 Welcome to today's lecture, this is on design analysis of gear pumps part 2, this is continuation of part 1. Now, last time what we have learned that in gear pumps external to the gear pump, if we consider the blue one is the driving one, then the flow is in here and then that is carried out to the other side of the pump and there must be the volume expansion and volume compression in this zone. Actually, we would say it can be shown that the volume displaced among three contacts, one contact with this wall, one contact with this wall and then contact between tooth. That space is will be in compressive here and this will be in expansion mode here. That is two contacts of two gears and casing and meshing teeth at center at both inlet and outlet. And the volume displacement is exactly equal to the summation of active amount of trapped volume. The active amount means if we look into this some oil is going back to the other side. So, active amount is the what is the volume displacement happening here. Now, also we learned that if we consider this is the external pump, in that case what we have to consider? We have to consider a control volume which is which constituted by the point here, the point at the mesh and the point on the other side of the wall. Now, what is happening when this is rotating in the clockwise rotation, then this is rotating in the anticlockwise directions. This point is going downwards, whether these two points are coming closer. We have to consider the contact point. I mean when this just left this wall from that point to the situation when this point again will leave this wall. So, in that way we can it can be shown that this area that is varying and this area varying in a nature that gradually this area is being compressed. So, that can be derived or that can be written in the equation form by considering a small amount of going in here and going in here and going out from this zone here. Now, this is for d theta 1 rotation of the drive gear. So, we can write down this equation in this form and then the final equation for the theoretical flow rate of the gear pump is in dimensionless form is obtained say Qd is equal to half into Ra 1, Ra 1 which is the addendum or tip circle radius of this gear drive gear. Rho 1 is the instantaneous radius at the contact point, Rp 1 by Rp 2 is equal to the ratio of this pitch circle radius of these two gears which is normally is 1 and rho 2 is the contact distance here. So, if we would like to equate in this way what we have to calculate each and every instant we have to calculate these two rho 1 and rho 2 because other dimension are fixed. So, if these two are varying naturally this dimensionless flow is also varying. We have to keep in mind this derivation is actually derivation of this area into the thickness of the pump which is constant. And we have also learnt in the dimensionless form what we did to make this flow is dimensionless form. We divided this into the Rb square and the width as well as divided by the speed to get this flow rate dimensionless flow rate. Now, to determine this rho 1 and rho 2, first of all we should analyze the gear mesh geometry with varying theta, theta is varying the shaft rotational angle which is shaft rotational angle and the instantaneous length of action and thereby the instantaneous flow rate is further determined as before that I would say that while we are doing some geometric analysis, first of all we must know what are the axis system we have considered. In this case if we consider capital X and Y this is fixed to the reference frame that is this is a fixed axis which as you can see this is X and Y capital X and capital Y and again that we should say that this it is ordinated by fixed angle with the line joining the centers of the gears. This is the center of line joining the centers of the gears from here this at this angle. Now, we must know what is this angle then we have considered another axis system XY which is fixed to the this gear the driving gear. Now, it is like that that XY small XY axis is through the middle of this one teeth which with which just the contact has started you see this is rotating in the clockwise directions. So, this is in the anticlockwise direction the contact has just started here at that position if we consider the half of this teeth then we get this line which is small X axis. Now, when theta is equal to 0 then this XY axis small XY axis coincide with capital XY axis that means by knowing this axis position with knowing the number of teeth etcetera, then we can find out what will be the capital XY axis we have considered. This is not a difficult task we can assign that and then what we do we consider this geometry we are trying to calculate row 1 and row 2 and then we consider this point is moving gradually and then we find out this length doing this pressure angle and from there we find out this length row 1 and row 2. Ultimately we find out the length of contact and ultimately we express the flow equations in terms of length of contact it is written here how this X axis are considered when theta 1 is equal to 0. Now, ultimately we can replace this row 1 and row 2 and the equation is expressed in terms of length of contact instantaneous length of contact which is the contact from the pitch point to the instantaneous contact we have to consider this length. If we can calculate this length then this will be expressed easily because other terms are constant. Now, what is instantaneous point of tooth contact that if we define that in terms of this contact radius and angle then contact radius is expressed in this form you can check this geometry and you will find that this is the expression and then we consider this beta 1 angle which was last time shown this can be derived this beta 1 angle in