 Today, we begin our lecture with the advantages of involute profile. Because of these advantages, we find involute profiles are most commonly used in mass produced gears. So, advantages involute gear tooth profile. In our last lecture, we have already seen that a pair of involute profiles can maintain the conjugate action, that is, it satisfies the fundamental law of gearing, that is, it maintains constant angular velocity ratio between the two shafts. So, first and foremost is maintains conjugate action. However, as I said earlier, this is not so much of importance, because if one is given any smooth curve as one profile, one can find another profile which will be conjugate to the given profile. So, this is not the unique choice of involute profile which maintains conjugate action. There are other advantages of involute profile over and above that it maintains the most fundamental requirement, that is, maintaining conjugate action. The second advantage is that operating pressure angle is constant. If you remember, in our last lecture, we denoted this operating pressure angle by the symbol phi, which is the angle between the line of action and the common tangent to the pitch circle, maintains constant pressure angle. Operating pressure angle is constant. We will get back to the same figure which we discussed last time to show what is the advantage if the pressure angle remains constant, what is the consequence of this pressure angle remaining constant. And the third advantage is that a pair of involute gears maintains conjugate action even if the centre distance between the gears are changed. That means we have a pair of gears, we can mount them with a little different centre distance even then this conjugate action will be maintained. Even if centre distance between the two gears, that is, between a pair of centre distance between the pair of gears changes. These two points, as I said, I will explain with reference to the figure that we discussed last time. And the last but not the least, the most important reason why involute profiles are so common is that ease of production. Why it is easy to produce involute gears? Because gears are produced, gear teeth are produced by a process called generation. That is, again we use the conjugate action but not between a pair of gears but between a rack and a pinion. And it is the conjugate rack of an involute profile is a straight tooth rack. If we have an involute profile, we will see this later that the conjugate rack tooth, if this is a rack whose tooth profile is straight, then it can maintain conjugate action with an involute gear and it is such a straight sided rack cutter is used to produce the involute tooth profile on a circular gear blank. This is, I can say, it is an involute rack which is used as the cutter and on the cutter it is much easier to produce straight size than any complicated curve and it is this involute rack cutter which can maintain conjugate action with an involute tooth profile of a gear. So this is called involute rack cutter. Why the involute of a rack is a straight line? This point again will be cleared in today's lecture a little later that the involute of a straight line is another straight line. Like we have seen, involute of a circle is a curve who if this radius of the base circle goes to infinity when the gear gets converted into a rack, then this also becomes a straight line. All these points will make clear in today's lecture. So let me get back to that figure where I showed the pressure angle and now will show why what is the advantage of this pressure angle remaining constant in a pair of involute gear tooth profile. Let us get back to this figure which we discussed in our last class. These are the two base circles of radius r b 1 and r b 2. These are the pitch circles of radius r p 1 and r p 2 and it is the common tangent to the base circle which we call line of action. And this line of action a b makes an angle phi with this common tangent to the pitch circle. This t t is the common tangent to this pitch circles of radius r p 1 and r p 2. And this angle phi what we called operating pressure angle. Now as we have already noted because the line of action just remains a b, this pressure angle does not change as if the profiles are involute. The pressure angles remain constant during the entire interval. Now what is the advantage that the pressure angle remains constant? If we neglect the friction force between the gear teeth which if it is oil and well lubricated friction force is not very large. The driving effort is along the common normal. If this gear is driving this gear then most of the force is acting through this along this common normal. So if this normal force we call say n then the torque that is transmitted is n times the base circle radius. Because this line of action n is along a b and this angle is 90 degree. So the torque acting on this gear 2 is n into r v 2. And if this torque is this is torque that is transmitted to gear 2. Now if this torque remains constant then n remains constant because r v 2 is constant the base circle radius. If n remains constant then you see both the magnitude and direction of this force n is remaining constant which means the bearing reactions here and here because it is the same n which is acting on this gear also. So the direction and magnitude of n both remains constant because phi does not change and if torque is steady torque if we are transmitting a