 Welcome to today's lecture. This is continuation of the analysis of an axial piston swash plate type hydrostatic pump. Earlier we have learnt the pressure repulse, the features, also the what will be the input torque of the shaft and volume displacement etc. In this lecture I shall discuss about the bulk configuration in a better way and thereby we also discuss about the pressure repulse and then one important aspects which is called swash plate torque. This means that the torque required to hold the swash plate in the position or in case of fixed displacement what are what are the torque experienced by the swash plate. Now this is the axial piston pump which we are studying already I have described that this is rotated the pistons are laid in a barrel which is rotated by an input shaft and then this the end of such pistons are with hydrostatic thrust bearing which is called slipper pad. This is moving on the swash plate which is inclined for fixed displacement this inclination is fixed in case of variable displacement this angle can be varied. Now as the pressure changes during the rotation the suction and compression as well as in this valve there are also transition zone depending on that torque on the input shaft as well as the swash plate shaft they vary. So while we are changing the angle particularly in case of variable displacement or when this is being used for a fixed I mean constant displacement or some constant pressure constant torque whatever may be the feature in that case we need to control that swash plate torque. So for that it is essential to know what the torque what is the torque acting on this swash plate at an instant. Now this is the geometric configuration which already I have discussed between these two pistons which are equispaced the angle is denoted by this one and this angle psi p is the valve angle total spread of the valve and the theta is the angle of rotation of the barrel that is rotation of the shaft. If we look from this side then this theta begins from the left hand axis the axis system is not shown here which already I have described in earlier lecture. Now this is the typical leakage shown I mean this typical figure for showing the leakage past the pistons. So for that we need to consider the pressure volume then bulk modulus density and also we should consider the area of the orifice flow pressure etc. Now the valve which is called kidney port. So this there is a valve plate which is not shown on the kidney port I mean on the valve these are the kidney port. If we consider from here to here and if we consider here to in this directions one obviously one is for the suction one is the for completion or delivery suction and delivery input and output. Now this here although it is shown apparently the cylindrical hole the other end of the barrel but usually this end is with also another kidney port this is kidney pattern it is written kidney pattern port. Now the dimension the or the angular spread between these two kidney ports is such that the length of this will exactly fit on this space. This means that momentarily there will be no flow in or out to a chamber to a piston and next moment it will go inside the this kidney port may be suction or may be discharge. Now look into this end say it is rotating like this then while it is entering whether it is discharge suction there is a groove is provided reason is that if it is exposed to this main port directly then rate of change of area is very fast and the pressure change in pressure in the fluid inside the piston becomes very much transient. First thing second thing is that there is a there will be huge noise when it will if it is directly connected to this if this groove is not provided. Therefore sometimes this groove is called silencing groove here is the geometry is shown what is the spread of this groove and this is the spread of kidney port on the barrel and this is the spread of this dead bent it is you should call dead bent known. Now this is again in some design these are not exactly equi spaced that means in one side it is less in other side it is more this also to reduce the dynamics because the barrel is moving the port is then coming into contact coming into this kidney port contact the if it is made equal then due to the dynamics actual time maintenance will not be equal or in other words you can say that this for the optimum performance it is found that if one angle is less made less another angle made higher the performance is better where this angle may be equal to this angle or this angle is slightly larger slightly larger than this angle. If we make this is less than this angle then what will happen there will be direct connection from discharge to input that output to input or vice versa. So, immediate pressure drop will be there and it cannot be controlled huge leakage will be there. Now if we think of the port that silencing groove now this is the main port on the valve plate and this you can say this is a through hole through which the oil is going from one side and from other side it is going to the pipelines. Now this silencing groove it may be a flat bottom and constant width. Now when this this is the kidney port of the barrel when this move gradually on that initially opening will be less gradually it will increase and then finally it will increase to the whole port will be open to this main port. Now it can be made simply like this which is say if you think of the manufacturing this can be made by a milling cutter end mill cutter like that. Now next one is the rammed bottom constant width. So, rammed bottom means this is an inclined bottom if you look into this the how to calculate area we will always consider this from this corner to this perpendicular area that is the minimum area open to this port. In this case also directly we instead of considering this area we consider this area perpendicular direction what is the opening