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International Compressor Engineering Conference School of Mechanical Engineering
1974 Calculation of the Hydrodynamic Lubrication of Piston and Piston Rings in Refrigeration Compressors H. Kruse Technical University Hannover
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Kruse, H., "Calculation of the Hydrodynamic Lubrication of Piston and Piston Rings in Refrigeration Compressors" (1974). International Compressor Engineering Conference. Paper 101. https://docs.lib.purdue.edu/icec/101
This document has been made available through Purdue e-Pubs, a service of the Purdue University Libraries. Please contact [email protected] for additional information. Complete proceedings may be acquired in print and on CD-ROM directly from the Ray W. Herrick Laboratories at https://engineering.purdue.edu/ Herrick/Events/orderlit.html CALCULATION OF THE HYDRODYNAMIC LUBRICATION OF PISTON AND PISTON RINGS IN REFRIGERATION COMPRESSORS
Dr. H. Kruse, Professor of Refrigeration Engineering Technical University Hannover I Germany
1. INTRODUCTION The calculation of the lubricating condi refrigeration compressors,the pistons of tions of a piston is, compared with a slid which'can ing bearing,much be more lubricated in comparison more difficult,because the to internal cqmbustion configuration of the engines, because oil film and the operat at least with oil soluble refrigerants ing conditions are much more complicated. the lubricating oil is not lost, but is circu Whereas the profile of the oil film in jour nal lated back'into the compressor, In spite of bearings can be described by eccentric the assumption circles, that of of fluid friction, the com a piston is essentially of plexity of the problem a more complicated form (Fig.1), has led to the situ ation where h¥drodynamic calculations for oistons (4) lS), and piston rings (6),(7), {a),(9),(1oJ,have been made almost exclu sive~y separately. Since however the two components form an interacting unit,it seems reasonable not to consider them separately in the calculation of the lubrication conditions. Eilon and Saunders (11) although assuming the unit of piston and piston rings, apply the hydrodynamic lubrication film theory only to the piston ring, Although they re gard the piston clearance as filled with oil and friction-producing, they do not take into account a hydrodynamic production of pressure in the lubrication film. 2. CALCULATION OF PISTON LUBRICATION Fig,l Configuration Here, as already indicated by the author of the oil film between (12) piston and liner the hydrodynamic calculations are used for the common lubrication film on The the pist shape of the oil film there is deter on and piston rings, which are derived from mined on the one hand by the surface profile the general Reynolds Differential Equation of piston and piston rings which are mathe for lubrication films, matically difficult to describe, on the other hand by their position in the cylinder, 3 3 a~(h ii> + ~(h ~) = 6D(U + whereby the piston rings are radially and 1 -U 2 )~~ 12DV (1) axially moveable in the ring groove relative Since there is no analytical solution to avail the piston. able for this equation, there are two pos As regards the operating conditions,the var sible methods for the calculation: iable gas pressure acting on the lubricant 1. Simplification of the problem, so that a film and the quantitatively unknown filling analytical solution is possible, and of the clearance between piston and liner 2, nonanalytical methods by changing this with oil,presents the greater complexity of equation into a difference equation and the problem,which leads to the question,to solving it with the help of a digital com what extent the theory of hydrodynamic lu puter. brication is at all applicable. To begin with the first method was used,as From measurements of internal combustion basically the same physical fundamentals re engines(!), (2), {3), boundary friction oc main and influence of the various geometrical c~rs in the region of dead centers at piston and operational parameters on the lubrication rlngs, whereas qver a wide range of the pi conditions of the piston can be studied more ston stroke liquid friction takes place. clearly and with less calculating effort, as Concerning the piston in scientific publi by an approximate solution through a calcu cations in general, the conclusion of a lation programme which demands a large memo greater degree of fluid friction is drawn ry and calculation time, from the slight wear, Later,in addition,such an nonanalytical so These assumptions can especially be made for lution was used to study parameters which
