Science, Juliana Engineering Javorova, & Education, Anelia Mazdrakova 1, (1), 16, 11- Squeeze film effect at elastoydrodynamic lubrication of plain journal bearings Juliana Javorova *, Anelia Mazdrakova University of Cemical Tecnology and Metallurgy, 8 Kl. Oridski, 1756 Sofia, Bulgaria Received 15 Marc 16, Accepted 31 October 16 ABSTRACT Te aim of te present work is to study te influence of te elastic displacements of bearing surface coatings on te ydrodynamic lubrication of dynamically loaded journal bearing wit finite lengt. Te problem is investigated for a Newtonian lubricant under isotermal and isoviscous conditions. Te bearing sleeve is covered wit a tin elastic coating wose radial distortions are of te same order of magnitude as te film tickness. Te elasticity part of te problem is studied in conformity wit te Vlassov model of an elastic foundation and comparisons wit oter models are carried out. Te numerical solution is done by finite difference metod wit successive over-relaxation procedure. Te presented results are obtained for prescribed loci of te saft centre wic corresponds of squeeze film effect. Keywords: elastoydrodynamic (EHD) lubrication, squeeze film effect, journal bearing. INTRODUCTION Nowadays, te need for iger speed and at te same time reliable operation of various types of rotating macinery continues to grow. An important factor in acieving tis goal is te ability to accurately predict te dynamic response and stability of te rotor-bearing system [1]. Eac rotating macine supported by one or more bearings, wose role is very important in te system as a wole, because namely te bearings are te components tat allow relative movement between te stationary and moving parts. As it is well known, tere are two main types of bearings wic are normally used in applications of te rotor-bearing systems. Tese are different types of journal bearings and rolling-elements bearings. Journal bearings are used widely and successfully for tousands of years and may be tey are te most used macine element in our civilization [1]. Currently large numbers and various types of sliding bearings are used in countless applications suc as small electric motors, ard disk drives, micro-electro-mecanical systems, automotive and aircraft piston engines, large steam turbines for electric power generation, etc. For dynamically loaded journal bearings are typical te squeeze film and dynamic film effects wic play significant role in te dynamic beaviour of te rotor. Since te tin oil film tat separates te moving surfaces supports te rotor load, it acts as a spring and provides a damping effect due to te squeeze film effect. Investigations in tis direction ave been made * Correspondence to: Juliana Javorova, University of Cemical Tecnology and Metallurgy, 8 Kl. Oridski, 1756 Sofia, Bulgaria, E-mail: july@uctm.edu 11
Science, Engineering & Education, 1, (1), 16 by many autors in recent decades [-7]. Te importance of te problem implies a penomenon studying in combination wit oter factors influencing te lubrication process of te ydrodynamic (HD) journal bearings. It is well-known tat elastic deformation of te material of te contact bearing surfaces, induced by te HD fluid pressure, can significantly cange te fluid film profile, modify te pressure distribution and, terefore, alter te performance caracteristics of journal bearings [8]. Te use of surface coatings (on bus or/and on journal) suc as wite metals and various elastomers can lead to significant deformations of te coated surface wic can be about te size of tickness of lubricating film. Tese coatings, used wit te aim of reducing wear, are generally caracterized by low values of te modulus of elasticity [9]. Te effect of te deformation of te bearing bus on te journal bearing performance caracteristics was reported by many investigators [8-13]. Many of tese papers are related to infinitely long/sort bearings and as wole te comparison between different formulas for elastic deformations calculations is absent. In tis work, we are interested in te study of te influence of te elastic displacements of bearing surface coatings on te HD lubrication of dynamically loaded journal bearing wit finite lengt. Te problem is investigated for a Newtonian lubricant under isotermal and isoviscous conditions. Te elasticity part of te problem is studied in conformity wit te Vlassov model of an elastic foundation and comparisons wit oter models are carried out. Te numerical solution is done by finite difference metod wit over-relaxation procedure; suc as te presented results are concerned toprescribed loci of te saft centre resulting from te squeeze film effect. ELASTOHYDRODYNAMIC MODEL Te cross section of a 36 degree plain journal bearing is presented in a Fig. 1. Te main components of te considered tribological system, 1 used in te ydrodynamic teory of lubrication, are te bearing saft (journal), te lubricant and te bearing busing wic is plain cylindrical sleeve wrapped around te journal. Te bearing bus is rigidly supported wile te journal moves and in our case it is dynamically loaded in a radial direction. About te journal motion it will be assumed squeeze film effect, i.e. a vibration velocity de / dt of te saft centre in a direction of te centre s line C L (Fig.1). Its frequency is equal to te angular velocity of te saft, wic is considered to be constant and equal to ω. Te amplitude of vibration will be about to te alf of te radial clearance c. Hence e= e +,6csinωt, as e =, c. Bot of te radiuses (of bus and of journal) are approximately equal because of wic it is possible to neglect te curve sape of te fluid film. An elastic liner wit elastic properties µ and E (Poisson s ratio and Young s modulus, respectively) is press-fitted in a rigid ousing. Te liner tickness d is assumed to be of te same order of magnitude as te lubricant tickness. Oter assumptions for te present analysis are: te journal is rigid; te lubricant wit Newtonian properties is incompressible fluid under isoviscous and isotermal conditions; te flow is laminar and te inertia effects are negligible. Fig. 1. Journal bearing geometry.