that means rather we have to consider the flow is going like this we have to consider this diagonal. In this case you can see this is more effective. So, it is gradually increasing. Now in comparison this we can have also it is like an pyramid. So, no this is flat bottom that means this area is increasing that gradually this is increasing this width is increasing. However depth remain constant. Now as you understand machining of such thing is very very difficult because here practically this is the vertex of a triangle and depth is like this. Probably I cannot think of machining with ordinary tools, but this is also explained that this can be used. But most common is that rammed bottom rammed width geometry that means this is inclined as well as this side is also increasing. I will show how to calculate the area one of that I will taken for an example and I will show that how the orifice area can be calculated. Now this is you can now see these are the four possible configuration and these are provided for the silencing groove. Now earlier I have already discussed that the instantaneous rate of change of pressure in the ith piston chamber is given by this equations. I am not repeating this because earlier we have discussed here we consider the sign depending on that direction of flow can be determined. So, this is the pressure drop and this is the leakage characteristics k theta. So, this is of earlier lecture if you remember in lecture 19 we have discussed this. Now again I am continuing with a actual pump. Actual pump means in this case we will consider the leakage and others to find out the pressure ripple. Now the port area remains at a maximum constant when it is completely over the discharge or suction manifold that means when this port kidney port will come over this main port then it will be maximum. In the transition region the port area gradually increases to a maximum from 0 when the piston is entering into the manifold zone from the dead zone that is from the top dead center or bottom dead center. In case of pump the oil is being exposed to high working pressure and in case of motor the oil is being exposed to low pressure if the BDC and vice versa if it is from top dead center. That means if we consider from it is bottom dead center then that means in case of pump it is completely full with fluid. Now then next moment it will be connected to a port in case of pump definitely it is a high pressure because it is going to discharge in case of motor it will be the reverse. So, to understand this language you have to think in that way. Similarly the port area gradually decreases to 0 from it is maximum opening when the piston is leaving the manifold zone to the dead zone. Now in earlier analysis I have shown that we consider the flat bottom and rammed with geometry. This is just to show the variation of the area, but this is not practical as I told the machining of this might be problem. Now at the beginning of the discharge manifold or suction manifold a silencing group is provided for which this is the area already I have discussed. And then these are the angle shown here different angle in earlier lecture I have already discussed. So, I am not repeating here. Now we are considering this area I mean this kidney port pattern this is the this one of this shape. Now area when we are calculating we are considering this area. Let the angle covered by the right end of the kidney port be theta. Also the kidney port of the barrel is considered to be rectangular kidney port of the barrel is considered to be rectangular that means there is slightly curved, but we will consider it is rectangular that means we will consider that this is an rectangle and this is a circle. This circle diameter will be width of this one and then length is from this center to this center. We will consider this actually it is curved by simple geometric principle the following results are obtained. It is assumed that this angle is greater than this angle. We have assumed and we have assumed also this angle is equal to this angle. And the small variation for the spread of this right hand side of this axis and left hand side of this axis is not considered in this analysis at all. Now then what is the area we consider? We in this case as it is a flat bottom in this case we consider whatever exposed of this area thus becomes clearly this is the height and this is the base and half of that do you understand my point. In case of the flat bottom and ramped width we simply consider the whatever area is opened here not in this diagonal direction directly we consider this area is the orifice area. So, it is calculated like this which already I have described in earlier lecture. Now the important factor is that what is the total opening from the barrel to the port that you can see that let the full area of the kidney port of the barrel is a k. Therefore, the variation of the port area of the ith chamber is given by the following relations. Now when this angle that is the lapping of this one we have considered now angle from this line this is the angle of rotation our actual angle of rotation we have considered theta. But while we are describing this area exposure we have considered that this is the zero position means the tip of this is here now this is increasing. So, when this tip is within this limit then area is given by this one this is a g is the area of the a g is the a g plus a g is the total area of this angle. So, it is given by this is total area of the silencing groove the flat area. So, then when it is after completing this it is going inside then for that we may consider this is the area. So, this area is already exposed that means this triangle is already exposed then a part of this area is going inside this kidney port. The width here is same this kidney port width and this kidney port width is same. So, this if you carefully just observe that these are the