutral steer, okay. Why is that people do not look at neutral steer? Only because look at that quantity, it is BC, alpha r and alpha f, alpha f and alpha r, what are they? They are the lateral slip stiffness, that is, as it is called, okay, lateral slip because lateral slip angle, lateral force is a slip angle, slope of that curve, so lateral slip stiffness. It depends upon so many factors, you know, it is a tire, so many factors and it depends upon the inflation pressure. If you are not maintaining inflation pressure, definitely your handling characteristics will be affected, okay, depending upon what is the inflation pressure, how that varies and so on, that will get affected and A and B are not strictly a constant, is for a car it may be a constant for even that we cannot say because there is a small difference can be there, how many people sitting, how they are sitting and all those things, okay. But for a truck, definitely for a bus, definitely it tends to affect, okay. So it cannot be straight away a neutral steer, so it will be slightly understeer, okay. So that is the important of understeer gradient. It becomes extremely important also in rear engine cars, rear engine vehicles. Many of the buses are rear engine buses, so then you have to be very careful in looking at this quantity, all right. So because it is a bus may not have that much effect because the passengers can be distributed and so on, but for a car, rear engine car, it becomes many of the electric vehicles for example, or small cars, very small cars, okay, or rear engine cars, in which case, okay, you will get this vehicle to be oversteered and hence, you know, that will be a problem. Then it comes to the fact that what happens to a Formula 1 car? Formula 1 car, the driver just does not have the time, okay, to corner, no, he cannot give so much of input, steering input in order that he corners the vehicle. So he would prefer it to be oversteered, okay. Yes, this is important, but these people are very well trained drivers, so they know how to handle it and so the vehicle will become an oversteered vehicle. One of the interesting factors in India, in India especially in the truck market is that they mix the tires, or in other words, they mix the radial tires with the bias tires. Many times they use the bias tires in the front, the radial tires at the rear and so on, okay. That will affect the handling characteristics of the vehicle because usually, the radial tires have two times more lateral stiffness than the bias tires and so mixing this is going to have an effect because of that factor. It can be a huge factor when it comes to the mixing, okay. Fortunately they have a radial at the rear, the bias at the front, okay, so the factor actually increases, so it is not a problem, but if it is the other way, then you will get into trouble. So it is better that when you mix it, you be aware that it will affect the handling characteristics. Of course, cars no one uses bias tires, so it does not matter, but for trucks in India especially because about only 20 to 22% of the tires are only radial tires yet. In fact, some of the survey one of my students did, we found that only about 6% of the trucks which he inspected recently, you know, they have radial in the front, the rest of them have only bias. So it is very important that we understand this factor. Yes. In case of a tractor trailer, how do you actually decide these things because when you are loading and unloading, there is a huge difference in the A and B. Yes, of course. So that is, that is why I said this would, this would vary, okay, when it varies, so you have to, you have to know the limits and accordingly adjust that. So it is not one factor, okay, that is why the handling characteristics depend upon the load, how it is placed and all those things will have an effect, okay. So that is important, right. So that is the key factor on understeer gradient. So as I said, the way it is defined is different, okay. You can define this in W and G out and so on. So be careful when someone tells you a value, okay, we will do some problems later. You have to be careful and ask him what exactly is your definition, okay. I have got into trouble like that before. So you have to clearly know how someone defines the understeer, okay, gradient. What is the, what is the formula he is using? So it can be K1 or it can be K2. For example, you can put W here and take this G out and then write this whole thing. Now delta is equal to, let me call that as K2 in that case, K2 into Ay plus L by R where Ay is the lateral stiffness, okay, expressed in terms of G, in terms of G. So radiance per G is what you would say is the unit, okay. So you have to be careful as to how you are writing that, clear? The questions? Of course, very important that user controls this, that is why he has to give an input. User controls that, but this factor is not controlled by him, okay. This factor is not controlled by him, that is the whole thing. So when he presses, in other words, which I should not say because we are looking at steady state, okay. So when the velocities are high, okay, his steering input has to be