 we are going to go a little bit more in details about this compliance ok, we have already seen its utility in the CDROM drive case ok. You remember that we had had this you know block diagram of the block which is you know supported by these wires and that was like hanging in the in the magnetic field and it had some kind of a coils when the coil the current passes then then block moves and it controls the lens of the CDROM drive ok. That system we have already seen and that system is used in CDROM drive for the fine positioning as you remember a fine positioning or fine tracking and then fine focus kind of a control ok. So, now let us see like know formally a little bit more formal kind of a like you know why compliance people want to use in the systems as you can guess like you know it can give you high precision positioning with a moderate range. So, not very large range that can be possible then it can have high speed possibility because when you use compliant elements typically the weight will go down. So, when the weight goes down you can kind of like know have higher speeds possible. But there is a flip side to it that because of the compliance you will introduce some kind of a vibrations in the system and one has to do a control good control strategy to kill those vibrations off. There is no backlash no wear and friction in the system ok. So, this nowhere in the system is because like you know there is no two elements rubbing against each other. So, you do not need any lubrication maintenance nothing you know that will be gone. Then there is very like low energy requirement because the mass is less. Then you have higher durability because there is no friction there is no rubbing of the elements and see this rubbing of the elements over the long duration of operations would really cause some kind of a you know the wear and tear of the system and that is where like know one needs maintenance and then the durability goes down and things like that. They are absolutely noise free again because there is no rubbing of things against each other. See you might have heard some motors when they move like you know they make make some kind of a noise mainly coming because of the bearings bearing and like imbalance in the system which creates some kind of a noise. So, since there is no friction here again no noise no lubrication ok. So, these are these are a lot of these different interesting advantages and that can help really in very high precision positioning which is required for this era I mean when we are talking about a nanotechnology micro technology already coming up in the market in big way in many many different devices like you know obviously like you know high positioning high precision positioning systems would have like no better prospects to go. And compliance like you know compliant mechanisms is one of the you know promising candidates towards towards that and people have started using compliant mechanisms based you know positioning systems nanopositioning systems. So, there are some kind of a basic motion elements that could be identified here where like you know you can have a rotation possibility. So, you you move these like you know some kind of a construction which will make the make the stage rotate without really having a rotary kind of a hinge joint in the system ok or you can have a translation possibility or you can have a you know screw kind of a possibility ok. So, these are kind of like you know some basic kind of a motion elements what we have seen in Cititon Drive is this kind of a translation element which was used. So, you you instead of like you know this kind of a plates you can think there are this kind of a three wires here one two and three here and three wires on the other side that is that will give you the same kind of a motion in the in the direction that is shown here. In addition like you know the because of this because these are wires see for example, this stage doesn't have a motion possibility in the perpendicular direction here as shown by the laser arrow. But because like you know because the stiffness of these plates in the in these direction bending is way too high. So, they can it is like a scale which you can a flexible link ok I can show you here the link which can which you can bend very easily, but you cannot like know if you want to bend in this plane it is not possible to bend it ok it is very difficult to bend in this this kind of a plane ok. So, this is you know basic elements of motion for compliant mechanisms. Now, I will just give you this linear kind of a motion stages in compliant mechanisms. So, this concept here is is is very simple. If you say ok I want this block to move in the linear fashion and it should be guided only by the compliant kind of a mechanism. So, if I attach one kind of a cantilever beam here then like know the motion is of this block as you see here but this cantilever system is is deforming like like this ok. So, it it has some kind of a deformation ok. So, so this block is this this block that is not really moving in a straight line fashion here it moving in some kind of a so to so to say I can some circular R kind of a fashion ok. So, this block also is going undergoing some kind of a if you see this bigger size here yeah this block also is undergoing some kind of a rotation ok. So, it was like this here and then like know in this it is tilted ok. So, this rotation is is is something which is not desired you want this block to move exactly like a like a straight line down below ok. So, for this now we introduce some two like you know two elements of the same kind like know you you you just do not have one single beam, but you have two elements ok I will show you in the in the block here ok. So, you