 Hello and welcome back, I would today like to give a brief introduction on to the process USM ultrasonic machining as we left yesterday. And then as we already have discussed before that the mechanical you know processes where mechanical energy is applied for material removal like USM or AJM abrasive jet machining. Once this whole fundamental process is clearly defined then I would like to go for the applications part of the process which would be in a subsequent lecture. So USM as we have already discussed is the short form the acronym of the process ultrasonic machining as the name indicates it basically involves the application of ultrasonic frequencies and as we have already discussed before the process came into existence at least it was proposed for the first time by J. O. Farer in 1945 on the basis of which the first tool of the USM was designed in this principle using the principle in 1954. So we have already seen that USM is applied mostly to electrically non conducting or semi conducting surfaces and particularly for brittle materials where the main mechanism of material removal is brittle fracture. The USM basically is a process again which involves a vibrating tool head and this was illustrated yesterday very clearly and this tool head is basically used over an abrasive slurry. You can see this is the region of the slurry, this is the work piece, this right here is the tool and so basically small amplitudes of motion about 15 to 20 microns are given to this tool head and it squeezes or compresses the abrasive particles which are coming between the tool and the work piece and this happens so at a very high frequency of about 20 kilo hertz in the ultrasonic range and this basically the movement or the small amplitudes of motion of the tool has a positive impact of plowing of the abrasive grain over the surface and this is basically the principle mechanism of material removal in a USM process. So we will try to explore different models for estimating how material removal rate can be characterized or how it can be organized and the first model which comes into you know mind when talking about particularly the material removal rate is what we also know as the M. C. Shaw's theory or M. C. Shaw's model of USM. So this right here of material removal in the USM process and the principle reasons for material removal are kind of illustrated here into this four categories or the first category is because of the hammering a direct hammering of the abrasive particles on the work surface by the tool. The second category is the impact of free abrasive particles on the work surface. So what we mean by free abrasive grains here is that when basically the slurry comes between the tool and the work piece there is a gap between the tool and the work piece which is almost always existing and also because we are giving small amplitude of motion to the tool the position of the tool changes from very close onto the surface to farther away from the surface and there is always a possibility of grains of small grain dia say about close to 15 microns or 20 microns to fly between the tool surface and the work piece surface because there is in one instance at least when the tool is farthest in its farthest position away from the work piece there is a lot of gap between where more than one abrasive particles can come it can strike or impact the tool surface and it can actually come back and impinge the work piece surface because of the impact of the tool surface or from the tool surface and so that mechanism also we need to consider or that aspect also we need to consider while considering the material removal rate estimation. So that is the impact of free abrasive particles on the work surface as illustrated here and then of course there is this effect of erosion and that principally comes because of cavitation. Now I would like to explain a little more on cavitation so what exactly is meant by cavitation is that whenever there is a high frequency object let us say there is a maybe a surface which is moving or which on which a fluid rests and it is moving at a certain frequency very high frequency with respect to the fluid over it where the fluid is resting on it. So what typically would happen is that the fluid because of a very large inertial component may not be able to move with that high a frequency and there is always going to be air gaps or gaps where air can be sort of pulled in or you know some degassing can happen of the solution also or the material which is over this surface and there are some cavities which are formulated because of the relative difference of motion between the high inertial fluid and the high frequency surface. So these cavities are basically nothing but either low pressure zones which is pulling in air where air can be out gassed from the liquid or you know in case there is an option of air to out gas sideways this air can come in and it can do this you know sort of small inclusions within as bubbles within the fluid and the work surface. So these bubbles actually get formulated hugely at a certain frequency which is also very high about 20 kilohertz in the USM process also and as formation of the bubbles would definitely lead to some kind of pull forces generating pull forces where the materials which are actually fractured using this brittle fracture theory on the surface would actually get pulled in by such a bubble and it would actually promote fast resolution of the debris which is formulated because of the USM process. So therefore erosion of the material due to cavitation or due to the formulation