Sensitivity analysis of size effect on the performance of hydrostatic bearing

Transcription

1 Sensitivity analysis of size effect on te erformance of ydrostatic bearing Cen Dongju 1a, Dong Liua 1, Zou Suai 1 and Fan Jinwei 1 1 College of Mecanical Engineering and Alied Electrics Teclogy, Beijing University of Teclogy, Beijing 114, Cina Abstract. For size effect on solid-liquid interface of ydrostatic bearing oil film ga flow in two-dimension, fluid dynamic metod is alied to investigate te influence of size effect on bearing caacity, dynamic stiffness and oter erformances. Wit te consideration of size effect, Reylds equation is modified by adoting velocity boundary condition into Reylds equation.te sensitivity factors are used to make a quantitative and qualitative analysis. Numerical simulation results sow tat size effect will affect bearing erformances to a certain degree and te effect curve of size effect on bearing erformances are given. Te four maimum oil film ressures reduce wit te increase of lengt. Te maimum sensitivity of bearing caacity is 81.94%. 1 Introduction Hydrostatic bearings are widely used in recision macines. Oil film of ydrostatic bearing lays an imortant role in balancing eternal load, reducing friction, eliminating wear and so on. Wit te consideration of te scale division of conventional liquid flow, te researc of oil film flow of ydrostatic bearing as entered into micro-scale. However, te erformance of liquid flow in micro-scale will be different wit te one in macro-scale. Size effect on te one and beaves of te ratio increase of fluid surface area to volume ratio, wen te scale is transferred from macro-scale (cm ~ m) to micro-scale (um ~ mm). Comared to te volume, te ratio of surface area increases from 1 m-1to 16m-1, wic will accelerate te rocess of eat transfer on te surface, tereby te temerature rise of te film will be reduced; on te oter and sows a velocity gradient effect, i.e. te sear stress, wic along te surface rmal is increased since size srink. At te same time, sear force imroves 1 times in micro-scale. If te sear stress in solid-liquid interface eceeds te limit va l u e, t e vel o c i t y s l i occu r s i n e vi t a bl y. Teoretically, oil film in micro-scale is more likely to in solid-liquid interface tan in macro conditions, and is validated by eeriments [1, ]. Velocity will affect te sae and ressure distribution of oil film [], tereby affecting oil film flow, bearing caacity, bearing dynamic stiffness and oter erformances arameters of ydrostatic bearing, and more seriously, oil film is rone to collase and failure [4-6]. Janez Luse et al [7] etended boundary element metod for te discretisation of te incomressible Navier-Stokes equations in te velocity-vorticity form. Paer [8] also oints out tat because of te size effect of oil film tickness, a Corresonding autor: djcen@bjut.edu.cn macroscoic tribology law in a way is longer alicable. In secific friction conditions, material conventional roerties cant be used as te researc basis in te analysis of tribology erformance, terefore, te influence of size effect on te material roerties sould be fully considered [9]. At resent, te researc of te size imact on friction erformance and temerature rise of ydrostatic bearings as t been conducted etensively. Tus, it is very necessary to ut teory to te erformance calculation of ydrostatic bearing. In tis aer, te influence of size effect on te erformance of ydrostatic bearing is researced. First, te Navier velocity- boundary conditions are alied to fluid continuous flow equation, tus te Reylds equation is modified in micro-scale. Net, according to te rincile of oil film flow conservation and modified Reyld, bearing erformances are obtained. Ten, comaring te result wit n- model, te influences of size effect on ydrostatic bearing erformances are given. Tis rovides a new metod for furter researc on liquid flow in micro-scale. Analysis of oil film flow caracteristics in micro-scale.1 Model te oil film flow Hydrostatic bearing can be divided into two basic categories: recessed ydrostatic bearing and unrecessed ydrostatic bearing. Tis article uts four ockets recessed liquid radial ydrostatic bearing as te investigated object and its structure is sown in Figure 1. Lubricating oil is ressured into te bearing ocket

