 Welcome to today's lecture, this is analysis of 3-way spool and flappern nozzle valves. This means that we shall consider 3-way spool valve as well as 3-way flappern nozzle valves. In last lectures, we have already learned about the 4-way critical centre valves. We have learned general design analysis of 4-way valves which is mostly applicable for servo valves. Now the figure 10.361, this is figure 1, it shows the 3-way spool valve and this must be used with an unequal area piston to provide directional reversal. Now what is 3-way valve? We know that what is 4-way valve, but 4-way valve normally which are used for the motion control that is we control the velocity, we control the position, we control also pressure there. In that case normally the direction reversal are not frequent or in other words, although directional reversal is done, the scope is there because it is a 4-way valve, but it is usually connected to rotary actuators and normally we do not go for reversal frequently. On the other hand, if we would like to go for reversal, in that case 3-way with the load cylinder, this arrangement is the best. Now again this valve may be critical centre or may be a open centre. Closed centre valve means overlap valves are not desired because of the longer bandwidth at the middle. However, critical centre valves are mostly preferred, although in true sense it is very difficult to maintain that condition, that critical valve, critical centre valve positions, usually it requires frequent null adjustment, null adjustment. However, we shall discussed about that, we will look into the critical centre 3-way valve. Now how it is working? If we look into this, this is the supply. Now instead of 4-way valve, in case of 3-vulve, supply is one to the valve, one end and other end directly to the rod end side of the pistons. So, if we keep this is closed, then this will simply move upward directions. Now let us consider, if we keep it open, then in that case what will happen? The flow is going like this and the flow is also coming in this way and if you think of the regenerative principle and you will find that this will move in this directions and as well as some flow is going out. This is for the control purpose, this is the insurbo valve, it always happens. Now the rod and head side, we call this is head side and this is rod side. Now if we look into this, this area is designated by AH and this area is designated by AR. Now the rod and head side areas of the pistons should be such that a steady state control pressure PCO is the control pressure of about half of the system pressure PS. Now this PCO or simply PC, we need half of that for the better control. This means that we can express this control pressure is equal to PS that that is the system pressure by 2 that is desired. Now for that what we do ideally it happens at area ratio half, what does it mean that the head area of piston AH is double the rod end area AR that means here the AR is the area is the annulus area, the head area means the full area. So, full area should be double of that annular area that can be done there is no difficult to do such things, but you should not confuse here that this does not mean the area sorry the diameter of the rod is half of the piston area, it is not the area is half you have to keep in mind. So, usually you will find that what usually you will find that this is also looking very thick because this more than half of that I mean this diameter is more than half of the piston area. Anyway this area ratio is maintained half ideally or in other words the rod area is half of the cylinder area this is just we can express in the formula like this AH is equal to 2 AR. These valves are made critical centre for better response in servo control, do you understand my point. I told that maintaining the this the critical centre is difficult this means that to make such valve critical centre both this groove dimension of the groove not only the width of the groove, but also the distance between these two groove should match with the distance of this pull. And in fact, I would say matching these two dimensions into different components is difficult and it is normally in practice if it is used for pull sorry the servo control valve in that case this pull and this this is called sleeve actually you will find in a body there is a separate sleeve is used for this outer member. In that case these are made match parts that means say 10 set of spool and 10 set of outer members are taken together and then each and every spool which is matched with the outer member and then it is examined that really how much critical centre it is it it best pair is taken from one valve then perhaps next one. In that way it may happen in a 10 sets of these two components this pairs five components may be rejected also. This is one thing second thing when it is in under operations then this adjustment with because this is driven by some actuator this might be electrical solenoid. Now, adjustment with this solenoid positions and this spool positions because this solenoid is not directly coupled it is it will be directly coupled but not a single component this are the solenoids spool and this spool is not directly not an integral part they are just connected. So, sometimes this it deviates from this critical centre positions and that can be adjusted of course. Now, it is found that critical centre because as I told this is for reversal operations that means it is some operations it is doing forward and backward. So, if there is a if it is a overlap valve then there will be difficult for position controlling or maybe this reversal time controlling. So, in that way critical centre valve is better. Now, for that critical centre valve if you recall for earlier analysis general