亚洲免费av电影一区二区三区,日韩爱爱视频,51精品视频一区二区三区,91视频爱爱,日韩欧美在线播放视频,中文字幕少妇AV,亚洲电影中文字幕,久久久久亚洲av成人网址,久久综合视频网站,国产在线不卡免费播放

        ?

        Calculation methods of lubricant film pressure distribution of radial grooved thrust bearings

        2012-06-21 01:58:18HUJibin胡紀濱LIUDinghua劉丁華WEIChao魏超

        HU Ji-bin(胡紀濱), LIU Ding-h(huán)ua(劉丁華), WEI Chao(魏超)

        (The State Key Laboratory of Vehicle Transmission,Beijing Institute of Technology,Beijing 100081,China)

        Grooved bearings are frequently applied in industries for better load capacities.For ability of bidirectional rotating,radical grooved thrust bearings are advantageous in certain working circumstances.For a finite-width,parallel step pad slider bearing,an analytical solution with a Cartesian coordinate system has been reported[1-2].Basu[3]solved the compressible Reynolds equation over a radial groove hydrodynamic section by both finite difference and finite element methods.Song[4-6]presented a finite difference numerical analysis of the pressure distribution of mechanical liquid face seal.Rahmani et al.[7]summarized relevant literature industrial significance of step bearings.Liu et al.[8]derived a set of analytical solutions for hydrodynamic lubrication of fan-shaped thrust step bearings considering in a cylindrical coordinate.

        In this paper,the hydrostatic pressure difference at surrounding boundaries is neglected.A ra-dial groove thrust bearing is considered in a cylindrical coordinate.The pressure distribution and load capacity of lubricant film are calculated by both analytical and numerical method (finite volume method),respectively.The precision and speed of numerical solutions are also discussed.

        1 Problem description

        In a cylindrical coordinate,a radial grooved thrust bearing with a step height ofhgis shown in Fig.1against an end face of a plane counterpart.Generally,the grooved bearing is assumed to be stationary and the end face rotates at an angular speed of ωwithrespecttotheorigin.Parameterh0istheminimumgapattheinterfaceandparametersaandbaretheinnerandouterradiusofthe annulusgeometry,respectively.Alongthecircumferentialdirection,agroovedregionisdefinedas0≤θ≤α,whilealandregionasα≤θ≤β.

        Fig.1 Schematic of a grooved thrust bearing

        Inthispaper,weonlyconsiderhydrodynamiclubrication.Forisothermal,incompressible,andNewtonianassumptionsforlubricant,the Reynoldsequationinthecylindricalcoordinateis writtenas

        whereμistheviscosityoflubricant,pisthehydrodynamicpressure,andhisthefilmthickness.

        2 Lubrication analysis

        ForaNewtonianlubricant,velocitiesoflubricantflowintheCartesiancoordinateareexpressedas[1]

        whereU= -rωsinθandV=r ωcosθaretheend face’svelocities.

        UtilizingtherelationshipsbetweentheCartesianandcylindricalcoordinates,thevelocitiesof Eqs.(2)(3)canbetransformedinthecylindrical coordinateas[3]

        The flow rate is an integration of the velocity with respect toz,therefore:

        The boundary conditions for the flow filed are as follows:

        By applying the mass conservations to link the two regions,the pressure distribution of lubricant film can be calculated as shown in the next section.

        3 Calculational methods

        3.1 Analytical method

        The radial grooved bearing has a constant lubricant thickness in each region.The Reynolds equation in Eq.(1)can be further reduced to Laplace partial differential equation:

        Thus,by applying exact solutions of Laplace equation combined with boundary conditions,the pressure distribution of film can be obtained[8]:

        In the grooved region,0≤θ≤α

        In the land region,α≤θ≤β

        whereR=a/b,λ=r/b,γ=nπ/lnλ,g=(H0+1)3coth(γα)+H30coth[γ(β-α)],H0=h0/hg.

        Theloadcapacitycanbeexpressedas[8]

        Although the analytical solutions are summations of infinite terms,normally only a limited number of terms are needed to achieve a given accuracy.A computer program can be used to get the solutions.A relative error is defined as a current term divided by the summation of previous terms.Whenever the relative error is smaller than an acceptable threshold,here 1×10-8for the load capacity and 1×10-5for the pressure distribution,the result is considered sufficient to approximate the true solution.

