 In the last few classes, we had looked at the quarter car model. Remember that we reduced this from a half car model. We said that we would be cutting it into front and rear. It is possible to put them also together and that we distributed the mass of this whole vehicle into two parts, the front and rear part and we had the, what are the two loads we have or two masses which we said, unsprung and the sprung mass okay. So with unsprung and sprung mass, we were able to get certain equations which included the natural frequencies okay, the two natural frequencies. Remember that we can look at this from a very traditional perspective from vibrational controls. We can also get, we can write down the characteristics equation. We can do all those things. In fact, we can express this whole thing in state space. Maybe this time we will do it in one of the later classes okay. So once we did this quarter car model, we also found out two natural frequencies, omega 1 and omega 2 and we said that this natural frequencies are spaced such that one of the natural frequencies, the first omega 1, the lower 1 is between 1 to 1.5 hertz. Typically they are 1 to 1.5 hertz and that is everything to do with the body of the vehicle. The second natural frequency, omega 2, we said that this is what is called as the wheel hop frequency closer to around 12 hertz okay and may vary between say 11 to 14 and so on. So these are the two natural frequencies. One of the difficulties we had, we wanted to now later go and look at and we also developed by the way, we also developed a simple equation for omega 1 and omega 2 taking into account un-damped system and when we talked about the damped systems, we had difficulty in arriving at the optimum damping as well. One of the things we found is that the damping does not have a very uniform effect okay throughout and that whatever be the damping, higher or low, some of them had better results in one range and others had better results in the other range and so on. So we said that we went back to the 50s and we had one optimum damping okay and that completes our one minute review of what we did. Then we moved over to what is called as the half car model right. So in half car model, we said that we will take a very simple case. We have say for example, we are going to remove in fact the unsprung mass. We will consider only the sprung mass and that the sprung mass m s, sorry that is not m s along with the moment of inertia, particular that is J theta would form the two variables or sorry two constants of the car of the vehicle and the variables are the bounds as well as the angle theta. We said that this is going to be supported on two springs, the front and the rear, the front and the rear and that we said we will remove also the unsprung mass and that this is the equivalent springs. When we take into account both the suspension as well as the simple model yes this is just to illustrate certain things. We can make it more complex use the well known MATLAB software in order to solve it and so on write down the governing equations either in the state space form by now you should be familiar as to how to write it in the state space form and then you can solve it in a much more complex fashion. But I am not going to do that because I am just going to derive certain important aspects from a simple model okay which will be useful to you which surprisingly gives very good results it is not that the results are not accurate the results are quite accurate and we can use a simple model to understand certain aspects. If we had quickly reviewed that let me write down the equation in the same form as I had done in the last class remember that that was A and that is B J y sorry we will put that is J y instead of J theta J y theta dot plus K f A into z plus A that is the moment balance equation so they these two together are the Newton Euler equations. Let us call following Wong let us call D 1 as 1 by m s K f plus K r D 2 is 1 by m s A K f minus B K r D 3 is equal to 1 by J y K f A square plus K r B square okay and right let us substitute this back into this expression all algebra so I am just going to write down only the final results substitute it and I am going to write this down something like this. Of course what we are doing is a free vibration problem because I am trying to find out the Eigen values in the mode shapes so I am as I told you in the last class I am going to look at the mode shapes so that is what I am going to do now. What I have essentially done is to calculate or replace J y by means of m into r y squared okay m r squared is the moment of inertia term okay so radius of gyration r y so I have written this in this fashion okay till now it is straight there is nothing just going to substitute it. Look at this two equations obviously it is coupled because let us call this as 1 and 2 because both the equations has z and the theta term so if I have to uncouple this equation if this were to be uncoupled which I said is not a very desirable quantity or desirable quality what is that I should do what is that I should do if I put D 2 to be equal to 0 obviously both these equations get uncoupled which means that a into kf is equal to b into kr then these two equations will get uncoupled. What is what is meant by uncoupling then in which case this body which is our the sprung mass would now start for the two frequencies which you would get and for two frequencies in for one frequency there would only be a pure bounce mode and for another there will be a pure pitch mode okay we said that this is not a very desirable thing because we are not going to sit at the center of gravity location and the front and the rear center of gravity location may be slightly behind okay behind the usually in the car it will be slightly behind the front seat okay but it may vary from car to car slightly behind the front seat or almost you know yes maybe one-third of the distance between the front and the rear seat we can