Effect of L/D Ratio on the Performance of a Four-Lobe Pressure Dam Bearing

Transcription

1 Vol:, No:8, 007 Effect of L/D Ratio on te Performance of a Four-Lobe Pressure Dam Bearin G Busan, S S Rattan, and N P Meta International Science Index, Mecanical and Mecatronics Enineerin Vol:, No:8, 007 wasetor/publication/6 Abstract A four-lobe pressure dam bearin wic is produced by cuttin two pressure dams on te upper two lobes and two relief-tracks on te lower two lobes of an ordinary four-lobe bearin is found to be more stable tan a conventional four-lobe bearin In tis paper a four-lobe pressure dam bearin supportin riid and flexible rotors is analytically investiated to determine its performance wen L/D ratio is varied in te rane 075 to 5 Te static and dynamic caracteristics are studied at various L/D ratios Te results sow tat te stability of a four-lobe pressure dam bearin increases wit decrease in L/D ratios bot for riid as well as flexible rotors Keywords Four-lobe pressure dam bearin, finite-element metod, L/D ratio I INTRODUTION HE present trend in te industry is to run te T turbomacines at i speeds Te ordinary circular bearins, wic are te most common type of te bearins, are found to be unstable at i speeds It is found tat te stability of tese bearins can be increased by te use of multilobes and te incorporation of pressure dams in te lobes Te analysis of multi-lobe bearins was first publised by Pinkus [] It was followed by Lund and Tomson [] and Malik [] et al, wo ave some desin data wic included bot static and dynamic caracteristics for laminar, as well as turbulent flow reimes Te experimental stability analysis of suc types of bearin []-[5] sowed tat te analytical stability analysis reflects te eneral trends in experimental data Te analytical dynamic analysis as sown tat noncylindrical pressure dam bearinsare found to be very stable L/D ratio is one of te important [6]-[9] parameters tat affects te stability of a bearin Te effect of L/D ratio on te stability of circular bearins was discussed by Lund [0], Badley [] et al and Hori[] Te effect of L/D ratio on te performance of two-lobe and tree-lobe pressure dam bearins was studied by Meta [] and Rattan [] respectively Te present study is undertaken to investiate Manuscript received October 5, 007 G Busan is wit te National Institute of Tecnoloy, Kuruksetra- 69, Haryana, India (pone: 9-7-6; fax: ; e- mail: aroraian@ yaoocom) SS Rattan is wit te National Institute of Tecnoloy, Kuruksetra- 69, Haryana, India ( ss_rattan@otmailcom) NP Meta is Director, MME, Mullana, Ambala, Haryana, India ( npmeta@yaoocoin) te effect of L/D ratio on te performance of a four-lobe pressure dam bearin supportin riid and flexible rotors II NOMENLATURE c : radial clearance c m : minimum film tickness for a centered saft,,, : oil-film dampin coefficients xy yx yy xy, yx, yy, : dimensionless oil-film dampin coefficients, ( ω c / W ) 0,,,, : coefficients of te caracteristic equation D : diameter e : eccentricity F : dimensionless saft flexibility, W / ck : oil-film tickness, c ( cosθ ) : dimensionless oil-film tickness, / c k : saft stiffness K, K, K, K : oil-film stiffness coefficients xy yx yy K K xy, K yx, K yy, : dimensionless oil-film stiffness coefficients, K K ( c / W ) L : bearin lent N : journal rotational speed O i : lobe center of lobe i (i,,,) p : oil-film pressure R : journal radius μnld R S : Sommerfeld no, W c V : periperal speed of journal W : bearin external load x, z : coordinates for bearin surface (xperiperal, z-alon saft axis) φ : attitude anle α : wirl rate ratio, α φ / ω International Scolarly and Scientific Researc & Innovation (8) scolarwasetor/07-689/6