2 MATEC Web of Conferences troug te restrict orifice (as sown in Figure (a)), until te oil ocket is full, and ten discarges from circumferential, aial sealing surface and is returned to cannel. Te lubricating oil flow in bearing clearance is just like te flow in two late ga sown in Figure (b), and te coordinate system is establised as sown in Fig (b).1, reresent journal and bus surface and V reresents line seed of rotating sindle. X, Y, and Z resectively reresent circumferential, radial and aial direction of bearing oil film. Figure 1 Te structure of ydrostatic bearing (b) Figure Oil film flow in ydrostatic bearing clearance and coordinates establisment (a) Assuming lubricating oil is Newtonian fluid, lies in laminar flow state, and te ressure along film tickness direction remains constant. So te Navier Stokes equation can be simlified as Eq. (1) and te isoviscous -D Reylds equation is derived from it. y u u y y y y (1) y yz w w z y y y y were is oil film ressure, y and yz is oil film sear stress along X direction and Y direction, u and w is oil film seed along X direction and Z direction, is fluid viscosity. It requires fied boundary conditions to solve Eq. (1), namely reflecting te effect of micro-scale velocity boundary conditions.. Velocity boundary condition in two directions Wit te imrovement of testing tecniques in microna level, it was found tat te size effect of micro-scale makes it easier to te occurrence of velocity in te solid-liquid interface. Currently, a lot of researces on te model of velocity ave been made, tey are: Navier model [1], nlinear model [11], ultimate sear force model [4] and so on. Navier model is suitable for te occasion of lower searing rate and te most widely used at resent. In tis aer, Navier model is adoted to analyze oil film flow in ydrostatic bearing. In Navier model, it assumes tat fluid velocity is roortional to te sear stress, tat is v v b y s wall () were v is fluid velocity along X direction, Y is te rmal direction of te interface, b is called lengt, wic is te distance between actual surface and assumed surface, sown in Figure. Te size of lengt is determined by some attribute arameters suc as material roerties, sear rate and gas layer eisting in solid-liquid interface, and tese factors cant be simly couled [1], so lengt is very difficult to determined, generally cant be obtained by numberical calculation but eeriments. Li Xinming used a metod of teory combined wit ractice to study te metod of lengt measurement and establised a model [1]. According to [14] te lengt in tis aer ranges from ~7µm. Figure Navier mode According to Navier velocity boundary condition, velocity boundary condition of oil flow film on te sealing surface can be written as: u at y, u u b, w w b y s 1 s u at y, u u b, w w b y s z w y w y ()

3 NUMIFORM 16 were u is line-seed of rotating sindle; b 1, b are te lengt in X and Z directions of bus; b, b are z lengt in X and Z directions of journal.. Modified Reylds equation by velocity boundary condition Taken boundary condition into consideration, oil film velocity equations in te circumferential and aial direction are gained by te integration of Eq. (1). 1 b1 u y y b ( ) b1 u 1 w y y b z were is oil film tickness in aial direction wen te sear stress is zero. (4) Te volume of fluid flow troug unit widt is gained by integrating Eq. (4) along te way of film tickness, q udy b b u 1 1 b q wdy z 1 z Te modified Reylds is from mass conservation equations [15]: (5) In order to carry out calculations, Eq.