calculate that in fact one of the exercises you can see is how to calculate the moment of or how to determine the moment of inertia from a simple experiment position of the moment of inertia a and b good experiment to do. So in which case if I am going to sit on either side okay and if it is going to bounce and pitch for any small disturbances whether in the front or the rear then I am not going to be very comfortable okay so this is not a very desirable mode all of you know how to calculate the natural frequencies I am not going to do that because of lack of time but we can write down the natural frequencies two natural frequencies for this by writing down okay for example for the uncoupled case uncoupled sorry yeah uncoupled case you can write down the two natural frequencies to be root of D1 root so that is the two natural frequency please do not confuse between this stiffness this a was multiplied in one of the earlier classes with another stiffness c alpha okay I do not get confused between the two okay this is a different k this is the stiffness in the vertical direction nothing to do with c alpha and c alpha okay so do not get confused with that right so that is the uncoupled case as I said not of great interest anyway if you want to calculate you can calculate it like that of course D2 is equal to 0. Now let us look at how to calculate the natural frequencies for this and all of you know it I am not going to write down too much on that okay I can assume a solution of the form theta is equal to theta a into e power i omega n t and z is equal to a e power i omega n t I am just going to substitute that into my 1 and 2 this is the thing and then write down the equation of the form D1 minus omega n squared z a plus D2 theta a is equal to 0 substituted you will see you will note you will note that when I differentiate this two times I get that omega n minus omega n squared that is the reason why I have minus omega n squared here and D2 by Ry z plus D3 minus omega n squared theta a is equal to 0 okay so what is the next step if I have to nothing difficult just substitute it. So what is the next step I want to find out the natural frequencies what is the next step yes take the determinant of this okay write down the determinant there is a characteristic equation write it down in terms of omega n power 4 and then take the two roots for omega 1 squared and omega 2 squared okay so that is straight forward and I am going to write down the final result for these two omega n squared and omega I will call this as omega n1 squared and omega n2 squared the two natural frequencies right any questions pretty standard stuff so I will write down omega n1 squared so you will half into D1 plus D3 minus root of 1 by 4 into D1 minus D3 squared plus D2 squared divided by Ry squared and omega n2 squared is equal to half into D1 plus D3 plus root of 1 by 4 into D1 minus D3 squared plus D2 squared by Ry squared right any questions so these are the two eigenvalues so the corresponding eigenvectors are the modes the mode shapes for these two natural frequencies okay can be calculated by substituting it and determining z and theta okay at the two two natural frequencies interestingly this is what is the result you get if I now get at omega is equal to omega n1 is equal to D2 by omega n1 squared minus D1 and is that by theta at omega is equal to omega n2 is equal to D2 of course divided by omega n2 squared minus D1 so in other words this gives that the ratio between the bounds and the pitch modes okay bounds and the pitch that gives the ratio in other words if I now just represent this by a straight line my vehicle you know the sprung mass and this happens to be the this happens to be the center of gravity location then the oscillations would be about a node for the first natural frequency something like this. Let us say that this distance is L01 as involved that is the distance okay where we have what is called as the node where the point does not move so the oscillation is about this how do I find out this point yeah no no from this I have already written down the mode shape okay I have substituted this is very standard 2 degree of freedom problem all of you have done this you know vibration course I am not going to repeat it so how do I find this out exactly so this I can substituted by or in other words this is equal to L01 multiplied by yeah so approximately theta okay divided by theta is equal to D2 divided by omega n1 squared right from which you can find out L01 right the same fashion you can get L02 by substituting it in that in the next one and in which case actually the oscillation is going to be okay something about another point okay that oscillation is going to be about a point which is at the rear that is for omega n2 right so this if I can keep it in such a fashion that the two nodes you know this this would get excited when the vehicle goes the rear wheel goes over a bump and the first one gets a second one gets excited when the front wheel goes over a bump and so on so if I can keep in such a fashion that that the front wheel bump does not affect the rear guy and vice versa then I have to have these nodes at the seat locations properly right the beam you can further extend it you can simplify it you can express this in terms of r y squared and so on I leave that to you because we have lot more to cover in this subject but this is the essential aspect of looking at it in a very simple terms from the point of view of a half car model okay right. Any questions this I am sure that you have done a very similar problem in your vibration courses so I am not that is the reason why I am not going to continue maybe the same problem you did okay now with all this you have a number of questions I know few of you asked me the question is that what is the excitation you are talking about natural frequency you are talking about other things how do I now characterize the excitation for this vehicle okay till now what you are doing is