利用被困体积提高轴向柱塞泵的容积效率.x

附件1:外文资料翻译译文利用被困体积提高轴向柱塞泵的容积效率研究分析结果显示，标准配流盘设计因为有不受控制地膨胀和压缩的流体发生经过插槽本身而产生一种容积损失。

通过去除这些插槽同时采用被困容积式，真正起到改善柱塞泵的容积效率的结果。

虽然目的并不在于研究适合所有柱塞泵的理想配流盘设计，但是该报告的确在被困容积的应用方面提供了理论依据，并且也对解决配流盘的整体设计中的问题进行了进一步的探索。

柱塞泵的工作和受カ在这一节中，推导出了和轴向柱塞泵操纵效率有关的方程。

注意：这里的效率在通篇中仅指和流体压缩损失有关的效率。

这次分析由泵的单一柱塞的机械和液压力图表展开。

利用该图表，分析计算了作用在柱塞上的机械力和作用在泵排油区一液体单元的液压力。

通过输出功率和输入功率的比值，推导出了泵的瞬时功率的表达式。

该表达式表明，为了计算泵的效率，必须考虑到必须的动力学、柱塞腔内的压力和流入流出柱塞腔的体积。

这些数值来源于本文接下来的章节中。

N个柱塞X周正方向的力Fn。

这个力是由于斜盘对滑靴的反作用力而使柱塞挤入。

同理，在柱塞排油的一腔流体上也作用了一压力Pn。

该压力驱使流体流出腔体或被认为是流体的排出力。

把输入的机械力Fn转换为输出的液压力Pn，是该柱塞泵的工作的基础。

液压力容积流量说明瞬时流线从第n个腔流出混入泵的排油腔。

用Q0表示泵的众多容积流量网合成系统的排油。

每个柱塞腔的压力是各不相同的，但是泵排油区一条流线上的压力是一个常数Pd。

液压系统排油区的压力为P0。

在以下的分析中，我们来考虑一个流体单元。

这个单元是封闭的从而可以代表第n个柱塞腔到系统排油区的流线。

液压力(Pn Po)An作用于此单元，这里Pn是第n个柱塞腔的压力，Po 是系统排油腔的压力，An是代表着从第n个柱塞腔流出的流线的流体单元的瞬时横截面。

图5是修饰后的配流盘的图解，它省去了最顶点和最底点的卸荷槽。

(intake port:吸入口 discharge port:排出口 kidney-shaped flow passage from a single piston chamber:从单个柱塞腔引出来的贤脏形状的流道）和图4同理，图5同样给出了从单个柱塞腔引出来的肾脏形状的流道配合着配流盘 上的弓形门状几何体。

当流道向θn =π/2位置移动是，事实上流道逐渐被此区域内的门状几何体所阻断。

当柱塞腔正好位于顶死点时，柱塞腔是关闭的没有流体的流 进和流出。

如图5所示，当柱塞向配流盘吸油区移动时这种封闭的情况依然存在。

在这种封闭的状况下，柱塞腔内的流体被困住，所以叫做被困容积泵的设计。

封闭区域的角度尺寸用ζ1表示。

在这种设计中，压力的转变并不是靠配流盘上的卸荷槽 来实现的，而是单独靠受控体积在柱塞腔内的体积膨胀来完成的。

穿过封闭区时，Kidney-Shaped Flow Passage from a Single Piston ChamberFig. 5 Trapped-volume valve plate design柱塞腔立刻与吸油区联通，流体从泵的吸油区流入柱塞腔。

当θn=3π/2柱塞腔靠近最底线时，也会有同样的状况。

在此区域内柱塞从吸油区移动到排油区，其封闭的角度尺寸用ξb表示。

在这个位置，压力的转变由柱塞腔内受控体积的压缩来完(图 6 piston pressures:活塞压强equation:方程式 angular position: 有角的位置）Fig. 7 Piston discharge flows(piston discharge flows:活塞流体流动）图5也在事实上考虑了柱塞泵中单一个柱塞腔的四个不同的区域的压力和流动分总结图6用这种泵的设计理论作为知道思想，把压力方程（27)和压力方程(36)做了比较。