(7) sould be dimensionless combined wit te yotesis of lengt. Caracteristic ressure is, caracteristic lengt and caracteristic oil film tickness are bearing radius clearance c, caracteristic aial lengt of oil film is alf te lengt of bearing L /, caracteristic aial lengt of oil film is radius R of bearing. Suose P, 6 w R, b bc, Hc, L z ( 1 1), R ( ), is osition angle of oil film, te dimensionless Reylds equation is as equation (8): 1 6b H 1 6b H P R P H L Calculation of bearing erformance.1 Te calculation of oil film flow (8) On te calculation of flow of ydrostatic bearing, te eisting studies do t take te imact of size effect into account, wic ave a great difference wit actual working conditions. To solve te roblem, tis aer takes orifice comensated ydrostatic bearing wit four ockets as researc object, and te main structural arameters is sown in Figure 4. Under te eternal load te sindle finally stays in te equilibrium osition O 1, te eccentricity of te sindle is e, deviation angle is θ, 1, is oil camber alf-widt and oil ad alf-widt, R is radius of sindle, P s is oil suly ressure. q q z z t (6) Assuming tat sindle rotated in steady state, modified Reylds equation is gained by bringing te equation (5) into Eq. (6): b b 1 u 1 b z 1 z It can be seen from te modified Reylds equation tat in te Z direction of journal is t introduced into te Eq.(7), tat is, te in te Z direction of journal does t affect oil film flow. However, some assumtions must be made about te equation for te furter study. Assumtions sould be based on current study and secific working conditions, suc as material conditions and fluid roerties alike. Sli lengt is assumed to be equal in isotroic material [15, 16]. Tis aer assumes b b1 b1 z b b z. (7) Figure 4 Structural arameters of ydrostatic bearing Wen te sindle occurs eccentric, oil film tickness is t constant, and it will cange as angular osition canges. Oil film tickness equation is given as follow: cos ecos 1 (9) 1 were is te initial film tickness, is te osition angle, is eccentricity ratio, e/.

4 MATEC Web of Conferences Combining wit te Figure 7, mass flow of eac oil ocket can be calculated: Q Q iao iro = Rwdyd 1 b =L 1 y y b dyd l b u u L 1 Q Q + Q io iao iro r z r l (1) were Q iao is volume of lubricating oil flow along aial sealing side, Q iro is te volume of lubricating oil flow along circumferential sealing side, L is te lengt of bearing, r is ustream midoint clearance of oil sealing side of a certain camber, l is downstream midoint clearance of oil sealing side of oil sealing side of a certain camber.. Calculation of bearing caacity Bearing caacity means wen oil film formats ressure, te syntetic ydraulic ressure floats te bearing and balances eternal load acting on te sindle, te syntetic liquid ressure is called load caacity. Te equation of load caacity is as follows: L W RP cos ddz L y Wy RPsin d dz (1) W W W were oil film ressure P is interval function of staging areas, wen te integration areas are te cambers( z, l,, 1 1,, 1 1,, 1 1, 1, 1 ), te film ressure P is P ri (i = 1,,,4), Wen te integration areas are return cutes, te oil film ressure P is zero, wen te integration areas are sealing surfaces, te oil film ressure P values as te linear distribution. According to above analysis, it is easy to determine bearing caacity. Tis aer rovides a convenient metod for designing te validation of ydrostatic bearing.. Calculation of bearing dynamic stiffness Figure 5 Oil flow out of te ydrostatic bearing camber Tis aer uses orifice restrictor. According to ydrodynamic equations, te flow troug orifice restrictor is [17]: d Qii = 4 s ri (11) were Qii is te flow tat flows into a certain camber troug orifice restrictor, is discarge coefficient (.6~.7), is te density of lubricating oil, d is te diameter of orifice restrictor. Assuming te oil cant be comressed. Te flow flowing into te bearing oil ocket troug restrictor is equal to te outflow troug sealing surface, tus te flow conservation equation is below: Qii Qio Qiro + Q iao (1) Dynamic stiffness of ydrostatic bearing is te ability to resist dislacement of te sindle. Bearing oil film stiffness is defined as: wen te ais as a tiny dislacement, te ratio of te increase of oil film force to te amount of dislacement. Te dislacement and is decomosed into two comonents (along X direction) and y (along Y direction), and te corresonding increase of oil film strengt can be decomosed into two comonents W and W. Four oil film stiffness coefficients are as follows: K K y W W y W, Ky, y Wy, Kyy y 4 Simulation of oil film erformance 4.1 Simulation of oil film ressure y (14) Te sensitivity factor of oil film flow 1 is defined as following:

