Transcript
Noah D. Manring Mechanical and Aerospace Engineering,University of Missouri—Columbia,Columbia, MO 65211 The Discharge Flow Ripple of anAxial-Piston Swash-Plate TypeHydrostatic Pump This research examines the idealized and actual flow-ripple of an axial-piston swash- plate type hydrostatic pump. For the idealized case, a ‘‘perfect’’ pump is examined inwhich the leakage is considered to be zero and the fluid is considered to be incompress-ible. Based upon these assumptions, closed-form expressions which describe the charac-teristics of the idealized flow-ripple are derived. Both the ripple height and the pulse frequency of the ripple are described for a pump with an even and an odd number of pistons. Next, the actual flow-ripple of the pump is examined by considering the pumpleakage and the fluid compressibility and for computing these results a numerical pro-gram is used. For both the idealized case and the actual case a comparison is madebetween a nine-piston, an eight-piston, and a seven-piston pump. From the idealized analysis it is quantitatively shown that the eight-piston design is less attractive than thenine or seven-piston design; however, the analysis of the actual pump flow reveals that the qualitative difference between all three designs may not be too significant. From a flow ripple point of view, the numerical results of this research show that a pump de-signed with an even number of pistons may be as feasible as one that is designed with anodd number of pistons. This is an unexpected conclusion. S0022-0434 00 00202-1 Introduction Background. Axial-piston swash-plate type hydrostaticpumps are used as the input power source for hydraulic circuitry.As the name suggests, these machines are comprised of a discretenumber of pistons that reciprocate in a sinusoidal fashion for thepurposes of displacing fluid. Due to the discrete nature of thedesign, the flow output of an axial-piston pump is not perfectlysmooth and tends to maintain some of the sinusoidal characteris-tics of the fluid displacement elements themselves. These sinu-soidal characteristics are usually referred to as the flow ripple of the pump and are often suspected for generating undesirable vi-bration and noise 1 .Within the last thirty years, significant research on axial-pistonpumps has appeared in the literature. Most of this work has beenconcerned with the control of the swash-plate of variable displace-ment pumps and the operating efficiency of the machine; how-ever, very little of the published literature has addressed the fluc-tuating aspects of the discharge flow. Among the publishedliterature on the topic of flow ripple is the influential work doneby Thoma 2 in which a graphical technique for describing theidealized flow ripple of the pump is presented as a function of thenumber of pistons in the design. The results of this work havebeen used to discourage engineers from designing pumps with aneven number of pistons due to the theoretically higher magnitudeof the flow ripple that is shown for an even numbered pistonmachine. Though this work is interesting, it only concerns itself with an idealized case of flow and does not set forth the derivationof a closed form equation that would prove most useful for un-derstanding its main conclusions. In another published document,Jun and Yi 3 have presented experimental work that has com-pared the flow ripple of an eight-piston design with a nine-pistondesign and the conclusions of their study show that a machinewith an even number of pistons may be as feasible as a machinewith an odd number of pistons.In the industry, where axial-piston pumps are designed, a cer-tain amount of confusion exists on the part of most designersregarding the tradeoffs between designing a pump with an oddnumber of pistons and designing a pump with an even number of pistons. In the srcinal US patent which describes the first axial-piston pump of its kind 4 the design is shown to have six pis-tons. Though it is not exactly clear when the development of apump with an odd number of pistons began, it is certainly the casethat today almost no one considers the design alternative whichuses an even number of pistons. In fact, the majority of today’spumps are designed with a total of nine piston and those remain-ing outside of this group are predominantly designed with sevenpistons. Pump Description. Figure 1 shows the general configurationof an axial-piston swash-plate type hydrostatic pump. The pumpconsists of several pistons within a common cylindrical block.The pistons are nested in a circular array within the block at equalintervals about the x -axis. As shown in Fig. 1, the cylinder block is held tightly against a valve plate using the force of the com-pressed cylinder-block spring. A thin film