英文原文
                                    Case Study
Theoretical and practical aspects of the wear of vane pumps
Part A. Adaptation of a model for predictive wear calculation
Abstract
  The aim of this investigation is the development of a mathematical tool for predicting the wear behaviour of vane pumps uscd in the standard method for indicating the wcar charactcristics of hydraulic fluids according to ASTM D 2882/DIN 51
389.
   The derivation of the corresponding mathematical algorithm is based on the description of the combined abrasive and
adhesive wear phenomena occurring on the ring and vanes of the pump by the shear energy hypothesis, in connection with
stochastic modelling of the contacting rough surfaces as two-dimensional isotropic random fields.
  Starting from a comprehensive analysis of the decisive ring-vane tribo contact, which supplies essential input data for the wear calculation, the computational method is adapted to the concrete geometrical, motional and loading conditions of thetribo system vane pump and extended by inclusion of partial elastohydrodynamic lubrication in the mathematical modej.
  For comparison of the calculated wear behaviour with expenmental results, a test series on a rig described in Part B was carried out. A mineral oil-based lubricant without any additives was used to exclude the influence of additives which cannot be described in the mathematical model. A good qualitative correspondence between calculation and experiment regarding the temporal wear progress and the amount of calculated wear mass was achieved.
Keywords: Mathematical modelling; Simulation of wear mechanisms; Wear testing devices; Hydraulic vane pumps; Elastohydrodynamic lubrication;
Surface roughness
1. Introduction
  In this study, the preliminary results of a newmethodological approach to the development of tribo- meters for complicated tribo sysLems are presented. The basic concept involves the derivation of a mathematical algofithm for wear calculation in an interactive process with experiments, which can be used model of the tribo system to be simulated. In this way, an additional design tool to achieve the correlation of the wear rates of the model and original system is created.
  The investigations are performed for the Vickers vane pump V104 C usedin the standard method forindicating the wear characteristics of hydraulic fluids according to ASTM D 2882/DIN 51 389. In a first step, a mathematical theory based on the description of abrasive and adhesive wear phenomena by the shear energy hypothesis, and including stochastic modelling of the contacting rough surfaces, is adapted to the tribological reality of the vane pump, extended by aspects of partial elastohydrodynamic lubrication and verified by corresponding experiments.
  Part A of this study is devoted to the mathematical modelling of the wear behaviour of the vane pump and to the verification of the resulting algorithm; experimental wear investigations represent the focal point of Part B, and these are compared with the results of the computational method derived in Part A.
2. Analysis of the tribo contact
  The Vickers vane pump V 104 C is constructed as a pump for constant volume flow per revolution. The system pressure is led to the bottom side of the 12 vanes in the rotor slots to seal the cells formed by each pair of vanes, the ring, the rotor and the bushings in the tribologically interesting line contact of the vane and inner curvature of the ring (Fig. 1). Simultaneously, all other vane sides are stressed with different and periodically alternating pressures of the fiuid. A comprehensive structure and stress analysis based on quasistatic modelling of all inertial forces acting on the pump, and considering the inner curvature of the ring, the swivel motion of the vanes in relation to the tangent of curvature and the loading assumptions, is described in Refs. [1-3]. Thereby, a characteristic graph for the contact force Fe as a function of the turn angle can be obtained, which depends on the geometry of the vanes used in each run and the system pressure. From this, the inner curvature of the ring can be divided into four zones of different loading conditions in vane-ring tribo contact (Fig. 2), which is in good agreement with the wear measurements on the rings: in the area of maximum contact force (zone n), the highest linear wear could be found [2,3] (see also Part
B).
3. Mathematical modelling
3.1. Basic relations for wear calculation
  The vane and ring show combined abrasive and adhesive wear phenomena (Fig. 3). The basic concepts of the theory for the predictive calculation of such wear phenomena are described in Refs. [4-6].
  Starting from the assumption that wear is caused by shear effects in the surface regions of contacting bodies in relative motion, the fundamental equation
  (1)
for the linear wear intensity Ih in the stationary wear state can be derived, which contains the specific shear energy density es/ro, interpretable as a material constant, and the real areaArs of the asperity contacts undergoing shear. To determine this real contact area, the de- scription of the contacting rough surfaces as two-dimensional isotropic gaussian fields according to Ref.
[7] is included in the modelling. Thus the implicit functional relation