Juliana Javorova, Anelia Mazdrakova Te governing equations for te elastoydrodynamic (EHD) lubrication of a dynamically loaded plain journal bearing wit finite lengt are: te equation for HD pressure distribution (Reynolds equation), equation for film tickness geometry and elasticity equation (for estimation of te elastic distortions of bearing liner). Derivation of te Reynolds equation Our interest in tis section is a derivation of an equation for te pressure field witin a dynamically loaded bearing. At neglecting te body and inertia forces te Navier-Stokes equations, governing te motion in te lubricant, for incompressible viscose lubricant flow are: = p+ η V, (1) were p is te ydrodynamic pressure, η is te lubricant dynamic viscosity, V is te fluid velocity. Te flow field model is completed by te incompressible fluid continuity equation V = () It is well known tat one of te main assumptions for te journal bearings lubrication teory is tat te fluid film is tin compared wit te journal radius and te curvature of te film is neglected. Ten te field equations (1) and () given in Cartesian coordinates can be reduced to te following: u = y η x = η y (3.a) (3.b) were uvw,, are te velocity components on te Cartesian rectangular coordinates xyz.,, Te boundary conditions for te velocities components are at y = : u = v = w = (4) at y = de dγ u = u = ωr+ sinθ e cosθ dt dt de dγ v= v = ωr + cosθ + e sinθ x dt dt (5.a) (5.b) w= w = (5.c) Here is te fluid film tickness, ω is te saft angular velocity, r is radius of te journal, e is te bearing eccentricity, θ = x/ ris a circumferential coordinate, γ represents attitude angle, t is te time. Integrating Eqns. (3.a) and (3.c) wit respect to y and employing te above boundary conditions, for te velocity components are obtained: u= y + C1( xzt,, ) y+ C( xzt,, ) η x (6) = + (,, ) + (,, ) η z (7) w y C3 xzt y C4 xzt were C 1, C and C 3, C 4 are integration constants. At satisfying te boundary conditions (4) and (5) te integration constants are determined as follows: u C1 ( xzt,, ) = η x ( ) C xzt,, = (8) w = y η z u v w + + = x x x (3.c) (3.d) C3 ( xzt,, ) = η z ( ) C4 xzt,, = (9) After substitution of (8) and (9) in (6) and (7) respectively, te expressions for te velocity components in direction of coordinate axes x and 13
Science, Engineering & Education, 1, (1), 16 z are received in te form: u( xzt,, ) = ( y ) y+ y η x (1) w( xzt,, ) = ( y ) y η z. (11) u is te sum of te flow due to circumferential pressure gradient p/ x and te flow due to te no-slip boundary conditions. Note furter tat te pressure gradients p/ x and p/ zare independent of y. For te considered case it is assumed a linear distribution of te radial velocity v( xyzt,,, ) using te formulation in [14] and [4]: (,, ) (,, ) v= C5 xzt y+ C6 xzt (1) At satisfying te boundary conditions (4) and (5) te integration constants are obtained as follows: ( xzt) v C5,, = = u 1 de dγ = ωr + cosθ + e sinθ x dt dt. C6 ( xzt,, ) = (13.a) (13.b) Tan te final form for tis velocity component is given by: v vxzt (,,)= y= 1 de dγ = ωr + cosθ + e sinθ y x dt dt (14) After integration of continuity equation (3.d) wit respect to y from y = to y = u v w dy + dy + dy = x y z (15) it is received u w dy + v + dy = z x (16) Applying Leibniz s rule for te differentiationunder te integral sign, ( ) x f ( y, x) dy = x ( ) x = (, ), f y x dy f x x ( ) d dx te first and te tird terms on te left side of Eq. (16) are transformed to: u dy = udy u x x x w dy = wdy w z z z (17) (18) Since w = from te boundary conditions (5.c) it follows tat (18) becomes: w dy = z z wdy (19) Substitution of (17) and (19) in (16) y ields. udy u + v + wdy = () x x z In te considered case of unsteady state motion of te lubricant te following equals are valid [4]: de dγ = cosθ + e sinθ t dt dt u v de cos e d γ + = θ + sinθ x dt dt o (1.a) (1.b) 14