higher, okay. In other words, you take a turn at 40 kilometers per hour, you take a turn at 80 kilometers per hour, your delta is different is a factor which is very important to understand, okay. So note that the problem also is that when you increase the speed, what happens to delta? It increases. That means that that much amount of steering input you have to give. So that is the time that you have to also consider when you look at understeering, clear? Yeah, because this whole, our whole, note that that is how we had to find this whole model, okay. So center of gravity location is the body centered coordinates, we have placed it at the center of gravity location, right. No, no, no, we were just looking at the forces, we did not write down the equations, we are just looking at forces, okay. Here we have put a coordinate system and we are looking at the equations based on this coordinate system, okay. Yeah, fine. Now we will, we will digress a bit, we will look at state space technique, okay. Remember that we had expressed before we go, we went into this expressions on steady state, we were, we were looking at it, looking at the stability, okay, from the state space perspective as well or in other words, we were looking, we were writing it before we went into the stability in terms of state space. Remember that we had written this as, what is the state vector, v and r, remember that we had written it in terms of v dot r dot is equal to a into v r, okay, plus some b into u, b is ac alpha f by m, sorry, c alpha f by m and ac alpha f by i, i and so on, okay. So state space becomes a very important vehicle to study vehicle dynamics. So in fact many of the controls that are used are based on the state space techniques, okay and it becomes very easy for us to even understand what happens for a given input or in other words how does the vehicle behave, quickly if you want to look at how the vehicle behaves for example, if I give a step input or some other input, so that becomes extremely important. So let us now get into this state space and let us now derive the expressions for or I would say brush your memory, okay, on state space. That will answer some of the questions which were asked as well. So let me write down the state space expression as note that when I put a squiggle at the bottom it means that it is a matrix or a vector, right, that is all. Remember that b for example, we just now said b is c alpha f by m and so in other words remember that b is c alpha f by m ac alpha f by i, i is what we mean by i z z because we removed that, okay and u is nothing but the delta, okay. Go back and look at that expression so that you can relate a is a matrix which we wrote you remember minus c alpha f plus c alpha r and so on, okay that is the expression for a. In other words, I am just abstracting that expression and writing down this form. So what is the first step now? The first step is to take the Laplace transform of this expression, okay. So taking the Laplace transform, so we are looking at the linear time invariant system so a is states as it is, right, okay. So s i minus a x s is equal to x naught plus b u s, clear? So that x s this is all which you studied is equal to clear, okay. Now in the same way as we give in the traditional control system space, let us name and name the inverse Laplace transform of psi s is equal to phi t, okay. I am just writing down a general framework from which we will get into specifics, right. From which I can write down the expression for the x t, okay. Just taking the inverse Laplace transform of 1, we can write down x t is equal to phi of t x of 0, sorry. I should write this as not 0, sorry, s x naught. Let me just x naught of s naught, x naught, okay. That is x at t is equal to 0, that is the input. So let me leave it as x naught, okay. X of x at t is equal to 0, that is what is meant by x naught. That is the initial condition, right, okay. So I can write it as x naught or x s but as long as you understand it, that is fine, okay. So I can leave it like this or I can put x naught but please understand that that is the initial condition that I am actually x of x naught is fine, okay. So just note that that is the initial condition, okay. So plus phi of t b u of... Now we will develop what can be the expression for psi of s or phi of t as well, right. Let us develop that expression. So how do you develop this expression? Is this okay? I deliberately wrote that, I wanted to check how much of Laplace transform you guys know. Is that okay? Is that okay? What is the Laplace transform of multiplication? You have heard about this term, convolution, yes. You would have seen in your earlier classes that convolution is given by a star, okay. So if I say phi of, say let us say phi 1 of t, convolution, phi 2 of t, we will see in a minute what it is. A Laplace transform of the convolution term is equal to in the Laplace domain, the convolution becomes a multiplication, right. So that would become s into phi 2 of s, right. So what is this term now? B is a constant of course. So it is a psi of s when I put, okay. This is a multiplication. So what should I get here? I should get a