with the weight function
 (2)
is found, which can be used to calculate the surface ratio in Eq. (1) for unlubricated contacts from the hertzian pressure Pa acting in the investigated tribo contact by a complicated iterative process described in Refs. [6,8]. The concrete structure of the functions F

and c depends on the relative motion of the contacting bodies (sliding, rolling). The  parameter a-
（m0m4）/m22represents the properties of the rough surface by its spectral moments, which can be deter- mined statistically from surface profilometry, and the plasticity index妒= (mOm4)y4(E'/H) is a measure of the ratio of elastic and plastic microcontacts.
3.2. Extension to lubricated contacts
  The algorithm resulting from the basic relations for wear calculation was applied successfully to unlubricated tribo systems [8]. The first concepts for involving lubrication in the mathematical model are developed in Ref. [8]. They are based on the application of the classical theory of elastohydrodynamic lubrication (EHL) to the microcontacts of the asperities, neglecting the fact that there is also a "macrolubrication film" which separates the contacting bodies and is interrupted in the case of partial lubrication by the asperity microcontacts. Therefore their use for calculating practical wear problems leads to unsatisfactory results [9]. They are extended here by including the following assump- tions in the mathematical model.
(1) Lubrication causes the separation of contacting bodies by a macrofilm with a mean thickness u. which can be expressed in terms of the surface
   roughness by [10]
 (3)
Where u0 is the mean film thinkness according to classical EHL theory between two ideally smooth bodies, which can be determined for line contact of the vane and ring by[11]

(2) In the case of partial lubrication, the macrofilm is interrupted during asperity contacts. A plastic microcontact is interpreted as a pure solid state contact, whereas for an elastic contact the roughness is superimposed by a microlubrication film. Because of the modelling of the asperities as spherical indenters, the microfilm thickness can be determined using the EHL theory for sphere-plane contacts, which is represented in the random model by the sliding number [8]
 (5)
(3) The hertzian pressure acting in the macrocontact works in two parts: as a hydrodynamic pressure pEH borne by the macrolubrication film and as a pressure pFK borne by the roughness in solid body contact.
(4) For pure solid state contacts, it is assumed that the limit for the mean real pressure prFK which an asperity can resist without plastic deformation can be estimated by one-fifth to one-sixth of its hardness
 (6)
Investigations on the contact stiffness in Ref. [11] have led to the conclusion that the elastic properties of the lubrication film cause a relief of the asperities, which means that the real pressure working on the asperity is damped. Therefore, in the mathematical model for lubricated tribo systems, an additional term fffin, which corrects the upper limit of the real pressure as a function
of the film thickness, is introduced p,EH =prFK[1 -fcorr(U)]    (7)
This formula can be used to determine a modified plasticity index {PEH for lubricated contacts according to Ref. [8].
  Altogether, the basic model for wear calculation can be extended for lubricated tribo systems by replacing relation (2) by
       (8)
(3)3.3. Adaptation to the tribo’system vane pump