simple geometric calculations now when it is further moving further that means this angle this angle is less than this and no sorry this angle is greater than this and less than this then we find this area is completely this area plus a part of that has come over that and then for the condition when it is fully on this port then this is the total area for this is the total area and this condition is that when it is just going out remember that it other end there is no silencing port if it is rotating in this direction silencing port is always when it is entering other end there is no silencing port. So, this angle is the total spade of this kidney port. So, when it is crossing then the angle is given by this one. So, my suggestion is that to calculate the angle and the calculate the area we should go through these equations and we need a practice otherwise we will not be able to find out what are the areas these are simple geometric relations, but it needs practice then the pump which we have considered here this is I have already shown earlier, but we consider the bulk modulus is 10 to the power 9 Pascal's and coefficient of discharge is respective of the size of the orifice is 0.62 angular extent of the barrel orifice 30 degree and angular extent of the silencing groove is 11 degree maximum opening area of the silencing groove is 2.25 10 to the power minus 5 meter square and maximum area of the each barrel port is 3.75 10 to the power minus 4 meter square as you see this is smaller than this is the if you consider the top of the silencing groove this is the area remember we will consider a g as this area for the orifice when we take this flat bottom type for the pyramid type that is ramped bottom and as well as the ramp opening for that we have to calculate such angle separately I will show that now leakage coefficient here we have considered k it I think it was given k theta there. So, we consider the leakage coefficient again when I was discussing about the leakage coefficient I was told that depending on the length of the stroke length of the opening this actually it would change, but if we find that stroke length is not very high because this was played angle for this pump is not very high in that case we may take for the analysis k is equal to constant where we have taken in this case now source plate angle alpha is maximum 18 degree and nominal volume of single piston chamber is 22.83 into 10 to the power minus 6 meter cube very small and r is equal to that is radius is 5.501 10 to the power minus 2 meter that is 55 millimeter dp is 110 millimeter rotational speed of the barrel say at a nominal speed we have considered 235 radian per second density is 850 kg per meter cube and we have also considered this angle two angles are same and the delta that is the deviation of the both the side distribution those are 0. Now with these values first of all we have this curve we have plot that is following the equation this is ideal the geometric displacement. Now what we have taken the number with the same diameter we have taken number of piston is 9 piston we say diameter of the piston is same, but we have taken this radius dp in such a way that stroke length is varying a stroke length is different for 7 chamber 8 chamber and 9 chambers. So, that total flow that is swept volume remains same for a same inclination. So, that is why you see this if you take the average through this that will be same, but if you think about the ripple in case of 7 the blue is 7 chambers. Now if you go for 9 then this ripple is reduced than 7, but in between that if you go for 8 ripple will be more you can examine from this equation as well as from our the flow ripple analysis which we have done by the phasor analysis. It can be seen that from the analysis of the ideal case flow the pump with even number of pistons has larger pulse time period and also larger amplitude than the odd number counterpart. As we have discussed earlier with phasor analysis such ripples are undesirable due to the resulting large fluctuation and noise. Now in this analysis what we have considered we have considered the variation of area while we are calculating this flow pressure etcetera. Now these are the data shown and what we find the valve float area for the flow in out is varying in this way. So, this variation due to the we can say that silencing groove and other end there is no groove. So, it is a straight forward. So, this is a sample calculations we have just calculated what are the areas and as you find for when the full kidney port is coming on the main ports the full opening is there. So, this area is you can compare with the area of the barrel kidney port. So, these are the area and then we plotted the pressure. Now this is for the now this is this as at the valve entry and this is the exit. Now as you see that as this area just from the dead zone when this area started opening momentarily there is a pressure rise. That means when the piston is coming on the it is full and it is pressure in case of pump it is in the delivery side and it is in the dead zone there will be momentary rise of this pressure. This depends on the area it is opening or in other words I would say that depending of different port you will find that this pressure rise will be different. However, if we provide that the delta has some value you will find there will be more pressure fluctuation there. So, then the question is that why we provide that we will find that overall losses and overall performance will increase although there may have a spike. But this actually I would say this calculations is done on the basis that apparently when it has reached the dead zone the initial opening of the orifice for a small angle. We have taken a small angle and area is too small, but in reality this pressure rise may not be there this will be truncated. Now here I have described that why this pressure rise which I