Juliana Javorova, Anelia Mazdrakova Considering (1.a) and (1.b) follows: = u + ν t x (1.c) Taking into account (1.c) te above equation () takes a form: + + = udy wdy x t z o At introducing te flow parameters wic are presented respectively as qx q, x () q z = udy (3) qz = wdy (4) te Eq. () takes a form: qx qz + + = x z t (5) Te expressions of te velocity components (1) and (11) can be placed respectively in integrand functions of (3) and (4); suc te flow parameters become: 3 3 x = = + 6η x 4η x q udy 3 1 p + u = + u 1η x 3 3 z = = = 6η z 4η z q wdy 3 p = 1η z (6) (7) Te derivatives of flow parameters wit respect to axes x and z are obtained at rendering into account (6) and (7): qx 1 3 p 1 = + ( u ) x 1η x x x (8) qz 1 3 p = z 1η z z (9) Te obtained expressions (8) and (9) subsequently are substituted in (5), wic is representing as: 1 3 p 1 3 p + = 1η x x 1η z z 1 = ( u ) + x t (3) For te first term on te rigt-and side of above equation can be written [4] ( ) u u = u + ωr x x x x Ten te Eq. (3) represents as 3 3 p p + = x η x z η z (31) 6ω = + 1 r (3) x t In te literature [3-5] an equation from te type of (3) is known as Reynolds type equation for pressure field witin a bearing or for ydrodynamic pressure distribution in te lubricant film of finite lengt journal bearing under unsteady state conditions. Tis equation differs from te classical Reynolds equation in te last term on te rigt-and side only. It takes into account te pressure canges due to te temporal variation of te film tickness. Film tickness geometry and elasticity equation Te approac used in te present study aims to superimpose te deformation of te layer on te bus (te oter components of te bearing and te 15
Science, Engineering & Education, 1, (1), 16 journal are treated as rigid), caused by generated HD pressure, onto te oil film tickness. Te gap tickness is ten modified in order to be account for te estimated elastic deformation as follows: = c+ ecosθ + δ (33) were c is a radial bearing clearance and δ is te surface points radial displacement of te bus liner. Te last term of tis equation takes into account te influence of te elastic layer deformations. In te current paper te liner s surface points radial displacements are determined according to te one of more precise models, namely te tree-dimensional model of elastic foundation suggested in [15]. Te solution of te elasticity part of te problem by Vlassov but for te case of tin layer is worked up in details in [16], because of wic only te final form is given ere, namely δ = ( 1 µ )( 1 µ ) E ( 1 µ ) d p (34) were d is a tickness of te bearing liner, µ is te Poisson s ratio, E is te Young s modulus. Furtermore it will be made a comparison of te results for elastic deformations calculated by oter formulas: one of tem by Higginson [8, 17] µ d δ = 1 p 1 µ E (35) and te oter by Kodnir (Winkler-Zimmerman ypotesis) [18, 19] ( µ ) 1 d δ = p (36) π E Load carrying capacity Te resultant force W, wic is balanced by te load applied to te saft, is given by: W = W + W 1 (37) werew 1 andw are te total components in radial (along te line of centers) and tangential directions respectively. Te Sommerfeld number, wic represents a coefficient of load capacity, may be express quantitative as W β S = ηω rl (38) L is te bearing axial lengt wile te clearance ratio. β = c/r is SOLUTION PROCEDURE Non-dimensional form of equations Te solution of Eq. (3) will be done by numerical metods. By reason of tis and also for keeping te conditions of geometric and kinematic similarity at te solution, tis equation must be represented in a non-dimensional form 3 3 1 ch Π 6ηωr + r θ η θ rc 3 3 1 ch Π 6ηωr L z 1 η z1 ( L ) c c H cω H 6ωr + 1 r θ τ = (39) by introducing te following dimensionless variables: Π= p. ( c/ r) /6ηω - pressure, H= c - film tickness, τ = t. ω /- time, z1 = z ( L/) - axial coordinate After transformations te dimensionless form of te equation for HD pressure distribution becomes: Π Π θ 3 3 H + α H = θ z1 z1 H H + θ τ (4) 16