have this now so you have this two elements here ok. So, this was a single element ok bending and now you have these two elements ok. So, you can see that ok this this bent like know it is moving parallel to itself maybe I can change the ok. We can see that this this moves you know in a in a way that the block is moving parallel to itself, but if you see on the on the carefully here on the slides here you see that there is still the parasitic deformation in the in the other direction. So, there is some axial motion that is happening here ok this this unwanted deformations are called parasitic errors here ok. So, to prevent that like you introduce one more such a system ok like this ok. So, you have introduced now two additional beams here and this stage is is additional stage here this is a this is a moving platform ok remember here it was a fixed kind of a you know wall, but now this is a moving platform here and a fixed ends are up here ok these are the ends which are fixed to the wall somewhere here. Now if you imagine like know this if this would have been fixed exactly the motion that you have seen earlier like know like this that motion happens would have happened ok. So, and then now if you just imagine like know that this this block is not there and only this block is there then if I like know give a force which is same force that I have I was there here then I will get deformation in the same way as was happening for this block, but in the opposite direction ok. The parasitic error that is produced will be in opposite direction the deformation is in the same direction in the same like know downward direction, but the parasitic error that will be produced like know the parasitic error in this case is going to be in this direction, but when you move this block down only without like having this block then the parasitic error produced will be in the opposite direction. Now when I see this in in combination that ok this block is applied a force the reaction force is transmitted here which will deform like know this this two outer beams ok. So, that like know gives this deformation in such a way that this block is is actually now having some kind of a parasitic error of motion which is compensating the parasitic error of this block and that is why this block will move completely in the straight line that you can see up here ok. So, this is a this is a mechanism where you can see that let me get used here clearly. So, see this you can see there is a four beams here 1 2 3 and 4 and then like know you you you see this block is moving in the exactly straight line ok. Actually you can put a pen here and see draw it on a on a paper and you will see that this motion is is is is in a straight line fashion ok because this block is is absorbing that parasitic error ok. So, like that you can have have a compliant mechanism with that and as you can see naturally like know this this this keeps on vibrating ok, keeps on vibrating a lot like know if I give this deformation this keeps on vibrating and like for long duration and this this block is made up of material called valium copper which is having a high fatigue strength material ok. So, so this is how like know one can get perfect straight line motion here and this can be utilized in our a lot of mechanical systems for for getting like know a straight line motion which is a kind of a accuracy ok and a like now bigger kind of a little bit bigger scale than our CDROM drives ok. So, CDROM drive has this principle only up to up to this point ok. So, CDROM drive didn't have this parasitic error compensation ok. So, there will be parasitic error in the CDROM drive case little bit, but it is it is ok that you know it will not bother if you see carefully this kind of error will not bother a any functioning of the CDROM drive application ok. So, this is the main kind of a concept for like know linear having a linear motion with compliant mechanism and there can be a lot of these possibilities many many different kind of a compliant mechanism possibilities could be there and nowadays because we can 3D print devices we can print these mechanisms very easily in polymer material for example. Metal is still difficult, but like know with EDM and other kind of a processes we can cut them in the metal, but with the 3D printing very interesting kind of a mechanisms can be like know readily printed in plastic kind of a material if that is ok for whatever application you are you are trying to drive ok. So, you can see some more links of compliant mechanisms here and get yourself a little bit more familiar with them so that you can think whether you want to use use them like know in your project or in any other application in the future you might be considering ok. So, we have this concept seen for CDROM drive that is again like know I am showing the picture here. So, this same concept as we had for this block up to this point ok that is acting here ok when this force is given like you are seeing like know when we deform this like know these beams undergo some kind of a deformation ok. So, just to recall you this same case is there in the CDROM drive ok. Now, there are many different kind of other possibilities. So, you can have a two stage compliant mechanism. So, this two stage double parallel kind of a mechanism or like you can have rotary joint with compliant mechanism. So, two stage means like know you have x stage motion possible and you have y stage in the same same way ok. You think about and know how this can be possible for having x stage motion and y stage motion in a similar kind of a you know the compliant elements are put in such a way that you have x stage motion also possible and for the same stage the y motion also is possible. See one of