        3.2 Numerical method

        Writing Eq.(1)in vector form as that in Ref.[9]:

        A lubrication analysis based on the finite volume method is derived by integrating Eq.(13)over the lubrication domain and then applying Green’s theorem,giving

        where nisanoutwardnormalvectorfromthe boundaryoffinitevolume.Thisequationexpressesmassconservationoverthelubricationregion.TheradialandcircumferentialflowratesareexpressedasEqs.(6)(7),respectively.

        Fig.2showsthefinitevolumediscretization overlubricationregion.Eq.(14)canberewritten asanalgebraicequation:

        Fig.2 Finite volume discretization

        Note that here the mesh does not coincide with the radial groove pattern.The groove/ridge boundary divides the finite volume into two parts with their proportions in grooved and land regions beingΨand 1-Ψrespectively.For solving Eq.(15)in the presence of clearance discontinuity,the block-weight approach technique introduced by Kogure et al.[10]is implemented in this scheme.The technique involves averaging the mass flow across the discontinuity by appropriately estimating the mass flow contribution from both parts.

        Then the mass flow terms in Eq.(15)can be written as:

        Thus,the differential scheme ofpcan be deduced from equations above,by which the steady pressure distribution in the grooved thrust bearing is determined by iteratively solving Eq.(15)until the pressure reaches a steady value.The load capacity is obtained by integratingpnumerically on the lubrication region:

        The ending condition of iteration is as follows:

        whereεis the iterative error from stepkto stepk+1,[ε]=1×10-6in this paper.

        In order to achieve the steady value as quickly as possible,Eq.(15)is solved by line relaxation with a relaxation factor.The modified equation is as follows:

        whereφis relaxation factor with range of 0<φ<2,pis the result of stepk,p~is iterative result from stepkto stepk+1,is the modified value ofp~.

        4 Results and discussion

        Conditions used in this section to obtain example results area=40mm,b=80mm,μ=0.1Pa·s,h0=1μm,hg=10μm, ω=16rad/s,α=60°andβ=120°.Thenumericalcalculational pressuredistributionofexampleisshownin Fig.3,inwhichtheareaisdividedinto40 (circumferential)by40 (radial)nodes.Anobvious effectofhydrodynamicexistsinthegroove/ridgeboundary,whichprovidesprimaryloadcapacityforthruststepbearings.

        Fig.3 3 D pressure distribution for example

        Analyticalsolutionandnumericalsolutionof anexamplearecomparedattwocrosssectionsin Fig.4aandFig.4b,i.e.,atθ=60°and atr=60mm.It is obviously that analytical solution and numerical solution are in good agreement in both radial and circumferential directions.

        The mesh density of differential scheme affects the precision and speed of numerical calculation.Therefore,mesh density should be large enough to insure sufficient precision of results.Meanwhile,it should be as low as possible to increase the speed.The solutions as the function of mesh number are shown in Fig.5,in which we take the maximum pressure and load capacity of analytical solutions as the exact solution.Fig.5 indicates that the solutions present higher precision when mesh number exceeds 70×70.

        Fig.4 2Dcross section of the pressure distribution

        Fig.5 Variation of solutions as mesh number increases

        Besides solutions precision discussed above,calculation speed is another important characteristic of differential scheme.For fast convergence,relaxation iteration is an effective method by adjusting relaxation factorφtoapropervalue.The computerprogramofdifferentialschemeinthispaperiswrittenwithMATLABandthecomputer CPUisIntelCore2.ThecalculationspeedversusφisshowninFig.6,wherethemeshnumberis100×100.Fig.6illustratesthatrelaxationfactorφevidentlyaffectsspeed.Furthermore,thefastestspeedof2.922sisobtainedwhenφequals1.94.Sincethespeedofanalyticalmethodsis muchfastercomparedwithnumericalmethod,it isnotdiscussedhere.