constant torque then this torque is also constant which means magnitude of n is constant because phi is constant means the direction of n is constant which means the bearing reactions while the gears are transmitting a steady torque remains the same which means the bearings are not subjected to any dynamic reaction the magnitude and direction both of the reactions remain same. So long a steady torque is being transmitted. This enhances the life of the bearings which are used to mount these two shafts on the foundation thus the pressure angle remaining constant implies that under steady torque the bearing reactions are not dynamic they are static torque that is the advantage of pressure angle remaining constant. The third advantage you discussed was that suppose this pair of gears right now the center distance is O 1 O 2 suppose this gear remains where it is only this gear is shifted a little bit upward then what we see that it is the same involute profiles will be useful to maintain the conjugate action. Let us see what is the center distance center distance O 1 O 2 is R p 1 the pitch circle radius of the first gear plus R p 2 pitch circle radius of the second gear and we have already noted that base circle radius R b is nothing but R p cosine of phi where phi is the operating pressure angle. So we know R p 1 by R p 2 is same as R b 1 by R b 2 because R b 1 is R p 1 cos phi R b 2 R b 2 is R p 2 cos phi cos phi cancels so this ratio remains same. So what we see that so long the gears are same that is the base circle radius remains same R p 1 by R p 2 remains same with the center distance changing it is R p 1 plus R p 2 this quantity is varying which means with varying center distance if the center distance varies the pitch circle radius R p 1 and R p 2 both change R p 1 and R p 2 both change but the ratio of R p 1 and R p 2 that remains same so it is the same angular velocity ratio omega 1 to omega 2 is maintained only thing what changes because R p 1 and R p 2 is changing means phi is changing because R b 1 is not changing and R b 1 is R p 1 cosine phi where phi is the pressure angle R b 1 is not changing because the gears are not changing but with changing center distance R p 1 is changing which means phi is changing because this common tangent if this circle is shifted a little bit upward the common tangent between these base circles will change that means this angle phi will change so pressure angle changes a little bit but the same angular velocity constant angular velocity ratio is maintained by the pair of gears even when the center distance changes a little bit that was the third advantage and the fourth advantage as I said will be explained later when we will be able to show that the involute of a straight line is a straight line that is for a rack to maintain conjugate action with an involute profile the took profile on the rack will be straight like trapezium as we have shown earlier on the board because involute profiles have all these advantages as I said earlier they are almost universally used in mass produced gears and as a consequence of this mass production of involute gears they have been standardized there are various standards but I can mention some typical standards which are more commonly used like bit is standard for involute profile so involute tooth profile some standard dimensions are the most common value of the pressure angle operating pressure angle phi is 20 degrees some old gears which are cast it also had a value phi equal to 14 and a half degrees but these days most gears involute gears will have an operating pressure angle equal to 20 degree similarly the value of the addendum which we defined earlier a is equal to the module where m is module the gear teeth is described in terms of the module of the gear teeth and the standard value of the addendum is equal to module similarly did end up which we denoted by b and which is always more than a the most common value is 1.25 times the module m is the module which is expressed in millimeter for metric radius and a is equal to the module m and b is equal to 1.25 m these 3 standard values we may take if unless otherwise specified as I said involutes are most commonly used and geometry of involute teeth is a very vast subject we are not going to get into all the details of involute teeth geometry but I will give you a little glimpse or a little basic idea of this curve involute which we call involutometry and the result that we obtain from this involutometry we will see will be very useful to determine various proportion of the gear teeth which are involutes. So we can say application of involutometry in gear tooth geometry I repeat we will discuss of course only the very basics of this gear tooth geometry you can always refer to hand books for the details that we like to know. So let me first say what is involutometry suppose this is the base circle with the center here and we start unwinding the string from this base circle from this point a and by unwinding the string I generate this involute. So at this this is the string length which is same as this arc length because this string was wound on to this cylinder and now it has un wound up to this point and this is the involute from this base circle. So the string is tangent to the base circle. So this radius r b base circle radius this is perpendicular to the string at this point the involute takes off radially. So this radius is tangent to the involute at