the very easy ways is to kind of like know say that ok oh my x stage is carrying my y stage and they are kind of like know in the series ok and they are like know the whatever motion that happens for the x stage the y stage motion will be like know y kind of a mechanism is put in a such a way that the motion produced by that y will be perpendicular to what the motion is produced by the x and the y stage is actually hosting or carrying the x mechanism ok. That is very easy to see, but like know if you want to kind of have the same stage but it will have a x and y motions possible in the same way as what we have for the single stage then like know it is a interesting case to think about. So, think about that and like know we will see if at all we have time we can discuss this little more. Then you can have a rotary joint with compliant mechanisms this is little I have shown you one kind of a case there are many different other kind of cases for that can be possible. One simple case is like know your cantilever ok. So, if you see this cantilever this motion that is happening like know one can kind of fit a circle to this motion and say that ok center of that circle will be like know my virtual hinge point ok and that appears to be like that ok. So, it moves in a fashion that ok it almost resist as like know some small little circle here ok. Then you can have this bi-stable mechanism this is very interesting kind of a case you might have seen this kind of a mechanism that we see. So, they are like know having two stable states ok. So, let me again show this here ok. So, can you all see this maybe I like to move from my side yeah. So, you can see that there is a beam which is here if I push it like know it goes and like make this click sound and goes down ok maybe I can bring it more closer here. So, I can show you this beam you can see this beam ok. So, if I push it it is now like know bent like that I push it like know from this side it goes on the other side and make this click. So, it is it is moving this two stable positions ok. So, maybe yeah maybe I will show you it in this angle. So, you push it it goes like that push it like that it goes down ok and these are like know one position is circle down and one position is circle like that ok. So, these are the positions in which it will go closer yeah you can see that. These are the bi-stable kind of a mechanism system. There are stable two positions, but middle position is not stable and they are very good for switches ok. So, switching application you can you are actually some of our like know home switches does have this, but they do not have the beams there, but they are spring based kind of a compliant mechanism systems are there. They are springs based, but not like not really this beam based ok. So, the spring actually kind of gets two positions. So, that is why like know use your switch at home know that has these two toggle positions like know you switch off and it toggles and switch in that position and like know you switch on and it toggles and switch into that is again like know that is a bi-stable kind of compliant mechanism. So, they find like know lot of usages in our day to day system as well. Ok. So, we will see little bit on the analysis of a simple analysis for getting these approximate solutions for this compliant mechanisms for the small deformation. Why approximate solution? Because they are the small deformation kind of cases. So, I will tell about like a little bit broader scope of this analysis and study and then like know we can focus on like just getting you know simple solution like know by using whatever theory we know little bit about a little bit only beam theory we may apply it and like know see some simple cases. And for more detailed analysis you have this CBCM that constraint beam this beam constraint model method or chain beam constraint model. So, this is a very recent kind of a development in the literature which we may not get into the details, but like know you need to kind of say that ok there are like know FE analysis techniques and there are some other kind of a simpler like know this is a competition little more efficient than FE analysis that is why this is a good method in the literature very very recently proposed. So, just to kind of sensitize you that ok there are lot of these different analysis tools that are available out there one can use it if you want to kind of like know do the analysis analysis more the simple analysis to get and then we can have a more detailed analysis. Then there is sorry for say buckling is still like an issue like know there is not a lot of lot of literature of you know buckled beams and post buckled kind of a beam analysis and things like that will. So, accounting for buckling is a little tricky issue in this and there is there is some literature and there is some more literature that is to be explored I mean it is coming some papers are coming in this area post buckled beam vibrations and things like that ok especially large deformation large deformation these problems are not yet solved ok. So, this analysis has lot of like know reach possibility of different problems to be to be looked at and solved and kind of analysis typically will be you know structure that will be used in this analysis will be like cantilever structure or like know with different kind of boundary conditions you can have multi layered structures and things like that ok. You can have a flexure hinges which are rotary kind of a flexure joints. So, one can have some kind of a approximation done in modeling saying that ok instead of cantilever I will model this as a some kind of a torsion spring there which is having like know is center some known center and then kind of go ahead with the