同样的道理，把流体流动方程（28)和（37)做对比，我们还能得到图7。

如图6所示，被困体积泵的设计中压力转变相对于标准柱塞泵的设计中的压力转变而言，有很大程度的滞后。

从图7可以看出，在配流盘压力转变区域内，标准柱塞泵设计中的容积流动受到了很大的阻力。

这种流体流动的阻力是由于在柱塞腔的最低点和最顶点流体受到了不受控制的膨胀和压缩而造成的。

在最低点附近的不受控制的压缩对柱塞泵产生了很不利的功率损失。

讨论因为以前的结果都是随时间变化的，为了出个方法解决这个问题，我们必须为每次压力转变的操作而设计一种新的配流盘的设计理念。

图8显示了随着压力操纵的改变，柱塞泵配流盘的设计也跟着改变，同时附表给出了基本柱塞泵参数的变化。

方程（40)和方程（43)分别描述了普通柱塞泵设计和被困体积柱塞泵设计的功率损失。

用附录中的参数，我们把这些方程描述在了图9中。

就如图9所示，相对于被困体积泵设计的功率损失而言，普通泵设计的功率损失要大。

这种结果可以用配流盘上的插槽来解释。

读者也许会记得，这些插槽分担了部分流体容积的流动，用来协调在最底部和最顶部压力跃迁的变化的。

在最底线那里，当柱塞进入排油口时，流体经过配流盘上的插槽进入柱塞腔内直到柱塞腔内的压力等于柱塞泵排油区的压力。

为了使得这些压力相等，柱塞腔内的流体受到了压缩，结果，一部分能量加到了柱塞腔的体积上。

在最顶部，配流盘上的插槽是用来缓解在最底部被压缩的流体体积的。

这种流体的缓解或者说是流体的膨胀导致通过插槽的流体流动释放了储存在流体中的能量。

这些被释放出来的能量因为柱塞泵吸油口的压力是一个恒定的压力源而永远也不能收回。

另一方面，被困体积柱塞泵的设计中不用为了在最底部和最顶部得到平稳的压力转变而开设插槽，所以流体中的能量不会以某种耗费能量的方式被储存和释放掉。

Fig. 8 Changing valve-plate dimensions as a function of operatingpressure(图8改变门状儿何面积作为压力转变的操作）Fig. 9 A plot of the power-loss equations(图9功率损失方程式）在被困体积的情况下，在最底线部位能量由于柱塞腔体积自身的机械变化而自动的补充到流体上。

同样的道理，从流体中释放出来的能量也因为柱塞腔容积体积的改变而被自动的吸收。

Fig. 10 A plot of the efficiency equations(图10容积效率方程式)但是，在这两种设计中能量都在柱塞泵的排油区和被考虑等于柱塞泵的吸油区的压力的液压系统的舱室的交界面上有了损失。