2 Vol:, No:8, 007 International Science Index, Mecanical and Mecatronics Enineerin Vol:, No:8, 007 wasetor/publication/6 β : squeeze rate ratio, β / ω : eccentricity ratio, e / c δ : ellipticity ratio, ( c m / c) θ : anle measured from te line of centers in te direction of rotation θ : oil-roove anle ρ μ ω v : fluid density : averae fluid viscosity : rotational speed : dimensionless tresold speed, / ω ( c / ) : ravitational acceleration const III ANALYSIS Te Reynolds Equation for te laminar flow is: x μ x z μ z x Te above equation is non-dimensionalized by makin te followin substitutions: Te non-dimensionalized equation tus obtained is: p x x p π α sin x Te various assumptions made in te derivation of te Reynolds Equation are tat te fluid is Newtonian, no slip occurs at te bearin surface, inertia terms are nelected, oil viscosity is constant and curvature is neliible Te Reynolds equation is analysed for a pressure profile usin te finite element metod[5] Te eometry of a four-lobe pressure dam bearin is sown in Fi A four-lobe pressure dam bearin is produced by cuttin two pressure dams on te upper two lobes and two relieftracks on te lower two lobes of an ordinary four-lobe bearin Te various eccentricity ratios and attitude anles of te lobes of te four-lobe pressure dam bearin are iven by: π β cos x D L x z πp c x, z,, p, D x x L ce,, μω φ R θ x α and β R D c ω ω p 6Rω φ sinθ e cosθ () δ δ cos( π / φ) δ δ sin( π / φ) p z z δ δ sin( π / φ) δ δ cos( π / φ) π x () () φ 5π / γ sin φ 7π / γ sin φ π / γ sin φ π / γ sin ( (sinπ / φ) ( (cosπ / φ) ( (cosπ / φ) ( (sin π / φ) Fi A four-lobe pressure dam bearin Eac lobe of te bearin is analysed separately Since te pressure profile as to be symmetrical about te bearin centre line, only alf of te lobe is taken for analysis Fluid pressures at nodal points are taken by applyin te Reynolds boundary conditions Te mes size used is x 5 for eac lobe Te resultin matrix is stored in a banded form Ten it is solved by te Gauss-elimination metod Stiffness and dampin coefficients are determined separately for eac lobe and ten added Te values of tese stiffness and dampin coefficients, saft flexibility, and dimensionless speed are ten used to evaluate te coefficients of te caracteristic equation, wic is a polynomial of te 6t order for flexible rotors Tis caracteristic equation as been taken from Han [6] and is obtained from te eneral case of an eccentrically mounted rotor on a flexible saft s s 6 Te caracteristic equation is: F v were 5 v 0 s v ( F F ) ( v F F0 v v F ) ( F ) s ( Fv v ) s v Step Relief track s 0 Lobe Oil supply R o φ o θ o o Step o o Lobe Lobe W 0 Lobe Relief track () International Scolarly and Scientific Researc & Innovation (8) scolarwasetor/07-689/6

3 Vol:, No:8, 007 o yy xy yx 0 K yy K K yy K yy K yy K yy K xy K yx K xy yx K yx xy (5) EENTRIITY RATIO L/D 075 L/D 0 L/D 5 International Science Index, Mecanical and Mecatronics Enineerin Vol:, No:8, 007 wasetor/publication/6 For a riid rotor, te value of F (dimensionless flexibility) is taken as zero Te value of F for most of te practical rotors vary from 05 to 0 and te same as been considered for te case of flexible rotors Te system is considered as stable if te real part of all roots is neative For a particular bearin eometry and eccentricity ratio, te values of dimensionless speed are increased until te system becomes unstable Te maximum value of speed for wic te bearin is stable is ten adopted as te dimensionless tresold speed Te present analysis as been done for te bearin wit te followin parameters wic ave been optimized for te best stability[7] S d 0, L d 08, L t 05, θ s 55, θ 0 Te ellipticity ratio ( δ ) 05 is selected for te present study Te value of L/D ratio is varied from 075 to 5 and te bearin is investiated for its static and dynamic caracteristics IV RESULTS AND DISUSSION Te effect of L/D ratio on te static caracteristics of a four-lobe pressure dam bearin is sown in Fis to 6 Te values of L/D ratios considered for tis purpose are 075, 0 and 5 It is observed from te Fis and tat wit te increase in L/D ratio, eccentricity ratio decreases wereas, te attitude anle increases for a particular value of Sommerfeld number Te minimum film tickness is observed to increase wit an increase in L/D ratio, wen considered for a particular value of Sommerfeld number (Fi ) Fis 5 and 6 sow te effect of L/D ratio on oil-flow and friction coefficients Tere is a considerable fall in te oil-flow, wereas tere is no sinificant cane in te friction coefficient wit te increase in L/D ratio wen considered for a particular value of Sommerfeld number Te effect of L/D ratio on te stability of a four-lobe bearin supportin a riid rotor is sown in Fiure 7 Te plots sow tat bot te zone of infinite stability and te minimum tresold speed increase wit decrease in L/D ratio Te zone of infinite stability increases from 06 to 0 and te minimum tresold speed from 9 to 865 wen L/D ratio decreases from 5 to 075 Tese effects are due to reduction in load carryin capacity of te bearin ATTITUDE ANGLE MINIMUM FILM THIKNESS Fi Effect of L/D ratio on eccentricity ratio L/D 075 L/D 05 L/D Fi Effect of L/D ratio on attitude anle L/D 075 L/D 0 L/D Fi Effect of L/D ratio on minimum film tickness International Scolarly and Scientific Researc & Innovation (8) scolarwasetor/07-689/6