this form and then finally with psi 1 is equal to pi by 2 z minus 1 that is this angle is equal to nothing but half of this angle half of this this is pi by z 1 is equal to tooth thickness basically I mean half one tooth angle and half of that ok. So, possibly this angle psi 1 is defined here no not defined but just consider this angle is calculated from the relation of the teeth number only. Then equation 12 and 13 must be solved numerically for a given theta 1 now we shall consider at starting this equation the rho 1 at starting will be expressed in this form and then contact length L s in dimension less form is also expressed as like this then we get this angle expressed by this expression where this is also at the starting and then the instantaneous length of tooth contact is defined as L is equal to rho 1 sin of this angle and say alpha. Now, the relationship between rho 1 and beta 1 is non-linear hence the numerical solution to this equation will yield the most accurate results for instantaneous length of action we can solve this numerically but what I would like to mention say this alpha alpha is the pressure angle now we have to take care of this alpha say here we consider that might be with generated by a standard pressure angle say 20 degree pressure angle then when they are meshing at if they mesh at their standard pitch circle radius then alpha remain constant that is working pressure angle remain same as 20 degree but if we change the centre distance definitely this working pressure angle will change. Now we should keep in mind sometimes the teeth number are taken less than the allowable teeth number to avoid the undercut and in that case working pressure angle changes. Therefore, we should always keep in mind this pressure angle is equal to the working pressure angle not the standard pressure angle. While we are trying to find out the base circle radius of such gear then we consider only the standard pressure angle there. So, we should not confuse with the working pressure angle and the standard pressure angle all such expression we have to consider the working pressure angle. However, a closed from approximation this I told that we can we should go for the numerical analysis I mean numerical solutions to find out the flow but recently wandering and his co-author is Kasar Gadha they have proposed a closed from solution. So, now to arrived into such closed from solutions first of all we have to consider all contacts point but before that I would like to mention in a methods which we have discussed so far what we have done we have considered a control volume and we have a control volume for that what we observe that basically the area is changing and the width remain constant. So, if we can repeat the change of area variation of area with theta then we can find out the flow variations. However, there is another method in which what is done we consider that those three points definitely but instead of considering this area what we consider we consider a beam joining this instantaneous tooth contact point and the point where this tooth is just separating the exposed zone to the trapped zone. Now, if I consider that as a beam say this point is point of contact may be here the casing is or my might be here anyway if you take these two point what we can consider this is we can consider a beam with uniform load which is equal to the pressure the load is equal to the pressure into may be the area that is uniform load by that way we can find out that how much resultant load acting on this line at the midpoint and we will find that from that midpoint we have a distance to the centre of the gate. So, therefore, if we multiply that distance into this load we will find a torque. So, that torque is acting on this gear similarly if we consider the other gear also we will find another torque which is acting on the other gear. Now, this total torque is transmitted by the shaft. So, in that way we can calculate what is the total power or total energy that definitely that is a rotational angle into that torque if you consider the power the rate of I mean speed in that case we have to consider speed and the torque. Now, this equation we are considering the energy not the power energy in that case what we consider that m 1 is the torque here instantaneous into the angle of rotation d theta 1 and m 2 is on the other gear and multiplied by d theta 2. So, this is the total energy. Now, again this energy can be expressed in form of the rate of change of volume and the pressure. Now, this pressure is the differential pressure that mean output pressure minus the input pressure in this case we consider the input pressure is 0. So, we have considered the output pressure here. Now, if we consider these two equations then we can easily calculate because the same pressure we are considering for this two. So, we can easily calculate rate of change of volume with the angle of rotation and from there again considering this change of this length that mean change in torque with of course, constant pressure because geometrically as the volume is being displaced here there is a compression in this side expansion then definitely this length is also changing accordingly. So, we have to find this length instantaneous length to find out the instantaneous torque. Now, again we have to consider the geometric through this geometric analysis we arrived into this equation with this energy method right. Then if we compare with this equation earlier which we have derived considering the control volume this equation more or less same only what we find that here the half is not there and here the instantaneous square of the instantaneous length where this is some value of this u. Now, we can relate this u with lay L with this relations while this was derived. In