   To apply the mathematical model for wear calculation to a concrete tribo system, all material data (specific material and fluid properties, roughness parameters) used by the algorithm must be determined (see Part B). Moreover, the model must be adapted to the mechanical conditions of the wear process investigated. On the one hand, this is related to the relative motion of the bodies in tribo contact, which influences the concrete structure of function f in formulae (2) and (8). In the case of vane-ring contact, sliding with superimposed rolling due to the swivel motion of the vanes was modelled
 (9)
A detailed derivation of the corresponding formulae for fsliding and f.olling can be found in Refs. [8,9].
  On the other hand, the hertzian presstire Pa acting on tribo contact during the wear process has an esseritial importance in the wear calculation. For the tribo system vane pump, the mean contact force Fe in each loading zone can be regarded as constant, whereas the hertzian
pressure decreases with time. The reason for this is the wear debris on the vane, which causes a change 'n the vane tip shape with time,leading to an increased contact radius and, accordingly, a larger contact area
   To describe this phenomenon by the mathematical wear model, the volume removal Wvl of one vane in terms of the respective contact radius Ri(t) at time t and the sliding distance SR(Rl(t》 is given by
 (10)
where the constants a and b can be determined by regression from the geometrical data of the tested vanes. The corresponding sliding distance necessary to reach a certain radius Ri due to vane wear can be expressed using the basic equation (1):
 (11)
Thus, applying Eq. (11) together with Eq. (10) to the relation
 (12)
it is possible to derive the following differential equation for the respective volume removal Wvll of the ring, which can be solved by a numerical procedure
 (13)
  The required wear intensities of the vane and ring can be calculated by Eq. (8) as a function of the contact radius from the hertzian pressures working in each loading zone, which are available from the contact force by the well-known hertzian formulae.
3.4 Possibilities of verification
  If all input data are available for a concrete vane pump run (the concrete geometrical, material and mechanical conditions in the cartridge used and the specific fluid properties, see Part B), the mathematical model for the calculation of the wear of vane pumps derived above can describe quantitatively the following relations.
(1) The sliding distance SR(RI) and, if the number of revolutions of the pump and the size of the inner ring surface are known, the respective run time t of the pump which is necessary to reach a certain shape of the vane tips due to wear.
(2) The volume removal W,.:uri(t) and the wear masses  WmW(t) of the vane and ring as a function of the run time t.
(3) The mean local linear wear Wl(t) in every loading zone on the ring at time t.
   Thus an immediate comparison between the calculated and experimentally established wear behaviour, with regard to the wear progress in time, the local wear progress on the ring and the wear masses at a certain time t, becomes possible.
4。Results
  In this study, the verification of the theoretical results obtained by comparison with experiments is based on a test series on a rig according to DIN 51 389 described in detail in Part B. The same mineral oil-based lubricant, without any additives, was used in each run to exclude the influence of additives on the wear behaviour, which could not be described in the mathematical model. As input data for the calculation, the mean values of all the quantities needed by the algorithm were determined from four 250 h test runs which were carried out under equivalent test conditions. The following results were obtained.
(1) The calculated temporal wear curve for the vanes, resulting from approximation (9), is in good qualitative agreement w:ith the measurements in Ref. [2] (degressive character and length of the inlet phase (see Part B》. Moreover, the calculated wear masses after a run time of 250 h correspond quantitatively with the experimental results (Fig. 5).

(2) For the ring, a degressrve wear trend was found by calculation, which is assumed to be a realistic result in association with the corresponding degressive trend of vane wear experimentally established in Ref. [2]. The calculated total wear masses, which represent the sum of the wear masses achieved in each loading zone at time t, conform with the wear masses measured in 250 h runs as well as short-time runs of 10 h (Fig. 6).
(3) The wear masses calculated for each separate loading zone on the ring are in quantitative agreement with the corresponding order of the contact force Fe (Fig. 6).
(4) The dependence of the wear behaviour on temperature during tribo contact, represented in the  mathematical model by the dependence of thelubricant properties on temperature, is suitably reflected by the calculation (Fig. 7).
  Further results, especially with regard to a comparison of the calculated and measured local linear wear on the ring, are dcscribcd in Part B.