of oil separates thevalve plate from the cylinder block which, under normal operatingconditions, forms a hydrodynamic bearing between the two parts.A ball-and-socket joint connects the base of each piston to a slip-per. The slippers themselves are kept in reasonable contact withthe swash plate by a retainer which is spring loaded using thecylinder block spring. A hydrostatic and hydrodynamic bearingsurface separates the slippers from the swash plate. For the pur-poses of this research the swash-plate angle, , is considered to befixed.While the valve plate is held in a fixed position, the cylinderblock is driven about the x -axis at a constant angular speed, .During this motion, each piston periodically passes over the dis-charge and intake ports on the valve plate. Furthermore, becausethe slippers are held against the inclined plane of the swash plate,the pistons undergo an oscillatory displacement in and out of thecylinder block. As the pistons pass over the intake port, the pistonwithdraws from the cylinder block and fluid is drawn into thepiston bore. As the pistons pass over the discharge port, the pistonadvances into the cylinder block and fluid is pushed out of the Contributed by the Dynamic Systems and Control Division for publication in theJ OURNAL OF D YNAMIC S YSTEMS , M EASUREMENT , AND C ONTROL . Manuscriptreceived by the Dynamic Systems and Control Division May 6, 1998. AssociateTechnical Editor: R. Chandran. Journal of Dynamic Systems, Measurement, and Control JUNE 2000, Vol. 122 Õ 263Copyright © 2000 by ASME piston bore. This motion repeats itself for each pump revolutionand the basic task of pumping fluid is then accomplished. Research Objectives. This research examines the idealizedand actual flow-ripple of an axial-piston swash-plate type hydro-static pump. For the idealized case, a perfect pump is examined inwhich the leakage is considered to be zero and the fluid is con-sidered to be incompressible. Based upon these assumptions,closed-form expressions which describes the characteristics of theidealized flow-ripple are derived. Both the ripple height and thepulse frequency of the ripple are described for a pump with aneven and an odd number of pistons. Next, the actual flow-ripple of the pump is examined by considering the pump leakage and thefluid compressibility and for computing these results a numericalprogram is used. For both the idealized case and the actual case acomparison is made between a nine-piston, an eight-piston, and aseven-piston pump and conclusions are drawn regarding the actualdifferences between the three designs. Idealized Pump Flow General. The idealized discharge-flow of the pump is deter-mined by considering a pump that does not leak while displacingincompressible fluid. In this case, it may be shown that the flowrate generated by a single piston is equal to the time rate-of-change of the instantaneous volume of the piston chamber. Math-ematically this is expressed Q n dV n dt , (1)where the flow out of the piston chamber is considered to bepositive. The instantaneous volume of the piston chamber is givenby V n V o A p r tan cos n , (2)where V o is the nominal volume of a piston chamber as if theswash-plate angle, , were zero, A p is the pressurized area of asingle piston, r is the piston pitch radius, and n locates the cir-cular position of the n th piston chamber relative to the negative z -axis. Substituting Eq. 2 into Eq. 1 yields the following resultfor the idealized instantaneous flow that is generated by the n thpiston: Q n A p r tan sin n . (3)In Eq. 3 the symbol describes the rotational speed of thepump. The total discharge flow of the pump is equal to the netflow from each piston chamber that is instantaneously positionedover the discharge port. The total discharge flow may then beexpressed as Q A p r tan n 1 n sin n , (4)where n is the total number of pistons that are instantaneouslypositioned over the discharge port of the pump. For a pump withan even number of pistons, this integer quantity is expressed n N e 2 , (5)where N e is the total number of even pistons within the pump. Fora pump with an odd number of pistons, this quantity is expressed n N o 12 , (6)where N o is the total number of odd pistons within the pump andthe 1 indicates that the number of piston positioned over thedischarge port is repeatedly fluctuating and depends upon the ro-tational position of the pump itself. Using the integral average of Eq. 4 , the nominal discharge flow of the pump may be expressedas Q ¯ NA p r tan 2 0 sin d NA p r tan . (7)Dividing the result of Eq. 4 by the result of Eq. 7 yields anormalized discharge flow from the pump which is given by Qˆ N n 1 n sin n . (8) Series Convergence. Since the pistons are spaced evenly in acircular array about the centerline