analysis for this spool valve then this can be written as the load flow can be written in the form of coefficient of discharge into a 1 is the area of this or if is and then 2 divided by this is the density of the oil and then P 1 minus P 2 here is the P 2 and here is the P 1 remember P 2 is not equal to P 0 P 2 is the pressure here and this P 1 is the pressure here. So, we can further write this equation that C d into W x v x v is the spool stroke it is called stroke the stroke is x v. Now, W is what is W? W is the peripheral diameter sorry periphery of the this you can say this spool land this diameter into 2 pi sorry pi into this diameter will give W. Now, this means that this is we are considering the rectangular port. Now, W is written as if you remember in general term sometimes instead of this continuous this group is continuous instead of this continuous group some rectangular holes are provided. If this is off with a rectangular port then rectangular holes say 4, 5, 6 holes over the periphery and that are usually spaced in that case also we can use this formula is equal to W because that W in that case width of that each groove each hole through the slabs plus the number of such grooves. Anyway this into the stroke length we will give the total area open there that is the orifice area and so this is the equation we can write for the load flow. But when this is in the opposite direction that is that means it is moving in the from downward directions in that case this equation simply written as we will consider the A 2 and this is again we will find that it is in the opposite directions and in that case only there will be the control pressure that means we are getting only control pressure over there. Now we can plot that what equations we have written there we can plot to study such a valve performance of a valve better to plot such graph and then we can understand that how it is performing. Now it is same as in case of 4 wave valve except the change in accessor scale if you remember in what we have already studied the 4 wave valve in that case you will find in this axis P c by P s is equal to 0 here and then it is varying in this directions negative and varying in this directions positive that is for the from left hand side to the right hand side of this spool in that case we simply use this value in this axis. Now here this is written in the dimensionless form this is also in the dimensionless form and we can observe these are the for different x v by x maximum. So, this ratio so all are in dimensionless form and what we find we should call this is the null positions this is the null position. Now at null q l is equal to 0 x v is equal also 0 no here it is the x v is 0 is here not that this is in the 4 wave valve you will find this is the 0 position whereas, this is the 0 position for flow and as well as for the no we have considered x v is equal to 0 that means this is the 0. So, this might be the curve at null positions. Now there what we find the control pressure p c is p s by 2 because we have taken the area ratio 2. So, at that condition p s by p c by p s is equal to 2 that means p c by p s is equal to 0.5 this means here what we consider that q l is equal to 0 and our control pressure is here at null position. Therefore, the valve coefficients at null positions of three way critical centre spool valve can be written as from this if you have attended the earlier classes then you can find that q l by del q l by del x v not theta l this is del q l this is del q l by this. So, this is flow gain that means how much gain will be there at null positions this depends on that critical centre or open centre and this ultimately can be written as from this condition in this form. Next we will look into the flow pressure coefficient. The flow pressure coefficient is written in this form this is w this also will be w not omega. Now the pressure sensitivity can be written in this form. Now what we find that at null positions due to looking into this curve this will be 0 whereas the pressure sensitivity which is actually this divided by this that will become infinity because this will have some value whereas this will become infinity which means pressure sensitivity at the null positions is infinity means highly sensitive at the null position. Now the flow gain is same if you look into the four way valve flow gain what we will find in this three way valve is also flow gain is same, but pressure sensitivity is half that of four way critical centre valve. This is from the equations you have to compare the equations of those I have not shown here and the equations of three way valve. Therefore the error to overcome loads will be more. Now this we should understand that as the sensitivity is half of the four way critical valve then the error will be more in case of three way valve. This means that again although the three way valve is simpler in construction and it is very good for using selecting such valve in case of the reversal operations that is for the mostly for the linear actuator, but what is there the error will be more that means to rectify such error we have need much stronger better control controller than the four way valve. It also can be shown that the dynamic load errors are almost double this due to this factor that limits the application of three way servo valve. So, three way servo valve you will find normally for the of course I would say the machine tools and at many places these are used, but if we look into the time response that means we need very quick actions. Suppose we are trying to position control very accurately and with almost no time real time control in that case possibly we have to go for four way valve even if for if there are frequent reversal operations. Now hydromechanical