 In the last class we were looking at brush model right and then we moved on to what we called as a magic formula model which is due to Praseka right and we saw that the basically the first model that was put forward in the Praseka model, tire model is to fit the experimental curves that were obtained during tire testing right and we will now slightly go into the details of tire models before we I think we have to quickly go to the lateral dynamics. We would not have time to go into completely the tire models, there were some questions on in fact what happens during combined slip we will spend a few minutes on combined slip before we go to the rest of the topics including the tire model but there were lot of questions on tire models of the class so let me let me explain this. This is from a paper in vehicle dynamics, a vehicle system dynamics this is the reference this is a very nice figure, volume 43 supplement 2005, 1829, Alex Eichenberger and Marcus. So this is the this is a paper just there is a reference for this and let us just quickly run through this so that you understand what we mean by tire models. As I told you earlier in the class it started that a lot of codes that are multi-body dynamics codes that are used and multi-body dynamics becomes extremely important because of the fact that there being lot of companies use it and that the companies look at how to optimize the vehicles and so on right so they use a number of softwares. The software that is used depends upon the reason or why they use this or in other words it is not one comprehensive software and one comprehensive model but there are a number of models and the models also depend upon the reason for which they are doing an analysis okay and a part of it is a is the tire model. Please understand that we are looking at dynamics of the vehicle, we are looking at oscillations be it cornering or be it ride or whatever it is we are looking at oscillations and the oscillations or the frequencies of oscillations that we are looking at covers a whole huge range okay and the models that you put in should be able to simulate okay or get you the results in that frequency range clear. So there is a frequency range and an amplitude yeah we will come to that a bit later but before that okay let us understand yes I understand what you are asking let us look at this figure then you will see what we want to say okay. So the vehicle oscillations can be classified in terms of frequency of oscillation okay and the amplitude of oscillations right. So yes when you talk about oscillations immediately you think that there is going to be a bounce you know vehicle goes over a rough road and you see that you know bounce and so on and immediately you have a tendency to sort of map this word oscillation to a vertical oscillation right but it is not necessary that it has to be vertical that is what we are going to see now okay and you will see that when we talk about oscillations it covers a gamut of things right it is like like the whole electromagnetic spectrum which we talk about when we talk about light we talk about sound and so on like that you have a number of you know the whole gamut of things that we are going to look at various frequencies and amplitude right okay. Now at the lowest end we look at what are called as quasi-static models where we study the suspension kinematics okay you have studied say for example 4 bar linkages and so on in your earlier class on mechanisms. So you use that kind of concept to understand actually when you give a steering input how say for example the steering linkages move okay what is the arc that it covers and all those kind of things okay. So you use the fundamental concepts in theory of machines or kinematics of machinery which you have studied in order to understand that okay so that is the quasi-static models right the amplitudes may be very large because you are looking at when you turn a steering how things move and so on and many times the models that you use for a tyre in this case is a finite element model you know simple static finite element model and so on right okay. We will come to this tyre models in a minute. Next in the range look at this you know this amplitude can be large. Next is what is called as the handling models handling models please note models simply means equations. Next we talk about handling models we are going to do that from this class may be starting from next class right we are going to look at the forces that are acting the accelerations and so on we have already seen this for the longitudinal case handling is more to do the lateral forces and the lateral dynamics and that we are going to see as the next step and so we are going to look at handling models. In these cases as you are going to see now you consider the chassis as one rigid body and look at how for example the chassis is going to yaw or the body of the vehicle is going to yaw is going to roll and so on right. So we do not go into the details and usually this kind of models look at a picture where we look at the whole chassis as one group of components yes good question. So what is that we are looking at there we are looking at various frequencies you will understand that because this depends upon see when we look at handling it depends upon the frequency of the whole vehicle of the vehicle at various modes or in other words in various directions in simple terms. You would see that we will come to this handling when we take a turn you would see that certain frequencies which are the result of the whole of the vehicle mass moment of inertia and so on are going to have a role to play and those frequencies are in this range or in this range. The frequencies which are going to play when we look at lateral dynamics handling is about 4 to 5, 3 and maybe 2, 3, 4 you know that is the range yes of course see the point is this is it that I am neglecting all this no what we are saying here is that we look at different frequencies to look at the effects look at the effects for example if you are looking at noise yes you may be cornering manoeuvrability if you are looking at manoeuvrability of the vehicle and so on the frequency range I am looking at is this but at the same time I am looking at the NVH the vibration characteristics