72 could_not be dealt with by simplified cal prof~le ~hrough the surface lines parallel culatlon methods, and to confirm to what ex a~d lncllned_to the cylinder wall results in tent t~ey :auld be considered by means of F~g,2 Df a plston section with one piston approxlmatlon_through an analytical solution, rlng, In order to Slmplify the Reynolds Differen tial Equation (1) it was at first assum~d y as with the other authors mentioned (4) (s) ( 6 ) , ( 7 )t ( 8 ) that ' ' a) the p~ston ~aves coaxially in the cylin der Wlth unlform velocity U2 = U and also the assumptions generally made are valid that ' b) the viscosity n in the lubricating film is constant, c) the cylinder is circular cylindrical, if d) the cross section of the piston is circu lar, and -U e) the piston rings posess an equal radial '-+---- A ------< pressure distribution over the circumfe rence, Fig,2 Sliding element of the piston Foran analytical solution of the Reynolds Differential Equation the lubricant film Here the real piston profile could be ap height configuration ~etween the liner and proximated not just by one but by a number the piston ring is described by straight li of straight lines of various positive a~d negative inclination rnA , in the same way nes parallel and inclined to the cylinder as ~he s~rfa:e o~ the piston ring also has wall with an inclination m, varlous lncllnatlons mB and m , ~iston ring pro~iles are also often approx 0 Between the piston and piston ring region l~ated mathematlcally by parabolic expres there occurs a step in the lubricant film, s:ons, yet wedg7 surface statementsas in (7) glVe results whlch agree more with measured Lord R~y~eigh (13) first proposed step sha values, ped sl:dlng elements for bearings, Although an real sliding surface profile on These ldeas were later taken up-by Archi pistons and piston rings can be better dealt bald_for thrust (~4) apd later for journal bearlngs and_are lntended here to be applied wi~h.by approxim~tion calculations by des crlblng the lubrlcant film configuration as t? the step ln the lubrication gap of a points in a height_array, however here they plston. By application of the equation (3) for each ought to be approxlmated by mathematical section A,B,C and D qnd the integration equations, in order to make clearer the in' taking int? account the boundary qonditi~ns fluence of the geometric parameters on the that the all film pressuresat both ends cor lubrication conditions, respond to the ambient pressure and should Under.the above mentioned conditions, the be equal at the boundaries of each region, t~o rlght ~and parts of both sides of equa one gets as a result the equations giving tlon (1) dlsappear, so that for the one di the load capacity and frictional forces for mensional problem without side leakage and the oil film pressure p and the shear stres radial movement it is simplified into equa ses T. tion (2) The film pressures in the different sections A,B,C,D are as follows d
73 into the ambient pvessure here,The shear n,U,and stresses are as follows PE as operating conditions,along with changing gas pressures on both sides 4 h of the lubricating film and (12) 1=nUCb - 3ET) the partial fil (13) ling of the lubrication gap with oil,which further influence the lubrication conditions, Pnuc.!!. - 31i...> For c c2 (14) 1;:nuci - 3~d- > (15) the investigation of these various in d fluencing factors with the integration constant the derived equations were fi as the rate progammed and the individual of relative oil flow, by equalizing of the parameters were pressures varied1>,in order to ascertain theiv effect Pc and p "at x=D resulting to on oil flow, equation (10): 0 pressure and friction in the lubrication gap of a piston of D=85 mm 0, chosen as an example,The initial values of calculation were: · Lengths: (16) . Thickness: A = 5,0 mm a 1 = 0,15 mm B = 1,0 mm a2 = 0,15 mm C = 0,5 mm b1-b2 It contains, like the above mentioned equa = 2•10-3 mm tions , the D = 1.0 mm d1-d2 = 2•10-3 mm unknown position represented by . Piston c , of the piston ring velocity U = 3 m/s in the oil film, Lubricant viscosity 2 which is also included in the values b and n = 0.1162 kps/m d .This position Piston ring tension PE = 1,3 kp/cm2 is determined by the equi Of special librium of the forces acting interest ~re the result~ of a va vadially on the riation of the ring, and is ascertained by application lubrication gap geometry in ca the region of piston and pacities Ps,Pc 1 and Pn in piston rings on the the region L=B+C+D lubrication conditions. 3,1,1. Variation of the Oil Film at the (17) Piston A variat~on of the piston clearance as in Fig,3 (18)