servos where greater errors are tolerable the source positioning is mechanical and quite steep compared with the force loads imposed by the spool. For this reason in case of such a system three way valve the force equation described in the spool motion is usually important and flow forces are not of interest. Usually what we have seen that flow force has significant roles on the controlling of the spool in servo mechanism. Now in that case the argument is that the as these are we are tolerating the errors and the source positioning is mechanical quite steep compared with the force loads. So, in that case we can go for three way valve where the flow forces not that significant. Now this is I have given the idea of three way spool valve. Now similarly we have to similarly there are three way flappen nozzle valve also. In flappen nozzle valve we will see some advantage over the spool valve. First of all it is less expensive, but mostly allowing higher leakage losses such valves are of low cost and less sensitive to dirt. This means that usually such orifice and you will find this flappen is moving thereby this flappen area is being controlled the dirt can easily go out. So therefore it is better than the three way spool valve. However always there will be leakage loss and that we should take care of and obviously the cost is because making of a spool is will be more expensive than such flappen valve. Flappen driven by torque motor this is usually you will find that a torque motor will be used there. What is torque motor? We know this motor principles in that case armature is such that with this armature it will generate some torque with the current. If we increase the current then the more torque will be there if we decrease the current. So there will be less torque. This is very common in the pilot stage of two flow control valve including servo valves. Now if you look into the four way servo control valve you will find in the first stage there is two stage actually because to drive the mains spool we need much force and sometimes what happens that there is a first stage valve which through which the flow hydraulic oil goes to control the mains spool. In that case you will find this first stage usually made of flappen nozzle. But here I would like to mention we will see that that flappen nozzle is normally it is by double flappen that means on the other side also there will be such nozzle and flow is coming this way and from the other way. But this is what I have shown here this is a single one and we will our basic analysis will be done only on the single one. Here also we have taken this cylinder with the same 1 is to 2 ratio. The pressure flow curves have better linearity. Also performance of these devices are quite predictable and dependable. Now we shall consider the analysis of a single jet flappen valve. It is also called sometimes the three way flappen valve because of this principle the same we have seen the same principle is used for the spool valve also. Now the flow through a fixed orifice of area A 0 in the upstream is modulated by moving the flappen to generate a desired control pressure P C. Now here we will find that a fixed orifice A 0 is provided. Now flow is going through this. Now this flow can further be controlled by moving this flappen. If we move this one this will be closed you will find that pressure will increase then due to this increase in pressure here. Here the system pressure is remain same there will be less amount of flow and this will be controlled. Whereas in this direction there is no orifice. So upstream means this we are calling the upstream flow. The curtain area is more important than the flappen nozzle orifice area. Now this what is called curtain area? Now this we would call this is the nozzle area and this area which is closing this and this is called curtain area. So this we have to know one is called curtain area and another is called simply orifice area. Orifice area is this one and curtain area is what I would say that if I consider the this periphery over here into this depth. So, this is initially X F O this height into the periphery that means pi into diameter here into this that area is important than this orifice area. The working range of the control pressure is estimated by obtaining the block load characteristics. Expression of mathematical model are shown in next slides and also we look into this block load characteristics. Now from the flow equations what we know that q 1 will be q 2 plus load flow. That means if we consider that this flow is q 1 then that must be equal to one load is going this way and another load is coming another flow sorry the flow is being divided into a part is for the moving the load a part from the from the just exert. Now just controlling this one we can control this flow also. So speed control is also done by controlling this flapper. Flows are expressed following the orifice equations as follows. Now the orifice equation is same only thing in this case what we consider that a 0 is the area this is pi by 4 d o d o is the diameter here. Now this is the system pressure and this is the control pressure over here. So this is the flow q 1 total flow. Now d 0 is the diameter of this orifice sorry a 0 is the orifice here. So we are writing the equations for this orifice which is simply this is obviously we are writing the equation we have considered a circular hole and normally it will be circular hole. There is no reason making of other shapes neither rectangular nor elliptical. So simply a drill hole you can consider. Now q 2 this flow is equal to a f area of we should call this is the cut end area and that becomes pi d n x f 0 minus x f that is the cut end height and c d f. The d n diameter is actually this diameter this is may be