【机械类毕业论文中英文对照文献翻译】轴向柱塞机械中阀片动态润滑油膜的有限差动算法
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【机械类毕业论文中英文对照文献翻译】轴向柱塞机械中阀片动态润滑油膜的有限差动算法,机械类毕业论文中英文对照文献翻译,机械类,毕业论文,中英文,对照,文献,翻译,轴向,柱塞,机械,中阀片,动态,润滑,油膜,有限,差动,算法
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轴向柱塞机械中阀片动态润滑油膜的有限差动算法潘华城，盛竟超，陆永祥液压传动与控制研究室浙江大学中国杭州3.10013在本文中，有限差分数值法用于轴向柱塞机械中润滑油膜在气缸体和配流盘之间的计算。 栅格层面和坐标变换技术被应用处理复杂的几何学的薄膜形状。 相对于配流盘的汽缸体中活塞的旋转在机器回转装置期间决定瞬时的润滑油膜形状的计算也在考虑之中计算结果与BHRA的同类结果和日本测试结果相较得出。1引言在本文中，有限差分数值法用于轴向柱塞机械中润滑油膜在气缸体和配流盘之间的计算。 栅格层面和坐标变换技术被应用处理复杂的几何学的薄膜形状。 相对于配流盘的汽缸体中活塞的旋转在机器回转装置期间决定瞬时的润滑油膜形状的计算也在考虑之中计算结果与BHRA的同类结果和日本测试结果相较得出。轴向柱塞泵和马达设计过程中最重要的问题之一是平衡缸体和配流盘之间的压力。 设计的关键要确定能够达到的临界压紧力，压紧力矩和分开力以及分开力矩。 在所有的力中，分开力的正确计算是最困难的。 因此，在最近 20 到 30 年期间，出版了许多关于这一个主题的文献资料。 一些大大简化了的可行方法为设计者们给了某些参考。但是因为临界载荷值对设计质量是敏感的, 因此，对于先进的轴向柱塞机械的设计方法，合理预测分开力的大小以及其他的流动薄膜参数是必需的。在 19世纪60年代早期，Franco (Ref.1)提出了一种计算缸体和配流盘之间润滑油膜的压力场的方法。他假定一个厚度不变的薄膜,忽略高低压力孔之间的压力场。他包含了离心力的效果但是在运转速度正常变化的过程中，这个力（离心力）常被疏视。 但是缺少了离心力的压力分布的结果与来自雷诺方程式的那些结果不一致。1963 年,Shute et al.(Ref.2)给出了关于假定某一厚度油膜以及密闭端面零压力下，配流盘横断面压力分布的正确表达。Shutes 的一个重要的贡献是那个藉由使用电位模拟方法在高低孔之间的端面获得压力场。在孔端面压力分布的结果以相等的等效孔外伸角的形式呈现。如此获得的外伸角在设计中能够容易地用来计算在全部分开力共同作用下的孔端压力。 因为,在某些情况下，孔端压力大约占全部分开力的5%，而且 在临界平衡 设计观念中的过载力大约是全部载荷的10到15%,所以这5%的力不能被忽略。J.McKeown et al., 1966(Ref.3) 年，作出了关于不同压力和温度情况下作用在油膜上的压力分布引起的不同黏性作用的详细分析。 他的分析在于某一假定不变油膜厚度的单向箱。他的结论是当输出压力不很高的时候,黏性变动的效果可以被忽略。 但是当输出压力在 2000 到 4000 C 磅/平方英寸或13.8 到 27.6 MPa的范围时, 那么由于使用一个假定的常数黏性，考虑不同的黏性计算的分开力可能存在10% 到 20% 的误差。Yamaguchi(Ref.4) 再1966年发表了第一个关于使用数字化的方法处理轴向柱塞机械的配流盘缸体的油膜的公开著作。他藉由解决极坐标下的雷诺方程式来获得了压力场。他还包括了由于薄膜厚度的不同所引起对楔薄膜的影响。为了避免由于孔口的圆倒角引起的计算的困难，Yamaguchi 假定孔底是平的来简化极坐标计算。同时，他给出了关于使用假定短轴承座的一个近似的分析压力分布。1975 年， C.I.Hooke 发布了 (Ref.5)一组数字化程序，藉由二选一的方法反复决定薄膜的压力来计算流体膜压力以及轴套的弹性变形。他还运用了短轴承近似值的方法来计算薄膜压力场，但是没有给出该方法的细节。1984 年，Yamaguchi (Ref.6) 出版了另一篇文献关于轴向柱塞机械中油膜的数字化计算。相比较 1966(Ref.4) 年他早期的工作,增加了雷诺方程式的不稳定期。 尽管如此, 为了避免使用极坐标时处理孔端的困难, 他假定了在底端之外高低孔之间存在一个 180度的外伸角。 由于已经引入了不稳定期,那么薄膜系数, 类似柔性系数和阻尼系数都是可得的。显然，只有当薄膜是在一个平衡的状态中，这些系数才是有意义的。Yamaguchi 还发布了关于使用液压垫片（Ref.7）和水力垫片（Ref.8）的轴向柱塞机械，其配流盘与缸体间油膜的计算方法。 这些全部使用了短轴承近似法。 与他完整的数字化计算 (Ref.6) 相比较 ,可观察到产生的分开力存在4%的误差 。潘(Ref.lO) 使用自然坐标下的有限差动方法来计算圆形孔实底面一定厚度油膜的压力分布。到目前出版的文献为止,我们还没有找到一个计算轴向柱塞机械缸体和配流盘之间的流体膜普遍的数字化计算方法。因此,现在的工作目标是, 建立一个能计算整个的薄膜面积的相当普遍的数字化代码, 包括圆角形底面, 以及同时在时间和空间层面下考虑不同的薄膜厚度。2. 自然坐标系下的雷诺方程式为了避免使用直角座标（笛卡尔坐标）或极坐标处理复杂的薄膜几何学时（可能）的困难,有必要介绍适合流体薄膜的界线的自然坐标系。因此雷诺方程式使用自然坐标系来表达。我们使用由直角坐标系( x, y)唯一变换而来的自然坐标平面 (， )。=（x,y） (1)=(x,y) (2)以及逆变换X=x(,) (3)Y=y(,) (4)在直角坐标系中存在雷诺方程式 (5)其中 (6) (7) (8)自然坐标系定义下的(1),(2),(3),(4)转换为自然坐标系的形式： (9)其中 (10) (11) (12) (13)由Jacobi解释 (14) (15) (16) (17)3栅格转换等式为了给出润滑薄膜形状与边界上晶粒分布的几何数据,使用 TTM 晶体分析法（x,y）能够自动地得到适合的自然坐标 (,) (Ref.11)。为此, 我们假定分别满足拉普拉斯方程式的不同的( ,)： (20) (21)在计算过程(,) 中，我们采用如下等价的等式(20) 和等式(21): (22) (23)某一确定的或的边界线与在x与y下的边界线是等价的。 这样，在物理范围 (x,y)中自动生成的网目 ( , ) 就变为了解决每个网点( , )对应的x,y值的等式（22）和等式（23）。4 计算中的一些注意事项自然坐标系中的雷诺方程式和晶粒产生的方程式之间存在矛盾, 需要使用有限差动方法来解决。由于篇幅的限制，这里就不给出具体的数据了,可以参考 Ref.13 。为了使上述计算成为一个简便实用的工具，编码时需要加入一些起预处理和后处理作用的子程序。下面将做简单介绍。4-1 薄膜形状变化的定义由于缸体内活塞相对于配流盘的位置不断改变，油膜形状也在不断改变。无论是否是液压油或者是高水位的溶液,油膜的流动性几乎无法压缩。由于油膜中的扰动导致的压力增幅速度至少比在一个旋转缸体某一固定表面高出2个等级。因此，我们可能可以近似将旋转某一时刻同样形状的油膜作为基本稳定的油膜来处理，如此在可能是雷诺方程式的稳定计算期间在油膜范围内设定一个稳定的边界条件或一个类似用对雷诺方程式输入的油膜厚度是不同时间使溶液基本稳定。因此在解决晶粒产生和雷诺方程式之前，在某一指定时刻,根据缸体旋转情况决定即时油膜形状的来作为子程序。如图1所示为某一油膜自动生成的网格。在这项研究中，座标每20个步长对应座标的120个步长被认为是相对密集的。