( 19) -
Po)od1= §!:JJ!_{h- 2c_ -In Do m!J 2c h cil} ( 20) in implicit form from the equation (21)
(21) With this, therefore the oil film pressures, and, by integration of the shear stresses, Fig.3 Variation of piston the clearance frictional forces in equations (22) (23), influences as according to equation (24),and 1 (6) 1 the (25) are able to be calculated in pressure of the oil film in the the separate piston piston re and piston ring regions, gion A,and thereby also the pressure PAa2 at the leading ring edge,which also,as seen in the equation (22) for the other regions,in fluences the oil film pressure acting there. The results of the variation of piston clear RBO= 3..JL.{41n .k -3/i bt -c} ance in Fig,4 show ms c bl"c (23) with increasing piston e c~earance a strong decrease in the pressure PAa2 on the leading edge of the Since ring, this pressure,supporting the force of (24) radial pressure,acts 6n the ring,a decrease means an uplift of the piston ring,represen R ted by the increase in the smallest 00=ll {4/n ..!!_ -3 ii -cj l~b~i mo { c c (25) cation film thickness c,Connected with it 3, STUDY OF THE INFLUENCING is an increase in the relative oil flow VARIABLES ON THE and a decrease h LUBRICATION CONDITIONS ON PISTONS in the friction force Rges As can be ascertained which can also be seen from equation (22) from the equations,the to ( 2 5) • relative oil flow fi,pressures p and friction 1) These calcutations have been carried out al forces R are essentially determined by Reometric· the in the diploma works of F.Wrede and-B. cil film dimensions and the values Wolter
74 1 R ;J~Ikp/mm] 1.5 c-10 ,tl·10'rmml 1,5 0 2 P, lkp/mm 1 01
0,5 D. 5
L_-~---,------~...,-----r------1-( ·0 -O,Q2 -0.01 0 ·11.01 m,!-l o,.o; 0,025 0.1 0.2 a,• a2!mml
Fig,4 Results of variations versus piston Fig. 6 .Results of variation versus pistol clearance profile A variation of the pisto-n profile .as in .Fig, 5
Fig.7 Variation of piston ring profile
.P., 10 !kp/mm2J 5
C•0.2L- C•O.H· C•0.5L- Fig.5 Variation of piston profile va-ri c" h ·C•O,ZL means a simultaneous medium clearance C" h~ C=D,4L ation and a variation in the inclination IDA· rc Fi, C=0.6L It can be Been (12)that when the medium ~le avance Ca1 + a2) I 2 ·remains the same ,a vari ation in inclination mA in vespect to the pressure PAaf at the leading edge of the ring and min::Lmum and medium oil film thick ness fi effects no difference, but only in re·spect to p·res·sure and load capacity in re R 9 ~ 5 C=O 2L C ~0-4l c~0.6L gi-on A, That means that the vesult in Fig.6 as re gards the above mentioned values,only reflects the effects of an increase in the c-leavance, L·e-., that with positive inclination and an 0 0 d J lmml .., Fl increase in the clearance,a decrease in the pressure a-t the leading ·edge of ·the rin-g Fig.S Results of variation vevsus piston PAa2 and an increase in the minimum and me ring profile ·dium t"hickness of the oil film -as a ra•te for C=0,2 -L 0,4L,0,6L the oil flow results, various middle flat land-s 1 ring only de force Rges changes only un7s the pressure in front of the The ~riction the effects sent:1.ally,for the change only -affects pH;ton creases insignificantl·y,whereas De·gio·n A,-whereas the friction in :the pis 75 the piston ring also improves its lubrica tion conditions, 1 3·16'-,-----..,.-....,------,------, 3,2, Variation of Operating Conditions p lkp/mm'l The effects of a variation of n,U and PE will not be dicussed in greater detail here, because it is easily seen and generally known,that increasing values of these three parameters lead to greater friction.They also result in greater oil film pressures, whereas the influence on the oil flow is different,for increasing internal tension of the piston ring throttles the oil flow,and increasing velocity and viscosity increase it,because with the increasing oil pressure the influence of the internal tension of the ring on the gap width,becomes relatively less, Of essential influence on the lubrication Fig.10 Pressure profile in the clearance conditions are the quantity of the oil in between piston ring and liner the clearance and varying ambient pressures oil film pressure zero in front of the as an significant expression ring, of the operat the oil film load,which stays constant ing conditions of the piston,Because un these der further reduction in the oil supply,de parameters are not clearly defined in size, mands equal hydrodynamic load capacities, the calculation of the lubrication of an expressed through the integrals of running piston is area of made more difficult, the pressure profile under the ring,which To investigate their influence,see