这中能量损失在方程（43）中被计算到了总的能量损失中，产生它的原因就在与当流体在经过柱塞泵排油区和液压系统舱室时，不受控制的膨胀造成的。

方程(41)和方程(44)分别描述了普通柱塞泵设计和被困柱塞泵设计中的容积效率。

利用附录中的参数，这两个方程被描述在了图10中。

如图10所以，被困体积柱塞泵的设计比普通柱塞泵的设计更有效。

造成这样的结果再一次说明了两种设计中不同的能量损失特征。

按照柱塞泵的设计和操纵压力，这种效率的提高可以达到5%。

从分析结果中可以得到，Vo提高了，使用被困体积设计柱塞泵的优势更加明显。

结论这篇报告试图说明一台柱塞泵的功率损失和效率可以通过改变配流盘通道的几何尺寸来得到提高。

特别是，这次研究对比了具有恒定面积卸荷槽的配流盘设计和在卸荷槽位置改用被困体积的流体压缩的容积损失。

在这次研究中，带有卸荷槽的配流盘因为流体通过最底部和最顶部是的不受控制的膨胀而产生了损失。

另一方面，具有被困体积设计的配流盘设计可以吸收流体从压缩到释放时的能量。

所以，被困体积柱塞泵设计比应用了卸荷槽的普通柱塞泵设计更为有效。

附录A b J = 16.0E-06 m2A p =789.2E-06 m2C d = 0.62N = 9P d = 35.0E+06 Pa Pi =5.0E+06 Pa r =67.3E-03 m V0 =43.6E-06 m3α =0.244 rad β =1.2E+09Pa ρ = 850kg/m3ω = 188.5rad/s专业名词和术语Ab, t 配流盘最顶部和最底部卸荷槽的恒定面积 An 包含第n个柱塞流线的流体单元的横截面积Ap 单个柱塞的有效压力面积Cd 柱塞腔外泄系数Fn 作用在第n个柱塞x轴方向的机械力M p 单个柱塞的质量N 柱塞泵中的柱塞数目n’瞬时连接到泵的排油区的柱塞的数目n 柱塞编号Pb 单个柱塞腔外的界限压力Pd 泵的排油压力Pt 泵的吸入压力Pn 第n个柱塞腔的流体压力Po 液压系统排油区的流体压力Qn 流出第n个柱塞腔容积流动速率Qo 流入液压系统的容积流动速率r 柱塞节圆半径sn 沿着第n个柱塞腔流线的坐标t 时间V无标注尺寸的柱塞体积Vb, t 顶部和底部的柱塞腔的体积Vn 第n个柱塞腔的瞬时体积Vo 单个柱塞腔的名义体积W 一般意义上的功Xn 第n个柱塞滑靴球连接在x轴上的位置a 旋转斜盘的角度β流体体积模数ξbt配流盘底部顶部卸荷槽的弧度值η柱塞泵的效率θ第n个柱塞的角度位置K 一般性的流动效率ξbt配流盘底部顶部被困体积的弧度值Π一般性的功率代号ρ流体密度Ψ肾型孔的角度尺寸ω泵的旋转角速度附件2:外文原文The exploitation surrounds a physical volume exaltation stalk to fill the capacity efficiency of pump toward the pillarIn the analytical result of this paper, it may be shown that the standard valve-platedesignintroduces a volumetric loss whichmay be accounted for by the uncontrolledexpansion andcompression of the fluid that occurs through the slots themselves.Bye liminating these slots, and utilizing a trapped volume design,it may be shown thatimprovements in theoperating efficiencycan be achieved. Though this paper does notclaim to providethe ideal valve-plate design for all pump applications, it doesprovide thetheoretical reason for utilizingtrapped volumes andlends general insight into the overallproblem of valve-plate design. Pump Work and PowerIn this section, the equations that govern the operatingefficiency of the axial-piston pump are derived. Note:throughout this research, the word efficiency will refer only to theefficiency that is associated with the compressibility losses of thefluid. This analysis begins by examining a diagram of mechanicaland fluid conditions that exist within the pump for a single ing this diagram,the mechanical work that is exerted on thepiston,and the hydraulic work that is exerted on a fluid column within the discharge chamber of the pump, are considered. Bytaking the ratio of output power to input power, an instantaneousexpression for the efficiency of the pump is derived. Fromthisexpression, it is shown that the kinematics of the piston, the pressurewithin the piston chamber, and the volumetric flow in and outof the piston chamber must be determined for the purposes ofevaluating the efficiency of the pump. These quantities are derivedinsubsequentsections of this paper.a diagram of mechanical and fluid conditionsthat exist for a single piston as it operates within the pump. In thisfigure,it is shown that the nth piston is acted upon by a force, Fn,which is shown to drive the piston in the positive x-direction. Thisforce is the input to the piston which is generated by the slipper’sreaction against the swash plate. Similarly, the fluid at the dischargeof the piston chamber is acted upon by the pressure withinthe nth piston chamber itself, Pn . This pressure tends to force thefluid out of the chamber and may be considered as theforcinginput to the fluid. Theprocess ofconverting the mechanical input,Fn,to ahydraulic input, Pn,is the fundamental operating task ofthe pump.Hydraulic Power.however, the bottom piston is shown to be the nth piston whichimplies that the number of pistons within the pump is generalized.the diagram of volumetric flow illustrates the instantaneousstream1ine of flow that is ejected from the nth pistonchamberinto the discharge chamber of the pump. The net volumetricflow from the pump discharge-chamber into the hydraulicsystem discharge-chamber is given by Qo . the diagramof fluid pressure illustrates that the pressure within each pistonchamber is generally different; but, that the fluid pressure alongthe streamlines within the pump discharge-chamber is essentiallya constant which is given by, Pd . The pressure within the dischargechamber of the hydraulic system is given by the constantpressure, Po .In the analysis which follows, a column of fluid within thedischarge chamber of the pump will be considered. This column of fluid will be chosen so that it will contain the streamlines offlow from the nth piston chamber to the discharge chamber of thehydraulic system. The hydraulic force