为了绘制清晰的视图，只需绘制坐标的一半。4-2 泄漏率的计算依照雷诺方程序，在高压力部分存在泄漏率:积分是在高压力口沿着确定的轴进行的。 同样地，我们也能获得流向低压力口的流动速率，以及从整个配流盘获得对于低压力口泄漏率作为 (24) (25)既然 Qh 是真正的漏流量率而且 Q 是漏流量率藉由测量测试的容器普遍测量，在给出作为输出时这两项的编码。4-3 分开力的计算分开力是作用在整个配流盘上的压力积分,包括高压力口: (26)Ah是高压力口的面积，包括由于缸体孔重叠导致的外伸口底部的面积4-4 分开距与分开力的中心绕x轴的分开距为 （27）绕y轴的是 （28）xh,yh 是高压口的几何中心。(包括由于缸体孔重叠导致的外伸口底部的面积)受力中心的坐标系如下 （29） (30)45 由于油膜黏度导致的轴转矩 （31）4-6 油膜厚度表达式薄膜厚度是目前计算输入数据之一。为了要清晰地描述薄膜厚度的分布,像其他的作者一样，我们假设固体表面是足够硬的，因此，厚度存在一个线性的分布: （32）实际计算过程中，ho, 和 都是输入数据5. 计算与讨论的结论计算在使用8087芯片的IBM/PC的个人计算机上进行。一次基本的计算过程大约要花费一个小时。本章中相比较 Shute et al (Ref.2) 采用的电位模拟法获得的压力场结果，以及采用Yamaguchi(Ref.12)方法的测试结果，我们计算得出了的二组结果。第一组计算结果根据参考2中的21个电位模拟结果的集合数据计算得出。图2所示 是在 b=0.4 ，=32 ,w=0.4的状态下计算压力的等高线 。改计算结果与参考2中的电位模拟法结果相相较。从图中可以看出现行的计算方法与电位模拟法的差别是非常小的。图3所示为参考2中21组数据得出的分开力与分开力系数。和它们比较，差别小到一些点几乎一致。 图4所示为孔底有效外伸角的计算,以及与参考2的结果相比较。比较的结果是差别少于一度。(外伸角一度的变化大约等价于1%分开力的大小变化)。第一组结果没有包括由于缸体旋转作用在油膜形状上的影响,也没有包括倾斜以及缸体相对于配流盘垂直运动所引起的油膜压力场的影响。因此，第二组计算用来与参考12中Yamaguchi的测试结果相比较。 这组计算使用参考12中V.P.3 测试模型的5组数据和V.P.4 测试模型的六组数据作为输入数据。在这里我们只给出一些典型的结果。在 Ref.12 中，给出了不同时间油膜厚度的动态变化结果，但是并未给出缸体中活塞的相对旋转后的位置。 因此,使用参考12中给出的油膜厚度的对时间的微分来作为当前计算的输入数据是不可行的。在参考12中，油膜厚度的测量被藉由在配流盘附近使用四个间隙感应器测出的。 在本计算中,空间上假定油膜厚度呈线性变化，等式32中的 h0 ，和从使用数据装卡法的参考12中的4组已测数据获得。只有某段时间测量的油膜平均厚度才可以作为输入数据。在本计算中，选择10个计数点的一系列不同形状的油膜来决定缸体中活塞的相对位置：T是与缸体(对于7活塞的机器)一次旋转时间的1/7等价的油膜形状变动的时间。图5给出了参考12中测试泵的压紧力和分开力的计算值。图6给出了机械设计中经常使用的x- y 平面中缸体中不同旋转的活塞下的受力中心。t = 0, 0.1T, 0.2T . 0.9T (33)如果空间上油膜形状与油膜厚度保持不变，那么分开力系数 Kf ，受力中心Xc和Yc也保持不变。 然而在具体操作过程中，由于存在倾斜以及 Kf, Xc 而且 Yc 也存在变化导致油膜厚度在空间中发生变化。 在图5和图6中,由于油膜厚度变化导致的 Kf ，Xc 和Yc 的差别小到在外形上很难区分。图7所示为在11种情况下计算得出的单位时间的平均泄漏率。对于这种形状， Yamaguchi(Ref.12)也给出过测量结果。对于测试模型V.P.3,计算出的泄漏率比参考12中的测试高出50多。对于测试模型 V.P.4, 计算出的泄漏率比参考12中的测试少于10。由于泄漏率与平均油膜厚度的立方成正比，输入厚度的一个微小变化就可能导致计算泄漏率时发生很大误差。在我们的计算过程中，采用参考12给出的油膜厚度数据。根据等式34 对于模型V.P.3 厚度数据应该减少12.5，也就是1.25微米，对于V.P.4 则需增加3，也就是0.18微米来适应试验时预测泄漏率。在参考12中，Yamaguchi也给出了他自己的计算，输入的油膜厚度对于V.P.3减少了6.2微米，对于V.P.4增加了2.8微米。与他的计算相比，我们目前的算法似乎更加合理。6. 结论本项研究中，推导出了计算轴向柱塞泵和马达中缸体与配流盘之间润滑油膜的数字化计算方法。采用了2种手段：(1) 有限元技术和坐标变换技术用于数字化地解决雷诺方程式,使的那些即使不满足笛卡尔坐标或者极坐标地某一确定形态地润滑油膜地直接计算变为可能。（2）在本计算中，在旋转期间活塞相对于配流盘位置地影响用来考虑决定某一瞬时地油膜地几何形状。这二个手段除去在其他的计算方法中出现的油膜形态描述的近似值,使目前地计算方法更合理。 由于上述提到地方法都是采用编码自动实行地，故而人本身的干扰性在实际操作中可以降低到最少。由于油膜厚度 h 以及它对时间的微分h/t仍然被当做输入数据,因此现在的计算方法还不是一个能够完全预测的工具，然而 如果我们让厚度保持为一个常数 (在这设计中认为是“过量”),那么 h 的数值将不会影响无量纲的分开力kf和力中心位置 xc 和 yc的计算结果。在这计算中，油膜厚度 h 实际上不是一个输入参数。因此,现在的方法就是要为在可接受的 过重 情况下的设计提供一个完整的预测方法。 既然现在的方法考虑了实际油膜形状和它的变化，那么对于实际的设计过程它将会是一个较正确工具。因为 h的微分式是输入参数而分开力和分开力矩是输出参数,现有的编码可以为油膜提供一部分动模拟编码。如此一个动态程序也必须包含不仅来自水力或者液压衬片的分开力的预测，也要包括配流盘表面直接由于弹性变形等引起的分开力的预测。结合了计算其他平衡力的这种现行方法有希望开发出一种不仅能够预测分开力和分开力矩，而且能预测其在时间和空间同时动态变化时候的厚度以及泄漏率等等。7 . 术语A 区域b 无量纲 配流盘宽度，2(w+z)/RaDx and Dy 栅格转换系数F 分开力Fr 压紧力h 润滑油膜厚度ho 润滑油膜平均厚度J 雅可比栅格转换等式kF 分开力系数kP 压紧力系数Mx 绕y轴的分开力My 绕x轴的分开力P 压力Q 泄漏率总量r 半径Ra 配流盘平均半径t 时间T (1) 润滑油膜变形周期 (2) 缸体上润滑油膜的扭距u,v 直角坐标系中的速度元件w 孔的半宽w 孔的相对宽度w/(w+z)x,y 直角坐标系z 密封件宽度 (1) 栅格转换系数 (2) 缸体的倾斜角 (1) 栅格转换系数 (2) 油膜厚度最大角 栅格转换系数 孔间角度 and 自然坐标系 (1) 栅格转换系数 (2) 极坐标角度 栅格转换系数 粘度 密度 栅格转换系数 栅格转换系数EQ 空间外伸等效角下标：1,2 缸体表面与配流盘c 受力中心h 高压力1 低压力8 . 感谢这项研究为浙江大学液压电力驱动和控制研究实验室赞助。感谢日本横滨国立大学的 Yamaguchi教授 ,对于他应本人的请求提供他测试模型的几何数据。