fact, this can be also derived within this form but here the geometric analysis was different from this axis considerations. So, in that way the formula is coming like but what is found that these are basically same or in other words now if we would like to find out the numerical solutions for the flow then whether we use this equation or use this equation we will arrived into same result. However, to for the close from solution we consider this equation 10 and we are trying to find out whether this can be solved. Now, close from solution means you have to find out you have to relate this L with theta with some approximation or with some solution. Now, let us say that what the monitoring at all have done for generating a closed from solution for the instantaneous pump flow a Taylor series expansion is done. In that case L which we have expressed in this form for very small value of theta I mean we have to consider in this case for such equations that theta 1 varying very discretely with a small amount. Then in Taylor series expansion we can express this instantaneous length of contact is in this form where the rho 1 s is the initial rho 1 that is at the starting s means at the starting point and as well because this angle is fixed it is not varying whether beta 1 is varying this beta 1 is varying as this gear rotates and theta 1 for which we are calculating the this one it is small we have considered and then this is the working pressure angle. So, we can express in this form obviously the other terms there is Taylor series mean there are other terms also which is negligibly small and we have neglected those. So, this simply expressed in this form. L is then express L s minus rho 1 s cos alpha beta 1 s that if you look into this L then L s minus when it is just moved we can calculate this part and L is expressed in this form. Then this of course could be considered, but again further approximation is done that this rho 1 s cos psi minus beta 1 s into say alpha is equal to 1. Why this is considered this from the geometry it can be proved this is very close to 1. We can say that this equation 80 18 is now reduced to this one and we may consider that there is some error which error can be found out by subtracting 19 from 18 and then it is found that at theta 1 is equal to 0 you can see that it is 0 error is 0, but it will increase slightly with the increase in theta 1, but remember this theta 1 we are considering what will be what should be the maximum theta 1 theta 1 is equal to 0 then what will be the maximum theta 1 just 1 pitch rotation 1 pitch rotations is means that 2 pi or actually not even 1 pitch 1 base pitch you can say and for that the total rotation is equal to 2 pi by z 2 pi by z and z is usually taken may be 20 at the lowest may be 10 not less than that at least I have not seen I do not know whether it can be 10, but you usually will find it in the range of 20. So, now you can find out 360 divided by 20 is 18 degree and you can examine that for 18 degree when theta 1 is 18 degree if you calculate these angles you will find that this error is not 0 slightly more, but still it is it will be within acceptable limit. However, if we with this closed solution if we plot the graph for flow ripple then we can only examine how much difference will be there. Now, the again if we use this general form of equations this L we have to express what we have derived and the maximum flow now we are trying to find out when the maximum flow will occur. If we examine this equation then when L is equal to 0 then there will be the maximum flow. Now, this maximum flow simply if we put this 0 this is expressed by q d max and we find in equation 10 this simply this term is not there. Now, again one interesting point I would like to mention that R p 1 by R p 2 usually you will find 1 that means they are of same size here, but in some design they are not equal. However, this number is not very big and what we will find this is close to 1 either it will be less than 1 or it will be higher than 1, but it is very close to 1 that means this term is always close to 2. Similarly, the minimum flow output of pump occurs when L is maximum that means when it is at the starting because we have in this geometry we have considered that from this starting point it is gradually approaching to this point and our total length is definitely at starting and then final length, but this length will gradually decrease and however this length will be maximum when it is L s and substituting that what we find that this will be the minimum flow. And this negative sign with this as a maximum value the total value will be minimum. Therefore, what we can find in repose a maximum peak amplitude from lowest point to the highest point definitely q d max by q d min which is again expressed by this one and this can be easily calculated because this L s will be fixed geometrically we can find out L s we do not have to go for any numerical solutions in this case. Now, if we would like to find out the average flow then we have to equate all such flow from starting point to the end point that means we can make this integral. Now here just keep in mind that here minus sign has been used because this minus sign is in the opposite from this pitch point actually this total length will be added that means we have to consider from the starting to the end point and which is which must be equal to what this is equal to the base pitch simply we can write here this is the base pitch length here. Now then this integration becomes like this where we have introduced a new term M which is expressed in this form in that case again we have to calculate L s and L f you can see this L s plus L f that is actually total length we are considering the actually total length. So, this is while you are