exerted on this column offluid is given by,(Pn2Po)An,where Pn is the pressure withinthe nth piston chamber, Po is the pressure within the dischargechamber of the hydraulic system, and An is the instantaneouscross-sectional area of the column of fluid which contains thestreamlines of flow from the nth piston chamber.Trapped-Volume Pump Design. Figure 5 shows a schematicof a modified valve-plate which has eliminated the slots near topand bottom dead-centers. Similar to Fig. 4,Fig. 5 shows a kidneyshapedflow passage from a single piston chamber which matchesthe arcuate porting geometry of the valve plate. As this flow passagemoves toward u n5p/2, the actual flow passage is graduallycut off due to the terminating port-geometry of the valve plate inthis region. When the piston reaches this point, the piston chamberis completely closed off and flow cannot be discharged or receivedby the piston chamber. As shown in Fig. 5, the closedportingcondition continues to exist as the piston moves towardthe intake port of the valve plate. In this closed-porting condition,the fluid within the piston chamber is trapped and thus it is calledatrapped-volume pump design. The angular distance of thisclosed porting is given by the dimension, z t . With this design, thepressure transition is accomplished, not by valve-plate slotting,but by the controlled volumetric expansion of the piston chamberalone. Once the piston chamber crosses the closed-porting zone, itquickly opens up to the intake port and begins to receive fluidfrom the intake side of the pump. A similar set of conditionsexists when the piston chamber is near bottom dead center whenu n53p/2. In this region，the piston is moving from the intakeport into the discharge port and the angular dimension of theclosed-porting zone is given by, z b . In this location, the pressuretransition is accomplished by the controlled volumetric compressionof the piston chamber.Again, the valve plate shown in Fig. 5 provides, essentially,four different regions to be considered in the pressure and flowanalysis for a single piston-chamber within the pump. Table 2 Trapped-volume value slate regionsRegion Angular Position Pressure Conditions Flow ConditionsThe pressure within the piston chamber is at dischargepressure.The discharge flow is equal to the displacement of theThe pressure within the piston chamber is betweenintake pressure and discharge pressure.The valve-plate porting is closed off and the dischargeflow is zeroThe pressure within the piston chamber is at intakepressure. The intake flow is equal to the displacement of the piston.The pressure within the piston chamber is betweenintake pressure and discharge pressure.The valve-plate porting is closed off and the intake tlowis zero.Within Regions 1 and 3，the pressure is approximated as a constant,either Pd or Pi,and the volumetric flow rate is given by the negative of the volumetric time rate-of-change of the piston chamberitself, 2V' n5Apr tan(a)v cos(un). In Regions 2 and 4，thepressure is changing as a function of u n and therefore some analysisis required to approximate the pressure characteristics withinthese regions.In Region 2 of the valve plate, the porting is closed off andvolumetric flow in and out of the piston chamber is no longerpossible. In this case, the time rate-of-change of the fluid pressurewithin the nth piston chamber is given bydPndt52bVndVndt, (29)where Vn is the instantaneous volume of the nih piston chamber.By eliminating dt from the denominator of both sides of this equation,the following separable differential-equation with its appropriatebounds of integration may be writtenEPdPndPn52b EVtVn 1 VndVn ,(30)where Vt is the volume in the nth piston chamber when u n5p/2.The solution to thisequation is given byPn5Pd2b lnSVnVt D'Pd2b SVnVt21 D, (31 )where Vn is given in Eq. 〜16! and Vt5Vo2Apr tan(a). Usingthese results yields the following simplified expression for thepressure within the nth piston chamber as the piston passesthrough Region 2 of the valve plate:Pn5Pd2b S12sin~u n!V21 D,(32)where V~ 5Vo /Apr tan(a). Note: is always greater than unity.Within Region 2 of the valve plate Qn50.To insure that theclosed-porting zone on the valve plate is designed sufficiently, itis important to note that when u n5p/21 z t,the pressure withinthe piston chamber should equal the intake pressure, Pi. Thismeans that the closed-porting zone on the valve-plate has effectively facilitated a full pressure transition from the discharge pressure,Pd, to the intake pressure Pi . By setting Pn equal to Pi ,and u n equal to p/21z t,Eq.