FINITE DIFFERENCE COMPUTATION OF VALVE PLATE FLUID FILM FLOWS IN AXIAL PISTON MACHINESPan Huachen, Sheng Jingchao and Lu YongxiangZhejiang UniversityHangzhou 3.10013, ChinaIn this paper, a finite difference numerical method is presented for the computation of the fluid film flow between the cylinder block and port plate in axial piston machines. Grid generation and coordinates transformation techniques are applied to treat the complicated geometrical film shape. Rotational position of cylinder block relative to the port plate is also considered in the computation for determining the instantaneous fluid film shape during machine rotation. Computed results are compared with BHRAs analogue results and Japanese test results.One of the most important problems in the resigns of axial piston pumps and motors is the balancing of the forces between the cylinder block and the port plate. The key in the design is to make sure that an adequate overbalance of pushing and separating forces and moments can be reached. Of all forces, the separating force is the most difficult to calculate exactly. Therefore, many contributions were published on this topic during last 20 to 30 years. Some available methods with great simplification can give certain guidance for the designers. But since the amount of overbalance of forces is sensitive for the quality of the design, a better prediction of the separating force and other film flow parameters are necessary for the advanced design methods for axial piston machines.In the early 1960s, Franco (Ref.1) proposed a method of calculating the pressure field of the fluid film between cylinder block and port plate. He assumed a constant film thickness, the pressure field between the high and low pressure ports were omitted. He included the effect of centrifugal force but also admitted that this force could be neglected in common change of operation speed. But his results of pressure distribution without the centrifugal force did not coincide with those from Reynolds equation.In 1963, Shute et al.(Ref.2) gave the correct expression of the pressure distribution across the dealing lands of port plate with the assumption of constant film thickness and zero pressure outsides of the sealing lands. An important contribution of Shutes work is the obtaining the pressure field at the end area between the high and low ports by Using the electric potential analogue method. The results of pressure distribution at port end area was then presented in the form of equivalent port extension angles. The extension angles thus obtained could be easily introduced into design calculation to estimate the effects of port end pressure on the total separating force. Since, in certain case, port end pressure contributes about 5% of the total separating force, and the excessive force in overbalancing design concept is about 10% to 15% of the total force, this 5% of force cannot be omitted.J.McKeown et al., in 1966 (Ref.3), gave a detailed analysis of the effects of the variation of viscosity due to the variation of pressure and temperature on the film pressure distribution. Their analysis is in one-dimensional case with a constant film thickness assumption. Their conclusion is that when delivery pressure is not very high, the effects of the viscosity variation can be neglected. But when the delivery pressure is on the range of 2000 to 400C lbs/ina or13.8 to 27.6 MPa, the computed separating force with