Effects of Micro-polar Fluids and the Tsann Roughness Model on Performance Characteristics of Two-lobe Bearings

Transcription

1 Vol- Issue- 6 IJARIIE-ISSN(O) Effects of Micro-polr Fluids nd the Tsnn Roughness Model on Perfornce Chrcteristics of Two-lobe Berings Rohollh Shfei, Mohd Hdi Shfei Mr., Engineering Deprtent, Yd University, Yd, Irn Mr., Engineering Deprtent, Shir University, Shir, Irn ABSTRACT In the lubriction theories, the bering surfces re generlly considered s sooth nd without ny roughness but ctully these re not ccurte. In order to find the bering perfornce preters it is required to solve the lubriction eqution. Without necessry siplifictions, solving such eqution is terribly difficult. By dding the roughness effects to the proble, the necessity of using nuericl ethods becoe ore obvious. The surfce irregulrities ply n iportnt role in the bering echnicl behviors. Such roughness, cuse conversion in lubricted fil thickness tht effect on lubriction preters. The considered fluids in this investigtion were incopressible icro polr fluids tht contin ingredients, which will endure the physicl stresses nd oents. In this rticle, using the finite eleent ethod, the effects of surfce roughness on the lubriction of two-lobe berings perfornce were proposed. Roughness nuericl odeling ws pproched by the Tsnn ethod. At first, the coputer progr hd been written for the sooth cse nd then the progr ws developed for the berings with roughness on their surfces. The proble ws resolved for two different surfce profiles nd the results were copred. The results indicte tht considering the thickness of lubricnt lyer with the Tsnn roughness theory will increse the pressure distribution nd lod crrying cpcity but it reduce the coefficient of friction nd side lekge flow. Keyword: surfce roughness, Tsnn ethod, finite eleent, icro polr fluids, two-lobe berings. Introduction So fr, different odels hve been presented for roughness. For exple, it cn be sid one-diensionl finite eleent odel for hydrodynic bering []. In this study, both the longitudinl nd trnsverse roughness re odeled. Fluid fil thickness ws pplied s rndo vrible, nd the issue hs been studied in stble condition. He chnged the Reynolds eqution bsed on defined preters to odeling roughness proble []. Prsd exined the perfornce of circulr bering tht ws ffected by teperture chnges nd the rough three-diensionl roughness []. Ferro fluids perfornce evluted in round short bering considering roughness []. The results show incresing the loding cpcity nd reducing the coefficient of friction. The first reserch of non-circulr berings ws been done by Pinkus [5] which presented lod cpcity nd energy losses in two-lobe berings. Then, Lund nd Thosen [6] obtined the non-circulr bering perfornce preters with the finite difference ethod. Their report is one of the ost vilble resources to design non-circulr berings in which vrious geoetricl preters were been noted tht ffect the perfornce of this berings. Recently, regrding this subject, the FEM [7] nd GDQ [8] nuericl ethods hve been used to solve the equtions without considering roughness

2 Vol- Issue- 6 IJARIIE-ISSN(O) The surfce roughness ws been investigted by tking Tsnn hypothesis nd using the finite eleent ethod in this pper. By substituting in the governing equtions, the lubricnt fil thickness is expressed in ters of Tsnn functions to chieve pressure distributions nd the lubricnt perfornce preters of two-lobe berings.. Mtheticl forultion of the proble Generlly, Nvier-Stokes equtions include stteents: inerti, pressure, ss, nd viscosity which solving such equtions is coplex nlyticl work. In series of issues, pressure nd viscosity re the only proinent preters which their flow re clled slow viscous otion nd the fluid lyer probles re ctegoried in this group. Therefore, the odified nd non-diensionl Reynolds eqution of berings for icro polr lubricnts is s follows [7]: N l h N l h L p R p h,,,, 6 () In which: h h N, l, h h 6N Coth l l l C C C, d, r N r N l h Diensionless thickness of fluid fil ( h ) on the clernce between the shft nd two-lobe berings is obtined vi geoetricl reltionships in ter of (fig ). Fig : Lubricnt fil thickness in two-lobe bering Tsnn Lin roughness ethod only ffects lubricnt fil thickness. However, jor chnges will ffect the nture of governing eqution due to the dding of new independent vrible in clculting the thickness of the lubricnt fil equtions. The lubricnt fil thickness is been clculted s follows []: H h () H depicts the lubricnt fil thickness with roughness, h describes the lubricnt fil thickness without roughness nd is used s sperity profile