of the bubbles in a USM is also another principle mechanism of material removal and then finally there is chemical action associated with the fluid used assuming if there is some kind of a softening of the material or some kind of a you know change in the material property by the fluid that is being used as a slurry for carrying the abrasive there is always a chemical action reaction which is involved on the surface which would actually loosen the material a little more and that can help in the material removal process. So these are the principally four mechanisms hammering direct hammering of the abrasive particles impact of free abrasive particles erosion due to cavitation and chemical action associated with the fluid that is used. So we would like to now illustrate a little bit on the numerically available predictions or theories or models which are associated with the USM process and while talking about the estimation of material removal rate in a USM the most widely used theory was proposed by MC Shaw and what is known as the Shaw model and we will try to discuss the whole material removal rate on the basis of the theory proposed by the Shaw's model. So let us see some of the assumptions which were made by the Shaw's model as here illustrated here in the next slide. So in this model the direct impact of the tool on the grains in contact with the work pieces taken into consideration and we as we will see later on when we do the numerical estimation of the MRR the free impact or the impact of free grains would actually not be substantially contributive of the material removal rate and whatever would be principally contributive would be the direct hammering action as we will see just a little bit later. So also some of the assumptions made are the following in the Shaw's model number one the rate of work material removal is proportional to the volume of the work material per impact which makes sense that whatever you know material gets removed by one set of grains coming in contact with the surface because of the direct hammering action would be the work material per impact and this somehow would determine the overall work material removal rate at least this should be proportional to the work material removal rate because the other factor that it will depend on is the operating frequency of the such plowing action removing the work material. The rate of work material removal also is assumed to be proportional to the number of particles making impact per cycle if supposing there is one abrasive grain and then subsequently in another case five abrasive grains. So the material removal is assumed to be five times the previous case for obvious reasons that at one impact or one cycle five grains are doing the plowing action as opposed to one grain which was done earlier and also the rate of work material removal is proportional to the frequency as I already instated before of the vibrating tool head which is the number of cycles per unit time and all these impacts are kind of identical and we assume that there is no non idealized situation. So the situation that we are assuming is that there is a tool which actually comes close to the surface and there are certain grains which are trapped between the tool and the work piece and you know when the tool moves apart there is always cavitation or erosion of the material which has been brittle fractured and the next set of abrasive grain comes when the tool again approaches. So this is the idealized condition of operation of the USN. So all impacts are assumed this way to be identical in the Schauss model and also one more important factor is that all abrasive grains are assumed to be identical and spherical in shape. So these are the five assumptions that the Schauss model makes and if you really go for a predictive estimation of the MRR or the material removal rate of an USM process this actually comes out to be of the form Q proportional to V Z nu where V is the volume of the work material removed per impact and we will calculate this what this volume is in terms of indentation. Z here is the number of particles making impact per cycle. So this basically at one impact how many particles are coming between the tool head and the work piece surface nu of course is the operational frequency meaning thereby it is the frequency of the vibrating tool head and Q is the MRR or the volume of the work material removal or volume rate of the work material that is being removed. So this is how the Schauss model predicts the Q or the material removal rate as. Now let us actually look into the various terms here like the V the Z the nu and try to derive at least an order of magnitude equation for different terms concerned in the USM process like for example maybe the grain diameter or the diameter of the impact so on so forth and try to arrive at this MRR to give us an estimate of at least the order of magnitude of the various parameters which are involved in the USM process for doing the MRR. So let us look at a little bit further by assuming that in a USM process now this is the work piece and there is an abrasive grain which is hitting the work piece it is coming down as if it is plowing the work piece and we assume that there are two different diameters in question one is small d as you can see here the small d is nothing but the diameter of the grain. So it is the abrasive grain diameter and there is another term capital D which is actually the diameter of the impact. So this right here is a circular region which is the affected region because of the plowing action of the