drawn in a wide way. And d n is the diameter here not this diameter diameter of this orifice. So d n. So pi d n this periphery into the current cut end height will give us the orifice area here. So q 2 is expressed in this way and here the p c is the pressure in the upstream and in the downstream is the exhaust pressure which is normally 0. That is why this is here we find two pressure term where only one single pressure term. Now x f 0 is the initial gap between the flapper surface and the nozzle when the flapper is in equilibrium. Now look at that we should should take care of the direction of s f x f. x f is the movement of the flapper which is positive in the upward directions. So this means when it is moving in the upward directions this cut end height is being reduced because this is positive and here we have minus sign. But suppose it is moving in the opposite direction do not forget to put x f itself is minus. So total cut end height will increase and total area will also increase. x f is the flapper displacement which is positive towards the nozzle. So, when there is no load there is no load flow that means it is no load flow then q 1 is equal to q 2 or in other words if there is this is not moving that means we want that it should not move, but still this is open then hold the well coming in that should go out. In that condition q 1 is equal to q 2 and p c by p s if we write in that way this will become this will be expressed in this form. The plot of above equation that is the block load characteristic it is called block load characteristics this can be presented in this form. So, if this is the this ratio is varying from 0 to 2.8. Now how it can be 0? The A f itself is 0 this area is equal to 0 how this means that this is completely closed that means x f in positive direction is equal to x f 0. In that conditions what we will find that p c is equal to p s this is 1. Now if we this extreme case what we have considered 2.8 that in that case of course this is the ratio remember see here to make this is 0 we can consider A f itself 0, but this ratio when this is this has some value then this ratio depends on the coefficient of discharge area etcetera. However what we find this is of course for a particular valve what we find that 2.8 of this ratio that p c by p s is 0.1 only that means control pressure is 10 times less than the system pressure. So, this diagram is useful for to find out the operating conditions. Now looking into the graph it is apparent that p c by p s is equal to 0.6 is a good value for stability point of view. Now how why we say this value is from stability point of view this is because of the reasons that if we move in this directions then definitely we will find that control pressure is much much lower than the system pressure and in that conditions we do not have much control over this load. So, what we find that possibly if we be here then this is of course this is simply a just a common maybe this was found by the analysis that at that conditions the operation will be the best. Whereas if we enter in this zone say this is 0.6 if this is 0.8 perhaps this is 0.6 or this is 0.4 this one must be 0.6 p c is equal to 0.6 is this one here it is 0.8 this is considered as a stable conditions stability point of view. If it is in this side what we will find that controllability will be difficult because it needs more accuracy whereas in that case controllability will be difficult due to the fact that the control pressure is too low. Considering this as a criteria for design the orifice ratio at the null point is derived as now this value we would like to put 1 because this pressure we want half p c by p s is half. So, that means this we would like to put 1. So, this ultimately derived into in this form and then it can be shown that it is good for it is a good objective for both 3 way and 4 way means double jet. Remember when there is another nozzle in the opposite directions which we call the double jet this will be the 4 way flapper valves. This criteria also depicts that the system with the piston and 3 way valve needs the working piston of area ratio 1 is to 2 for best performance and optimum drift. Now, this about this drift we should go to the to know more about this drift we can consult the reference to which I will show later. But what is drift? Drift means that while we are using such 3 valve 3 way valves then we are moving the load both in plus minus directions. So, this means that it is operating something and in most of the cases you may find this is operating a another spool. Now, what is drift? You will find that when this is at the central positions this spool this piston cylinder what we have called, but this is basically controlling a spool that should be in such a position that that spool is also in the null positions, but due to the leakage and error what it happens when this is in the mid positions this may be what it is controlling that is in not in the null positions. So, this null position and that null positions may not match. So, that is called drift. Now, there is also technique that we can control the drift also, but this is important. However, we are not analyzing the drift this to know about this drift we should consult these references. The equation for pressure flow curves can be derived by equation 8 and substitute equation 9 and into 10 and then these applications this equations become like this. So, this is for the pressure flow curve we derived this equations you just substitute this and you can arrived into these equations. Now, similarly substituting 12 and 13 we get these equations. Now, we can perhaps if we look into this equation then this is plotted here in this curve and this plot of pressure flow characteristics finally, expressed in 14 and this