 Let us begin with this next module which is on friction modeling ok. So, think of a friction what what comes to your mind ok and let us see how how this friction can be modeled. So, see in the day to day life we experience this phenomena of friction. You have a block lying on the surface you start try to move it it is there is some friction resistance that will be paid. You try to move it in opposite direction friction is turning into opposite direction that is a simple thing that everybody has experienced in their life. So, what it what comes to your mind as a simple friction model like that we will start kind of you know thinking about friction with some common sense understanding and like making it little more deeper and deeper. So, let me switch to the slides mode ok and get my pointer also right let us see yeah pointer is here. So, so that is a model that you have been like familiar with probably in your JEE mechanics and other kind of courses is based on like say ok friction friction is equal to mu n ok that is what you say ok. Now can you represent really in mathematics this friction is equal to mu n how do you represent it in a mathematical form ok think about it write down the equation of friction model ok. So, while while say say you have a block that is resting on the on the on the surface and it is applied by some force F and it is opposed by say some friction force F F and small f then how do you write the dynamics of this system. So, you now of course, they know the additional force F F needs to be getting considered here. So, the F then will be equal to m times acceleration plus this additional force F F. Now how do you express this F F as a mathematical equation ok that is a question ok. So, now we will see see that different different kinds of our scenarios here. This is a scenario that we are we are just talking about we have a force it is getting applied on the mass m as this F force and there is a friction as F and small f is its you know symbol and now we want to represent this force ok. So, we know that as I move the block in this direction ok. So, the block is if it is moved in one direction then the friction force is in this direction. If the block gets moved in other direction then the friction force is in the opposite direction that is what we know. Now, if I say the block moves in this direction can I capture that intent of that by looking at x? Probably not because even if I am at some other different x value I can be moving in the other direction ok. I can be at a positive x, but moving in the moving towards the zero value ok. So, so x may not capture that intent, but x dot may capture that intent right because if x dot is positive that means I am moving in this direction my direction of velocity is this ok. If x dot is negative my direction of velocity is opposite ok. So, let us get pen here. So, if x dot is positive I am moving in this direction ok. I need to change the color of the pen also. So, so x dot changes x dot this is the positive x dot here ok. So, this x dot is positive here and then we have this direction of x dot that will be negative ok. So, so the friction will occur in the direction opposite to the direction of x dot ok right. So, x dot if it is moving in this direction the friction will be in the opposite direction if x dot is moving in this direction friction will be opposite to that direction ok. So, that is how like now you need to capture the friction in the in the equation ok. And a magnitude of friction say for simple when you have this mu n kind of a way of representing the system you know typically you say ok or the friction force here is mu times n ok. So, so that is mu times n only after the block starts moving ok. Before the block starts moving friction is equal to this f force ok. So, we can see ok whether that can be also captured in the model. So, think about what is that way we can write now the equation for f of f here f f here ok the friction force here. So, one can use what kind of a functions mathematical functions one can use to kind of get this direction the directionality dependence represented. So, if I plot in the with respect to x dot the friction force then the friction force is positive value here ok as x dot is positive and then friction force changes its direction and becomes negative at as x dot becomes negative, but for any other value of x dot friction is is still remaining the same that is like a Coulomb friction model ok. So, this mu n you say ok no matter what is the velocity of x dot you say ok your friction is always mu n ok there is some maximum like no limiting value of the friction that you consider. We do not consider any dependence on x dot for the for the friction in our normal kind of a you know Coulomb friction kind of a model or normal like what we have studied in mechanics. I mean the simple j e kind of a think classes ok. So, write down this equation you develop this equation and then like you posit here and develop this equation and then like you know think see that. So, this f f is f c sign of x dot ok. So, f f will be actually should be negative here ok you can put this sign negative here because it is in the opposite direction to force f here ok. So, negative of this signum of x dot. So, x dot is in this direction then friction is is in the in the negative direction to x dot ok. So, that may change this also in the opposite direction by the way ok. So, r l is like we can consider this as a f f and in the equations you can use my negative f f or whatever. So, so the the intent is important here you can kind of correct little bit integrity of these equations, but when x dot is a positive direction then like you know this friction should have a negative direction ok. So, that is a that is a way you can represent the friction. Oh sorry this may not be there because I am considering this. So, this is a positive only here because f f itself is considered in this direction that is what is is being considered here ok. This is already taken care of here. So, do not worry about the the sign is really positive only here because I am direct actually drawing this force in the in the negative direction here anyway. So, that that sign is taken care of because I am kind of like you know representing this force in the opposite direction. So, when I am drawing the free world diagram that time also I need to