5. Conclusions
  The mathematical algorithm for the calculation of wear on vane pumps presented in this study enables the experimentally established wear behaviour of the tribo system investigated to be retraced qualitatively and quantitatively. Thus the extensions introduced to cover partial elastohydrodynamic lubrication have proved a success and represent an essential improvement of the results achieved so far [9]. In this way, the preconditions for the development of a mathematical tool for wear prediction and for simulation of the wear behaviour of a tribometer for the tribo system vane pump have been created. For further qualification of the mathematical model to achieve a real forecast of the wear behaviour, theoretical investigations combined with experiments must be enforced, espeaally with regard to the following topics:
(1) inclusion of the inlet phase of the wear process in the model (so far, the mathematical modelis related only to the stationary wear state; an algorithm must be created which is based exclusively on the input data obtainable before starting the wear process and which can successively adapt the data used by the calculation to real wear progress);
(2) extension of the model to practically important lubricants with additives (this can be achieved in a first step by using a heuristic relation to describe the influence of additives on the wear behaviour, derived from corresponding test ScrieS with scVeral lubricants).

中文译文
在理论和实践方面叶片泵磨损的研究（A部分）:为适应预测磨损计算模型
R. Gellrich a, A. Kunz b, G. Beckmann ‘, E. Broszeit b
a大学科技，经济和社会科学Zittaul/Gorlitz，数学和自然科学学院，Th.- Kiirner-阿利16处， 02763齐陶，德国
b材料科学研究所，达姆施塔特技术大学，Grafenstr。二，64283达姆施塔特，GermanyPetersilienshz二维，03044科特布斯，德国
1994年3月29日收到，1994年11月1日接受

摘要
   本次调查的目标是预测用于判断液压流体的磨损特性的叶片泵的磨损行为的一种数学工具的发展，根据ASTM D 2882/DIN 51标准方法389。
   相应的数学算法的推导是基于合并后的描述和磨料粘着磨损现象发生在环和假说的剪切能在叶片泵连接，与随机建模为二维各向同性随机粗糙表面接触领域。
   从环叶片摩擦接触的决定性全面分析开始，为磨损计算，适应具体的几何，运动和摩擦系统叶片泵的负荷条件，被部分弹流润滑延长列入数学模型的计算方法提供了必要的输入数据。
   对于磨损性能的计算与试验结果的比较，对钻机的一系列测试在B部分的叙述中会提出。不含任何添加剂的矿物油基润滑剂，采用排除添加剂的影响，不能在数学模型描述出来。在计算和实验之间的一个良好的定性关系随着时间磨损过程和计算磨损质量的量达到了。
关键词：数学模型 ;磨损机理模拟;磨损试验装置，液压叶片泵;弹流润滑;表面粗糙度
1.简介
在这项研究中，对于复杂的摩擦系统摩擦计的发展的一种新的方法的初步结果被提出来了。这个基本概念涉及到一个在实验的相互影响的过程中的的磨损计算的数学算法的起源，它可用于预测的摩擦磨损性能系统的力学模型可以模拟互动的过程推导。这样，一个额外的设计工具，实现了模型和原系统的磨损率的相关性创建。调查是执行了用于判断按照美国ASTM 2882/DIN 51 389 D型液压流体的磨损特性威格士叶片泵V 104的标准方法。在第一个步骤，一个以磨料和粘结磨损现象的描述剪能量假说为基础，包括随机粗糙表面接触模型的数学理论，是适应了现实的叶片泵摩擦磨损，延长了部分问题弹流润滑和相应的实验验证。
这项研究的一个部分是专门用来对叶片泵的磨损行为的数学模型，并由此产生的算法验证; 实验磨损调查代表了B部分的焦点，这些都是与计算方法和在A部分所得结果进行比较
2．分析部落接触
   威格士叶片泵V 104C是一个每转流量不变的泵。系统压力导致了在转
子槽12个叶片的底部来封住由每一个叶片 环 槽盒和叶片的线接触和环的内部弯曲组成的单元。同时，所有其他的不同的和定期交变的流体压力叶片面也都被强调了。一个在泵惯性力作用的所有准静态建模，考虑到环内曲率，和切线曲率和装载假设相关的叶片旋转运动在文献中有描述，从接触力F的特征图上，作为转角度的功能可以得到，这由每次运行和系统压力所使用的叶片几何形状而       定。由此可见，环内弯曲可分为叶片环摩擦接触（图2）
这与上环的磨损测量吻合分为四种不同的负荷条件区：在最大接触面积部分（第二区），最高的线性磨损可以发现[2,3]（见B部分）
3．数学建模
   3.1．磨损计算基本关系
   叶片和环形显示联合磨料和粘结磨损现象（图3）。预测磨损计算现象的理论的基本概念在文献中有描述。 【4-6】。从磨损是由在具有相对运动的接触表面的高剪切效应引起的这个假设开始，基本方程如下：（1）
在静态磨损状况中的线性方程组的磨损强度I可以得到，，其中包含具体的剪能量密度，可作为材料常数解释，和真正的A区粗糙的接触发生型剪切，为了确定 这个实际的接触面积，作为二维的基础粗糙表面的描述根据文献中的高斯领域。[7]是包含在建模。因此，隐函数关系
被发现，它可以用来计算式中的表面比。（1）从赫兹压力作用的研究摩擦接触，由一个复杂的迭代过程每年在文献中描述无润滑接触。 [6,8]。对混凝土结构的函数F和c取决于身体的接触（滑动，滚动）的相对运动。参数：u=通过它光谱表示了表面粗糙度的特性，也可以由表面的轮廓决定，塑性指数是弹性和塑性比例的量度。
3.2  扩展到润滑接触
  这个算法由摩擦计算的基本关系被成功应用于无润滑摩擦学系统引出[8]。对于涉及数学模型中的润滑的第一概念在文献中被开发。[8].他们是基于弹流润滑的经典理论，以粗擦的微接触的应用，忽视了也有一个微膜隔开了这个连接机构在微接触的部分润滑情况下被打断这个事实。因此用它们来做的磨损问题的实际计算不理想。他们在这里延伸，包括在数学模型的以下假设。
（1）润滑导致了通过平均厚度为U的微胶卷的接触机构的分离，它可以依据通过（10） 的表面粗糙度表示