the consideration of a varied viscosity may have a difference of 10% to 20% with that computed using a constant viscosity assumption.The first work found in open literatures dealing with the port plate-cylinder block oil film in axial piston machines using numerical method was published by Yamaguchi in 1966 (Ref.4). He obtained the pressure field by solving the Reynolds equation in polar coordinates. He also included the wedge film effect due to the variation of film thickness. To avoid the difficult in computation caused by the round shaped ends of port, Yamaguchi assumed rectangular ends to easy the computation in polar coordinates. He also gave an approximate analytical pressure distribution using short bearing assumption.in 1975, C.I.Hooke reported (Ref.5) a set of numerical procedure to iteratively determine the film pressure by alternatively calculating the fluid film pressure and the elastic deformation of casing and shaft. He also used short bearing approximation to calculate the film pressure field but did not give details of his method.Yamaguchi published another paper on the numerical computation of oil film in axial piston machines in 1984 (Ref.6). Compared with his earlier work in 1966 (Ref.4), unsteady term was added into the Reynolds equation. Still, to avoid the difficulties in treating port end area using polar coordinates, he assumed that high and low ports had a 180 angle of extention without the end area. Since the unsteady term was introduced, film coefficients such as elastic coefficient and damping coefficient was obtained. Obviously, these coefficients are of significance only if the fluid film is in a condition of balance.Yamaguchi also reported computations of the fluid film between the port plate and cylinder block of axial piston machines with hydrostatic pads (Ref.7) and hydrodynamic pads (Refl8) and (Ref.9). These works were all using short bearing approximation. Compared with his full numerical computation (Ref.6) , 4% of difference in resulted separating force was observed.Pan (Ref.lO) computed the pressure distribution of the film of constant thickness with the real end areas of round shaped port ends using finite difference method with natural coordinates.From the literatures published so far, we have not found a general numerical computation method for the fluid film between the cylinder block and port plate of the axial piston machines. The aim of the present work is, therefore, to establish such a general numerical code which can compute the whole film area, including the round shaped end area, and also considering the variation of film thickness both in time and space.2. REYNOLDS EQUATION IN NATURAL COORDINATESTo avoid the difficulties in treating the complicated film geometry using Cartesian coordinates or polar coordinates, it is necessary to introduce the natural coordinates to fit the boundaries of the fluid film.Therefore Reynolds equation should be expressed in natural coordinates.We use plane natural coordinates (， ) which have a unique transformation from Cartesian coordinates ( x , y )=（x,y） (1)=(x,y) (2)X=x(,) (3)Y=y(,) (4) （5）Where (6) (7) (8)From the definitions of natural coordinates (1),(2),(3)and(4),Reynolds equation (5)can be transformed into natural coordinates form: （9） （10） （11） （12） （13） （14） （15） （16） （17）3GRID GENERATION EQUATIONSFor given geometrical data of fluid film shape and grid points arrangement in the boundaries,boundary-fitted natural coordinates (, ) can be automatically generated in (x,y) domain using TTM grid generation method (Ref.11).To do that,we assume variables (, )satisfy Laplace equation in physical domain (x,y) respectively: （20） （21）In computational domain (, ),we have the equivalences of Eq.