this instant and at this configuration this is the string which is perpendicular to the involute. So the tangent is perpendicular to the string this is tangent this is the tangent at this point say b. This radial line is tangent to the involute at a and this line which is perpendicular to the string at the point b is the tangent at the point b and this angle between these two tangents I call the roll angle theta theta is called roll angle this line does not look like perpendicular so let me draw it correctly this is o so this line is perpendicular to the string this radius is also perpendicular to this string which is tangential so this is parallel to this so if this angle is theta this is also theta the roll angle that is the string has unknown up to this point b while the roll angle is theta this particular point on this involute this is the involute this distance ob is the radius vector let me call it if this circle this difference the base circle of a gear and the gear is rotating about the point o then this particular point has a velocity which is perpendicular to ob this is the direction of the velocity and this is the normal to the involute so that is the line of action so at this point this angle between the normal and the direction of the velocity normal really indicates the direction of the force at this particular point this angle is called psi which will call involute pressure angle this is the angle between the line which is perpendicular to ob and the string at this configuration when I have reached up to the point b please note that this is not the operating pressure angle which we discussed in case of a pair of gears this is only one gear we are talking of one involute we are talking of and this particular angle between the direction of the velocity and the direction of the normal to the involute I call it involute pressure angle now this line is perpendicular to the tangent and this line is perpendicular to the direction of velocity so this angle is also psi involute pressure theta is the roll angle psi is the involute pressure angle theta is the angle between these two tangents at a and b which is same as these two radius at a and corresponding to b and psi is the involute pressure angle so let us say base circle radius is r b and this is the string length is the radius of curvature of the involute at this point b if this is the involute at every instant the radius of curvature is nothing but the string length which has been unknown from the base circle so row is if I call this point c then b c is the radius of curvature row so what we see row is given as square root of r square minus r b square where r is this radius vector of the point b measured from the center of the base circle and if I use this angle as a polar angle to define the involute curve what is this angle that is theta minus psi this angle is theta minus psi and we also see this row is same as this arc length ac because this is the length of the string and this was the original length of the string so ac is nothing but row which is same as r b into theta and also you can see tan psi is row by r b if this angle is 90 degree this is row this is r b and this is psi so tan psi is row by r b we can write tan psi is row by r b and from here we see row by r b is nothing but theta so I can write theta minus psi as tan psi minus psi theta is row by r b and row by r b is tan psi so this angle that the line ob makes with this original radius o a that angle is tan psi minus psi and this is given a name which is called involute function of psi and we can also see r b is nothing but r of cosine psi so we can write base circle radius is related to r as r cosine psi so everything has been found in terms of this involute pressure angle psi which keeps on changing at various points the distance from o I can get as r b by cos psi and this angle I can get as tan psi minus sign psi which is called involute psi is very easy to see that if one gives me the value of psi I can calculate involute psi very simply this is called involute function just like sin cos we call it involute of psi given the value of psi it is easy to calculate involute of psi but not the other way round so for this tables are available just like tables of sine cosine tan log I can get the value of psi if you give me the value of involute of psi by consulting that table now at this stage I will be able to show that what happens to this radius of curvature of the involute profile as r b increases as I said when r b goes to infinity the gear gets converted into a rack so under that situation what is the radius of curvature of the corresponding involute profile so let me now say what is the radius of curvature of the involute tooth profile of a rack rack means where r b goes to infinity we have already got rho equal to square root of r squared minus r b squared where r is the instantaneous value of this distance so if we start from a and go up to some point by changing initial value of r is r r b and then I go to r b plus delta r let us say as I start from a I go a little away from this when r changes from r b to r b plus some delta if I substitute it there what we get we get twice r b into delta r plus delta r squared to the power half this is the value of rho as we go out from this point a now if r b goes to infinity it is obvious that rho also goes to infinity and if the radius of curvature goes to infinity which means this profile has become a straight