have the friction also coming in the similar kind of a direction then things will be fine ok. So, that is how things will go and this f c will be actually you know your mu c into n ok. So, this Coulomb friction coefficient of friction will will come into picture. Now, we want to kind of little bit complicated matter ok. So, now we know already that you know this x dot is also going to affect like know your friction force because we know that viscous component we have talked about in the in the motor model ok that bearing has some viscous friction in it. But typically people normally consider only this viscous part and like they they do not consider Coulomb part because viscous part gives nice you know x dot relationship without this Coulomb component if x dot that relationship is is linear in nature it is not it does not have this nonlinearity. So, you see this like you know sudden jump from here to here for the friction as x dot changes is is is some kind of a nonlinearity in the system or if you see in terms of this signum function signum function itself is a non-linear function that is why like you know this becomes a non-linear equations. So, your system becomes non-linear with this kind of a functions. So, again one can think about what is this friction force you pause here again think about and write the equations considering now this viscous part also and you will find you come back to this. So, this you will find now that ok this friction force is equal to now this f c signum x dot plus now c x dot. So, this is additional component corresponding to the velocity or viscous friction that will get added ok. So, this is how like you know you will represent this you know Coulomb plus viscous friction kind of a model. Is that sufficient? What do you think? Think about this part is this sufficient yet you see I have you remember we have done this small like I asked you to kind of take out your mobile and keep it on the surface and apply a force to control ok. So, when you apply a force does it starts start moving immediately or you feel some some stickiness ok. Especially if you have a you know cover on your mobile you will feel that stickiness with that rubbery kind of a material. So, that stickiness comes as an additional kind of a friction part in the model. How do you consider that it is called a stickiness ok that is there only to begin with ok. Once motion starts then you do not feel that much resistance ok that is called a stickiness or stickiness in the in the system, stickiness is a term that is used for that. So, with the stickiness how do you this how this picture for the model will change this picture basically. How do you represent stickiness in this part ok that you can think about and draw that and then like you need to see what is a corresponding representation for the equation again. So, this is a stickiness. So, you have momentarily very large kind of a force called FS coming up here and then like now it decreases suddenly and you have this again our Coulomb and a physical friction continue ok. So, this is like now little more enhanced model with the stickiness considered. Now again how do you represent this into how do you represent stickiness now in the in the mathematical form. Pause again for a while write your own equations you go and you create that kind of a space in your mind to understand what what is being said here. Otherwise like you know you just read this slide sort of go through this as if you are like you know recording go through this recording as if you are just kind of listening to some video things may not make sense unless you do some stuff ok. So, please like you know write down what is a what is that you think as a equation for this this case and then like you will see this equation coming up ok. So, now this FF will be FC signum X dot plus C X dot that is what we had previously, but that that is now only valid for X dot is greater than 0 ok. It is not valid when X dot is equal to 0. So, what is what that is valid at X dot is equal to 0 what you should represent there. So, now X dot is equal to 0 you cannot use signum of X dot, but still there is a change in the sign of this FS value ok. FS is positive here, FS is negative here. How do you kind of consider that ok what kind of thing that you can use think about this again see how can you represent that and then like you will see this. So, what you need to do is you need to take F applied here now see the the system is not moving, but you are applying force if F applied itself is 0 then again like now this FF is going to be 0, but if F applied is equal to value which is more. So, F applied is 0 then signum of FS signum of F applied will have some kind of you know strong like a positive value ok. So, till the time F applied becomes equal to you know F applied becomes equal to FS, F applied is what is going to kind of be the friction force ok. Once F applied becomes equal to FS ok. So, F applied is equal to FS then like this friction force will reach its limit value and things will start moving once the motion happens then like now you switch from this condition to this condition and then like now the friction value is now coming down to suddenly to Fc you know. So, that is how like now this this would be represented in the model form ok. So, till the time F is applied F applied becomes exceeds FS value by something and then the system starts moving X dot changes its value then like now you get into this equation before that you are into this equation ok. So, F F applied has to like now go beyond FS to get into this form that is what it means. So, you see this model representation. So, one can simulate this system also and then check out like now say a simple system like mass resting on the surface with this kind of a thing how the system is going to behave one can check ok. By the way think about this in simulation can you represent it like that and hope to simulate. Do you see any difficulty that may happen here? So, think about these issues they are like not very trivial issues that ok they will you may expect some difficulty when there is this kind of a non-linearity or discontinuousness in the system. It has something to do with some basic conditions that need to be satisfied for the solutions of differential equation to exist ok. So, we will see that ok. Now, again is this enough think about whether