essentially extending your knowledge and vibration regular your one-on-one course on vibration it will shift and we will see and enter into the realm of what is called as random vibrations and actually that forms the basis for understanding the noise vibration in VH of the vehicle okay. So we have to go through what is called as the random vibration analysis and that is what we are going to do now I am going to pick some of the concepts which probably you would have studied in your random vibration course sorry I do not think many there is an elective but I do not think you would have done a random vibration course so maybe some probability course which you have done okay you would have learned some of the things that I am going to pick up right okay so what is that we are trying to study now assume that you do an experiment you have profiler meter okay you go and look at the road profiles across say a number of sections say for example you go to one of the highways and then measure the road profile go for a small length some length possible length and then go to another place measure it go to another road measure it and so on. So when you start measuring these things okay then you would notice that obviously suppose I plot say the heights of what you have very accurately measured by say a laser profiler meter or whatever it is so you will have for the first measurement you would be it would be something like this the second measurement would be something like this and so on okay. So maybe a third measurement would be something like this in other words what essentially in the from the language of probability you have done is to do an experiment and you are in the process of what is called as observation right so how do I in other words the key factor here is how do I express this surfaces from a random processes perspective in other words we have to understand what is a random process how do I express this as a random process and more importantly what are the statistical parameters which govern the random process no two roads can ever be the same right because the gravel that is used the stones that are used are not of the same size everything is not the same okay any surface for example whatever be the surface you have tally surf for example if you want to measure the surface roughness okay you will get something like this you are not going to get yeah smooth one line okay of course depends upon what is the magnification you go to right so I am that is why I said that we have a laser profiler meter which is used and get this you will never get a straight line obvious because of so many undulations that are there which are randomly distributed okay yes good question why am I studying this I have expressed all these things okay why am I studying this because this is going to be my input okay this is going to be my input certain statistical parameter which parameters which I am going to now explain will form the basis of input to my system which I would call as the linear time invariant system you would notice that we are going to develop certain statistical parameters and that statistical parameters of the input which is the road and the output for example the seat location accelerometer okay acceleration of the seat location for example would be related through a very simple frequency response functions okay so in order to do that I have to study what is the input in simple words what it simply means is that you would have done that in ergonomics course and all of you took an ergonomics course right and you would see that there is a tolerance level okay or perception of tolerance to a human being or the perception of fatigue and so on is proportional to the acceleration levels that you are given right and depending upon the acceleration you can withstand that for a longer time shorter time we will go into all those details a bit later okay so in other words if I have to look at right comfort ultimately that is my area of interest if I have to look at right comfort I have to look at this right comfort from the point of view of these statistical parameters which will form an input into my linear time invariant system and what is my output they are the acceleration levels okay at various points in the car and that would be related to my well-being and that is how we are going to study the right comfort so in other words you would have noticed that if you drive in a very what we call as a kachar road okay then you are the vibration levels are going to be higher you are going to feel you are not going to travel in that road for example 4 hours say for example from here to Bangalore you have a road which is a totally kachar road you know by the time you end up there okay your bones are all not intact you know you will feel so tired on the other hand the right now the beautiful road that you are traveling you can keep you need not even stop anywhere you can travel for the next 5 hours in order to reach Bangalore if the traffic is quite benevolent to you right so in other words this input is going to affect your right comfort okay and the whole idea of this design of suspension system for example is to see to it that you isolate okay to a great extent or make your life more comfortable we would not be able to go through the complete process in this course because as I said this is this forms the basis for NVH lot more needs to be done it is not only just the vibration levels but also noise which is generated by the road okay the road contribution for noise is very high okay so 25 decibel of the total the 60 decibels or something like that may be from the road in other words the road contribution is extremely high right it is 25 decibels and I have a PhD student who works on noise and his major aim is to find out for example what is the vibration level at the knuckle position and related to what is the noise that one hears inside okay so these things are not only going to affect the vibration but also the noise more