Fig,9, leads to higher maximal pressures at decrea sing lengths. Plotting the results over of the relative lengths SA and SB, Fig,11, 2 7·10 rr------.., h (mml c !mtnl R11 ull 5·10' 2 ,.,If' Pma.l' -~---- 1•1Ci 2 ~------~~----· Fig.9 Variation of oil film lengths and gas 0,1 0.2 O.J 04 0.5 pressure 0 O,,:zS 0,25 Q.J75 O,!i ti,Ei25 0,75 0,117~ 1.0Sa •• the quantity of oil in the lubrication gap Fig.11 Variation and the in the quantity of oil in gas pressures,p11 in front,and PH be the hind the lubrication gap oil film 1 were varied,in which case the given equations had to be slightly ex with an increase in the quantity of oil un tended. der the ring SB a greater up-lift of the piston ring occurs,expressed through the 3,2,1, Variation of the Lubrication Gap Fil- minimum oil film thickness c,connected with a greater · tlfng · h · f oil flow and less friction,If,how A varla lon ln t e quantlty o u 1 b rlcan· t was ever,the simulated clearance in region A is also fil by systematic variation of the re led with oil over lative length a length SALthen the fric SA in piston region A,relative tion increases again,This,however,is to the length not L of the piston ring, and the only because of the length Se relative now additional piston 1 to the length B under the friction,but mainly piston rlng. because the minimum lu brication gap width is again The qualitative pressure reduced, profile in the lu This reduction in the width brication gap shown of the lubricat in Fig,9 varies for the ion gao c also means a reduction different lengths,as in Fig.10 in the re 1 so,that the lative-lubricant flow h through maximum pressures with the lubri decreasing length SA cation gap,This means,that unstable at first decrease and increase condi with further tions occur,when there is an decreasing length SB• accumulation of oil in front of the ring,to the extent that The initially falling pressure in front of the piston gap tends either to fill up the ring and also above the ring means or a fall to empty,because the ring gradually closes, off in oil film load under the ring,But at as the oil supply increases,and opens up,as 76 statement ap 10 the oil supply decreases,This p lkp/mm 2 1 pears essential for the study of lubricat~ng conditions on pistons and has to be checked 1 10' by experimentation 1 which is why an experi mental set up is at the moment being devel _, a 10 oped, in Gas Pressure _, 3,2,2, Variation 6 10 A variation in gas pressure on both sides of the oil film was chosen of a type where their _, 4 10 difference'S alone were studied 1 so that the excess pressure in front of the lubrication gap Py 1 or that behind t:he gap pH was varied J 10 ' from the other opposite film pressure, pre-pressure pre At positive values of Py 1 0 dominated1at positive values of PH after 0 0.2 0.4 0,6 0.8 1,0 pressure,whereas at the values of zero, equal pressure on both sides was present. to the effect of the Attention must be paid fig,13 Pressure pro£ile in the clearance in pressure vaiation of these differences between piston ring and liner at of the piston ring in on the axial position higher pre-press~res the ring groove,Whereas under the predomi nation of pre-pressure as in fig.12 the ring must be equal,as long as there is no accumulation of oil in front of the ring. Under conditions of a limited oil supply this leads to higher maximal pressures,and vice versa,under increasing lengths SB 1de creasing maximal pressures,When there is additional oil accumulation SA in front of the ring together with the consequently rai sed pressure on the edge of the ring,higher pressure,and thereforea load on the oil film occurs above the ring in addition to the gas pressure and to the internal tension of the ring,so that the maximal pressure must rise again in order to maintain the equilibrium of load, If the pressure behind the lubrication film is so much greater than the pre-pressure that the ring changes its axial position(as in fig,12)counteracting to this,the oil pres fig,12 Possibilities of axial piston ring sure at the leading edge PA 2 and the fric position tional force RKR of the pis~on ring;then the the piston ring in the left hand part of the oil film pressure profile between the piston figure lies ,as in previous studies with ring and cylinder changes as in Fig.14 its trailing edge in the ring groove when looked at in the direction of movement,by a of after pressure over the oil p l kp/rnrn 2 l predomination 1. 