equating you have to be a little careful about this plus minus sign. Now usually the teeth numbers of driver and driven gear are same in common gear pumps. However, pumps with equal as well as different numbers of gear teeth are considered for the purpose of comparing and finding the optimum flow ripple characteristics. Now this exercise is done by Malring to find out whether is there any advantage to optimize the flow ripple by changing the gear ratio from 1. Now to make a clear comparison among pumps of different teeth pairs the average flow rate of each pump is maintained constant in the design process. Now the advantage of dimension less analysis is that we can find out the non-dimensional flow and then we can multiply with actual dimension to get the flow or in other words depending on the flow rate what we done suppose we select the teeth numbers. We select then the speed of whatever it might be or centre distance working pressure angle all we fix up. So, we can equate for non-dimensional flow. Now then if you would like to have more flow from this pump simply what you can do one is of obviously you can flow rate if you want more flow rate one is that you can go on increasing the speed, but otherwise for a fixed speed what we can do we can simply change the module we can simply change the size to have more volume output. So, this means that depending on the flow rate usually say speed depending on the accuracy of this pump manufacturing accuracy of this pump there is a speed limitation say for example in usually gear pump you may not find more than 2500 rpm. If you go for very high speed then there will be very high frequency noise will be there. Again of course that how it is the teeth are ground all such things depends on what is the casing dimensions how is the error in centre distance all such things will depend to have the flow. So, then depending on that depending on the purpose what we do we finalize that what will be the speed then if we want to increase the flow rate with that speed maximum flow rate what we do we go for higher module instead of 4 module we can go for 5 module like this. Now in other words q d average as expressed in equation 24 is held constant for all pump designs. So, here I shall show you some results or typical results of such analysis in that case we have considered this is fixed. Now what we have considered the result of flows form approximation we have considered to find out the solution that is we have considered l is equal to l s minus theta 1 this expression from the approximation we have found and then substituting this we get the starting length of action l s is defined in equation 15 the final length l f is derived as derived in equation 17 and then with this we get l f is equal to is expressed like this. Now this separately we have to calculate now theta 1 starting and theta 1 final that you can say as I told that is simply 2 pi divided by number of teeth of the driving gear which we are driving or gear 1 we would say. So, this calculation will not be difficult and we know this formula and from there we can calculate we can calculate the final length of contact means this one where this expression for the final one is given by this one and this is also as I told that we can find out this and then theta 1 f 2 pi by z 1 as I told and psi 1 is pi by 2 z 1 that is the half tooth angle. Then from the geometry what we find R p 1 is equal to R p 1 cos alpha. Now here I would like to say that if we have considered this R p 1 because R p 1 is something fictitious if you ask if you given one gear if you ask measure its pitch circle radius can you measure that you cannot measure you do not know what is pitch circle. Pitch circle is the circle when two gears are missing and if you divide the centre distance by the gear ratio then only you will find the pitch point and then only you will find the R p 1 ok. So, if we derived in that way R p 1 then we have to consider the working pressure angle if we consider this as a standard then we should consider the standard pressure angle. Anyway this expression is that R p 1 is equal to say alpha with that R p 2 is given by this one and the addendum circle radius of each gear is followed by the American Gear Manufacturing Association standard that is Agma standard in which according to their recommendations for the gear pump it is taken as the R a 1 is equal to Z 1 plus 2 divided by Z 1 in R p 1 which means the addendum factor is 2 by 2 is 1 normally in standard gears we follow that even for the gear pump the same standard is followed. Now, in design at first the number of teeth of gears are selected that means in non-dimensional analysis also we will be first tempted to take Z 1 is equal to Z 2 and we can start with say teeth number is for 20 degree we select the 20 degree pressure angle in that case we will probably we will take this Z 1 Z 2 very close to 17 which is the minimum teeth number for 20 degree pressure angle and probably we will compromise from starting from 20 to this side may be 16 not less than that. 16 teeth without gear correction is possible if we use the pinion cutter suitable pinion cutter, so in that case we may consider from 16 to 20 like that. Let us consider we have considered say 16 so 16 for Z 1 and Z 2 is also 16. Now, the average flow rate of pump in this case in this calculation we are going to show some results Q D average is very close to 0.3. So, it is taken 0.297 this is an arbitrary value one can one can take 0.1 also, but we have to be careful that from some absurd value all the data will be absurd. So, now we consider again this equations and M is expressed like this and in that way typical now we get the some result which is shown in the next slide in and 12. Now, there may be a questions for 12 number of teeth do