Numerical analysis of grease thermal elastohydrodynamic lubrication problems using the Herschel-Bulkley model.pdf

else jier science tribology intemclfional vol 30 no 6 pp 401408 1997 0 1997 elsevier science ltd printed in great britain all rights reserved 0301479x 97 17 00 o oo pii so301 679x 96 00069 2 numerical analysis of grease ermal elastohydrodynamic lubrication problems using the herschel bulkley model jin gyoo yoo and kyung woong kim grease thermal elastohydrodynamic lubrication tehl problems of ine contacts are analysed numerically the effects of temperature and rheological parameters on grease tehl are investigated using the herschel bulkley model as a rheological model of greases the pressure distribution the shape of grease film mean film temperature and surface temperature of solid wall in ine contacts are obtained it is found that thermal effects on the minimum film thickness become remarkable at high rolling speeds the effect of yield stress of the herschel bulkley model on minimum film thickness is negligible while the flow index and viscosity parameter have significant effects on minimum film thickness 0 1997 elsevier science ltd keywords grease thermal elastohydrodynamic iubrication herschel bu kley model introduction elastohydrodynamic lubrication ehl theory is the interpretative theory of lubrication phenomena in rol ling bearing gear and cam mechanisms this theory is used in the determination of minimum film thickness which is a very important factor in the design of machine elements to avoid metal to metal contact the appropriate minimum film thickness must be main tain ed sicce the first numerical solution considering elastic deformation and the viscosity pressure relationship was obtained by dowson and higginson in 1959 many theoretical and experimental researches in ehl have been carried out and the results have been applied to the design process of many kinds of machine elements practically in most ehl analyses the lubricant has been considered as a newtonian fluid and the tempera ture effects on the density and the viscosity have been departn ent of mechanical engineering korea advanced institute of science and techrlology 373 l ku revised 23 septe accepted 14 0c ober 1996 co respondence should be addressed to prqf kyung woong kirn e ruxil tnelzo cais knist ac kr ignored assuming an isothermal process however in the case of a high speed rolling bearing lubricated with a non newtonian fluid both the temperature effects on the density and the viscosity and the non newtonian effect become significant and thermal non newtonian ehl analysis2 is needed for prediction of a more accurate minimum film thickness grease which is widely used in many applications for example electric motors household electric appliances measuring instruments etc is a typical non newtonian lubricant although the ehl theory of oil lubrication is well documented the ehl theory of grease lubri cation is not well established because of the complexity of its rheological properties kauzlarich and greenwood3 published the first theoreti cal analysis of ehl with grease in 1972 they formu lated the reynolds equation with the herschel bulkley model and examined the validity of this model wada et al obtained a numerical solution of grease film shape and pressure distribution using the bingham model jonkisz and krzeminski fredasz6 zhu and neng and cheng8 obtained a numerical solution of ehl problems with herschel bulkley model grease and compared it with experimental results dong and qian9 obtained a numerical solution of an ehl prob lem with bauer model grease however all of them tribology international volume 30 number 6 1997 401 assumed isothermal conditions because of the com 2 dpldx 0 where e is the coefficient of thermal expansivity of the grease k pn 8 x mm the boundary conditions for the reynolds equation are expressed in dimensionless form as p 0 at x xmin p 0 and g 0 at x xend 9 subscripts min and end mean inlet and exit location of the film respectively the inlet location xmin is chosen at four times the hertzian half width from the centre of the hertzian contact region the temperature distribution within the grease film is determined from the solution of the energy equation under appropriate boundary conditions the following assumptions are made in the development of the energy equation 1 the thermal conductivity k specific heat c and coefficient of thermal expansivity e are inde pendent of both pressure and temperature the variation of k c and e with pressure and tempera ture were not considered in our paper because of lack of data 2 since the grease film thickness is much smaller than the length of the contact region the heat conduction in the rolling direction x is much smaller than that in the cross direction y and can be neglected in the temperature calculation 3 for line contact the cross heat convection term is much smaller than that in the rolling direction12 and therefore can be neglected in the tempera ture calculation 4 to make the problem simple the viscosity and the density of grease take constant values across the film corresponding to the mean temperature t across the film and that dt ax can be replaced by dt l x where t is defined as t x h z vw dy 10 h 2 applying the above assumptions substituting the velo city distribution into the dimensionless energy equation an 3 integrating this equation twice over the range from 1112 to h 2 we obtain the following energy equations 1 1 dp dx 0 h h hp m 3 m 2 m 3 m f 4 3 h hp m 4 h h hj 3 2 dp dx 0 11 t t h i ka