grain and we assume another parameter D capital D which is actually the diameter of the impact or in other words it is the diameter of the crater which is formulated because of the plowing of this abrasive grain on this work piece surface. We assume here by the Schaas theory that this grain here is a complete or a perfect spherical shaped structure. So we assume the grain to be a perfect sphere. Now if we really find out the D to be or if we define the D to be diameter of indentation. So if the D is the diameter of the indentation caused by such an impact at any instance let us say instant of time and this small h here as you can see is at the depth of penetration of the grain. So and at that particular instance of time h is the corresponding depth of penetration of the grain. So we have a geometrical relationship between all this the capital D the small d and the h right and it can be written down as capital D by 2 square is actually equal to small d by 2 square minus small d by 2 minus h square just by using Pythagoras theorem. You know that this height here is small d by 2 minus h this is capital D by 2 this size here is capital D by 2 and so therefore, this is the right angle triangle and we are just simply applying Pythagoras theorem here. So it becomes d by 2 square minus d by 2 minus h whole square. So calculating it further square of the D can be represented as 4 d h minus 4 h square typically we do assume in USM process that the grain penetration depth h is very very small in comparison to the diameter of the grain D. So grain diameters can be something like typically about 20 to 25 microns and in a USM process typically this h could vary between anywhere between 1 and 5 microns. So it is quite small in comparison to D and so therefore, this equation can actually be approximated as 4 d h we can neglect the h square term square is negligible. And therefore, a relationship exists between the impact dia D and the grain dia as D is approximately equal to twice root of small d times of h where small d is the grain dia h is the penetration depth of the grain in question. So if we assume that the volume of the material dislodged is sort of proportional to the cube of the D. So as you know that this actually is a diameter of penetration of the grain and if we assume that this grain is sort of hemispherical or the penetration is actually it is sort of hemispherical then we can safely say that the penetration diameter D the cube of the penetration diameter is estimating the volume of material removal even otherwise if supposing h is too small and we cannot equalize capital D to small d the grain diameter. In that case also if the capital D signifies the impact diameter or the diameter where the grain has penetrated at least the cube of this dimension should be able to estimate in some proportion should be able to estimate the overall volume of the material that has been work material that has been fractured because of this impact. So let us assume that and let us say that the volume so we assume here the volume of material dislodged per impact to be proportional to the cube of this diameter. So we get the cube from the Shaw's theory earlier is proportional to volume of the work material removal per impact which is actually cube of this D for obvious reasons discussed before times of the number of grains or number of particles coming to deliver the impact per cycle times of the cyclic frequency nu. So here we can substitute the value of D as d h under root. So this becomes d h to the power of 3 by 2 times of Z times of the tool frequency nu. Let us make this equation equation 1. Now let us also understand the impact of the force on such a grain diameter because typically the grain diameter of the grain is being pushed using a certain force by the tool. So for bringing home a good model you should be able to somehow correlate the force per unit area coming on the impacted area of the work piece to be equal to the flow stress of the work material. So it is a cut off threshold which when reached would actually promulgate deformation and sometimes fracture as in this particular case. So the ultimate flow stress of the material needs to be at least reached it can be more than that the impact force can be more than that per unit area. But then at least it should be the flow stress of the material for the material to start coming out or start getting fractured. So the assumptions behind the model really is to somehow be able to predict from the applied force what amount of force per unit area is coming on one grain as a matter of fact on many grains and that has to be somehow equated to the ultimate flow stress of the material that you are releasing or removing of the surface. So let us look at it and see what this estimate is. So since the mean speed of the tool is low we can assume that the mean static force F applied to the tool must be equal to the mean force of the tool on the grain. There is hardly any inertial component of the grain also which is involved the grain itself is too small for involving any inertial component of the grain as such in this theory. So therefore, we say that since the mean speed of the tool is low the mean static force of feed of the grain let us call it F here applied to the tool must be equal to the mean force of the tool on the grains. Now on the duration of such an impact is assumed to be some small value delta T let us say and the maximum value of the impact force can be something like F i max the nature of variation of F with time can be plotted and seen and the behavior can be estimated