                     (3)
   其中u是根据两个非常光滑机构之间经典的弹流理论的平均膜厚，可以通过[11]的风向标和环的线接触决定
           （4）

（2）在部分位置的情况下，微膜在粗接触时被中断。一个塑料微接触被解释为纯固态接触，而对于一个弹性接触表面粗糙度是通过微润滑膜叠加的。由于作为球型压头的粗糙的造型，微缩胶片的厚度可以由使用的球平面接触弹流润滑理论确定，代表了随机滑数模型。[8]
（3）                           (5)
（4）在微接触的赫兹压力工作分为两个部分：作为动水压力通过微润滑膜承担，并作为固体机构表面粗糙度所承担的一种压力。
（5）对于纯固态接触，假定这个平均压力的限制，一种能抵抗塑性变形的粗糙能通过它的硬度的五分之一到六分之一估算出来
在文献中接触刚度的调查， [11]已经得出结论，该润滑膜的弹性性能导致了一个粗糙减轻，这意味着关于粗糙的真正压力工作被抑制，因此 在润滑摩擦系统的数学模型下，一个附加项纠正了作为膜厚度功能的真正压力的上限，引入
                                         p=p[1－f]              (7)          
               这个公式可以用来确定一个可修改的可塑性指数和r作为润滑接触根据
文献[8]
总而言之，磨损计算的基本原型可以延长润滑摩擦系统通过替换关系（2）
                                 (8)
3.3叶片泵摩擦系统的调整
    把数学模型应用到具体的摩擦系统的磨损计算，所有的材料数据（具体材料和流体性质，粗糙度参数）使用的参数必须确定（见B部分）。此外，这个模型必须适应磨损过程调查的机械条件，一方面这关系到机构在摩擦接触中的相对运动，影响到功能f在公式（2）和公式（8）中的具体结构，在叠加 滚动叶片被制成叶片环接触模型，在叶片环接触，由于叶片的旋转运动而滑动被仿照的情况下
                         （9）
一个相应的公式推导和可以在文献中找到[8,9]
另一方面，赫兹压力和作用在摩擦过程中的摩擦接触在磨损计算中有一个必不可少的重要性。对于摩擦系统叶片泵，平均接触力F在每个载重区域可视为一个常数，而随着时间的推移，赫兹压力减小。其原因是叶片上的磨屑，导致了叶片尖形随时间的变化，进而增加接触半径及较大的接触面积。（图4）
通过数学磨损模型来形容这个现象，体积去除一个叶片的，依照在时间是接触半径R（t）和通过式子给出的滑动距离
                                  （10）
因此运用式子（10）和式子（11）的关系
                            （11）
推导出下面能被数值程序解决的环的各自体积量的微分方程是可能的