(20) and Eq.(21) as follows: （22） （23）The equivalent boundary conditions are the x and y values at constantor constant boundary lines. In this way, the automatical generation of mesh (, ) in physical domain (x,y) becomes the solution of Eq.(22) and Eq.(23) to get the x and y values in each grid point of (, ) mesh.4. SOME COMPUTATIONAL CONSIDERATIONSReynolds equation in natural coordinate and grid generation equations are discretized, and solved by the finite difference method.Due to the restriction of the paper size, details of the numerical technique are not given here,they can be found in Ref.13.To make the present computation an easy-use and practical tool,some subroutines performing pre- and after-processing of the data are included in the code.They are briefly mentioned as follows.4-1 Determination of Film Shape VariationThe film shape is continuously changing due to the change of the position of cylinder block relative to the port plate.The fluid in the film, whether is hydraulic oil or high water base solution,is nearly incompressible. The propagation speed of the pressure disturbances in the film is at least two-order of magnitude higher than the speed of a fixed surface point in the rotating cylinder.Therefore, we may approximately treat the simultaneous film shape in an instant time of the rotating as a quasi-steady film shape, thus setting a steady boundary condition around the film during computation which may be steady solution of Reynolds equation or a quasi- steady solution with time differential of film thickness as an input for Reynolds equation. Therefore before the solution of grid generation and Reynolds equation, a subroutine is called to determine the instant fluid shape according to the information of cylinder rotation at a given time. Fig.1 shows an automatically generated grid for an instant fluid film shape. In this study, relatively dense mesh is arranged with 20 steps in coordinate and 120 steps in coordinate. For clear plotting,only half of the constant lines are plotted.4-2 Calculation of the Leakage Flow RateAccording to the Reynolds equation, we have the leakage flow rate from the high pressure port:where the integration is along a constant line around high pressure port. Similarly, we can also obtain the flow rate towards the low pressure port , and the net leakage flow rate out of the whole port plate as Q = Qk - Q! (25)Since Q is the real leakage flow rate and Q is the leakage flow rate commonly measured by measuring container in tests, the present code gives both of them as the outputs.4-3 Calculation of the Separating ForceSeparating force is the integration of pressureon whole area of the port plate, including the high pressure ports:where Ak is the area of high pressure ports including