convolution, okay. I should get a convolution, right. So be very careful when you write Laplace transform, okay. So many people would leave out this term, x of 0, okay. Let me call that as x of 0, I am very comfortable with that. So x of 0, that is the, so that x 0 of s is not there, x of 0, okay. Many people leave this out when I say Laplace transform immediately x dot, I know x dot is equal to s of x, x of s. That is one more term. Like that you just cannot, you know, blindly write it like this and you have to look at when it is a multiplication, you have to write down, okay, the convolution term. How do you write the convolution term? 0 to t phi of t minus tau B u tau d tau, that is the convolution term. So that is the inverse Laplace transform, okay. And I will not say this again and you have to make sure that you remember this, right. So now we will develop a specific, I suggest that you go back and revise your Laplace transform, maybe look at the Laplace transform table once more, okay, because we may be using Laplace transform when I give an input, okay, different input, we will be using that, so you should be knowing how to convert it. So get back to your control scores and look at these things once in order to understand this, okay, right. So I want to develop this and let me look at a simple differential equation or ODE, which is the first order differential equation and write down that expression. So x dot of t is equal to A of xt with an initial condition say x0, what is the solution for this equation? X of t e power at, okay, x0, right. Now let us define a matrix, okay, which I would call as e power at which I would call as e power at. In the same fashion as I would define e power, okay, at when a is a scalar quantity. What is e power at, what is e power at? 1 plus at plus a square, 2 square by 2 factorial plus a cube, t cube by 3 factorial and so on. Let me define e power at to be like that. That is now identity matrix plus at plus a square, t square by 2 factorial and so on. No, no, no, that is the expression. If I am, if this is the expression, okay, if that is the, no, no, I have not come back to this. Just wait a minute, right, I have not yet looked at it. We will, we will, this is only to tell you like what e power at looks like, right. What is the, yeah, so that is the velocity at which delta becomes 0, right. Remember that we had k1 into ay for example or m into bc alpha r minus ac alpha f divided by lc alpha f alpha r into u squared by r plus l by r, you know. This term, that, that what is there in the bracket multiplied by u squared by r, that term if it becomes negative, okay and beats this l by r then only it becomes unstable. So the point where both the terms are the same, that is the point. No, no, no, this, please note that we are cornering, we are cornering. So imagine like this, I have a understeered, sorry oversteered vehicle, okay. I keep on, I keep on increasing the speed, I, it is say for example it is a, it is a round tunnel, okay, which you see in the street, say for example you are going around, right. So you start with the velocity, then you give a steering input, right and then you go around, then you increase the velocity, okay, forget for a moment steady state, then do not get confused in that. Let us say that you increase the velocity. The steering which you have given, you have to now make it smaller, now further increase the velocity, okay, the steering input, again you will make it smaller. At one point of time, you are not going to give any input to the steering, okay and the vehicle will go around at one velocity, okay. You increase the velocity further, you will have to give the steering this side in order that you go in the clockwise direction, okay. So that point at which you give zero input to the steering is that critical velocity, clear. I suggest that there are a number of videos in the YouTube, okay. Go back and look at this YouTube videos where you know steering input is given for an understeer vehicle, an oversteer vehicle and so on, okay, it will become very clear. So we will have a course next semester on vehicle dynamics lab in which you will also be studying or you will be doing simulations as well as some testing as well, right, okay. Let us get back to this, e power at, why are we defining like this? The interesting fact is that this matrix has all the properties of e power at, right. So if I now differentiate what is d by dt of e power at, a into e power, okay. Similarly when you differentiate this matrix, d by dt of e power at is equal to differentiate that and you would see that it is the same, right, okay. So we are going to use this fact, this fact, okay, in order to understand what can be expression for psi of s and also the fact that e power at, the inverse of the e power at is equal to e power minus at, right, we are going to use that, we will do that in the next class when you are going to substitute and give an expression, clear, okay. Any questions on this, okay, anyway we will continue it once it is, the direction will take one more class then we will be completely on board, right, we will stop here and continue the next class.