3 Vol- Issue- 6 IJARIIE-ISSN(O) By substituting H for h, the governing eqution is odified s follows: N l H N l H L p R p H,,,, 6 () In which H H N, l, H H 6N Coth l l N l H Both of nd directions hve cosine curve in Tsnn sperity profile. The geoetric sie of the sperity is defined by the height, H, nd the hlf width both in the direction, L, nd in the direction, L. The sperity profile is expressed theticlly s H, cos L cos L L L () Surfce roughness preters re shown in the following fig. Fig : Three-diensionl Tsnn sperity odel [] Diensionless pressure boundry conditions re shown in the following for P (, ), P (, ), P (, ) dp / d(, ) (5) By setting ll negtive pressures, the pressure of the leding edge boundries, nd the loction of the strting of cvittion one equl to ero, the Reynolds odified eqution is estblished. relted your reserch work Introduction relted your reserch work Introduction relted your reserch work Introduction relted your reserch work Introduction relted your reserch work Introduction relted your reserch work Introduction relted your reserch work In. Bering perfornce chrcteristics With the pressure field known, the bering perfornce clcultions cn be crried out. The diensionless rdil nd tngentil lod coponents re found in the fro π Wx cosθ Pi W y sin θ dθd (6) The lod cpcity is written s

4 Vol- Issue- 6 IJARIIE-ISSN(O) x y W W W (7) Attitude ngle is n ngle fro fixed line connecting the center of the bering with the center of the journl to line, which is in the direction of the lod vector. This ngle is defined bsed on the ide tht the input lod is just supported in verticl direction nd is clculted by n itertive ethod. The ngle is obtined s follows: X W rctn rctn j x Y W j y (8) The diensionless friction force is given by l λ θ λ θ i h θ (9) i F Adθd A dθd h Where i i i λθ λθ i h p A θ N Nl h h tnh l () Consequently, the friction coefficient is [9] f R / C F () W The non-diensionl side lekge flow rte is [] π R p L () Q ψ N, Λ, h dθ. Nuericl solution of equtions Approxiting functions depend on the geoetry nd the nuber of nodes in ech eleent. Needed lgebric functions for finding unknown coefficients will be derived by Reighley-Rit nd weighted residul ethods fter extrcting the pproxiting functions. Generlly, the in finite eleent steps in nlying proble re s follows - Discretiing the doin re into eleents include the nubering of nodes nd eleents, nd, driving required geoetric properties (such s coordintes, res, etc). - Deriving the equtions for every eleent: - Foring vrition functions for the differentil eqution. b- Considering n u c N s dependent vrible nd substituting in the for of K u F j j j order to ccessing eleent equtions. c- Deriving the interpoltion functions nd eleent trices. - Cobining eleent equtions to obtin the generl eqution of the proble: - Creting continuity between the eleents in the priry vribles. e e e in

5 Vol- Issue- 6 IJARIIE-ISSN(O) b- Creting blnce between the eleents in the secondry vribles. - Applying boundry conditions. 5- Solving the cobined eqution. Usully flow chrts regrding pplying nuericl solution re exhibited s schetic of the solution procedure nd wy of chieving nswer. The flow chrt concerning this pper is s follows (fig ): Fig : Nuericl solution flow chrt 5. Results nd Discussion Firstly, to prove the ccurcy of the written nuericl odel, the diensionless coefficient of friction s function of, ε, nd N ws copred with the published results [7] for two-lobe berings

6 FRICTION COEFFICIENT, f, DIMENSIONLESS Vol- Issue- 6 IJARIIE-ISSN(O) Newtonin Present work Result of Ref. [] N l, DIMENSIONLESS Fig : Diensionless friction coefficient s function of chrcteristic length for both Newtonin nd icro-polr fluids in two-lobe berings ( L / D,.5,.5) ò. Fig shows tht there is close greeent between the results nd [7]. Press b Press Fig 5: Pressure distribution for icro-polr fluids with the sperity height of ( H.) in two-lobe berings when ) L / 5, L. b) L /, L.5 (,.5, ò.5 l, N.5). There re two iportnt preters to deterine the effects of icro-polr fluid lubriction on two-lobe loded berings. The first diensionless preter is the coupling nuber, which specifies coupling fetures of icro rottionl fluid prticles in liner nd ngulr oentu equtions. Higher vlues N represent ore strong effect between liner nd ngulr equtions. When the nuber pproches ero, the icro-polrity effect decreses nd fluid is considered s Newtonin. The second diensionless preter is chrcteristic length, which describes the reltionship between the bering geoetry nd boundry lyer thickness; oreover, it is function of fluid oleculr sie. Lower vlues of this correspond lrger chrcteristic length respect to the clernce nd when it pproches infinity, lubricnt will be close to Newtonin fluid.,