importantly tire is going to interact with the road and that is going to create noise tire noise is a major issue okay major major issue apart from the apart from the engine the tire you know these two are the culprits for noise and so how this is going to have an effect on on the tire and so on all these things become important in other words unless I characterize the input I would not be able to study the output okay so that is what I am going to do so we will go into a short very short introduction into random vibration with random processes before we look at the other things okay let us now define certain quantities which are common in probability from where we will take off into random processes we will be slightly more formal okay I can take off from here because you have done a course in probability so I will be slightly more formal in this we had already defined what is an experiment what is an observation the result of an observation is what is called as the outcome the result of an observation is what we will call as the outcome the set of all possible outcomes is what is called as the sample space so I do an experiment okay I observe and then I look at an outcome okay and the set of all possible outcomes is the sample space okay let us call the sample space as yes and it can be continuous okay or it can be discrete let us define a very simple continuous space sample space yes let us say that we start the class at t is equal to 0 okay so students start coming in they are free to come in for 50 minutes not in this class but nevertheless usually people walk in okay from for 50 minutes so I am going to now observe okay and outcome my outcome is the time that people start coming in so let me write down that from that let me write down the sample space sample space now consists of all possible outcomes tau where tau is assuming that the students do not come before the start of the class which has never happened okay 0 to 50 okay so this is actually the sample space this is a continuous sample space right we call as event a subset of the sample space okay an event is the subset of the sample space yes for example I can say that a 1 is an event where a 1 covers tau which is less than equal to 5 minutes so a 1 is an event all that which is within the first 5 minutes I can define a 2 to be tau which is 5 less than tau less than equal to or greater than tau is greater than 5 less than equal to 50 so these are events or events can be this continuously as tau itself. Now a random variable a random note this carefully a random variable is a function random variable is not a variable in the strictest sense but is a function which actually maps these events let me call that as the event space there are number of events and so on random variable is one which maps this event to a real line so basically random variable is a function is a function but though we loosely later use random variable to be the to be a variable but actually in strictest sense of the word random variable is a function. Now we can assign the next step is we can assign a probability to this random variable in other words from here I can assign a probability many times probability look at this there are three steps one is an event event to a random variable then from here to a probability so I can say that the probability of this event happening is 0.5 and this event happening is 0.5 so the total probability of course is equal to 1 clear so random variable is that function and then from here I can map it to a probability through a probability function instead of that it is also possible to look at your distribution instead of defining it through that you can also put probability directly to the event as well to the event as well. So now let us extend this let us extend this and see how we are going to deal with the profiles that we had measured in the road the reason why we did this is because we can we can apply the rich random processes the mathematics behind the random processes to a road profile and understand random vibration from a very mathematical sense. Let us say that I have taken a number of readings let me call this readings by Zeta say R S where S denotes the distance from the origin I start I start a strip you know I take readings from the strip that is the S and I take another strip and another strip and so on. So and I take a number of such readings so that this I would call as Zeta 1 S this I will call as Zeta 2 S let me not put that in the bracket Zeta 2 S and so on. So I get there can be infinite such Zetas which are which have two things two notations one is R and other is S. So this is what is called as a random process in other words a random process is a family of random variables it is a family of random variables family of random variables and are usually like in this case have two indices R and S length it can be time for example you can look at it as a time but you can look at it as this actually gives the roughness of the road okay roughness of the road and note that note that if I take any let us say that is S1 and that is S2 and so on okay just because I measure at one location the surface roughness say for example if I know S1 it is not possible to predict or to say that this would be the same as S2 or it is 4 times I mean sorry if I measure here it is I cannot say that this would be some 4 times S1 or the same as S1 and so on okay. So I have to now bring in a statistical basis for this what I am measuring okay at these positions it is loosely called that psi what I am measuring is what is loosely called as the random variable okay what I am measuring here is what is called as the random variable right okay. So now let us put this together let us just develop this so in other words though random variable is a function we are going to call what I measure as the heights at various positions as the random variable so that okay there is a mapping in the sense I make an experiment I make you know the observations and