                                                       （13）
  该叶片和环的磨损强度要求由作为来自各个装载区工作的赫兹压力的接触半径的功能的公式（8）计算
3.4 可能性核查
如果所有的输入数据对于具体运作的叶片泵（具体几何，材料和黑盒使用的机械条件和特殊的流体性质，见B部分）是有价值的，对于以上叶片泵的磨损的数学模型可以定量的描述以下关系
（1）滑动距离和 如果泵的转速和内部环表面的尺寸已知，各泵的运行时间对于因磨损而达到一定的形状是必要的。
（2）体积去除和叶片的磨损量和作为运行时间t功能的环
（3）在时间t时环上的每个加载区域的局部平均线性磨损度
在计算和用实验确定的中，关于在一定时间内的磨损过程，环上的局部磨损过程和一定时间t的磨损量的磨损行为之间的直接的对比成为可能。
4.结果
 在这项研究中，以通过DIN51389对一台机器的一系列的检测为基础的通过实验对比得出的理论结果的核查在B部分中有详细的描述。同样的矿物油及基润滑剂，不含任何添加剂，被用到排除关于磨损行为的添加剂的影响的每一次运行中，不能用数学模型来描述。至于计算输入数据，所有通过算法的数量需要的平均值由在同等测试条件下进行的4个250小时的测试运行决定。得到下面的结果
（1）对于叶片的时间磨损曲线的计算近似（9），是在文献2（递减的性质和进阶段（见B部分）的长度）中良好的定性协议下。此外在250h运行后的磨损量的计算定量的对应实验结果（图5）

（2）对于环，一个磨损递减的趋势通过计算被发现，代表了一个在时间t内每个装载区域完成磨损量的总和的总磨损量的计算与在250个小时运行和10个小时运行的测量想符合（图6）。
（3）对于在环上的每个分离的装载区域的磨损量的计算和相应的接触力F的命令相一致
（4）在摩擦接触过程中磨损性能对于温度的依赖，适当的通过计算表示出来（图7）
进一步的结果，特别是考虑到计算和环上局部线磨损测量之间的对比在B部分由描述。

5.结论
   在这项研究中，关于叶片泵磨损计算的数学算法被提出来了，使用实验的方法确定摩擦系统中的磨损性能成为可能，并追溯到定性定量的分析。因此 对于覆盖部分弹流润滑的扩展介绍证明是成功的并在迄今为止取得的结果中占有必要的重要地位，这样，对于磨损预测的数学工具的开发和对于叶片泵摩擦系统中磨损性能的模拟的先决条件已经被建立起来了。为了对完成一个真正的磨损性能的预测的数学模型的进一步限制，与实验相结合的理论研究必须执行，尤其是以下主题
(1) 在模型中磨损过程的入口阶段的内含物（到目前为止，这种数学模型只描述 了固定的磨损形式，一种在磨损过程开始之前依据能得到的输入数据和通过估算出这个真实的磨损过程而能成功的运用这个数据得算法必须被创造出来。）
(2)对于实际上很重要的使用添加剂的润滑的模型的扩展（这个能在第一步通过用启发式的探索对描述添加剂在磨损形式中的影响来完成，从一系列对于润滑的测验中导出