4 Vol:, No:8, 007 FLOW OEFFIIENT L/D 075 L/D 0 L/D reduced wit te increase in flexibility of te rotor wile tere is no cane in te zone of infinite stability 00 0 Stable Unstable L / D 075 L / D 0 L / D 5 International Science Index, Mecanical and Mecatronics Enineerin Vol:, No:8, 007 wasetor/publication/6 FRITION OEFFIIENT Fi 5 Effect of L/D ratio on flow coefficient L/D 075 L/D 0 L/D Fi 6 Effect of L/D ratio on friction coefficient FOUR-LOBE PRESSURE DAM BEARING 0 0 Fi 7 Effect of L/D ratio on te stability of a four-lobe dam bearin supportin a riid rotor (F0) L/D 075 L/D 0 L/D 5 pressure Fis 8 and 9 sow te effect of L/D ratio on te stability of a four-lobe bearin supportin flexible rotors Te results are found to be similar to tat of te bearin supportin a riid rotor It is also observed from tese plots of te stability tat for a particular L/D ratio te minimum tresold speed is 0 0 Fi 8 Effect of L/D ratio on te stability of a four-lobe dam bearin supportin a flexible rotor (F05) 00 0 Stable Unstable L / D 075 L / D 0 L / D 5 pressure 0 0 Fi 9 Effect of L/D ratio on te stability of a four-lobe pressure dam bearin supportin a flexible rotor (F0) V ONLUSION Te eccentricity ratio decreases wile attitude increases wit an increase in L/D ratio Te minimum oil-film tickness increases wit an increase in L/D ratio Te oil flow coefficient decreases wit an increase in L/D ratio Te friction coefficient remains almost uncaned wit an increase in L/D ratio 5 Bot te minimum tresold speed and te zone of infinite stability increase wit decrease in L/D ratio for a four-lobe bearin supportin riid rotor as well as flexible rotor Terefore te stability of a four-lobe pressure dam bearin increases wit decrease in L/D ratio International Scolarly and Scientific Researc & Innovation (8) scolarwasetor/07-689/6

5 Vol:, No:8, For a particular L/D ratio te minimum tresold speed reduces wit increase in te rotor flexibility wile te zone of infinite stability remains uncaned International Science Index, Mecanical and Mecatronics Enineerin Vol:, No:8, 007 wasetor/publication/6 REFERENES [] O Pinkus, Analysis and aracteristics of Tree-Lobe Bearins, ASME Journal of Basic Enineerin, vol 8, pp 9, 959 [] J W Lund, and K K Tomson, A alculation Metod and Data for te Dynamic oefficients of Oil Lubricated Journal Bearins, Proceedins of te ASME Desin and Enineerin onference, Minneapolis, pp, 978 [] M Malik, Maes andra and R Sinasan, Desin Data for Offset- Halves Journal Bearins in Laminar and Turbulent Flow Reimes, ASLE Trans, pp -0, 98 [] J Nicolas, L E Barrett, and M E Leader, Experimental Teoretical omparison of Instability Onset Speeds for a Tree Mass Rotor Supported by Step Journal Bearin, Trans ASME, Journal of Mecanical Desin, pp -5, 980 [5] R D Flack, M E Leader, and E J Gunter, An Experimental Investiation on te Response of a Flexible Rotor Mounted in Pressure Dam Bearins, Trans ASME, Journal of Mecanical Desin, pp 8-850, 980 [6] N P Meta, A Sin, and B K Gupta, Dynamic Analysis of Finite Half-Elliptical Pressure Dam Bearins wit Rotor Flexibility Effects, ASLE Trans, vol 9, no, pp 6-66, 986 [7] N P Meta, and A Sin Stability of Finite Ortoonally-Displaced Pressure Dam Bearins, ASME Journal of Triboloy, vol 09, no, pp 78-70, 987 [8] N P Meta, and S S Rattan, Performance of Tree-Lobe Pressure Dam Bearins, Wear, pp 8-85, 99 [9] N P Meta, S S Rattan, and G Busan, Static and Dynamic aracteristics of Four-Lobe Pressure Dam Bearins, Triboloy Letters, vol 5, no, pp 5-0, 00 [0] J W Lund, Self-excited stationary wirl orbits of a journal in sleeve bearins, PD Tesis, Rensselaer Polytecnic Institute, NY, 966 [] R H Badley, and J F Booker, Turborotor Instability: Effect of Initial Transients on Plane Motion, Trans ASME, Journal of Lubrication Tecnoloy, pp 65-6, 969 [] Y Hori, A Teory of Oil Wip, Trans ASME, Series E, pp89, 959 [] N P Meta, A Sin, and B K Gupta, Stability of Finite Elliptical Pressure Dam Bearins wit Rotor Flexibility Effects, ASLE Trans, vol9, no, pp , 986 [] S S Rattan, Stability analysis of pressure dam bearins usin finite element metod, PD Tesis, Kuruksetra University, 995 [5] G Busan, S S Rattan, and N P Meta, Hydrodynamic Lubrication Analysis of Four-Lobe Pressure Dam Bearins by Finite-Element Metod, Proceedins nd International and 9 t National onference on Fluid Mecanics and Fluid Power, IIT Roorkee, pp 8, 00 [6] E J Han, Te Excitability of Flexibility Rotors in Sort Sleeve Bearins, Trans ASME, Journal of Lubrication Tecnoloy, pp 05, 975 [7] G Busan, S S Rattan, and N P Meta, Stability Analysis of Four-Lobe Pressure Dam bearins, Triboloy Letters, vol, no, pp -7, 00 International Scolarly and Scientific Researc & Innovation (8) scolarwasetor/07-689/6