10° film pressure and the frictional force,can its position,that it lies to the so change 1 groove with its leading edge. 8 10 In the first case 1as in fig,13,at high pre 1 pressure1the lubrication gap produces a si 6-16 mil~r pressure profile to that seen previous (Fig,10),the only ly without pre-pressure 1 1.. -16,_ deviation being,that the pressure profile the leading edge is raised all in a12 by on 1 p~ 2 -10 a gas pressure of,in this case 120 kp/cm 1 and the hydrodyndmic oil film pressures are higher. 0 considerably 04 C.5 o.a 1,0 12 x/L The raised gas pressure in front of the ring 0 0.~ edge,also has an effect above the ring,and there supplements the internal tension of the be ring in pressing the ring against the cylin Fig,14 Pressure profile in the clearance at higher der wall. tween piston ring and liner This must be compensated for by means of an after-pressure improved bearing capacity,produced by higher In this forward axial position the after pressure in the oil film, pressure pH also has an effect above the In order to generate this load capacity,the ring and intensifies the internal tension of area integrals of the pressure profile under 77 the ring as a constant value.This film constant of oil under the ring load demands,once more,constant load (SBs1),With additional capacities at the various oil accumulation in front of the ring lengths under the the friction SA>O piston ring s ,so that here also decreases further,if the after pressures 8 the maximal press4re PHalready decrease with an increasing oil predominates,a.nd that as accwr,ulation. Compared expression of the p:rev:i,ously with the previous fi provement described im gure,however,they further in the lubrication conditions,as decrease,if the long as the oil supply is so great,that an accumulation ring stays in the rear position. of oil also occurs If the ring changes its position,the in front of the ring SA pressure after because the consequently generated additionally loads the rirg from namic hydrody above and leads furthermore pressur~ on the leading edge of the to an increase ring as a result of in friction, its forward position can The minima not have an effect above the ring of the curves not shown,therefore, fore and there are evidence of the film load stays const~nt, a change in the position of Because the ring. of the higher pre-pressure on the leading edge,the It is recognized that with an pressure profile in this length increasing case under the ring is fuller,so SA,increasingly gre~ter after-pres that the sure~ are necessary,to maximal pressure for equal load capacity effect this change comes less. be in position of the ring. Furthermore A similar case also occurs,when,although it is ascertainable from the after-pressure the diagram,that with is greater tnan the pre-pres increasing length under sure~but the ring s ,the still not sufficient to change 8 friction at first decreases, axial position the to later increase with a further of the ring.Then the raised in increase after-pressure only a_cts the length SA in the piston region upon the lubrication front of in film,but is sealed off from the ring-a phenomenon,which has al the the part above ready been shown ring by the rear position of the and explained in Fig.11. As a result ring. The reason lies in the it raises the pressure level of various effects of the the oil film under oil accumulation under the ring and the ring,without,atthe same of the in front time,loading the ring additionally ring on the uplift of the ring.These bove from a influencing factors the ring,In this position this result can be seen in Fig,16 is an improvement of the friction because conditions, of greater load capacity and thereby greater thickness of the oil film.This can be seen case in the results of variation of the gas pressures,in Fig,15 in respect the frictional of force,and in Fig,16 in con nection with the oil flow. Fig.15 shows,how the frictional es with force chang the variations in gas pressure ac ting on both sides of the oil film,whereby under the pressures Py and PH•excessive sure pres compared with· the opposite film boundary pressure are understood, 0 0.5 0.5 0 J 0,2 R~;~es 0,1 0,1 0.2 lkp/mm l 2 O.J 0.4 0,5 e-w·' PHfkp/mm 1--I----Pv fi(p/mm 2 1 Fig.16 Oil flow versus gas pressures in which the oil flow resp. the medium_lubri cant thickness h are plotted pressures against. the gas 4-16 in front of and behind film,In the oil general it is recognized,that with increasing gas pressures in front hind of or be the lubricant film,the oil flow through the gas decreases. The reason lies in the additional gas pres sure load on the ring,which 0 supplements the o+.s---o~A---0.~3--Pc~2~,.-w-~7~-,~--,o--~~c1•_o_/m~~~~-, internal tension in pressing the 1 -o~.3~~o~.