this is enough again ok. So, this captures most of the things, but you know the main thing that is happening that this FS going from suddenly becoming FC that is somewhere like you know it is not in the nature the things are not so, sudden ok. So, we for this to understand little more about this we need to like really go into microscopic level of like you know what is happening into this how this friction is coming up in the first place and understandly different phenomena that are happening there and then we can enhance this model further to see that ok. This FS does not go to FC in that is kind of a sudden fashion here X is equal to 0 and X is greater than 0 you have some sudden value coming up here. So, there is like more like a smooth pattern that may be happening here ok. So, this smoothness that is happening here is the next thing to capture how this smoothness will come is here you can see this has at a microscopic level you can see. So, see some differences that is like you know this is a the asperities of this are in contact with each other undulation or asperities on the surface that engage with each other and what will happen when the force is applied in the side wise direction if this asperities on the surface will start deforming and then they will start kind of like you know breaking and at some other point the contacts will happen or some other point the asperities will start coming up ok. So, when they will move relative to each other like you know say right now this is a picture of the contacts of asperities are happening at many different places. As I start moving this contact will break from these places and it will start making up some other places ok. So, the contact areas will get deformed to some extent. So, this kind of a phenomena that is happening will make that transition to be not so like you know sudden it will make that transition to be little bit smoother ok. So, this actually also explains like this breakage of the contact happens and then now new contacts coming like again they will break again new contacts will coming again they will break that is the kind of a thing that will happen continuously here. But once first time the contacts are broken like that is the highest force that will need for breaking the contacts. Next contact is made and like you know immediately it is broken it is not allowed for the for the settling of the system to in that contact position. So, that is why that force will be little lesser than you know initial contact breaking force. So, that is what you need to see that and slowly as the velocity increases you will find this force will come gradually down ok. So, this phase is coming down to F c is in a gradual manner. So, you can use conceive some model for the for that gradual change and that will typically be based on some experimental observations ok. So, you get the experimental profile and then like based on that experimental profile one can kind of think about this kind of a model ok what kind of model can fit this profile ok. So, this is a experimental profile for example like you know the the F s will come down to F c and then like you know you will have this slowly the coulomb the viscous component of the friction kicking in to kind of get here get this again higher value ok. So, this is a kind of a gradual change that will happen in the actual scenario. In some systems like you know metal to metal contact this this this may not be the key situation. So, there so, you need to kind of see these are the situation where is like you know it is talked about in general for the entire model of the friction. Now, some phenomena may be dominant in some cases some phenomena may not be dominant in some cases ok. So, we we need to consider that appropriately. Say for example, in in your system like distinction and F c will be there, but there is this this change may not be so, gradual ok. Then you it it it may be more of a sudden change also possibility ok. So, the slope of this this coming down here let me make a point right. The slope of this coming down here may be may be a very very large slope here ok. So, that may be possibility and then it will rise again ok. In some cases it will come like a very gradually down and then go like that you may have different different possibilities depending upon like you know the surfaces and materials that we are we are using. But I am giving you all these different kinds of a model possibility. So, that one can choose one of these models shooting to to the system at hand ok. And that is how like you know one can think about and and get appropriate model to to appropriate details. You may not be see if you are always using motor at little bit higher speeds you are not bothered about this this part at all ok. You are not stopping and starting motor multiple times and you are not interested in like very fine positioning into the extent. So, you you see if you are if you have a positioning system you definitely stop, but you are not interested in like no fine kind of a positioning then like no you do not worry about this part ok. You leave this part to to to your integrator module to kind of handle if at all ok. But if you want to kind of capture this material this monotony gritties you have these models available. And they will be useful in the cases when you want to go extremely high precision positioning considering the friction into account ok. But this scenario will not remain for so long because as as a system wears and tears like no this curves are going to shift or they are going to change ok. Now you think about this part like no what will happen if this C is very very high. And now what is the governing equation for such a kind of a behavior. So, you then you need to know ok what is how gradually you are coming down from Fs to Fc ok. Whatever that gradually you are coming down that part will be factored into your equations in some way. And then so you assume say some Gaussian kind of a way of behavior coming down ok. So, you think about if it is a Gaussian variable you know the equation of Gaussian curve or if you do not know you just Wikipedia may search curve you will get the equation of Gaussian curve ok. It has some e to the power negative x dot square kind of a relationship in this case coming up something like that. So, if you want to use that kind of a model to go from Fs to Fc. So, you