as how the force would vary from the stage that the grain is coming in contact with the tool to the stage that the grain is getting plowed because of continuous hammering of the tool to the stage when the material gets dislodged and the grain becomes a free grain to move along with the slurry. So, this whole event or sequence of events can be recorded as a force applied by reaction force applied by the grain on the tool in just the opposite or the reverse to the action that is happening on the workpiece and plot that F the force on the tool with respect to time. So, let us say that the duration of the impact is some value delta T small and the maximum let us say the force of impact is represented by the term F i and the maximum force of impact is represented again by F i max maximum value of F i at a certain instance of time. So, if we really look at the variation of the force F i with respect to time it looks something like this. So, as you can see here these are several stages in the plot where the whole grain is being hammered and crushed between the tool and the workpiece surface. You start with time t equal to 0 here for example. So, this is a case when the grain the abrasive grain is free there is no direct contact of the tool of the grain and let us say that is where the tool has started to oscillate. So, this is the start of oscillation of the tool. So, the tool does not have to find out a grain unless it moves closer to the surface by a certain distance and this time here let us say for example, this can be t 1 is spent in coming in contact with the grain which is already in contact to the work surface on the other side. So, this is the tool let us say and this is the surface that we are trying to machine and there is a grain which is resting somewhere here abrasive grain which is resting somewhere here. So, the time t 1 is really spent by the tool in covering this distance here from the initial position of the tool to this position where contact with grain takes place which is already in contact with the work piece surface. So, that is represented by time t 1. Now, the tool has come and it has started to do the plowing action the tool is now somewhere here and it has started to just about do the plowing action and therefore, there is always a reverse reaction force which is available from the grain on to the tool which is being plotted as a force in the force time diagram here and therefore, you know really there is no deformation which happens until a critical value of stresses reached at this level here where it can start to penetrate inside the work piece at this particular interface. So, the force is continuously increasing all the way to F I max and the F I max per unit area of indentation is really the ultimate flow stress of the material. So, F I max per unit area of indentation can be equated safely to the ultimate flow stress of the work material. So, that is about the time let us say time t 2 which is spent in reaching from direct contact which starts here start of contact all the way to F I max. So, corresponding to this point here which is the force needed for causing the material to flow and then of course, once the material has started flowing and the grain starts going inside the force on the grain comes down it comes down and the tool really you know after this point does not face much force because the material is in the flow state. So, the force comes down and then after a while the tool leaves contact. So, the force is 0. So, it comes down here and somewhere the tool leaves contact and from here to here. So, that happens let us say after time t 3 and after t 3 is passed then the tool leaves contact. So, the tool leaves contact from this point and goes back to its original position or the mean position because it is a oscillating tool. So, this is the whole time cycle for the tool and the force versus time plot. Now, if we assume this characteristics because typically impacts are over very short durations the delta t that is the time needed for the whole impact and flow process. So, which is also equal to t 2 plus t 3 as in this case you can see is very very small. So, delta t is very small and it is obvious to assume that in this kind of a situation if the delta t were large or the delta t is very small slowly you know the different time values as you can see here from all gates the curve to go from a round shape all the way to a shape of a triangle. So, if this time gap is very very small delta t then it can you know and the force is pretty high because it is an impact force which creates the stress. So, it will not be very much erroneous to assume that the nature of variation of f i is triangular particularly when this delta t is very very small. So, therefore, if we look at the average force it can really be a time average force meaning thereby that if this whole time for the vibrating tool head time period for the vibrating tool head is t capital T then the average force can be written down as integral 0 to t f i as a function of time let us say d t divided by capital T right. And this typically is nothing but the area under the curve the force time curve area under the force time curve. So, as I have illustrated here already the the area can be estimated can be estimated particularly for short time periods to be triangular. So, this is the therefore, the total force here can be approximated by 1 by time period t times of area of this triangle here right one whose height is f i max and whose base is delta t. So, half base altitude is the total area of a triangle. So, we write half