and so on and then I would comfort for example I would look at these frequencies I am looking at actually acoustics I would look at higher frequencies it is the vehicle is moving it is that you are artificially distributing it at these frequencies and studying the effect at these frequencies go and study when I say that go and study the frequency at this range which means that you are looking at handling and that is the result of the characteristics of the vehicle when it takes a corner right and the ride for example the next set of frequencies that you are looking at is the ride please note why am I saying this please note that the model should be capable of capable of representing the vehicle in these frequencies it should be able to represent the vehicle in these frequencies okay suppose you take a simple model what we are going to see but for example bicycle model for handling okay then if you tell me that I want to study the acoustics okay with this kind of model you would not be able to study okay because it will not produce the kind of frequencies the natural frequencies of modes or whatever you call it we are going to give some names to it okay these things will not be represented by that mathematical model okay so there is no point in having a simple model which we are going to say I mean study and say that I am going to study this because that model will not represent the behavior of the vehicle at these frequencies does that point make it clear but as a vehicle yes as a vehicle it has various spectrum of behavior as a vehicle okay but many times I would not be able to take everything together and start working and my interest also may not be on all these frequencies right it is not yeah yeah yeah okay the frequency range may be maybe this side that side but it is it is around less than 10 okay let us look at the next set of frequencies I will take the question after I finished this okay then we look at the next set of frequencies we are looking at a ride okay looking at ride ride usually is divided into two categories okay one is what is called as the primary ride and the other is what is called secondary ride purely in terms of the frequencies the lower frequencies at 10 12 hertz okay you would call this as a primary ride okay people vary when they define what is actually the primary ride what exactly is the frequency and so on but usually we will look at the primary ride as one due to the oscillation of the suspension system okay this is because of road induced oscillations and so on 10 12 14 hertz is the type of frequencies that you would call when there is when the vehicle is subjected to these kind of oscillations okay at those frequencies we call that as the primary ride but then ride is a topic which is not only affected by note this carefully there is not only affected by by the road okay today ride encompasses a much larger things okay so for example that can be oscillations of the engine transmission and so on which may be at a much higher frequency may be about 30 hertz okay so there may be oscillations of of the of the engine okay which may be transmitted and your engine mounts may not be very good there may be transmission of oscillations due to engine and so on okay then the frequency of excitation due to these factors will be at a higher level or higher frequency and that is what is called as the secondary ride right then comes harshness okay what we call as loosely as NVH that is comfort okay that is at a higher frequency at a much higher frequency so in fact if you look at the total ride how comfortable I am in the vehicle then it actually covers the whole thing okay you can also see that as I go down the frequencies actually the amplitude comes down very well known case right the amplitude comes down now you move to a much higher frequency okay which is above 100 hertz okay strictly speaking going up to 20,000 hertz we are not interested up to that because that is the acoustic range but 20 to 20,000 but we are not interested in that kind of large ranges but the acoustic frequencies which is of interest to us may extend up to about 2000 hertz yes the quasi static is where I am looking at the suspension mechanics or mechanism okay sorry steering mechanisms so suspension kinematics steering kinematics and so on yes so these are oscillations okay these are when I take say for example when I give a steering input okay how does actually what is the range to which this with the steering has an effect on the wheel and that whether Ackerman steering is completely followed or there may be error what is Ackerman error scrub radias all those things are there all those things no they are not that is why we have we are saying that it is it is at a much much lower frequency that is why we call this as quasi static very near zero it is though it is not actually static we call that as quasi static okay right we will come to that you know these are frequencies when the vehicle is subjected to that kind of frequencies okay the frequency that whatever happens is that frequency range we call that as NVH it is just for us okay to understand and give a name and I am going to talk why we have we have done this as well now I have to have these are the frequencies at which I am going to study okay I have to have models as I said which will depict these the behavior in these frequencies and one of the most important model here is what is called as the tire model the tire model okay let us go to the extreme range yes yes yes yes yes these are the frequencies okay which will excite the vehicle various components of the vehicle vehicle as a whole various components of the vehicle okay we will we will okay good question let me go into that with the tire model let us go to the extreme case okay now I want to study noise say for example I want to study tire noise okay which means that I am looking at frequencies which are above hundred hertz okay yeah that is where the usually that is a big topic usually the range are usually the range of hearing is from twenty to twenty thousand and very interesting factors have come about life for example your ear is very sensitive to frequencies of about thousand hertz okay so that is where you are you are very sensitive to this and so there are the magnitude