kb t g kch3 kd j h hp l 4 h h hp m 3 m 2 m 3 m 4 3 m h3 h hp m ke g h h 3 m 3 h h h 2 kf uc zom s i uc so f x 4 3 2 1 0 1 2 x fig 5 solid surface temperature for diflerent rolling speeds uc ph 0 4 gpa ryo 139 8 pa qso 21 98 pas n 0 63 der increase the mean film temperature rises to a maximum value near the entrance to the hertzian contact region while the surface temperature takes its maximum value in the hertzian contact region the mean film temperature drops in the hertzian contact region figures 6 and 7 show the effect of yield stress ryo on the pressure distribution and the film shape in thermal sol i i i i i 4 3 2 1 0 1 2 x fig 6 pressure distribution uc io m s ph fig 9 film shape uc io m s p 0 4 gpa rro 0 4 gpa r 21 98 pas n 0 63 139 8 pa n 0 63 fig 7 film shape uc io m s ph 0 4 gpa qso 21 98 pas n 0 63 4 3 2 1 0 1 2 i a fig 8 pressure distribution uc 10 m s p 0 4 gpa ryo 139 8 n 0 63 ehl analysis for the values of uc 10 m s p 0 4 gpa respectively according to figs 6 and 7 where all curves corresponding to any value of ryo nearly coincide with each other the effect of yield stress ryo on minimum film thickness is negligible as in the case of isothermal ehl analysis the yield stress has some effect on film thickness when the yield stress is high and viscosity parameter is 10 table 3 and figs 8 and 9 show the effect of the viscosity parameter qso on the minimum film thickness 406 triboiogy international volume 30 number 6 1997 grease thermal ehd lubrication problems jin gyoo yoo and kyung woong kim table 3 isothermal minimum film thickness and thermal minimum film thickness uc 10 m s fh 0 4gpa ryo 139 8 pa n 0 63 viscosity parameter rlso pas io 21 9 30 40 dimensionless dimensionless isothermal thermal minimum minimum film thickness film thickness hmin hmin 0 8802209 0 8070543 1 635797 1 400858 2 070231 1 732164 2 555876 2 079722 ratio of thermal to isothermal film thickness 0 9 1688 0 85638 0 83670 0 81370 table 4 isothermal minimum film thickness and thermal minimum film thickness uc 10 m s p 0 4 gpa t 139 8 pa qso 21 98 pas rheological index n 0 60 0 613 0 66 0 70 dimensionless isothermal minimum film thickness hmin 1 i37542 1 635797 2 295712 3 358974 dimensionless thermal minimum film thickness hmin 1 031371 1 400858 1 886916 2 601375 ratio of thermal to isothermal film thickness 0 90667 0 85638 0 82193 0 77446 the pressure distribution and the film shape in thermal ehl analysis for the values of uc 10 m s and p 0 4 gpa respectively as the viscosity parameter qso becomes larger the film thickness increases similarly to the case of isothermal ehl analysis and the difference between the minimum film thickness obtained by iso the ma1 ehl analysis and that obtained by thermal ehl analysis increases table 4 and figs 10 and 11 show the effect of flow index y1 on tehl performances in the same manner as table 3 and figs 8 and 9 as the flow index n becomes larger the film thickness increases as in the case of isothermal ehl analysis and the difference between the minimum film thickness obtained by iso 1 5 a n o 60 ii nzo 63 2 nzo 66 2 l 9 n o yo 2 22 z i7 5 2 a 0 4 3 2 1 0 1 2 x fig 10 pressure distribution uc 10 m s ph 0 4 gpa 7y0 139 8 pa rlso 21 98 pas n 0 60 n 0 63 n q 66 n 0 70 e go 1 1 i i i 4 3 2 1 0 1 2 x fig i i film shape uc 10 m s p 0 4 gpa ryo 139 8 pa qso 21 98 pas thermal ehl analysis and that obtained by thermal ehl analysis increases the above results show that minimum film thickness obtained by thermal ehl analysis is less than that by isothermal ehl analysis and the qualitative character istics of the effects of the rheological parameters on the ehl performances in thermal ehl analysis are similar to those in isothermal ehl analysis the flow index n and viscosity parameter rig0 have significant effects on the pressure distribution and the film shape of the ehl problems with grease although the effects of yield stress 7y0 on the pressure distribution and the film shape are negligible tribology international volume 30 number 6 1997 407 grease thermal ehd lubrication problems jin gyoo yoo and kyung woong kim conclusions a numerical analysis method for grease tehl has been developed using the herschel bulkley model and numerical calculations are carried out by means of the method the findings are summarized as follows 1 thermal effects on the tehl performances can not be negligible and at high rolling speeds the effects become very significant 2 at high rolling speeds the yield stress of the herschel bulkley model has negligible effect on the tehl performance 3 the flow index and viscosity parameter of the herschel bulkley model have significant effects on the tehl performance references 1 dowson d and higginson g r a numerical solution to the elastohydrodynamic problems j mech eng sci 1959 1 h 15 2 salehizadeh h and saka n thermal non newtonian elastohy drodynamic lubrication of rolling line contacts asme tribal 1991 113 mi 491 3 kauzlarich j j and greenwood j a elastohydrodynamic lubrication with herschel bulkley model greases asle tmns 1972 15 269 277 4 wada s hayashi h haga k kawakami y and oka jima m elastohydrodynamic lubrication of a bingham solid bull jsme 1977 20 110 115 5 jonkisz w and krzeminski freda h pressure distribution and shape of an elastohydrodynamic grease film wear 1979 55 81 89 6 7 8 9 10 11 12 13 14 15 jonkisz w and krzeminski freda h the properties of elas tohydrodynamic grease films wear 1982 ii 277 285 zhu w s and neng y t a theoretical and experimental study of ehl lubricated with grease asme j tribal 1988 110 38 43 cheng j