5 NUMIFORM 16 1 Pma Pma P (15) ma were, Pma is te maimum ressure of oil film wen te occurs, Pma is te maimum ressure of oil film wen te does t aen. Figure 6 sows te variation of te dimensionless maimum ressure of four ockets P 1, P, P, P 4 wit te eccentricity in different level. It can be seen from Figure 6 tat, wit te eccentricity increasing, te maimum ressures of te oil ockets P 1, P, P 4 decrease, and te maimum ressure P increases. Wen te eccentricity is constant, te four maimum oil film ressures reduces wit te increase of lengt, but te reduction is constantly canging, te sensitivity of maimum oil film ressure can be used to describe te reduced degree of ressure and quantitatively analyzes te influence degree of on te maimum ressure. Figure 7 sows te variation of te ressure sensitivities of four ockets Δ P1, Δ P, Δ P, Δ P4 wit te eccentricity in different level. Te maimum ressure sensitivities are 59.1%, 1.14%, 7.54%,.1%. Comared Figure 6 wit Figure 7, it can be concluded tat wit te increase of lengt, te four maimum oil ressure sensitivities will increase accordingly. 4. Simulation of oil film flow were Q Q Q Q (16) is te oil film flow wen te occurs, Q is oil film flow wen te does t aen. Q 1, Q, Q, Q 4 are te dimensionless oil film flow of four ockets of te bearing. Figure 8 sows te variation of dimensionless oil film flow Q 1, Q, Q, Q 4 wit te eccentricity of te journal in different level. It can be seen from Figure 8 tat, wit te eccentricity increasing, te flow Q 1, Q, Q 4 increase, and te flow Q decreases. Wen te eccentricity is constant, te four flows increase wit te increase of lengt, but te increase rate is constantly canging, te sensitivity factor of flow can be used to describe te increase degree of oil film flow. Figure 8 Oil film flow along wit eccentricity ratio Figure 6 Maimum cavity ressure along wit eccentricity ratio Figure 9 Oil film flow sensitivities along wit eccentricity ratio Figure 7 Maimum cavity ressure sensitivities along wit eccentricity ratio Figure 9 sows te variation of te oil film flow sensitivities of four ockets Δ Q1, Δ Q, Δ Q, Δ Q4 wit te eccentricity in different level. Te maimum flow sensitivities are 84.14%, 18.%, 7.4%, 18.% and te minimum flow sensitivities are 4.67%, 4.81%,

6 MATEC Web of Conferences 16.9%, 5.1%. By comaring Figure 8 wit Figure 9, it can be concluded tat wit te increase of lengt, te four oil film flow sensitivities will increase accordingly. Figure 1 sows te relationsi between te total flow Q of te ydrostatic bearing and te rate of eccentricity in different case. Under different cases of, te total flow increases wit increasing eccentricity. Te largest relative increase rate of total flow is 57.78% wen te does t occur and te lengt is 7um, largest relative increase rate is.%, altoug te does t cange te relationsi between eccentricity and total flow, te aen of weakens te degree of increase of total flow. Wen te sindle eccentricity is constant, te flow increases wit increasing lengt, maimum total flow sensitivity was 1%. Tis means tat te occurrence of accelerate flow in bearing clearance, wic is conducive to ig temerature lubricant raid outflow of bearings, tere is more likely to ave a dynamic ressure effect on seal and imrove te bearing caacity. and tis sows it is necessary to researc te mecanism and influencing factors of in oil film clearance flow. Figure 11 Dimensionless bearing caacity along wit eccentricity ratio 4.4 Simulation of bearing dynamic stiffness Figure 1 Total flow along wit eccentricity ratio 4. Simulation of bearing caacity Te sensitivity factor of oil film flow is defined as following: were W W W W W (17) is bearing caacity wen te occurs, is