Juliana Javorova, Anelia Mazdrakova were α = r/ L - diameter to lengt ratio and H = εsinθ θ H ε γ = cosθ + ε sinθ τ τ τ (41) Te Eq. (33) for film tickness can be rendered dimensionless troug te relevant substitutions and written as: H = 1+ ε cosθ + δ (4) Here ε = ecis te eccentricity ratio and δ are dimensionless displacements. Furtermore te radial distortions of te bearings liner surface points /Eq. (34)/ are presented in non-dimensional form as: ( 1 µ )( 1 µ ) 6ηωr d δ = Π 3 c E ( 1 µ ) (43) Te ydrodynamic film forces acting on te system are expressed in terms of dimensionless quantities by 1 π W = Πcosθ dθdz W 1 1 1 1 π = Π sinθ dθdz 1 1 (44) were W1 and W are radial and tangential components. Ten te resultant dimensionless load carrying capacity W can be calculated by β W = W1 + W = W (45) 6ηωrL Te Sommerfeld number in present analysis is equal to: S = 6W (46) Numerical solution Te computation is done using te finite difference metod, as te numerical procedure is iterative and te successive over-relaxation metod is used to improve te convergence rate [4, 1, 16]. About te pressure are used te Reynolds boundary conditions; suc in te circumferential direction it is assumed tat te positive pressure terminates at * θ were te pressure gradient angle is zero. Prior to starting te iteration process, te film pressure is initially determined considering te bearing liner as rigid one. Wit te elp of tis pressure te non-dimensional deformation is computed. Wit tis deformation te Reynolds equation (4) is solved again to get a new film pressure distribution. Tis process is repeated troug iteration tecnique till convergence is acieved. Wit te elp of tis film pressure after convergence, te bearing performance caracteristics are obtained. NUMERICAL RESULTS AND DISCUSSION Te presented results are obtained at different elasticity parameters ( ). Te later are related 11 to four separate cases: E =.1 Pa, µ =, 5 (steel 8 - rigid case); E = 1, 63.1 Pa, µ =,38 (elastomer - 7 soft case 1); E = 7, 33. 1 Pa, µ = 4, (elastomer 7 - soft case ); E = 4, 7.1 Pa, µ =, 41 (elastomer - soft case 3). In Figs. and 3 are given te longitudinal centre-line distributions of te HD pressure, film tickness and radial displacements of te bearing liner surface points. As it was expected, for te soft cases a significant reduction of te maximum HD pressure is observed wile for te radial displacements te effect is reversed. Tis influence is a more visible for te softer materials of te elastic liner. Under deformability conditions te fluid film geometry is canged in te bottom; suc a increase of H in te middle section is observed. δ 17
Science, Engineering & Education, 1, (1), 16 Fig.. Distribution of HD pressure Π and film tickness H under different elasticity conditions. Fig. 3. Displacements distribution at te bearing centre line. Te results in Figs. 4 and 5 about time variation of HD pressure (caused by te vibration velocity de / dt of te saft centre - squeeze film effect) sow te same tendency for reduction of maximal values of Π as above. Te grapics presented in a Fig. 5 give results for pressure obtained by different formulas for elastic deformations calculations. Te influence of te elasticity layer deformability on te Sommerfeld number is displayed in a Fig. 6. Analogically as te pressure te S values becomes smaller for te soft case; suc tis tendency can be recognized more clear for te maximal values of te coefficient of load capacity. Fig. 4. Time variation of Pwit radial velocity under different elasticity conditions. Fig. 5. HD pressure versus time calculated by different formulas for deformations. 18