of the pump shaft, the circularposition of the n th piston may be expressed n 1 2 N n 1 , (9)where 1 locates the position of the number 1 reference piston inthe machine and N is the even or odd total number of pistons.Using Eq. 9 it may be shown that n 1 n sin n sin n N csc N sin 1 n 1 N . (10)For a pump with an even number of pistons, Eqs. 5 , 8 and 10 may be used to express the instantaneous flow rate of the pump as Qˆ e N e csc N e cos 1 N e , (11)for values of 1 between 0 and 2 / N e . For a pump with an oddnumber of pistons, Eqs. 6 , 8 , and 10 may be used to expressthe instantaneous flow rate of the pump as Qˆ o 2 N o csc 2 N o cos 1 2 N o , (12)for values of 1 between 0 and / N o . The Shape of the Flow Pulse. Equations 11 and 12 de-scribe a single pulse of the discharge flow ripple for a pump withan even or odd number of pistons respectively. This pulse is cy-cloidal in shape and repeats itself continuously as the pump shaftrotates. Figure 2 illustrates the shape of a single flow pulse as itvaries with . In this figure, Qˆ shows the normalized height of the flow pulse and T shows the period of the pulse width. Usingthe result of Eq. 11 , it may be shown that the normalized heightof the flow pulse for a pump with an even number of pistons, isgiven by Qˆ e N e tan 2 N e . (13)The period of the pulse width for an even numbered piston pumpis Fig. 1 General pump configuration 264 Õ Vol. 122, JUNE 2000 Transactions of the ASME T e 2 N e . (14)Using the result of Eq. 12 , it may be shown that the normalizedheight of the flow pulse for a pump with an odd number of pis-tons, is given by Qˆ o 2 N o tan 4 N o . (15)The period of the pulse width for an odd numbered piston pump is T o N o . (16) Actual Pump Flow General. The previous discussion of pump flow has beenbased upon the idealized assumptions that the pump does not leak and that the fluid is incompressible. Since neither of these as-sumptions are actually true in practice, it is worth examining themore complicated problem in which leakage occurs and the fluidcompresses due to the pressure which is instantaneously applied.Figure 3 shows a schematic of the piston chamber from which thefluid is being discharged. In the analysis that follows, it will beassumed that the discharge flow out of the n th piston chamber ischaracterized by a high Reynolds number and that the classicalorifice equation which is derived based upon the Bernoulli equa-tion may be used to model the flow. This result is given by Q n sign P n P d C d A o n 2 P n P d , (17)where P n is the instantaneous fluid pressure within the n th pistonchamber, P d is the discharge pressure of the pump, C d is thedischarge coefficient, A o n is the instantaneous cross-sectional areaof the discharge flow, and is the fluid density. See Fig. 3. Thetotal discharge flow of the pump is equal to the net flow generatedfrom each piston chamber instantaneously positioned over the dis-charge port. Dividing this sum total by the nominal result of Eq. 7 yields the following result for the normalized discharge flowof the pump: Qˆ NA p r tan n 1 n sign P n P d C d A o n 2 P n P d ,(18)where n is given in Eqs. 5 and 6 for an even and odd numberof pistons respectively. Instantaneous Pressure. The instantaneous fluid pressurewithin the n th piston chamber is described by the standard pres-sure rise rate equation which is derived based upon the conserva-tion of mass within the chamber and the definition of the fluidbulk modulus. This result is given by dP n dt V n Q n Q leak n dV n dt , (19)where P n is the instantaneous fluid pressure within the chamber, is the fluid bulk modulus, Q n is the discharge flow given in Eq. 17 , Q leak n is the leakage that occurs due to the clearances be-tween the piston and bore and/or any other leak paths that mayexist in the design of the piston chamber, and V n is the instanta-neous volume of the chamber given in Eq. 2 . If it is assumedthat the leakage out of the piston chamber occurs at a low Rey-nolds number, then the leakage may be modeled as a laminar flowgiven by Q leak n K P n , (20)where K is the leakage coefficient. Substituting the results of Eqs. 2 , 17 , and 20 into Eq. 19 yields the following result for thepressure within the n th piston chamber: dP n d n sign P n P d C d A o n 2 P n P d K P n A p r tan sin n V o A p r tan cos n . (21)Note: in Eq. 21 it has been recognized that d n dt for aconstant angular velocity, . Equation 21 must be solved nu-merically for the instantaneous position of the n th piston and theresult of P n must be substituted into Eq. 18 to yield the instan-taneous discharge flow of the pump.At this point, it may be worthwhile to observe some of themodeling inconsistencies of Eq. 21 . The two primary equationsthat have been combined to get Eq. 21 have been Eqs. 17 and 19 . As previously mentioned, Eq. 17 is based upon Bernoulliprinciples which assume that the fluid is incompressible and thatthe flow is steady. On the other hand, Eq. 19 is generally derivedfor a fully compressible and time-varying situation. When thesetwo equations are combined, at best it may be said that the finalresult of Eq. 21 is only valid for slightly compressible flows.Furthermore, since the Bernoulli equation is steady, it would beunreasonable to expect Eq. 21 to produce results that wouldinclude the effects of water-hammer. Justification for using Eq. 21 is primarily found in the mathematical expediency it provides Fig. 2 Flow-pulse shapeFig. 