7 Non-diensionl Pressure Non-diensionl Pressure Vol- Issue- 6 IJARIIE-ISSN(O) Press b Press Fig 6: Pressure distribution for icro-polr fluids with the sperity height of ( H.5) in two-lobe berings when ) L / 5, L. b) L /, L.5 (,.5, ò.5 l, N.5). Pressure distribution ws obtined by solving odified Reynolds eqution vi nuericl finite eleent progr for icro-polr fluids with considering the Tsnn roughness theory. It cn be seen tht the pressure distribution profiles re chnged considerbly when the roughness odel is pplied. The effect of the vrious preters regrding Tsnn roughness odel on pressure distribution re shown in the figs. Copring figs 5 to 6 nd figs 5b to 6b depict tht incresing the diensionless height in Tsnn odel leds to incresing the pressure distribution profile nd kes those teeth-shped s well. As sperity length nd width in prticulr sperity height reduces, the pek of the pressure distribution decreses nd becoe ore uniforly (fig 5). Incresing the sperity height, this effect is ore visible (fig 6)., N=. AND HC=. N=. AND HC=.5 N=. AND HC=. N=. AND HC=.5 N=.6 AND HC=. N=.6 AND HC= Thet Fig 7: pressure distribution with considering Tsnn theory for icro-polr fluids in the direction of the θ in the centrl line of two-lobe berings ( L / D,.5, l, N.5) N=. AND HC=. N=. AND HC=.5 N=. AND HC=. N=. AND HC=.5 N=.6 AND HC=. N=.6 AND HC= (Nondientionl) Fig 8: pressure distribution with considering Tsnn theory for icro-polr fluids in the direction of the / in the centrl line of two-lobe berings ( L / D,.5, l, N.5) Figs 7 nd 8 represent the pressure distribution, which were selected in the direction of the θ nd in the centrl line of the two-lobe berings

8 Friction Coefficient Nondientionl Lod Cpcity Vol- Issue- 6 IJARIIE-ISSN(O) Generlly, when the sperity height nd the diensionless coupling nuber re incresed, the pressure distribution is incresed s well. H/C= (Sooth) H/C=. 8 6 Micropolr (N =.9) Micropolr (N =.5) Newtonin Eccentricity Rtio Fig 9: Diensionless lod cpcity s function of eccentricity rtio for both Newtonin nd icro-polr fluids considering Tsnn theory in two-lobe berings ( L / D,.5, l 9). 5 H/C = (Sooth) H/C =. 5 Newtonin Micropolr (N =.5) Micropolr (N =.9) Eccentricity Rtio Fig : Diensionless friction coefficient s function of eccentricity rtio for both Newtonin nd icro-polr fluids considering Tsnn theory in two-lobe berings ( L / D,.5, l 9). All of the bering perfornce preters cn be chieved fter obtining pressure distribution. Figs 9 nd portry the bering perfornce preters chnges in ters of eccentricity rtio in prticulr chrcteristic length with respect to presence of Tsnn roughness odel. Grphs show when coupling nuber is incresed in prticulr eccentricity rtio the diensionless lod cpcity is incresed too (fig 9), lthough, diensionless friction coefficient is decresed (fig ). Incresing eccentricity rtio with sperity height of. leds to increse the diensionless lod cpcity for both Newtonin nd icro-polr fluids (N =.5 nd N =.9). This feture will intensify when icro-polrity properties re incresed so tht s the fluid becoe ore non-newtonin, the effect of incresing the sperity height is becoe ore obvious regrding diensionless lod cpcity ter (fig 9) nd there is 6 percent deduction in diensionless friction coefficient when eccentricity rtio is incresed in the two lobe-berings (fig )