line that is what we said that if we have an involute rack that the tooth profile from the base circle as it comes out it comes out in the form of a straight line so the tooth profile of an involute rack is a straight line and which can maintain conjugate action with an involute gear or involute pinion and that gives the advantage that I can use such a rack cutter to generate the involute tooth profile on a circular gear blank let me now repeat what we have just now discussed with the reference to this figure this is the base circle this is the center of the base circle and involute is being generated starting from this point a when the involute if I consider a point I here then the string at this configuration corresponds to this line I b which is tangent to the base circle O b is the radius of the base circle which we denoted by r b instantaneous polar radius of this point I I denote by r the tangent at a to the involute is this radial line and the tangent at I is this line the angle between this tangent at I and the tangent at a this angle theta we call the roll angle theta was defined as the roll angle if this I take as a gear and the gear is rotating about the point O then the velocity of this point I is perpendicular to O I say this is the direction of the velocity of the point I if it happens to be a point on the gear and the string which is normal to this gear tooth profile of the involute profile that is the normal to the involute at the point I and the angle between this velocity direction and this normal which is nothing but B I I call psi this is normal and this is the tangent. So, this angle is 90 degree this is also perpendicular to the string B I which is tangent to the base circle. So, this line the tangent at I and this radius O b are parallel. So, this angle is psi then this angle is also psi similarly if this angle is theta this this angle is also theta because this line is parallel to this line. So, theta is the roll angle and psi is this angle now the polar angle of this line O I from this vertical line that is the radius through a initial point this angle which is theta minus psi we defined as involute of psi involute of psi is this angle which is theta minus psi and what we have seen that this length A B is nothing but R B into theta and the string length B I is also equal to A B. So, this row which is the radius of curvature of the involute profile at I the string length is the radius of curvature row this row is nothing but R B into theta from this triangle O B I I can write tan of psi because this angle is 90 degree tan of psi is also row by R B. So, you can write row is R B tan psi which means theta is equal to tan of psi. So, this involute psi which is theta minus psi I can write is as tan psi minus psi then we also found that expression of row in terms of R and R B which we wrote as row equal to square root of R square minus R B square as we draw this involute from starting from this base circle the R changes from R B it increases from R B say by a value delta R then substituting in there cancelling R B what we found that expression of row turned out to be twice R B into delta R plus delta R whole square. So, this clearly shows if R B tends to infinity then row also tends to infinity that means right from the point A as soon as R B R changes from R B to plus small delta R the radius of curvature becomes infinity that is the involute of a straight line when R B goes to infinity this base circle becomes a straight line that is the gear is converted to a rack and the involute profile becomes a straight line because radius of curvature is infinity which tells us a rack involute rack tooth profile looks like a straight line. Just now we have discussed the basic geometrical relationship between an involute profile and its base circle these relations are very useful for studying gear tooth geometry as I said earlier and here we discussed an example. Suppose this is an involute tooth profile which has been generated from this base circle suppose the thickness of the tooth as we see keeps on varying from the base circle up to the addendum circle suppose the thickness of the tooth at any point x which is defined by the polar radius R x this point x is defined by this distance O x and the thickness at this level I denote by T x then when this polar distance changes from R x to say R y I get to this point y and I want to find what is the thickness of the tooth at this level that is T y so T x is given I would like to find T y and R x and R y are given to do this first we note when the involute was at x the string is d x this is the length a d which has become d x and when we come to the point y the string is represented by b y this is the tangent b y is same as the arc length a b so the angle between this O x and O d we defined as the psi so I write it as psi x corresponding to x d is the x d is tangent to the base circle and the angle between O d and O x I call the involute pressure angle at x which is psi x similarly the involute pressure angle at y is the angle between O y and the tangent to the base circle from y which is y b that is what I called psi y we have already seen the base circle radius R b this is also base circle radius R b and this can be written as R b is nothing but R x cosine psi x it is also equal to R y cosine psi y so if you are given the values of R x and R y I can get psi x and psi y now how T x and T y are related for that let me consider