f i max times of delta t and this can be given a subscript f average. So, we see here that the average force really is actually equal to half of f i max delta t by time period. So, the motion of the tool for the various positions tool positions along the whole cycle is indicated in this particular figure here. And this we have already discussed, but I would just like to diagrammatically repeat the whole statements we have made earlier for the just for the sake of repetition. This is the tool here the work piece and there is an abrasive grain in between. And the tool goes between positions C which is the top most position the other extremity of the tool further away from the surface. And nearer to the work piece the position which is also the bottom position that is B. Now, the mean position of the tool is illustrated by the point O here. So, typically the force really starts to get executed from the point A goes all the way up to B. And then after the tool releases the contact it is 0 force and the tool comes back and that is how the plot of the force and time had been arrived at in the last slide. The total time delta t for this force to grow from 0 to f i max all the way to 0 is nothing, but the force starting from the position A up to the position B. Also it is important to assume that if the total time period of the tool for this whole oscillation where it goes from O to B back to O to C back to O. So, there are in total 4 times needed for translation between O and B B and O O and C and O and C and O. In other words for one of these if the total time period of oscillation were t the total time needed for only one of these motions or one sector of this motions is t by 4. I would also like to illustrate what happens when the abrasive grain starts getting crushed between the tool and the work piece as illustrated here. So, you can see here that there is a certain depth of indentation on the work piece surface and a certain other depth of indentation on the tool surface given by h w and h t as can be indicated here. And these are the top and the bottom you know surfaces of the tool and the work piece respectively in case a grain is engraved on to both. And there is an engraving on the surface engravement in the tool side as well because the tool also has an ultimate flow stress and although by design we make the flow stress to be of the tool to be higher than the flow stress of the work piece. But still you know during the process of the impact etcetera there is a general tendency because of a very small area of the grain interface with respect to the tool. The pressure can really the force per unit area can really go to the level of the flow stress of the tool material also. And therefore, there has to be an indentation on the tool side there has to be an indentation on the work piece side. So, coming back to this theory again with these assumptions we move forward and therefore, one thing is very clear that because there is a depth of indentation both on the tool side and the work piece side in this case for sake of convenience we are representing it by h t and h w. So, the total depth of indentation h as represented by h t plus h w where obviously, h w is the work piece indentation and h t is the tool indentation on the tool side. Supposing a is the amplitude of oscillation actually predict the motion the velocity of the tool by just looking at how much time is needed for the tool to move between the various sectors. Just for the sake of repetition I would like to bring your attention to this last slide again. So, as I told you the whole path motion of the tool is represented by a motion from o to b o to b. B back to o o to b to b to b to b to b to b to b to b to c and c back to o and the amplitude of motion really is this one sectoral movement let us call it a that is how you define amplitudes in case of simple harmonic motions. So, the total velocity of this tool assuming a to be the amplitude of oscillation of the tool. So, the average velocity of the tool head is basically the amplitude a by the time period in one quarter cycle that is how the definition of amplitude comes into picture. So, it is four times of amplitude of the tool divided by the total time period of motion of the oscillating tool head. So, if supposing we have already seen that the total depth which is moved because of indentation on the tool side and the workpiece side is equal to h how much total time will be needed to cover that particular depth h. It should be equal to the depth h divided by the velocity of the tool head the velocity has already been estimated as 4 a divided by t. And therefore, the total time which again is actually delta t that is needed to move a distance h which was actually equal to h t plus h w the total indentation should be nothing, but h divided by the velocity of the tool velocity has already been calculated as 4 a by t. So, it is h divided by 4 a by time period t. So, the average force which has been earlier predicted as half f i max times of delta which is same as this delta t right the amount of time needed for the total indentation to proceed divided by t can be actually now represented as half f i max times of 1 by t times of the value of this delta t which is equal h divided by 4 a times of capital T. And this h again as you know is nothing, but the indentation on the tool side and the indentation on the workpiece side. So, here we are able to generate an expression for this maximum force on the tool head which is actually equal to 4 times amplitude of motion of the tool times of 2 here. So, it is