of noise okay expressed in decibels are now adjusted and you have scales like a scale which adjusts to the sensitivity of your noise and so on so you are you are sensitive as a as a person see you perceive vibration okay due to various parts of your body right I mean so in fact we will see later when we go to right dynamics okay the last part of this course where we will look at right dynamics we will look at what is the amplitude which your body can withstand without fatigue how long it can withstand what is the frequencies what is the corresponding amplitude for example how much you can tolerate all these things we would look at it later in the course but if you look at acoustics you are what your ear is sensitive to those frequencies are about this okay your ear is very sensitive to frequency that is why you are able to enjoy music whose frequencies okay what we call in Indian classical music the Swara what what is that there is nothing but the frequency okay it comes from the Archimedes Archimedes principle of dividing the frequencies into different octave and so on let us not go and get into those details let us let us look at this model you know let us come back and look at the models okay now to answer this question like we are looking at acoustics let us see actually physically we that is been a practice in this course so I have a tire okay now the tire goes over some some rough surfaces that is snow then the tire we saw in one of the earlier classes say for example there are belts that are available in the tire okay so the belt starts now vibrating it is something you know you have done some course in mechanics it is something like a shell okay stiffened shell or an orthotropic shell okay that starts now vibrating because it has gone over a rough surface right so when it starts vibrating let us say that it vibrates like this it vibrates like this okay now this vibration of the belt okay which is actually the sum of various modes or mode shapes of the belt okay now these vibration of the belt what does it do it has it displaces so it starts vibrating like this so it takes a inner and outer shape goes up and down so when it goes up and down like this what does it do it excites or passes on this to the air okay molecules which are near this okay that is one of the things so it starts oscillating so the air molecules near it gets I mean into this oscillation on those pressure changes and this acts as a source okay for the pressure change and at certain distance you hear that as noise because these guys are displacing the air above and you know below and then that is the pressure wave that starts that is the source that is one thing the other thing is that when they vibrate okay when they vibrate they also change the forces that are acting or in other words if you now plot a force or acceleration at the axle level they also change okay so in other words with time the accelerations change and these accelerations now start okay travelling inside the vehicle and if you are sitting in the driver location you know you are going to hear that as well so in other words the noise that is produced okay passes on through the air and from the air it may be it is translated or transmitted to the door and it starts oscillating and you are going to hear all those noise so tyre for example tyre noise is can be classified into what is called as the airborne noise and this structure borne noise through the axle and gets you gets it inside and then you hear this. Now if I have a model since you ask this question now if I have a model which cannot actually capture this this kind of belt vibration you know what we are looking at is also a brush model simple model I am not able to capture this then whatever I do obviously I won't be able to capture noise hence that is why models become important so what model you use unless I have a model which I am able to capture this which is the source of vibration or excitation at that frequencies I am not going to produce that frequency in order that I hear this right okay so our simple brush model for example will be enough to do things here but then I need to keep innovating and improving it in order that we do it further yes. Why are we interested in only your sound and noise characteristics no no no I am not saying that we are why are we interested in noise I am saying that noise is one of the factors okay it is not that I do not want to make too much noise right yes every every frequencies of interest okay so if you are looking if you are going to make a car and sell it okay you are going to be worried in this whole spectrum of frequencies you are going to your car is not going to sell if you say my handling is very good but only thing is you cannot sit inside the noise is very high right engine noise is very high when you sit inside but I can handle my car handling is very good this won't work exactly so at different frequencies so that has to be now taken care of isolated so that you do not hear or you do not feel whatever is your sensory perception is of these frequencies this is very important to understand this yes this is well known fact in the industry but your students it is very important that you understand okay what we are talking about and the macro picture is extremely important for you to appreciate the course on vehicle dynamics okay I would have given this first but you would not understand now that we you know what is a tire model now you know you you are able to understand that very well okay so this is this is very this is very important for us to look at right so the tire models that are used are now different at at different in the different frequency ranges you have today a set of tire models which can handle different frequencies okay. You have for example a model called Swift model that the name itself is like that short wavelength intermediate frequency tire model right so that that may be about 70 80 Hertz and that's the frequency which it can accommodate and that's what you see here so most of these models are yes we saw an empirical model but there are semi-physical models as they are called in other words the tire is modeled as a consistent consisting of a number of springs okay mass systems and so on and that approximates the behavior