are on the part of the Gaussian curve coming down ok. We are not you were worried about a negative part of the Gaussian curve only positive part of from Fs high value we are coming down with the Gaussian variation to Fc value that is what we are we want to do. Then how do you kind of like know write the equations of motion equations of friction here. So, think about that ok what is this if I am going to be under such case pause write down your equations and come back ok. So, what is Ff is coming up here is you see this some function additional function that is introduced here is this gx dot. Now this gx dot function will come like you know get the force from Fs down to Fc value ok and then you have the cx dot come as it is ok. And then this gx dot term is like is given in this fashion ok you have some Fc some kind of a like level constant it is a shifted kind of a Gaussian thing and then Fs to Fc difference is here like know this Fs to Fc difference and that is kind of coming becoming lower and lower as as x dot increases. And then Vs is a characteristic velocity called Strabeck velocity that is be used used for this is like a fixed property of the two surfaces. And as this Vs is high or low you will change the slope of this coming down nature that is what will happen here. Now again is there further anything to be considered ok. And the answer is yes if you think of a surface is like a very soft rubber ok. So, very soft rubber when you are you are having these two surfaces in contact like very soft rubber will first have some motion the rubber will get deformed kind of a motion will be there and then like know the motion actual motion of the of these two surfaces relative to each other and all these friction and other effects are going to happen. So, this deformation that is happening before this you know other things come into picture is what is missing right now ok. One can understand that based on this simple you know video that is there for video I need to get out of this pointer option automatically here. And then like know you have this video you can observe. So, you can see this is a toothbrush which is running on the surface you need to carefully observe this area. Now like know the direction is reversed. So, the bristles are getting you know deformed in the other direction and then they start kind of making and breaking contacts on this paper surface straight ok. So, look at this video carefully it is moving in the in this one direction here you can see that there is a some kind of a bristles which are making and breaking contact up here and then they are kind of you know when I reverse the direction they stop and then like know unless my deformation happens beyond certain limit they will not like you know leave that contact point ok. So, this is a kind of a phenomena that is going to happen ok when you reverse the direction here you see that I have some deformation happening without anything happening at this and in this position if I leave this brush ok if I leave the force the brush will come back to its original state because these are cantilever kind of a deformation they have some stored energy that will push this thing back ok. So, this is a kind of a phenomena that is going to happen for the cases where you have a very very soft surfaces here are in contact with hard surface for example. And this needs to be getting modeled for a very high precision positioning applications ok. So, I gave you this example of soft surfaces, but even in the hard surfaces if you are interested in nanoscale kind of a precision in control then you will need to have this effect also taken into account ok. So, that is what is a is a is a important thing. Nanoscale is difficult it will be difficult to kind of realize because there are some uncertainties that will govern this process, but people have developed this kind of a models in the in the in the literature use as a mechanics engineer typically will not have to use this kind of a depth of model rather for the for a very high precision positioning more better solutions exist in the in the form of these compliant mechanisms or like you know piezo based actuators and things like that ok. So, we will not use probably this advanced part of the model for the friction, but I mean this is like you know good for you to kind of know and it is I am not expecting you to kind of think about the model for this friction I will just give you the model in the next slide to see and think about ok how people have captured these mathematics in a very elegant way into some of these model equations. This is maybe development a few decades ago kind of a ok. So, this is a model that is called Lugrem model for two universities Lund University and Grenoble Institute. These two come you know some joint research might have happened and this is called Lugrem model of the friction and it captures like you know all these effects that we are talking about so far ok. And how it captures and you know what how these states are like you know defined in this kind of a fashion and all I mean it needs a good amount of mathematical you know deciphering of things that people have to pose this model ok. So, we may I mean you do that mathematical I mean those were like you know really intrigued by some kind of a mathematics these are challenge for you to kind of really figure out how this is representing you know what effects and how this is really the meanings of all these variables are given here. This chi is a friction state which is corresponding to these deformations of the bristles that are that we showed for the brush or that kind of a small deformation and then like you know this corresponding spring stiffness is sigma 0 ok in the pre-sliding regime this sigma 0 is a spring stiffness and that is that is this is a force that will come into picture when you have x dot is equal to 0 ok. So, think about this these effects and you know this model how it is capturing like you know what we have said the intent of this crystal based deformations and further continuation into normal friction regime that is sliding regime. So, think about that this is good the full for your brain ok all analytical sharp brains here. So, give a give a nice thought and like you know understand if you have any questions like maybe we will discuss in the discussion session. So, we will end for now here.