the extented port end areas due to the superimposed cylinder ports.4-4 Separating Moments and Separating Force CenterThe separating moment around x axis is .That around y axis is where x , yk is the geometrical center of the highport ( including the extented port end parts due to the superimposed cylinder ports). The coordinates of force center are given by.4-5 Shaft Torque due to the Film ViscosityThe part of the shaft torque caused by the film viscosity is given by.4-6 Expression of the Film ThicknessFilm thickness is one of inputs of the present computation. To facilitate the description of the distribution of the film thickness, we assume, like other authors, that the solid surfaces are rigid enough, so that the thickness has a linear distribution:In actual computation, ho , and are the input data.5. RESULTS OF THE COMPUTATION AND DISCUSSIONSComputations were performed in an IBM/PC personal computer with 8087 chip. A typical run took about one hour.In this paper, we give two sets of the computed results, which are compared with the electric potential analogue results of the pressure fieldsobtained by Shute et al. (Ref.2), and with some of the test results made by Yamaguchi (Ref.12).The first set of the computational results are computed according to the geometrical data of 21 electrical potential analogue results in Ref.2.Fig.2 is the computed pressure contours under the condition of = 0.4, =32 , w = 0.4. The computed results are compared with the electric potential results in Ref.2. From the figure, it can be seen that the difference between the present computed result and that of electrical potential analogue are very small. Fig.3 shows the computed separating force coefficients of the 21 cases and their comparisons with the results in Ref.2. The differences are so small that some of the points are almost identica. Fig.4 shows the computed effective extended angle of port ends, and their comparison with the results from Ref.2. The differences are less than one degree. (Variation of one degree in effective extended angle is equivalent to about 1% variation of the eparating force).The first set of the results does not include the cylinder block rotation effect on the shape of the film, nor the effects of inclination and vertical movement of the cylinder relative to the port plate, on the pressure field of the film. Therefore, a second set of computations were conducted to compare with Yamaguchis test (Ref.12) results.This set of computations were using the conditions of six cases of V.P.4 test model and five cases of V.P.3 model in Ref.12, as the inputs data. We only give here some typical results.In Ref.12, dynamic variations of film thickness with time are given, but the subsequent relative rotational positions of cylinder block are not given. Therefore,it is impossible to use time differential of the film thickness given in Ref.12 as inputs for the present computation. In Ref.12, film thickness measurements were made by using four gap sensors around port plate. In the present computation,assuming a linear thickness variation in space, h and in Eq.