bearing caacity wen te does t aen. Figure 11 sows te relationsi between te dimensionless bearing caacity and te eccentricity. It can be seen from Figure 9, wen occurs, bearing caacity increases wit increase of eccentricity, and te relationsi cant be canged by velocity occurred in solid - liquid interface. But wen te eccentricity is constant, bearing caacity increases wit te increase of lengt, te maimum sensitivity of bearing caacity is 81.94%. Terefore, considering te in te design of ydrostatic bearing is different from does t consider, Dynamic stiffness of ydrostatic bearing is te ability to resist dislacement of te fleible saft moving. In tis aer, an incremental metod is used to solve te four bearing dynamic stiffness K, K y, K y, K yy. Figure 1 sows te relationsi between bearing dynamic stiffness and eccentricity wen te does t occur. It can be seen from Figure 1, te four bearing dynamic stiffness as te same relationsi wit eccentricity; wit te eccentricity increase, te four dynamic stiffness decreases and roves te cross stiffness K y and K y are equal. Wen te eccentricity is constant, K is te largest, followed by te cross stiffness Ky and K y, K yy is te minimum. Te sensitivity factor of oil film flow 4 is defined as following: 4 D D D (18) were D is bearing dynamic stiffness wen te occurs, D is bearing dynamic stiffness wen te does t aen. Figure 1 sows te relationsi between four dynamic stiffness K, K y, K y, K yy and lengt wen te eccentricity is.. Four stiffness increase linearly wit lengt decreases, te maimum sensitivities are.%,.14%,.%,.6%, troug quantitative analysis sows tat te velocity as a certain influence on dynamic stiffness. Terefore, sould be taken into consideration in te design of ydrostatic bearing, esecially in te secial working conditions te dynamic stiffness is very imortant.

7 NUMIFORM 16 Ackwledgements Tis researc was funded by te National Natural Science Foundation of Cina Grant No , te Cina Nova rogram of Beijing and Natural Science Foundation of Beijing Municiality Grant No.145. References Figure 1 Dynamic stiffness of along wit eccentricity ratio Figure 1 Dynamic stiffness along wit lengt for ε=. 4 Conclusions Tis aer analyzes te roerties of two-dimensional oil film flow of ydrostatic bearing in micro-scale, and successfully gives flow regularities using Navier model. Te Reylds equation is modified by definite boundary conditions in te consideration of velocity. Finite difference metod is used to solve Reylds equation, and oil film ressure distribution, bearing caacity, oil film flow, bearing dynamic stiffness are gained. Sensitivity factors of oil film ressure, load caacity, flow and dynamic stiffness are deduced and te variation of tem wit te eccentricity in different lengt is sowed, and te result of te velocity reduces bearing dynamic caracteristics and stability to some etent is obtained. Tis roosed te sould be taken into consideration in te design of ydrostatic bearing, esecially in te secial working conditions te dynamic stiffness is very imortant. 1. Ol Vigradova, International Journal of Mineral Processing, 56,1-6 (1999).. G Zang, P Huang, etc., Tribology, 18,4-47 (1998).. G Zang Gu, HR Meng etc., Lubrication. Lubrication Engineering, 4,48-51 (1999). 4. C Wu, Journal of Dalian University of Teclogy,, (199). 5. P Huang,J Luo,S Wen, Acta Mecanica Sinica, 1, (1999). 6. Z Liu, Journal of Mecanical Engineering,47, 14-11(11). 7. J Luse, L Skerget, J Ravnik, Journal of Mecanical Engineering, 61,54-64 (15). 8. Q Xue, J Zang, Tribology, 14,6-69 (1994). 9. Z Zou, Cinese Journal of Mecanical Engineering, 9,-6 (). 1. C Navier, C Navier, Mem.acad.r.sci, 6,89-44(18). 11. P.A Tomson, S.M Troian,etc., Nature, 89,6-6(1997). 1. C. Neto, DR Evans., E Bonaccurso, etc., Reorts on Progress in Pysics, 68, (5). 1. LI Xin-.Ming, F.Guo, W Pat-Lam, Mocaue Xuebao/tribology,,4-9 (1). 14. L Erenwort, Journal of Fluid Mecanics. 489,55-77(). 15. CY Cen, QD Cen, WL Li, Tribology International, 61, (1). 16. WL Li, HM Cu, MD Cen, Tribology International, 9, , (6).