of the tire we won't be able to go into the details of it we have another course actually on tire mechanics where we would talk a lot more about tire we are not going to do that in this course but I want you to understand where we talk about and I mean what we talk about and what we are looking at and so on right so this is what is the importance of time models right and this what we call as multi-body dynamics this is multi-body dynamics the the range of multi-body dynamics applications also looks at dynamic loads the effect of dynamic loads for example when you come to automotive structures you would look at this carefully you would look at dynamic loads what is a load that is transferred for example to various parts of the vehicle for example to the chassis frame okay what is the effect of this dynamic load it's affecting in other words what is the durability and all those things also will be studied what's beyond this is what is called as crash okay crash is a crash covers a gamut of frequencies very high frequencies and amplitudes are also very large so that is the that's the crash part of it okay it's a nice picture it gives you the whole thing and let's get back to what we were doing in the last class right wait for some time to get an answer so what is handling what is your how does it have an effect all those things will be studied in this course so in other words you will get into models here okay I won't be able to get into these models beyond this but I will get into models which cover this and this you know these parts are part of this course so you have to wait for some time in order to understand you know what I mean by the second thing right we will do that yes so different it's not different component of the vehicle you know this is we are talking about we are talking about the vibration of the vehicle okay induced due to various components okay for example for example when you are looking at the primary right okay you can treat this whole body as a rigid body and we are going to look at that as a rigid body right at those frequencies so in other words the frequency is dominated by the by the spring dashboard system of the suspension okay so when when I go to higher frequencies then there are contributions at higher frequencies and that may be due to bending of the chassis frame okay so in other words the source for these vibrations and the way they are transmitted and so on okay would be different right at different frequencies say for example when I am I am here I am looking at the body as a whole controlled by the suspension characteristics right wait for some time you will understand more about this as we go along okay yeah we are looking at suspension kinematics we know handling is affected by by these frequencies so wait for some time you know we will we will come into those things okay we will we will look at these things why am I looking at this this frequency why is it because this frequency when I take for example when I take a turn and we are going to look at this from a very simple bicycle model and then if I take a turn then the frequencies which are excited okay is in that range right okay we are going to look at the mathematics behind it and then we will we will understand why these frequencies why this frequency of 12 hertz okay you will see that there is what is called as a wheel hop frequency and the wheel hop frequency is 12 hertz you would understand that there is a body you know the frequency of the the whole body that would be at 2 hertz you know you have to wait you know please be patient to understand these frequencies understand this alone that that vehicle as a whole as a gamut of frequencies which can be excited okay which can be excited and so you have to understand the behavior of the vehicle at all these things in other words in other words just to just to complete this it is not that I have to excite all these frequencies in other words I can isolate frequencies okay by different means at different levels for example I can isolate completely the noise due to the engine there are ways and means by which you can isolate this noise completely but that does not mean that the technique that I use in order to isolate this noise would all make my handling better so you have to look at what you so the idea of studying this is because the solutions as a design procedure or a design problem is different at different levels so when you say that this vehicle in other words when you say simply put it when you say that this vehicle handling is fantastic but my ride is not very good or vice versa okay what is that you are saying you are saying that the vehicle behavior or you are characterizing the vehicle behavior at different frequencies that is all okay you are characterizing them that is it okay let us move on let us now look at combined slip I am not going into the details I thought I would do it but I think we are running out of time we have to move to the next topic on lateral dynamics I just want to point out that when we have combined this slip we are looking at longitudinal as well as lateral okay both of them are acting together okay in other words the forces in the longitudinal direction and the lateral direction are going to now compete with each other in order to win over the mu into f or fn right so if I now plot say for example the force sorry fx and the force f y okay if I now plot this graph this word okay because of certain reasons of anisotropy and so on would become an ellipse something like this in other words what it simply means is that this is due to longitudinal say braking or acceleration and this is due to cornering okay when I corner when I do a cornering without any lateral acceleration I am going to get the full benefit of my mu this is the boundary after which okay I am going to it is going to stop and after which I have reached the mu into fn or I have cast all that is that is in the bank called the friction forces so once I do not have any braking or acceleration okay that is the complete force that can be used in order to generate f y and that is the end okay where I would generate the complete or I would use the complete frictional force in order to get this yeah because this is nothing but the force okay qx plus qy squared okay