is presented here. Now, if we look into this in this case also this is same as the spool valve. So, abscissa is same as it is different from the four wave valve, but here only we have considered only one side of this curve. In that case obviously, this flow is this is 0 because the load flow can take in both the directions and what we find that X f by X f 0 is here. So, this is basically the null positions no sorry this is coming over here this is the null positions where as for the movement of X f in both the directions we get different curve. So, from there we can find out the valve coefficients. Now, if we differentiate 14 and evaluate at X f is equal to Q l is equal to 0 that is for the null positions and P c by P s is equal to 2 sorry P c by P s is equal to 1 by 2 then we get the valve coefficient at null positions and these are we can find we can express in this form and null flow pressure coefficients is expressed by this as well the pressure sensitivity is expressed by these equations. Now, this is important to know the null for each and every valve for 3, 4, a. So, null coefficients are important to understand how better is the valve in performance. The leakage or center flow at null is given by these equations. Now, here in it is important although it is a center sorry this is a critical center valve, but still there will be some flow some leakage flow if you we have earlier discussed about the central flow. The central flow is important for a valve because that affects the flow gain, but this leakage can be expressed in this form. Obviously, in case of flappen nozzle there will be leakage because we should not say the critical center flappen nozzle valve because these two nozzles by no means we will sit on this flapper because there will be some gap this means that there will be leakage. So, knowing this central flow is important for this flapper valve. Now, the striking force by the flow from the nozzle can be derived as in this form we shall also discuss in the next lecture. Now, what is striking force? The striking force means this when the flow is coming on then this is striking on this flapper which is ultimately required to estimate how much torque is required. So, this f1 is estimated by this formula. With very small gap if we make this gap is very small that is in the null positions or maybe in the even in the operations while we are moving this xf in this directions this gap is being reduced. Then for very small gap what we will find f1 is equal to p c into a n. What is a n? a n is the simply this area not the curtain area. It is observed that with the increase in xf0 xf minus xf that is that that gap the force f1 increases due to the z force and may become as large as twice p c into a n. Now, look into this normally if the gap is small this force is p c into a n whereas if this gap is increased this force become double. So, all in other words I would say this we are equating here, but in other words I would say this flapper valve are designed in such a way you will find that with small gap f1 is equal to this and with the maximum displacement of or the maximum with the maximum gap with the maximum increase in this this force become double or in other words within this range we design the torque motor. So, torque motor is selected in such a way that there will be twice the torque from the minimum torque motor required to operate this valve. However, in practice that xf0 by dn dn is the diameter here is usually 1 by 16. So, suppose if dn is equal to say what I would say 1.6 millimeter diameter of the or if this nozzle is 1.6 millimeter in that case xf0 is only 0.1 millimeter 0.1 millimeter 0.1 millimeter you can imagine is very small. So, and again we are giving the motion of that. So, this motion is very small and very precision and f1 remains close to pc into an. Differentiating equation 19 at a null point we get this expression df1 by dx0 is this much then this is equivalent to the spring coefficient of a fluid spring. If you compare this you can if you consider the mechanical spring this is as if a fluid spring with this much of stiffness, but it is a negative spring what does it mean what is negative spring. So, means normally with the increase in when this is the gradually when it is being compressed mechanical spring the load is increasing here it is opposite. This cause a destabilization due to this negative nature a destabilization is observed. However, the effect is small and flapper drive is made sufficiently robust to handle this situation. This means that while we are designing this torque motor we have to consider this part and it should be robust so that it can handle such destabilization. For balancing providing a small piston or spring on the opposite side is common. Sometimes a spring is used in the opposite size that is for the flapper nozzle what happens to if for the if there is double nozzle in that case you will usually find a springs are used here in both directions. In case of single nozzle you can use this spring directly opposite otherwise you can use also a coil spring for both directions. In double jet 4 way flapper valve balancing is done by opposite jet in this means that in case of opposite jet if there is another nozzle in the opposite direction sometimes we do not need any spring at all because this can be balanced by the opposite one. Now, we shall continue our lecture in the we shall discuss more about this flapper nozzle valve in the next lecture. For this lecture the mostly I have taken from the this merits book which is hydraulic control systems also particularly to know about the drift this paper is important. So, you can go through this paper and there are some general idea from the blackburn book of blackburn and refer and say thank you for listening.