3 Schematic of a single piston chamber Journal of Dynamic Systems, Measurement, and Control JUNE 2000, Vol. 122 Õ 265 and the fact that it has been used conventionally for fluid-powerresearch in the past. No one would object if a better model werepresented; however, to date, Eq. 21 is the best model that wehave for the research that is typical of this paper.Figure 4 shows a typical numerical result of Eq. 21 as n goes from /2 to 5 /2 one pump revolution . Note: both pres-sure, P n , and port area, A o n , are plotted in this figure. As shownin Fig. 4, the typical numerical solution to Eq. 21 demonstratesrather uninteresting behavior for the pressure, P n . As the pistonbore passes over either the intake port or the discharge port of thevalve plate, the port area, A o n , remains at a maximum constant.Within these regions, the pressure within the n th piston-bore alsoappears to remain fairly constant i.e., P n P t or P d . The twoports on the valve plate are bridged by transition regions where A o n goes from a maximum value to a minimum value, slowlygrows within the transition slot, and then quickly returns to thesrcinal maximum value. As the n th piston-bore passes over thetransition regions, the pressure changes almost linearly from oneport pressure to the other. Actual measured-data has shown thesesame basic trends 1 .Figure 5 shows another result of this study where the pressuredrop between ports has been reduced. Figure 5 represents a runusing one sixth of the discharge pressure of Fig. 4. From Fig. 5 itcan be seen that a lower pressure drop between ports tends tocreate pressure spikes within the transition regions of the valveplate. This phenomenon is strictly a result of the volumetric com-pression and expansion within the chamber. In the first case, thechamber volume decreases at a rate faster than the fluid cansqueeze out through the port. If the boundary pressure is not suf-ficiently large compared to the starting pressure, volumetric com-pression of the fluid will cause the pressure within the piston boreto overshoot the approaching boundary condition. In the secondcase, the chamber volume increases at a rate faster than the fluidcan enter the piston bore. If the boundary pressure is not suffi-ciently small compared to the starting pressure, the pressurewithin the piston bore will undershoot the approaching boundarypressure. In either case, the pressure relaxes itself back to theappropriate boundary condition once sufficient flow is permittedby an increase in discharge or intake area. Results Ideal Pump Flow. The results of Eqs. 11 and 12 areshown in Fig. 6 for a nine-piston, an eight-piston, and a seven-piston design. Figure 7 shows an FFT of Eqs. 11 and 12 for anine-piston, an eight-piston, and a seven-piston design. Note: thefrequency shown in Fig. 7 has been normalized by the piston passfrequency of each design. The piston pass frequency is given by f N 2 , (22)where N is the total number of pistons within the pump and isthe rotational speed of the pump. Figure 8 shows the results of Eqs. 13 and 15 multiplied by 100 to yield the pulse height as apercentage of the nominal pump flow. Actual Pump Flow. Figure 9 shows the numerical results of Eq. 18 for a nine-piston, an eight-piston, and a seven-pistonpump operating at an angular speed 235rad/s, a dischargepressure P d 20MPa, and an intake pressure P i 2 MPa. Figure10 shows an FFT of these results with the frequency normalizedby the piston pass frequency which is given in Eq. 22 . Eachpump with nine, eight, and seven pistons, respectively was de-signed to displace 250 cm 3 /rev and the valve-plate design wasessentially the same in each case. The design parameters for eachpump are given as follows: Fig. 4 A numerical pressure-profile for the n th piston exhibit-ing essentially no overshoot or undershoot in the transitionregionsFig. 5 A numerical pressure-profile for the n th piston exhibit-ing overshoot and undershoot in the transition regionsFig. 6 Idealized flow rippleFig. 7 FFT result of the idealized flow ripple 266 Õ Vol. 122, JUNE 2000 Transactions of the ASME