〜32! may be solved to determine theproper length of the closed-porting zone on the valve-plate. Thisresult is given byz t5cos21S12Pd2Pib-V~21! D. (33)Similar analysis can be done for Region 4 where the pressuretransition being achieved is between the intake pressure, Pi,andthe discharge pressure, Pd . In this region, the pressure within tenth piston chamber is given byPn5Pilb SIlsin〜u n!V~ll D. (34)Again, within Region 4 of the valve plate, Qn50.It can be shownthat the appropriate closed-porting length in Region 4 is given byz b5cos21S12Pd2Pib-V~11! D. (35)To summarize the approximate pressure results of this section,the following piecewise equation is presented for the instantaneouspressure within the nth piston chamber:Pn5! Pd z b2p2,u n，p2Pd2b S12sin〜u n!V~21 D p2，u n，p21z tPip21 z t,u n,3p2Pilb SIlsin〜u n!V~11 D 3p2,u n,3p21z b.(36)The approximate volumetric flow results of this section may besummarized using the following piecewise equation for the instantaneous discharge-flow from the nth piston chamber:Qn5:Apr tan〜a !v cos〜u n! z 62p2,u n，p20p2，u n,p21 z tApr tan〜a !v cos〜u n !p21 z r，u /i，3p203p2，u /?，3p21 z /?.(37)Summary. Using the pump design information in the Appendix,Fig. 6 has been generated for the purpose of comparing thepressure equations 〜27! and 〜36!. Similarly, Fig. 7 has been generatedfor the purpose of comparing the flow equations 〜28!and〜37!. As shown in Fig. 6, the pressure transition of the trappedvolumedesign significantly lags the pressure transition of thestandard design. From Fig. 7, it can be seen that the volumetricflow of the standard design experiences significant spikes in thetransition regions of the valve plate. The flow spikes of thestandarddesign result from the uncontrolled expansion and compressionof the fluid at top and bottom dead centers. At bottom deadcenter, the uncontrolled compression of the fluid causes an undesirablepower loss for the pump.Standard Pump Design. Substituting the results of Eqs.〜13!，〜27!，and 〜28! into Eqs. 〜10! and 〜12! yields the followingresults for the output and input power of the standard pumpdesign:P —out5Pidea/Hcos2Sj b2D2DPb〜V~11 !4 J,(38)P ~in5PidealH12cos~j t!j t2 112cos〜j b!jb2 2DPb-V21 !4 J,where the ideal power transmission of the pump is given byPideal5NAprv tan〜a !DPp. (39)In these equations, DP5Pd2Pi. Subtracting the output powerfrom the input power yields the power loss of the standard pumpdesign. This result is given byP -lo ss5PidealH 12cos〜jt!j t2 1l2cos〜j b!jb 2 2cos2Sj b2 DlDPbl2J.(40)The efficiency of the standard pump design is given byh5P ~outP -in5Hcos2Sj b2 D2DPb〜V~11!4 JH12cos 〜j t!j t2 112cos〜j b!jb 2 2DPb-V21!4 J . (41) Trapped-Volume Pump Design. Substituting the results ofEqs.〜13!，〜36!，and 〜37! into Eqs.〜10! and 〜12! yields the followingresults for the output and input power of the trapped-volumepump design:P _out5PidealH12DPb〜VI1 !2 J, P - in5PidealH12DPbV~2 J.(42)Subtracting the output power from the input power yields thepower loss of the trapped-volume pump design. This result isgiven byP -loss5PidealHDPb12J. (43)The efficiency of the trapped-volume pump design is given byh5P ~outP ~in5121 S2bDP2V~D. (44)Fig. 6 Piston pressuresJournal of Dynamic Systems, Measurement, and Control DiscussionTo make plots of the previous results as they vary with pressure,a new valve plate needs to be designed for each operatingpressure. Figure 8 illustrates the changing valve-plate designs asthey vary with operating pressure for the basic pump parameters given in the Appendix. Equations 〜40! and 〜43! describe the power losses of the standarddesign and the trapped-volume design respectively. Theseequations are plotted in Fig. 9 using the parameters given in theAppendix. As shown in Fig. 9，t h e power losses are greater for thestandard design as compared to the trapped-volume design.Fig. 7 Piston discharge flowsThisfact may be explained by the slots on the valve plate. The readerwill recall that the slots are used to provide a flow passage whichaccommodates the pressure transitions at top and bottom deadcenters. At bottom dead center, when the piston is entering thedischarge port, fluid flows through the valve-plate slot into thepiston chamber until the fluid pressure within the piston chamberis equal to that of the fluid pressure in the discharge port of thepump. In order to make these pressures equal,the fluid in thepiston chamber needed to be compressed; and, as a result, energywas added to the piston-chamber volume. At top dead center, thevalve-plate slot is used to decompress the fluid that was compressedat bottom dead center. This decompression or expansion of the fluid results in a