that is the that is the force say for example force per unit unit length should be less than equal to mu into qz per unit length is what you know qz is a normal force per unit length okay and if you if you give it like this actually it should be a circle but qx and qy need not be the same you know like what happens in the longitudinal in the lateral direction forces that need not be the same because it is the forces are split in the ratio of the of the slip and so on so the practically this would not be a circle but it would be sort of a distorted ellipse yes yeah so that is the because the way it is shared is different here so ultimately say for example if you are doing both cornering and braking that is the type of you know forces that you would generate yeah this is the effects and f y see this is a this is an ellipse like this so you can say that that is an ellipse like this okay so you are putting only the one part of it actually the qx and qy is given by and I will just write this quickly when it is in the slipping region per unit length I already said that force per unit length okay so and you can derive this and I am not going to as I said go into the details you can look at Pasekka's book so derive this similar same way as you had done it for the longitudinal force now include longitudinal and lateral force okay and then you can derive this and the effects now becomes look at that I will quickly indicate after this how we got this that is fx just a minute let me let me finish this I will explain you know let me let me write down this equation and I will explain this and this would be here assignment how we are going to get this though the expression looks formidable it is not very difficult to derive this it is quite simple this is seen to sigma x sorry that is the whole thing multiplied by sigma x the whole thing multiplied by sigma y and m is it okay so essentially what is that what is that you do essentially how did you get this it is too much of noise that is going to have an effect yeah essentially how did we how did we get this we got it in the similar fashion as we did before remember how did we get this kind of expression for a longitudinal force we looked at we looked at the deformation okay remember that we looked at the deformation we looked at how much the the tread which was a brush deform as you move in into the contact patch and then at a particular point of time we said that the force is not good enough to sustain the frictional forces and it starts slipping this is exactly what you would do you would now look at the brush movement both in the x direction and the y direction so the brush movement in the x direction so you define sigma x and sigma y as given here and you would this is exactly what you did previously into sigma y now which now becomes the basis for the force in other words these are the displacement of the brush remember that we looked at the brush okay in the brush model that is the we had those brushes so now please understand that there is going to be a longitudinal as well as a lateral right both are acting so the brush that brush the bristle will actually be displaced in the x direction as well as in the y direction this is a displacement right go back and look at it this is what we did before this would this displacement would result in a force which I would call as qx okay and the force is because of this displacement multiplied by the stiffness cpx in the x direction displacement multiplied by the stiffness per unit length you know gives you the force per unit length so that would be cpx into a minus x into sigma x and qy is equal to cpy into a minus x into sigma y right these are the forces now when it slides root of qx squared that is this this whole sum of these forces okay should be equal to mu qz so in other words the sliding distance xs is determined by root of cpx into a minus xs into sigma x okay this whole whole thing squared because qx squared plus cpy into a minus xs into sigma y whole squared is equal to mu into qz remember qz we got it from the parabolic distribution in our earlier class so you can substitute for qz solving which you will get xs then what do you do you find out the total force okay because of deformation plus slip sliding rather so the rest of the region it is going to slide up to xs it is going to stick to form after that it is going to slide try this out if there is any question I will answer that in the next class okay please try this derivation see whether you get this it is not very difficult because you just want to add then integrate it you are going to get this okay so we will not go further on this unless there is a question which I will answer in the next class try this out okay fine now we will we now know that combined slip is a it is more dangerous because whatever lateral force that is required is not given becomes dramatic the case of motorcycles gives rise to what is called as high side fault okay becomes really dramatic so in other words when you take a turn in a motorcycle breaking the motorcycle and suddenly release the brake we will explain that maybe in the next class starting that so you will understand lateral forces suppose you are taking a turn braking and suddenly release the brake what really happens now it is a it is a very sharp turn young blood going fast take a sharp sharp turn right suddenly you see a kid coming across and just press the brake okay still taking the turn and then suddenly release it situation is very dangerous why is it because when I break it think about it we will continue that in the next class okay when I break it I have I am at this position you know sudden breaking right when I release it when I release it my effects now comes to zero okay immediately effects comes to zero but my f y is not is not actually is not the same as what it was before f y now actually increases suddenly so in other words whatever friction which was which the longitudinal forces holding now gets transmitted to the lateral force so lateral force becomes high lateral force should be just enough to compensate for my centripetal acceleration the lateral force becomes high whole vehicle okay turns over over turns we will explain that in the in the next class before we go to the lateral dynamics.