Juliana Javorova, Anelia Mazdrakova REFERENCES Fig. 6. Variation of Sommerfeld number wit time under different deformability conditions. CONCLUSIONS On te base of Navier Stokes and continuity equations te generalized Reynolds equation for lubrication of dynamically loaded journal bearing wit finite lengt is derived in details. Te numerical results for squeezed film conditions sow tat te maximum pressure for elastic bearing liner is smaller tan tat for bearing wit rigid layer, and tis effect is more significant for te softer liner materials. A similar trend was observed by oter investigators for steady-state or transient problems of journal bearings. Deformability of te layer modifies and film sape; suc appreciable increase in minimum film tickness for more flexible bearing liner is observed and it is also a favorable effect from te designer s viewpoint. Te differences between te results obtained by different formulas for elastic deformations calculations are not so ig (till 8%) and all tese metods are applicable and ave own significance. 1. U.Yucel, Calculation of Dynamic Coefficients for Fluid Film Journal Bearings, Journal of Eng. Sciences, 11, (3), 5, 335-343.. J. Lin, Squeeze Film Caracteristics of Finite Journal Bearings: Couple Stress Fluid Model, Tribology International, 31, 4, 1998, 1-7. 3. A. Szeri, Fluid film lubrication, Cambridge Univ. Press, 1st Ed., Cambridge, U.K., 1998. 4. J. Javorova, P.D. Tesis, One solution of te non-stationary problem of te elastoydrodynamic teory of lubrication, University of Transport, Sofia, 1998, (in Bulgarian). 5. B. Hamrock, S. Scmid, B. Jacobson, Fundamentals of Fluid Film Lubrication, nd Ed., Marcel Dekker Inc, N.Y., 4. 6. Z.Guo, T. Hirano, R. Kirk, Application of CFD Analysis for Rotating Macinery - Part I: Hydrodynamic, Hydrostatic Bearings and Squeeze Film Damper, J. Eng. Gas Turbines Power, 17,, 5, 445-451. 7. G.J. Reddy, C.E. Reddy, K.R. Prasad, Effect of viscosity variation on te squeeze film performance of a narrow ydrodynamic journal bearing operating wit couple stress fluids, Proc. of Inst. of Mec.Eng, Part J, J. of Eng, Tribology,,, 8, 141-15. 8. G. Higginson, Te teoretical effects of elastic deformation of te bearing liner, Proc. Mec. Eng., vol.18, 1966, 31-38. 9. L. Dammak, E. Hadj-Taieb, Finite Element Analysis of Elastoydrodynamic Cylindrical Journal Bearing, Fluid Dyn. Mater. Process, 6, (4), 1, 419-43. 1. T. Osman, Effect of lubricant non-newtonian beaviour and elastic deformation on te dynamic performance of finite journal plastic bearings, Tribology Letters, 17, (1), 4, 31-4. 11. Y. Ma, Performance of dynamically loaded journal bearings lubricated wit couple stress fluids considering te elasticity of te liner, J Zejiang Univ Sci A, 9, (7), 8, 916-91. 19
Science, Engineering & Education, 1, (1), 16 1. H. Attia, S. Bouaziz, M. Maatar, T. Fakfak, M. Haddar, Hydrodynamic and elastoydrodynamic studies of a cylindrical journal bearing, Journal of Hydrodynamics,, (), 1, 155-163. 13. S.Gua, On te Steady-State Performance of HD Flexible JB of Finite Widt Lubricated by Ferro Fluids wit Micro-Polar Effect, Int. J. Mec. Eng. & Rob. Res., 1, (), 1, 3-49. 14. A. Kelzon, J. Tcimanskii, V. Jakovlev, Dynamic of rotors in elastic bearings, Nauka, Moscow, 198,(in Russian). 15. V. Vlassov, U. Leontiev, Beams, plates and sells on elastic foundation, Moscow, 196, (in Russian). 16. J. Javorova, V. Alexandrov, K. Stanulov, Static and dynamic performance of EHD journal bearings in turbulent flow, Proc. of te 3rd Intern. Sci. Conf. Power Transmissions 9, Tessaloniki, Greece, 9, 453-46. 17. D. Dowson, G. Higginson, Elastoydrodynamic Lubrication, nd Ed., Pergamon Press, Oxford, 1977. 18. D. Kodnir, Contact ydrodynamic of macine details lubrication, Moscow, 1976, (in Russian). 19. V. Tiscenko, Models of layer deformation of an elastic alf space, Elsevier Journal of Soviet Matematics, 65, (1), 1993, 1396-14.