have already told. Now initially I have shown the what will be the flow rate, but if we consider the leakage if we consider the pressure fluctuation which all such considerations your actual ripple may come like this. Experimentally it is also shown that it is not a smooth curve there will be fluctuation like this. So, this is of course not the experimental on considering this curve is drawn on the basis of considering the leakages considering the pressure rise considering the change in area when the port is opening and closing and this will be the theoretical curve. This may not match with experimental curve because in experiment while we are conducting the experiments there are some more unknown factors which may not be considered in theoretical analysis. Now pressure in transient region in this case to analyze the pressure what will be the pressure variation while a piston moving from 0 degree to 360 degree that we will analyze first. Then that pressure is considered for the further analysis of the flow pressure etcetera. Now in this case we have considered this V angle is 90 degree that means if we consider this group silencing group this is the we have considered the V shaped that means this is ramped in this direction as well as ramp in this directions. Now this angle is 90 degree that means this angle is considered 90 degree. Now for that if you look into this geometry then this altitude will become this angle. Now what the orifice area and its relation to the angular position theta is given by this angle where in this case again we have considered the theta angle with respect to the valve rotation. So this angle is considered as a total angle theta minus theta 1. So this means that from a reference point the position of this tip is theta 1 and our total rotation is theta. So when we know this theta and we know this positions we will use this angle whereas this angle is the angle of inclination which is not shown. Angle of inclination that means this angle with respect to this surface this angle is gamma. Now therefore this area is can be calculated simply using this equation. So this is the height and as it is 90 degree this is the base half base if we consider this side. So this is the area of one side and multiplied by the by 2 will give the total area. Now look at this in this case we are considering this area as the orifice not the surface area in case of this configuration of the silencing group we have to calculate in this way. Now the area is in this case can easily be calculated we do not consider the area of the valve barrel port. So now we substitute this into the earlier equation this was the earlier equation what we used. So to find out the flow q 1 1 here is the gamma angle is shown this is the gamma angle that means this angle is the gamma. Now let us calculate the force we will now consider the force plate torque. Now we consider here is the pressure. So we consider a force F p along the pistons this is simply the pressure into the area of the piston. F p is the cross sectional area of the piston. Now what will be this force reaction force offered by this first plate this will be definitely F pn divided by cos alpha pn is the n is the nth piston. So for the nth piston this will be the force. When we are calculating the force of a pistons then we consider that particular position of the piston and the pressure the instantaneous pressure at the piston. So this is simply why this is we have divided by the cos alpha because force acting in this directions. So definitely in this direction the force is opposite to this to generate I mean to offer that much resistance in this directions the inclined plate plate has to offer this much force which is the which we have to find by dividing by cos alpha this is from the simple geometry and force analysis. Now so this already I have described. So this much force is acting over there then torque arm this is the torque arm that means we are calculating the torque about this point. So we have to consider this force and this arm length to find out the torque about this point. Now here one interesting point is that we have considered that this pivot point of this first plate not intersecting the axis of the barrel shaft or axis of the input shaft in case of pump output shaft in case of hydraulic motor. Now this eccentricity if the if it is asked why this eccentricity is provided again I would say which is very small eccentricity it has been found that pump performance or motor performance improve for some cases not all cases due to some range of the operations this eccentricity is beneficial that is why here the eccentricity is shown while we are calculating this arm we have considered the eccentricity but in our final calculation we have neglected that that means we have considered E is equal to 0. So RS can easily be calculated this is R sin theta if this is R then sin component this is R. So this is sin component of that then A this is the length of the actually I would say that here although so it is shown no it is right we have to consider this length from this pivot from the centre of this ball joint this is the separate plate. So this is a ball joint from here to the pivot point this length is A. So we have to take this component also then this length E divided by cos alpha is this arm we consider this length we consider this length and then this length divided by cos of source plate ventilation give us the torque arm. So we combine the earlier equations and these equations and we finally get this is the T i in this case again we have considered ith chamber not the n was for the force analysis sorry this is a mistake we would have considered they are also ith chambers. So this is only for one pistons one piston this will be the torque now similarly we have to consider all the pistons and here instead of this force we directly we are considering the instantaneous pressure and the