flow through the slot which releases thestored energy in the fluid. This released energy is never recoveredsince the intake port of the pump is modeled as a constant pressuresource of fluid. On the other hand, the trapped-volume pumpdesign does not utilize slots for achieving a smooth pressure transitionat top and bottom dead centers; and, as a result, the energyin the fluid is not added or released in an uncontrollable fashionthat dissipates energy. In the trapped-volume case, the energyadded to the fluid at bottom dead center is added mechanicallythrough the volumechange of the piston chamber itself. Similarly,at top dead center, the energy releasedfrom the fluid is recoveredmechanically since it is achieved through the volumetric changeof the piston chamber as well. In both design cases, however,energy is lost at the interface between the pump discharge chamberand the hydraulic system chamber which is considered to be atthe same pressure as the intake port of the pump. This energy loss amounts to the total energy loss shown in Eq.〜43! and is due tothe uncontrolled expansion of the fluid as it crosses the boundarybetween the pump discharge-chamber and the hydraulic systemchamber.Equations 〜41! and 〜44! describe the volumetric efficiency ofthe standard design and the trapped-volume design, respectively.These equations are plotted in Fig. 10 using the parameters givenin the Appendix. As shown in Fig. 10，t h e trapped-volume designis more efficient than the standard design. Again, this is due to thedifferences in power-loss characteristics of these two designs.This efficiency improvement can be as high as 5 percent dependingupon the pump design and the operating pressure. It can beshown from the analytical results of this study that, as Vo increases,the advantages of using a trapped-volume designbecomemore apparent. ConclusionThis paper has attempted to show that the power loss and efficiencyof a pump can be altered by changing the porting geometryof the valve plate. In particular, this research has compared thevolumetric losses due to fluid compression between valve-platedesigns that have constant-area slots and ones that utilize trappedvolumeregions in the place of slots. In this research, it has beenshown that valve plates with slots generate losses that result fromthe uncontrolled expansion of fluid which occurs through the slotsat top and bottom dead centers. On the other hand, valve platesthat are designed with trapped-volume regions can mechanicallyrecover the energy change that occurs from compressing and decompressingthe fluid. As a result, trapped-volume designs aremoreefficient than the standard pump designs which utilize slotson the valve plate.Pd5 35.0E106 Pa Pi5 5.0E106 Pa r5 67.3E203 m V o 5 43.6E206 m3 piston chamberFn 5 mechanical force exerted on the nthAppendixAb,t 5 16.0E206 m2 Ap 5789.2E206 m2 Cd5 0.62 N59 NomenclatureAb,t 5 constant slot areas onthe valve plate at bottom andtop dead centers An 5 cross sectional area of the fluid column containing thestreamlines of flow from the nth pistonchamberAp 5 pressurized area of a single piston Cd5 discharge coefficient of a piston chambera 5 0.244 rad b51.2E109 Pa r 5850 kg/m3 v 5188.5 rad/spiston in the x-direction Mp 5 mass of a single piston N 5 total number of pistons within thepump n 5 piston counter; i.e., the nth piston n8 5 total number of pistons instantaneously connected to thedischarge side of the pump Pb 5 boundarypressure outside a singlePd 5 discharge pressure of the pump Pt 5 intake pressure of the pump Pn 5 fluid pressure within the nth piston chamberPo 5 fluid pressure with the discharge of the hydraulic systemQn 5 volumetric flow-rate out of the nth piston chamberQo 5 volumetric flow-rate into the discharge of the hydraulicsystemr 5 piston pitch-radiussn 5 coordinate along the streamline of flow from the nthpiston chambert5 timeV~5 dimensionless piston volumeV b,t5 piston-chamber volumes at bottom and top dead centersV n 5 instantaneous volume of the n t h piston chamber V o 5 nominal volume of the single piston chamberW5 the general symbol for workxn5 position of the nth piston-slipper ball joint in thex-directionthe valve plate atbottom and top dead centersh 5 pump efficiencyu n5 circular position of the nth pistonK 5 the general symbol for flow and capacitance-type coefficientsj b,t 5 valve-plate slot angular-length at bottom and top deadcentersP 5 the general symbol for power r 5 fluid densityc 5 angular dimension of a kidney-shaped flow passagev 5 shaft speed of the pumpa 5 swash-plate angleb 5 fluidbulk-modulusz b,t 5 trapped-volume angular length on。