9 Nondientionl Side Leckge Flow Rte Attitude Angle Vol- Issue- 6 IJARIIE-ISSN(O) H/C = (Sooth) H/C = Newtonin Micropolr (N =.5) 75 L Fig : Attitude ngle s function of chrcteristic length for both Newtonin nd icro-polr fluids considering the Tsnn theory in two-lobe berings ( L / D,.5,.5) ò Newtonin Micropolr N = H/C = (Sooth) H/C =. L Fig : Attitude ngle s function of chrcteristic length for both Newtonin nd icro-polr fluids considering the Tsnn theory in two-lobe berings ( L / D,.5,.5) ò. Figs nd show plots in conjunction with the bering perfornce preters in ters of incresing chrcteristic length fro to for both Newtonin nd non-newtonin fluids in prticulr eccentricity rtio. Reduction in side lekge flow rte nd ttitude ngle re cler in the fig nd fig, respectively. 6. CONCLUSIONS The in purpose of this study ws to evlute the effect of Tsnn roughness odel on icro-polr fluids in twolobe berings. Micro-polr fluids re subset of icro fluids, which deonstrte icro rottion properties; this ens tht the fluid by hving icrostructure is ble to rotte independently on icro-volue without ny chnge. Finite eleent ethod ws used for solving diensionless icro-polr fluids eqution in the two-lobe bering nd the conclusions re s follows: - As the coupling nuber is incresed, icro-polr fluid properties re becoe ore proinent, oreover, the fluid viscosity is incresed, nd pressure distribution is intensified in prticulr sperity fetures. - Pressure distribution in inly influenced by sperity profile height. - Incresing roughness preters, especilly sperity height, the teeth-shped profile cn be seen in pressure distribution. - Chnging the width nd the length of sperity profiles, pressure distribution does not vry regrdless of whether fluid is Newtonin or icro-polr. 5- On prticulr chrcteristic length, incresing eccentricity rtio, the pek of pressure distribution is incresed nd this feture cn go further if Tsnn roughness odel is considered

10 Vol- Issue- 6 IJARIIE-ISSN(O) On prticulr eccentricity rtio, incresing chrcteristic length, the pek of pressure distribution is decresed nd this feture intensified if Tsnn roughness odel is considered. 7. List of Sybols C Bering clernce, c, c Angulr viscosity coefficients, Ns/ d C D F f H H h h L L L l Miniu clernce when the center bering nd shft re concentric, Lobe bering dieter, Diensionless friction force Diensionless friction coefficient Diensionless lubricnt lyer thickness Diensionless sperity height The lubricnt lyer thickness regrdless of the Tsnn roughness odel, Diensionless lubricnt lyer thickness regrdless of the roughness odel, (h/c) Bering xil length, Diensionless hlf sperity width in direction, Diensionless hlf sperity length in direction, Diensionless icro-polr fluid chrcteristic length Prelod of the bering, (C /C) N Coupling nuber, (N= / + ) / r r P P Q Hydrodynic pressure, N / Diensionless hydrodynic pressure Diensionless side lekge flow rte R Shft rdius, W Diensionless lod cpcity W, W Diensionless lod cpcity coponents x y in x nd y direction x, y Subtitles for coponents Coordinte xis in xil direction, Diensionless xil length, (/L) Greek sybols Asperity profile, ò Eccentricity rtio The ngulr coordinte esured fro the x-xis Diensionless chrcteristic length Newtonin lubricnt viscosity, Ns / Prtil rottionl viscosity, Ns / r Attitude ngle

11 Vol- Issue- 6 IJARIIE-ISSN(O) REFERENCES []. Turg, R., Sekhr A., nd Mjudr B., 996, Stochstic FEM odel of one-diensionl hydrodynic berings with rough surfces. Wer, p. -7. []. Lin, T.-R., 996, Hydrodynic lubriction of journl berings including icropolr lubricnts nd threediensionl irregulrities. Wer, p. -8. []. Prsd, E.S., Ngrju, T., nd Sgr, J.P, Therohydrodynic Perfornce of Journl Bering with D- Surfce Roughness nd Fluid Inerti Effects. []. Hsu, T,, Lubriction perfornce of short journl berings considering the effects of surfce roughness nd gnetic field. Tribology Interntionl, p [5]. Pinkus, O., 956, Anlysis of ellipticl berings. Trns. ASME, p [6]. Lund, J. nd Thosen, K, 978, A clcultion ethod nd dt for the dynic coefficients of oil-lubricted journl berings. Topics in fluid fil bering nd rotor bering syste design nd optiition. [7]. Rhtbdi, A., Nekoeiehr, M., nd Rshidi, R.,, Micropolr lubricnt effects on the perfornce of noncirculr lobed berings. Tribology Interntionl, p. -. [8]. Rhtbdi, A., Zre Mehrjrdi, M. nd Fel, M., Perfornce nlysis of icropolr lubricted journl berings using GDQ ethod. Tribology Interntionl, p. -9. [9]. Ds S, Guh SK, Chttopdhyy AK., "On the stedy-stte perfornce of isligned hydrodynic journl bering lubricted with icropolr fluids". Tribol Int,p.. []. Wng XL, hu KQ., "A study of the lubricting effectiveness of icropolr fluid in dyniclly loded journl bering". Tribol Int,p