        Fig.6 Variation of consumed time asφincreases

        5 Conclusion

        Inthispaper,thepressuredistributionand loadcapacityofradialgroovedthrustbearings havebeencalculatedwithanalyticalandnumericalmethods,respectively.Theblock-weightapproachisusedtodealwithnon-coincidenceof meshandradialgroovepattern.Theresultsof thetwomethodsareingoodagreement,which validatesthenumericalmodelproposedinthispaper.Thenumericalsolutionspresenthigherprecisionasmeshnumberexceeds70×70,andtherelaxationiterationofdifferentialschemepresents thefastestconvergencespeedwhenrelaxationfactoriscloseto1.94.Althoughanalyticalmethodis recognizedtobemoreexactandfaster,itismore difficulttodealwithcomplicatedgeometrycomparedwithnumericalmethod.

        [1]Hamrock B J.Fundamentals of fluid film lubrication[M].New York:McGraw-Hill,1994.

        [2]Chi Changqing. Hydromechanical lubrication[M].Beijing:National Defence Industry Press,1998:201-212.(in Chinese)

        [3]Basu P.Analysis of a radial groove gas face seal[J].Tribology Transactions,1992,35(1):11-20.

        [4]Song Pengyun.The liquid film characteristics of hydrodynamic mechanical seal with a spiral grooved face[D].Chengdu:Sichuan University,1999.(in Chinese)

        [5]Song Pengyun,Chen Kuangmin,Dong Zongyu,et al.Numerical analysis of the pressure on the face of a radial groove mechanical seal for gas[J].Journal of Yunnan Polytechnic University,1999,15(3):1-6.(in Chinese)

        [6]Song Pengyun,Huang Zecheng,Dong Zongyu,et al.An analysis of a radial groove mechanical seal for liquid[J].Journal of Sichuan Union University,1999,3(4):152-158.(in Chinese)

        [7]Rahmani R,Shirvani A,Shirvani H.Analytical analysis and optimization of the Rayleigh step slider bearing[J].Tribol Int,2009,42(5):666-674.

        [8]Liu S B,Chen W W,Hua D Y.Analytical solution to the hydrodynamic lubrication of fan-shaped thruststep bearings[J].ASME Journal of Tribology,2010,132(2):024504.

        [9]Castelli V,Pirvics J.Review of numerical methods of gas bearing film analysis[J].ASME Journal of Lubrication Technology,1968,99(4):777-792.

        [10]Kogure K,F(xiàn)ukui S,Mitsuya Y,et al.Design of negative pressure slider for magnetic recording disks[J].ASME Lubricated Technology,1983,105:496-502.

        (Edited byCai Jianying)

        九九99久久精品在免费线97| 狠狠躁夜夜躁人人爽天天古典| 人妻献身系列第54部| 国产午夜无码视频免费网站| 国产免费人成网站在线播放 | 亚洲一区二区三区少妇| 久久久久久久综合综合狠狠| 国语对白做受xxxxx在线中国| 无码熟妇人妻AV不卡| 中文字幕色资源在线视频| 无套中出丰满人妻无码| 疯狂做受xxxx高潮欧美日本| 91精品综合久久久久m3u8| 小黄片免费在线播放观看| 国产精品久久国产精品99 gif| 人人狠狠综合久久亚洲| av少妇偷窃癖在线观看| 亚洲国产一区二区视频| 亚洲熟女综合色一区二区三区| 中出内射颜射骚妇| 国产精品国三级国产av| av在线免费观看麻豆| 人妻仑乱a级毛片免费看| 亚洲精品夜夜夜| 国产一区,二区,三区免费视频| 青青草视频在线观看网| 国自产精品手机在线观看视频 | 国内精品视频在线播放不卡| 久久久精品3d动漫一区二区三区 | 欧美变态口味重另类在线视频| 99热在线播放精品6| 久久99国产综合精品女同| 久久婷婷五月综合97色一本一本| 国产精品白浆一区二小说| 男女在线免费视频网站| 人妻中文字幕在线中文字幕| 狠狠色噜噜狠狠狠狠米奇777| 亚洲色成人WWW永久在线观看| 精品一区二区三区牛牛| 无码aⅴ精品一区二区三区浪潮| 内射后入在线观看一区|