area of the pistons. So this is in more general form now in general the pivot axis intersects the barrel that is the shaft axis and E is equal to 0. However in some cases the small amount of eccentricity for overall performance. Now next phase we have to consider the pressure distribution at different zone system pressure P 2 along the discharge port of the valve plate this is P 2 is the system pressure. So this is the P 2 region. So when it this is the delivery side this is the delivery side that means force plate is inclined in this way this is coming out of the screen this is going inside the in so that while the piston is moving from this point to this point it is gradually it is being compressed. So this is the high pressure zone region then P 1 along the suction port. So this is the low pressure zone next the transition region between the discharge P 2 and the suction P 1 in case of pump the pressure at region will reduce from P 2 to P 1 and is termed as P C 1. So this is the P C 1 region only this region then next phase we consider that this is the P C 2 region where the pressure is gradually increasing. So in equation while we are calculating the pressure each for each and every piston when they passing through this region we have to carefully consider that pressure that means while we are calculating for a particular pistons we have to consider what are the pressure there. Some pistons have a P P 1 some piston is having P 2 some piston having P C 1 some piston is having P C 2. So while we are calculating the swash plate torque all for all pistons we have to calculate separately and they we have to sum up them. So therefore precisely at various values of theta 1 the acting pressure P 1 will have different values as shown like this we can now we can divide into different zones and we can write it down what are the pressures over there what I have described here this is written in the equation from here. Then in addition to the pressure forces transmitted by the pistons to the swash plate this swash plate exerts inertia force on pistons causing them to reciprocate. Do you understand my point say this pressure force are there as well as there is a inertia force for the pistons. So that is to be considered for the particularly for the transient regions while we are thinking the control of swash plate torque by the what are the actuation system we have to control this torque to keep the swash plate in a position or to move the swash plate from one position to other positions. So we consider this force first of all we calculate the longitudinal displacement of the pistons ith pistons then this you can calculate look at this it is rotating like this. So first of all we will consider say this is the theta angle we will consider this is r and then this is the sin theta then this is inclined. So, we consider the tan component of that. So, this becomes we are calculating while we are calculating the swash plate torque we have to calculate this height perpendicular height this becomes first this value and then with the tan alpha we get the real vertical this torque arm positions from this pivot point. Look at this we have here also neglected the e there is no eccentricity. Now the velocity of the ith pistons then if we differentiate with respect to time it gives this value and as well the accelerations we find we can calculate further differentiating with respect to the time right. So, therefore, the inertia force of the ith pistons is expressed by this equation this is the acceleration term and this is the mass of the pistons. Now adding this force component to the expression the earlier expression the combined torque that is swash plate control of the ith piston is given by this expression it is acting over here this is only for ith piston. Now this reciprocating group of the pump includes a holding plate which holds the piston shows against the swash plate. Now actually here say if we consider the how this pistons are moving forward motion is possible because this is rubbing on that, but there is no physical connection between these two only this is the if the compression force is there then it will move. So, while it is moving in the other directions how this will be pulled out actually there is a plate which is sometimes called retainer plate that retainer plate is held by a spring here against the barrel. So, this always keep these seeper pad in touch in contact with the swash plate. Now we must consider the dynamics or the inertia of this retainer plate also. Now while we are considering this accounting for the inertial of the plate related to the swash plate pivot adds another torque the ith piston component expressed as follows. This is equal to we can that moment of inertia and r omega square t alpha the derivation is not shown, but as if we have considered the mass of that and this is acting at a if we consider that this is the polar moment of inertia, this is radius of gyration and omega square tan alpha will give us the that torque. Now we add this torque with also the original torque what we have calculated. So, now the summation of this that is the for each piston inertia plus the pressure force torque due to this part plus the torque due to the retainer plate. So, this is precisely the equation for the torque of controlling or holding the swash plate in a position or to rotate the swash plate at an instant. So, these are the reference I suggest that you should read this paper one particularly one and then four, but if you would like to know the details about the dynamics you should also follow this two paper if you look into this this was published may be in the same year, but in other issues I make a both no this is this was published in ASME this was in other journal, but first they found out this a nominal torque and then they did the dynamic analysis whereas to know the kinematics details you should read this paper. Thank you.