then I call this heights as the random variable and the random process is the collection of all these things the ensemble of all that is possible to be done okay. Let me put this together and let us see what we mean by each of these terms and how we are going to statistically understand this process okay. Let us now put this in one go in other words one yes right okay one of them would be like this first one second one would be like this the third one would be like this and so on okay those are the zeta 1 zeta 2 zeta 3 and so on clear. So let us call let us give zeta R yes R is equal to 1 2 and 3 is what is called as the sample function it is a sample function when I put this into a bracket R that means it is the ensemble all of them sometimes it is called as also as the realization of sample function or realization. The collection of all these things is what is called collection of all those Rs is what is called as the ensemble okay. So yes yes absolutely so the beauty of that that is what we are coming to that so look at that this heights vary along the road in other words in one of this sample function okay they are not constant they are varying so for example that shows how it varies and also varies okay across roads okay. Now how do I study this that is the whole process the random process okay so what are the statistical parameters that I can use it. Let us now put down let us give some notations so right now our most important thing is random process random process is a collection of all these things okay and the sample function is one of them and that ensemble is the complete collection of whatever you know that is possible is what is called as the ensemble okay. Now let us say that let zeta j indicate the value okay of this roughness say in terms of height for a particular distance S j so as you rightly said the question is what is S j say for example it can be S 1 the question is how is the roughness okay distributed over this over I mean at every point in that whole ensemble okay so in other words that variable this variable has a statistical properties number one. Number two that is one thing the other thing the statistical property I am interested in is as I vary yes how does this heights this roughness changes with that length okay. So these are the two things I am going to study right I am also going to look at certain important characteristics of random processes like stationary whether it is a stationary random process and whether it is going to be what is called as ergodding right. Let us now introduce what is called as probability density function all of you have heard about this density function which I am going to use in order to characterize this random variable which I called as zeta at the position S j okay. A number of statistical or probability density function is possible one of the most popular probability density function is the Gaussian function is a Gaussian distribution function going back to the definition of the probability density function the probability that this roughness value which I would now start calling as a random variable is between the values of A and B is given by how do I write this is given by what is the what am I looking at what is the probability that it is between say 3 mm and 5 mm right. So is given by integrating the probability density function okay in that range and then finding out what is the area under B and A. So in other words I can plot the probability density function say for example as a normal or a Gaussian distribution okay normal Gaussian distribution whatever it is I can plot this as a Gaussian distribution we will put down the equations in a minute. One of the major things is that clear this is quite straightforward okay. So probability that in other words this is nothing but the area under the curve from A to B gives you the probability. The density function gives you the probability versus the tau j and then you take that area between say A and B and then say that what should be the what is the total probability and from which you get the probability distribution function okay as you keep integrating okay. Suppose I integrate this value between A and B between minus infinity to plus infinity I get the total probability to be equal to 1 okay. So integrating this is what you would get a probability distribution function you can also look at probability density function as the differential of the probability distribution function okay. So we are not going to too much of niceties because we have lot more to look at okay we are going to look at this whole thing this whole from the point of view of what is called as expectations okay. Now in order to understand this expectation let us do a small experiment for the next one minute and then we will extend this to look at what is called as the expectations okay. We cannot complete a probability you know discussion without a dice and a you know coin can you coin. Suppose I take a dice okay and then I am going to throw the dice okay say maybe 6 times I am going to throw the dice what is the in your I mean in the general language what is the average value that you would get like I throw this look at what first time I throw it is 2 second time I throw it is 1 second and so on okay how would you calculate now what is the expected value as it is called average of the mean value how do you do that what is the probability that in that dice okay it is an unbiased dice and throw it okay what is the probability that I will get 1 1 by 6 2 3 4 is 1 by 6. So the probability that I get one of these numbers is 1 by 6 tell me what is that value would get how do I calculate it yes exactly okay sum it up and you will get expected right. I am going to use that in order to define my expectations okay and I am going to distinguish a slightly between what you had learnt earlier say for example as an average and I mean or in other words the sample in the population okay we will start from expectations in the next class.