basically which is basically 8 times f average times of the amplitude of motion a divided by h w plus h t. So, that is how you calculate the maximum force coming on the tool head and then try to do something to equate these force per minute area of the indentation to the ultimate flow stress of the material. So, let us actually try and do that by looking at what is the force at least per grain which comes between the several grains which are there between let us say the tool and the workpiece surface. So, it should be mentioned that during the period delta small t or delta capital T where the indentation is happening it is not a single grain which is doing this indentation. So, if we assume that there are z grains which are there between the tool and the workpiece surface at that small instant of time delta t where the indentation is happening and they are all placed at equal height with respect to the surface and the tool is a perfectly flat surface. So, therefore, all these grains are together going same distances in terms of its ploughing action on to the workpiece. So, in that case if supposing z number of grains are simultaneously in contact. So, force per grain is represented as if I max by z for obvious reasons and if we suppose estimate that the approximate area of contact of the work surface per grain is 0. So, pi by 4 capital D square remember D was actually the indentation diameter. So, this can also be estimated as pi by 4 times of 4 D h where this small d is the grain diameter is the grain diameter and this h here is the indentation of the workpiece surface. Mind you this h is really not equal to h t plus h w because while considering the geometry of the penetration of the grain on to the workpiece only the indentation on the workpiece side was being considered and not the grain. So, this h in the equation for the in the relationship between the indentation diameter, the grain diameter and the traverse h there the h actually is the depth of indentation on the workpiece surface. So, h w so that is how we can relate all this and then we can mention this to be equal to pi d h w. Simultaneously the maximum stress imparted by the grain on the workpiece surface is given by the total force per grain which is available per unit area which is pi d h w from the previous slide. So, this stress should be actually equal to the flow stress of the material sigma w. So, this is the flow stress of the material for the fracture to happen the brittle fracture to happen or for the material to get deformed and the crater would thus start to formulate in that situation. So, therefore, the sigma w here is represented by the earlier equation as 8 times of force average times of amplitude A divided by pi z d h w times of h w plus h t on the previous equations. So, we are just simply substituting the value of f i max in this particular equation for sigma w. So, it is quite reasonable to assume that the depth of penetration is inversely proportional to the flow stress of the material as long as the load in the indenting spheres diameter remain the same. And we can always always say that if h be the indentation depth this h can be inversely proportional to the flow stress of the material meaning thereby that if a material has a higher flow stress for a certain force level it would have a lower depth of indentation and vice versa. So, if sigma t and sigma w are the flow stresses of the tool and the work surface then the ratio of h t that is depth of indentation of the tool on the tool side and h w that is the depth of indentation on the workpiece side can be represented by the ratio sigma w by sigma t which this where this is basically the ultimate flow stress of the work material and this here right here is the ultimate flow stress of the tool material. Let us suppose that this ratio is equal to some constant lambda particularly because we are not changing the tool material or the work piece material. So, it is really a it is a material property and that ratio is represented by this factor lambda here. So, therefore, the maximum the sigma w the ultimate yield stress which is developed can slightly be modified as 8 f a divided by pi z d h w square times of 1 plus h t by h w which actually is signifying this constant lambda. So, we can write this down as 8 f a divided by pi z d h w square divided by 1 plus lambda. So, as of now we will continue this in the next lecture, but as of now we have come to know that there is a way that you can actually relate the ultimate flow stress of the work material with respect to the force the average force on the tool the amplitude of motion of the tool the number of grains per impact or impacting the surface all at one go between the tool and the work piece. The grain diameter and the penetration depth on the work piece is square times of this material property which is the ratio between the ultimate flow stress of the work and the tool with respect to each other. So, with this we come to the end of today's lecture and we will continue this in the next class and try to find out how we can put a value for z in terms of numbers per unit volume of the particle that would be a composition of the that would be indicative of the composition of the slurry the abrasive slurry and then try to find out what is the MRR based on all these different parameters on at least an order of magnitude basis then we can also estimate certain plots and trends from the actual experimental methods to this theoretical model and try to ascertain whether they are in unison or they are in consonance with each other. Thank you.