[Music] we define a mechanism as a combination of interconnected links or rigid bodies which can move so this is the basic definition of a mechanism so the natural question that arises is can any combination of rigid bodies or interconnection of rigid bodies give rise to a mechanism that means a combination that can move well to know that we need to calculate what is known as the degree of freedom of kinematic chain so in today's lecture we'll start with the calculation of degree of freedom or DOF we'll look into in the course of the lectures we'll look into the calculation of degree of freedom of planar and spatial kinematic chains with examples so the question first that arises is how do we define degree of freedom so here I have written out the formal definition is the minimum number of independent coordinates or variables that need to be specified to fix the configuration of a mechanism so minimum number of independent coordinates and configuration of mechanism these are the keywords so configuration by configuration I mean how does the mechanism look like when it is fixed now when we do this calculation one link of the chain must be grounded now why this requirement suppose I have a plier I have a plier I can take this flier from here to another place but that does not constitute the degree of freedom of that plier the degree of freedom of the plier is what the relative motion of the links of the plier can have but relative motion they can have so I must hold one of the links and move the other links with respect to that so we must ground one link so that is the requirement now let us look into how we calculate degree of freedom let me start with a simple thing a planar body so a body on a plane as you know this link this body has three degrees of freedom what are the three degrees of freedom it can move along this coordinate let me call that as X it can move along Y and it can rotate in the plane so these are the three degrees of freedom so we will say that the degree of freedom of a rigid body in a plane is three let us look at another example suppose there are two bodies now again in a plane now this body can have this kind of motion this body can also have this kind of motion so individually they have three plus three so the total system of these two rigid bodies is six now we look at this example we have two lengths now connected by a kinematic pair so this forms a chain as we had discussed I will ground one body if I ground one body immediately it loses all its degrees of freedom so right now I have this rigid body this link which can move so without this kinematic pair let us see without this kinematic pair this body has three degrees of freedom this body had three degrees of freedom but I have grounded it so what I will do is I will say that there were two rigid bodies I will subtract one from it because I have grounded it and multiply it with three because in a plane now there is one rigid body which can actually move out of these two rigid bodies which I have one is grounded so I subtract this one from the total number of rigid bodies and multiply it with three so that is the degree of freedom that this rigid this link let me call this as 1 and this as two so link two can have but no I have a kinda matic pair here let me say that this kind Ematic pair takes away or every kinematic pair takes away three degrees of freedom that means it is completely immobilized it completely immobilizes link two but again it doesn't completely mobilize what does it do it allows the degree of freedom afforded by that kinematic pair for example if this is a hinge then as you know a hinge has one we're variable so it has one degree of freedom so I add that one to it so therefore finally I have degree of freedom of this system as one let me take another example this is a kinematic chain with three links so let me write number of links is equals to three as discussed we are going to ground one now we have degree of freedom as so I had initially three links - one is grounded so it has lost all degrees of freedom so now if I multiply this way three I get so this is one two three I essentially get the degrees of freedom of links two and three but there are two kinetic pairs let me say that each kinematic pair takes away three degrees of freedom that means it completely mobilizes so I must subtract because it is taking up a degree of freedom it is taking away there are two Cano Matic pairs it is taking away three degrees of each kinematic pair is taking away three degrees of freedom so two will take away six degrees of freedom but then these kinetic pairs must have their pair Baria Buhl's so it must have some degree of freedom so I will add to it the degrees of freedom of the individual kinematic pairs so suppose again these are hinges so there are two hinges so one each hinge has one degree of freedom each hinge has one degree of freedom so one plus one so therefore the degree of freedom is 2 in this case now you can keep increasing the number of links change their interconnections and do this calculation now if you keep doing it you can then generalize this so that is what I am going to show you now so this is the generalization so if the number of Link's is n L number of kinematic pairs or joints is NJ and degree of freedom of the ayats joint is Fi then the degree of freedom of a planar mechanism can be given as F is equal to 3 times NL minus 1 so this term gives you the degree of freedom of the movable links so because I have grounded 1 so I subtract this 1 so total number of Link's minus 1 are the movable links so NL minus 1 are the number of movable links multiplied by 3 because on a plane they individually can have 3 degrees of freedom then I subtract from that 3 times the number of joints why because first I say that each joint takes away 3 degrees of freedom it completely immobilizes but it doesn't so I add the sum of degrees of freedom of individual kinematic pair or individual joints that gives me the degree of freedom of a planar mechanism now I would like you to look at this 3 why did we have 3 why did we have 3 because in a plane a rigid body can have 3 degrees of freedom so from there comes this 3 now if it is a body in space then what happens for a body in space it has got six degrees of freedom as you know a rigid body in space has six degrees of freedom three translation and three rotation about these three axis so three translational motions say about a along X along Y and along Z and three rotational motions along X along Y and along Z so you have six degrees of freedom for a rigid body so therefore for a special mechanism again the definitions remain the same number of Link's is an L number of joints is enje degree of freedom of the IH joint is fi NL minus one because one will be grounded so NL minus 1 will be the movable links now we multiply this with six because they will have 6 degrees individually they will have six degrees of freedom so number of movable links into six will be the total degrees of freedom - we subtract what the did now the the number of joints times six why we say