modeling and thing like that. So, there are lot of this models people have proposed in literature and they are available ok. So, this is kind of a table which captures somewhat like know what kind of analysis and theory that is required to be used for say static and dynamic analysis for different kind of cases that you have. So, for example, like cantilever beams single or multi layered you know those kind of simple structures you will do static analysis some beam bending theory mostly Euler Bernoulli beam bending theory with a you know small deformation analysis. Then you may have thermal analysis also possibility for the cases where like know you have expansion in the structure to be considered in addition you know. So, the thermo mechanical kind of analysis one can do ok. Buckling theory we can use for compression kind of analysis ok and for dynamics like again you can use Euler Bernoulli beam with with a simple something called assumed modes method ok. So, we may not be having time to go through this kind of a technique, but this technique is good for many control kind of applications. Of course, you can do FE analysis for dynamic analysis study, but FE modeling is not amenable many times for a for a good controllers to be developed based on the FE analysis have known because that model becomes really really you know many degrees of freedom kind of a model where assumed modes method can be having only two or three degrees of freedom kind of a model can be produced with the assumed modes method ok which will which will suffice for like not of like no small deformation kind of analysis of beams. Then in the torsion you can have torsion theory static rotation of micro mirrors and so these are some of the applications given here ok comb actuators or biosensors in all these compliant mechanisms we talked about till the time we have like no small deformation analysis ok then micro mirrors with these attach to so AFM cantilever is there like no in AFM mechanism ok. So, those kind of things can be analyzed nicely with you using simple oil and Bernoulli beam theory and then you can have a plates or you know rectangular or two dimensional kind of compliant elements and they can have different ways of like no FE typically FE way will be there for the finite element method will be used for static analysis here ok. So, this is a kind of a way typically like no the structures will be analyzed ok we will not get into all these details like no if you are interested in some more components of these we can go through, but will some small analysis will focus on all Bernoulli beam analysis to kind of we get some kind of a flavor of some of the compliant mechanisms how one can apply such whatever the things that you have done in oil and Bernoulli beam theory ok. So, this is what we will do analysis for by using oil and Bernoulli beam. So, you know all like no this formula I am presuming that that this comes from all Bernoulli analysis of you know planar section remain plain small deformations are like you know the deformations are relatively small as compared to the length of the beam. There are no shear effects in the length wise direction of course, there is a shear in the cross section shear forces there, but in the length wise direction there are no shear effects. So, these are kind of very you know what you say a lot of assumptions which are maybe working for small deformations only they are not really large deformations this kind of assumptions will not work, but the people will do that to kind of get some kind of you know theoretical base to kind of analyze at least small deformation cases very well and for large deformations one can build on this theory further to incorporate large deformation analysis I think like that ok. So, let us kind of like now apply some of these all Bernoulli beam techniques here. So, you are given this length to be say 500 mm and then thickness to be 2 mm and this is a machine out of 35 mm thick slab by using EDM process is what is given that means like you know this you start off with the with the 35 mm thick block and like you know you start cutting this block like you know along these lines to to get this you know material from here removed so that you get finally this mechanism ok. So, this is this material from here here and here is removed in some way and like you know you get finally this mechanism to be in place and so the width of this like the beams here is going to be 35 mm ok and the aim is to determine say some activation force needed to move this mass by 5 mm and think like that ok. So, you can think about here pause here and think about how will you apply that all Bernoulli formula here to kind of see like you know how do you get these deformation ok. So, one can think of like you know what are the boundary conditions for the for the formula to apply and what is the moment that will be coming on the system ok either you can work with the moment or you can work with the with the you know shear force either way is going to be giving you the same thing only thing in like you know you need to make sure that you consider the boundary conditions really really well. So, I would say you can think about this analysis and do some stuff I mean I think this is a good it will be good if this is done and then we discuss it more then like you know whatever doubts you have can be clarified much better ok. So, I have given some kind of a guidelines here to see and you do not like look at these slides unless you have done something on your own ok and got stuck somewhere or you have got solution also then no problem then you can have a look at and check whether your solution in these matches or what you have missed. So, this is what is the exercise that you would do.