we need a correction in general we need correction, but in case of gear pump the tooth load is not that high for power transmission gear for reduction gear box usually tooth loads are high if we use the steel and then for 12 teeth without any correction the root is weak, but in this case root is not that weak only problem is that we have to analyze the contact. So, that if the contact try to approach beyond the involute then there will be some problem again if a cutter is designed properly that is the pinion cutter we are considering the state tooth or in mass production definitely of cutter then it is also possible that with 12 teeth uncorrected the root fillet is not severely undercut. So, therefore, in this case because this is dimension analysis what we can consider simply we can say that this is the module used for the flow we will not consider the correction of gear teeth because with the correction of gear teeth then definitely this volume we have to while we analyze this contact point that will vary we must introduce that correction part in that analysis, but here in this case we shall not consider or this calculation in made without considering that these are corrected gear this is uncorrected gear. So, 13 and 12 then in that series that Z 1 we have kept 13 up to this point and it is varied from 12 to 16 then gradually we have increased the teeth number and we have that increase that driving teeth number up to 16 whereas driven from 12 to 16 that means in this case only say this is one case where both the teeth are equal similarly there is another case where both the teeth are equal and in case of this also 16 teeth there are 16 16 teeth. So, 13 13 then 14 14 then 15 15 and then 16 16 these two are of equal teeth if we only consider say these two as we find that definitely center distance factor is changing. However, if we consider the what are the flow ripples there we will I will show the graph, but before that I would like to take I have expressed that R p 1 and R p 2 may not be the standard one just mentioning the center distance we have to consider that R p 1 and R p 2 may not be the standard one just mentioning the center distance we can dimensional and say center distance we can mention what might be the volume because that will directly give us the non-dimensional flow rate we can compare with this non-dimensional flow rate with the center distance. Now say this is the center distance and this center distance non-dimensional center distance we have varied from approximately from 1.5 to 2.6 and then what we find with different combination of teeth say if we consider the blue line then that is with 13 teeth in the Z 1 is the 13 and Z 2 is varying from 8 to 20 we have calculated from 8 to 20 we have extended this graph, but in this chart was 13 something 13 was their minimum teeth number, but what we find that this is with the driving gear when it is 13 and this curve when driving gear is 14 and this is with 15 and this is 16. So, from this chart we can have a conclusion that the center distance increases strongly as the number of teeth on the driven gear increases and decreases weakly as the number of teeth on the driving gear increases. This means that you can judge this from this graph what we find that if we increase the number of teeth of the driven gear then this is this increases rapidly, but on the other hand if we when we increasing 13 to 16 then for a given number of driven gear this variation is not much whereas the if we change the driving gear one then there will be lot of change. So, from there we can have a some assessment what pair we should select for the pump, but again I would say that this normal practice is still equal Z 1 is equal to Z 2 and there is really not much reason that this number to be varied if there is reduction of noise flow ripple etcetera that is nominal not much. This results say that physically smaller pump of the same displacement per revolution may be designed if the number of teeth on the driven gear is decreased while the number of teeth on the driving gear is in increased. So, if we make the driven gear this is small and this is big the size of the gear pump will be smaller, but I would say this is not much still for the sake of optimization the size will be small. However, we are not considering what really will be to the noise and dynamics. The instantaneous flow ripple of the pump is given by the solution of equation 7 that is this is this equations and then we find that this is the flow ripple for different combination of teeth and this angle you can directly you can calculate by 2 pi divided by the number of teeth of the drive gear say Z 1. So, for example, if we divide 2 pi divided by no this is for half of the angle. So, pi by 13 how much it will be 3 divided by 13 3 divided by 3.14 divided by 13 will be approximately 0.25 or so no. So, this is 2 pi by 13 1 full cycle yes 2 pi by 13 and then 2 pi by 14 like this. So, I think this one for 16 teeth 16 teeth is 0.4 whereas, for the 13 teeth it is around 0.5. Anyway, this calculation is not that difficult, but you can see this ripple the ripple is increasing with definitely if the teeth number driving gear teeth number is small. Now, we should also consider the flow pulse amplitude which is expressed by this equation and then this we can if we plot this value with the driving increase in the number of teeth here and in this axis this amplitude then we find for different teeth Z 1 is 16 here and Z 1 is 13 and definitely amplitude will be higher for in case of small number of teeth and what the observation that strong dependence on the number of teeth on the driving gear. Otherwise, this flat is not this this curve is almost say flat you can say with the increasing number of teeth of the driven gear. So, we end here.