(32) can be obtained from the four measured data in Ref.12 using data fitting methods.Only time average measured film thickness are used as inputs.In this computation ten specific times in a period of film shape variation are chosen for determining the relative positions of the cylinder block: t = O, O.1T, 0.2T . 0.gT (33)where T is the period of film shape variation which is equivalent to 1/7 of the time for one rotation of the cylinder block (for 7-cylinder piston machines).Fig.5 gives the computed results of pushing and separating forces of the test pump of Ref.12. Fig.6 shows its force centers in differentrotational positions of the cylinder block, in the x-y plane, a form which is often seen in engineering designs.Separating force coefficient Kf, force center and Yc remain constant, if film shape remains constant and film thickness is constant in space.Since in the actual cases, film thickness varies in space due to the inclination, Kf, Xc and Yc have certain variations. In Fig.5 and Fig.6, the differences of Kf , Xc and Yc due to film thickness variation in space are so small that they cannot be distinguished in the figure. Fig.7 shows the computed results of time average leakage flow rates in 11 cases. In this figure, measured results of Yamaguchi (Ref.12) are also given. For the model V.P.3,computed leakage flow rates are about 50% greater than those from tests in Ref.12. For model V.P.4,computed leakage flow rates are about 10% less than those found in Ref.12. Since the leakage flow rate is propotional to the cube of average film thickness Q hi (34)a slight change of inputed thickness may cause large variation in computed leakage flow rate. In our computation, thickness data from Ref.12 are used. According to Eq.(34), thickness data for V.P.3 model should have a 12.5% reduction, i.e. 1.25 um, those for V.P.4, a 3% increase, i.e. 0.18 um, to make predicted leakage flow rates consistant with experiments. In Ref.12, Yamaguchi also gave his own computation, in which inputed film thicknesses for V.P.3 had a reduction of 6.2 um, those for V.P.4 had an increase of 2.8 um. Compared with his computation, the present one seems mote reasonable. 6. CONCLUSIONSIn this study, a numerical computation of steady fluid film between the cylinder block and port plate of an axial piston pump or motor is conducted. Two measures are aken: (1) Grid generation and coordinates transformation techniques are applied to the numerical solution of Reynolds equation, making it possible for the direct computation of a fluid film in real shape which need not be approximately treated to fit the requirement for Cartesian or polar coordinates; (2) In this computation the effect of the position of cylinder block relative to port plate during rotation is considered for determining the instant geometrical shape of the film. These two measures eliminate the approximations in film shape discription appeared in other computational methods, making the present computation methods more reasonable. Since ail above mentioned measuures are automatically conducted in the code, the human interferences are minimized in practical operations.Since the film thickness and its time differentia h/t are still treated as inputs, the present computational method is not a complete prediction tool, However, if we let the thickness be a con
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