that each joint or each kinematic pair takes away all the six degrees of freedom but they don't so I have added this third term which is a summation of degrees of freedom of individual joints now let us see what a mechanism must be in terms of its degrees of freedom as I have mentioned right at the outset that a mechanism is something that can move so it must be able to move it must have at least one degree of freedom so from this calculation we can now determine the degree of freedom of a combination or an assortment of links connected by certain kinetic pairs now if this assortment or this combination has to move then it must have at least one degree of freedom so a mechanism must have at least one degree of freedom now what is a structure a structure is something that cannot move so a structure must have zero degrees of freedom now if you calculate the degree of freedom and find it to be negative it can happen in certain cases we will see very soon that we calculate the degree of freedom and you find it to be negative it means that it's an over constrained structure now we'll take some examples and do some calculations of degree of freedom the first example that I have here is like this so let me first count the number of links that is what we need in the degree of freedom calculation the number of links the number of joints and the degree of freedom of individual joints so let me start this counting as I have mentioned the ground is always 1 2 3 4 5 6 so we have number of Link's as 6 the number of joints 1 2 3 4 5 now here you see there are three readings rigid bodies connected at a hinge as we have discussed these this is a special case of two hinges because there are three links connected at this so there are two kinetic pairs here so 5 6 7 8 so number of joints is 8 now let us look at the degree of freedom of individual joints and sum them up because we require the summation of degrees of freedom of individual joints now this has one degree of freedom this is a hinge 2 3 this is a higher pair lower pairs this is a lower pair contact as you can see this at this contact there can be both sliding and rolling of body 6 this is the lower pair contact with point contact so this has 2 degrees of freedom as we have discussed previously so 3 four five six now here there are two kinetic pairs so seven eight nine so summation of degrees of freedom of individual joints is nine so therefore the degree of freedom calculation is three times six minus 1 minus 3 times 8 plus 9 so if you do this you will get the answer to be zero so 15 plus 9 is 24 and this is minus 24 so the answer is 0 what does this say about this assortment of links it says that this is a structure so just a combination of link need not move like this one it cannot move so this is a structure of course we assume that this lower pair contact is maintained otherwise if this lifts off then this kinematic pair goes out of our calculation so maintaining this contact this combination of links cannot move it is a structure next we look at this steering wheel mechanism of a car so let me again start counting this I will ground this link is connected to the body of the car and number this as one this as to the coupling link as three this as 4 so 2 and 4 are connected directly to the wheels so number of links is equals to 4 number of joints as you can see here there is one can Americ bear 2 3 4 summation of degrees of freedom of individual joints so each is a hinge therefore 1 2 3 4 so the summation is 4 so therefore degree of freedom 3 times number of links minus 1 three times number of links minus one - three times the number of joints plus summation of degree of freedom of individual joints so this has so this is 9 + 4 13 - 12 so this is 1 so degree of freedom of this mechanism is 1 which is the steering wheel mechanism and as we expect it to be we we operate the steering wheel just by rotating that our steering wheel in the car and it has got only 1 rotational degree of freedom so but just by one degree of freedom we are able to steer the car next we look at this crimping tool let me start by counting the things I will ground this one and number as one so it is already numbered so this is 2 this is 3 and this one is 4 so number of Link's is 4 number of joints is 1 2 3 4 summation of degree of freedom of each joint they are all hinges there are 4 hinges each has one degree of freedom so this is 4 so degree of freedom is 3 10 number of joints plus summation of degrees of freedom of individual joint so that turns out to be 1 so this has again one degree of freedom this is an ice engine remember that this is the crankshaft so we will start counting 1 2 so I will count this as 2 three is the connecting rod and four is the piston so number of links is for number of joints is one two three and there is a prismatic pair here so for summation of degree of freedom of individual joints so they are all one degree of freedom is a hinge or a prismatic pair they are all one degree of freedom kinetic pairs so degree of freedom is three times number of links minus one minus three times number of joints plus four that happens to be one again this is the surgeon's tool first I will quickly draw the kinematic diagram as we have seen so one two three and four number of joints as you can see one two three four summation of degree of freedom of each joint they are all one degree of freedom joints so four so a degree of freedom turns out to be one let us look at this scissors again I need to draw the kinematic diagram so this was the kinematic diagram this is a ground hinge so ground is one two three four five six is the slider so number of Link's is six number of joints so here there are two kinetic pairs 2 3 4 5 6 & 7 is the prismatic pair then we have the summation of degree of freedom of individual joints so here there are two kana Matic pairs of one degree of freedoms each so two three four five six seven so degrees of freedom so that turns out to be 15 plus 722 minus 21 is 1 so this has one degree of freedom let us move over to this transfer device let me draw the kinematic diagram first here you can see there are two ground hinges here there is a an actuator here there are two more ground hinges this one is connected like this and from here is connected like that so this is the kinematic diagram so let me count the number of links so ground is one two three with the two links which have the prismatic pair for the ternary link has five six seven and eight so number of Link's is eight number of kinematic pairs one two three four five now here again two so six seven eight nine ten summation of degree of freedom of individual kinetic met animatic pair they are all one degree of freedom except so here 1 2 3 4 5 6 7 there are two kinetic pairs 8 9 10 so degree of freedom is so that 31 - 30 so this also has one degree of freedom and that is actuated by this actuator so I have discussed the kinematic diagram as well as the degree of freedom calculation of this transfer aid so finally I will leave you with the summary of what we have discussed today and close this lecture