this angle A O c O c is the mid line of the symmetric line of this gear tooth O c and the angle that O c makes with O a I call this angle so angle A O c I can write as T x by 2 divided by R x that I am getting this angle this angle is T x by 2 divided by R x and this angle is nothing but involute of psi x at the point x the angle that O x makes with this starting line O a I defined as involute function of psi x so angle A O c is nothing but T x by 2 divided by O x which is R x plus involute function of psi x and if you remember involute function is tan psi x minus psi x involute of psi x tan psi x minus psi x and A O c the same angle I can also write as this angle which is T y by 2 divided by R y so T y divided by 2 R y plus this angle this angle that is the angle between O a and O y which is nothing but involute of psi y this angle is involute of psi y so this plus involute of psi y so if R x and R y are given I can find psi x and psi y from this relation one psi x and psi y are known I can find involute of psi x and involute of psi y because involute function is tan minus the angle involute function of any angle psi x is tan minus psi x minus psi x of course psi x must be measured in radian so from this relationship if T x is given R x is given R y is given I can find T y which is the only unknown because psi x and psi y I have already found out from here and in using the involute function I can find involute of psi x and involute of psi y so thus for the get to geometry as we see if you give me the thickness at any level I can find the thickness at any other level of the same involute tooth profile for continuous transmission of rotation from one gear to another it is obvious that is imperative that a pair of teeth must remain in engagement before at least until the next pair of teeth comes into engagement it should not happen that one pair of teeth has lost his engagement and the next pair of teeth has not come in engagement because then there will be no transmission so to maintain continuous transmission of rotation from one gear to another it is imperative that a pair of teeth must continue to be engaged must continue to remain engaged at least until the next pair of teeth has come into engagement this phenomena is studied in terms of a geometrical quantity which we call contact ratio let me now define the contact ratio and try to get the expression of this contact ratio so we study contact ratio for involute tooth profile we use the symbol m c indicate the contact ratio let us say this is one base circle and center of this gear is at o 1 and the center of the other gear is at o 2 this is the other base circle base circle 2 if you remember that it is the common tangent to this pair of base circles if I draw the common tangent to this pair of base circles that defines the line of action this is a and this other point of tangency is b so we know the contact between a pair of gear teeth will always lie on this line a b now let us talk of one tooth of this gear suppose this gear is rotating this way and to the teeth action by teeth engagement it is rotating this gear in this direction so if we consider a tooth on this gear the first it comes into engagement at its outer circle of the addendum circle and because the contact point must lie on this line when the addendum circle when the addendum circle comes here in contact with this line a b that is where the contact starts on this face of the gear tooth this is one gear tooth now how long the now this gear starts rotating this way how long this tooth will remain in contact with the tooth on the other gear gear number one so long the outer circle of this gear is in contact so this is the addendum circle of the gear one let me call it radius r o 1 that is the outer radius of gear one and this is b one so the contact will as this tooth rotates this tooth comes here and when it the tooth on this gear which is pushing it let us say here this tooth comes and rotates and when this outer circle leaves this line the contact is lost with this particular gear so if I draw this tooth here this is where the contact is lost so this particular pair of teeth is in contact as this contact point moves from this point say e to this point say f the contact between this pair of teeth this one and its mating tooth starts the contact when the addendum circle of this lower gear gear two this is the addendum circle of gear two and this is the addendum circle of gear one when the addendum circle intersects line a b at the point e the addendum circle of gear one intersects the line a b at f so this pair of teeth is remains in engagement from e to f what we should ensure that before this point is reached the next tooth in this gear must come in engagement must come in engagement that is outer circle must intersect here for the next tooth to do this let me define what we call base pitch we consider this is the same tooth we are considering but now let me consider two consecutive tooth on one gear if you remember that we define the circular pitch on this pitch circle going from one point to another identical point on the adjacent tooth measured along the pitch circle this is what we defined as pp which is 2 pi rp where rp is the pitch circle radius divided by the number of tooth similarly you can define base pitch that is this point and the identical point on the next tooth but on the base circle this distance is called the base pitch we call it pb base pitch that is 2 pi base circle radius divided by n and if you remember rp is nothing but rp cosine phi where phi is a operating pressure angle now what is the movement along the base circle of this particular teeth from the start to the end of the engagement that is given by this circular measure on the base circle this is the rotation that is the distance covered by the this particular tooth from the start of its engagement to the end of its engagement the rotation is such that it covers so much distance on the base circle to maintain contact between two pair of teeth that is before this contact is lost the next pair must come in engagement this distance must be more than this base pitch this distance if I defined is that pb then this distance must be more than pb such that the next tooth has come up to this position because the distance along the base circle between this tooth and the adjacent tooth is the base pitch when this point comes here that point must come above this point that means this circular distance must be more than the base pitch let me call this point s and t so to maintain contact all the time between these two pair of gears between this pair of gears s t must be more than base pitch and it is this ratio s t by base pitch which I defined by pb which I have defined just now must be more than one and we defined m c as s t by pb what is s t that is the movement of this tooth along the measured along the base circle from the start to the end of the engagement to calculate s c we proceed as follows this is what we call the pitch point p where the common tangent the line of action intersects this line of centers o 1 o 2 now because this is the involute I can say and this is the tangent to the base circle b t is nothing but b e b t that is the distance along the base circle is same along the length of the string b t is same as b e same way I can say b s is same as b f this is the distance along the base circle and this is the length of the string so I can say b s is same as b f if I subtract I get s t is same as e f and the contact ratio m c which was defined as s t by pb I can write this as e f by pb the expression of pb we have already written p b is base pitch 2 pi r b by n where n is the number of teeth and r b is the base circle radius which is same as 2 pi pitch circle radius cos phi where phi is the operating pressure angle that is this angle and also this angle is same as this angle so we have got the expression of m c in terms of e f by p b p b is given by that expression now let me try to write e f in this way as e f plus a e plus e f plus f b I have added a e e f f b so I subtract all these three so minus a e minus a e plus e f plus f b e f is this e f I have added a e e f f b I have subtracted a e e f f b e f plus a e as we see e f plus a e is nothing but a f similarly e f plus f b is nothing but b e and a e plus e f plus f b is nothing but a b what you can see what you can see what is a f this angle is 90 degree because this is tangent this is radius so a f I can write square root of o 1 f squared minus o 1 a squared and what is o 1 f nothing but r o 1 this is the random circle that is how we determine the point f so o 1 f is nothing but r o 1 and o 1 a is r b 1 so a f I can write square root of r o 1 squared minus r b 1 squared the same way I can write b e what was e that was the random circle of this gear so o 2 e is nothing but r o 2 and this is r b 2 so b e I get r o 2 squared minus r b 2 squared minus a b a b is this angle is 5 so o 1 f minus a b is this angle is o 1 p sin 5 is a p similarly o 2 p sin 5 is b p so o 1 o 2 sin 5 will give me a b o 1 p plus o 2 p is o 1 o 2 that is the center distance and a p plus b p which is a b if you take sin 5 o 1 p sin 5 you get a p o 2 p sin 5 you get b p so this is the center distance which I write as c is sin 5 where c is o 1 o 2 so I get e f in terms of base circle radius and the outer circle radius random circle radius similarly base circle radius random circle radius of both the gears center distance between the gears and the operating pressure angle and this divided by p b which again I express in term the operating pressure angle number of teeth and the pitch circle radius this I should have taken second gear so 2 pi r b 2 n 2 so 2 pi r p 2 n 2 and r p 1 and n 1 it is the same so r p 2 by n 2 is same as r p 1 by n 1 so it is the same expression we have to take the pitch circle radius and put the number of teeth of the same gear so that gives me the expression for the contact ratio the recommended value of the contact ratio for a pair of involute gears is 1.4 what does that mean this is a not an integer what does it mean that during some phase of the contact only one pair of teeth is in contact for certain other phase of contact at least two teeth is in contact because there must be a region when two teeth must be in contact so that continuous transmission is ensured and on an average 1.4 pair of teeth remain in contact if I consider the entire cycle that is the physical meaning of this contact ratio as 1.4 that on an average during the entire cycle 1.4 pair of teeth remain in contact that means sometimes there is only one pair of teeth in contact and just when the contact is being lost between one pair other pair comes into contact before that so for sometime at least two pair of teeth remain in contact so this contact ratio gives you the average number of pair of teeth that remain in contact during the entire cycle.