• --~o.s st ring again 0 1 the cylinder wall,The greater thickness Fig.lS in the oil film Frictional force versus gas pressures at less after-pressure,when the ring is still in its rear position It is seen,that with decreasing son for (rea pre-pressure the reduction of friction),is con py,the frictional force decreases nected with an this 1 because increase in the relative oil gas pressure loads the ring in addition flow fi. to its internal tension,and Tbe maximal by removing the clearance h and the maximum oil load,a greater thickness of lubricant flow are reached the ring,together under at the moment when the ring with less friction~occurs. changes its position,That is This reduction in friction the case at goes up to pre greater lengths S ,and again at greater pressure py=O,when there is only after-pressures,wfiereas accumulation when oil is only pre- 78 sent under the ring cs 6s1) the ring actually cond term on the rignt hand side of the changes its position w~th the change in pres equation was ignored1which takes into ac sure;Further it can be seen from the figure, count a radial movement of the piston or the that when the trailing edge of the ring lies ring,just as an alternating piston velocity against the groove 1 represented in the r:(.ght was also ignored.Changing piston velocities hand arm of the curv~ 1 the oil flow 1with in with the piston in a concentric position crea~ing l~ngth sa under the pist~n ring,be were studied by Furuhama with respect to ra comes ~t f1rst greater)and then Wlth further dial move~ent of the ring, increasing length in front of the rings SA, Here are the essential results of the ap less qgain, . proximation solution for eC¢~ntric and in This is a~ expression of the instability of clined pistons: the oil film,alreadY, explained in fig.11 1 If the piston runs eccentrically in the cy which leads to piston gap either fillingi linder,then the pressure at the point of when the oil supply increases,or emptying, minimum oil thickness increas,es to a greater when it decreases,On studying the course of extent at the cir~timferencs 'than it decreas the left arm of the curv~ 1 it c~n be seen es on the side -of the maximum gap.In medium that with the forward position 'of the ring clearance in the vicinity of the piston pin in tbe groove,stabie conditions are pres~rit hoies,which has the same clearance as the as,with increasing lengths SA and SB 1 th~ ~e piston running concentrica~y 1 the lubrication dium oil thickness h and thereby the oil film pressure is obtained at-the value of flow,becomes greater,so 'that ~he i~ngths now the one dimensional solution,Similar state at a standstill stabilize in the lubrication ments about the minimUm ·lubrication gap gap,because ~ greater oil supply c4uses ·bath width,the oil flow,and the friction,can be a larger ring up•lift and a greater oil f+,ow made,The question is,what value the one di through the lubrication gap, mensional calculation ha~ for'trunk pistons, From ~his,conclusions about the oil flow ano in particular,if the lubrication condi through the ring sealing system of a compres tions,at various singl~ points on the cir sor can be dra~n. cumference1paying attention to the oil film In the compression and discharge stroke there ttiickness present there 1so calcul~ted,give is qefinitely only an oil supply present on sufficiently exact results, the liner,so that probably no oil accumula A comparison of such a one'dimensional cal tes 6n the rings.Then 1with increased oil sup culation with a two dimensional approxima ply1the ring would open and hinder the pas7 tion solution results in deviation of less sage of oil into the discharge chamber,where than 1 % in respect of tb~ oil film pres as with a decreased oil supply,the ring clo sure and the minimum oil th~ckness,if the ses,so that the oil necessary for hydrodyn~ ch~mging oil film thickness on the piston mic lubrication of the ring cannot flow a~ay, circumference is substituted into the one In the reexpansion and suction st~o~e,the · dimensional calculation, ring,at first in the forward positiop and The result of this,is 'that the one dimen with an increased oil supply,would allow sional calculation can be applied to trunk more oil pass through and leave an oil film pistons with-good accura¢y. on the cylinder wall for the following stroke 5, REFERENCES However;after changing position in"t~e ~our'e 1 POPPINGA 1R,Deutsch,Kraftf,Forsch,H,54 of the stroke,it would have a scraping effect 2 COURTNEY-PRATT,Proc,Inst,Mech,Eng, 155 and prevent an ·overgreat oil flow into the 3 FURUHAMA 1 S,Bullet,Jap,Soc.Mech,Eng,3 14 system, · · 4 LISCHEWSY,Energomaschinostr.s (1958) The calculations described and presented here 5 WASSILJEW,B,N.,Kraftf.T~chn; 5 (1955) and their results are valid for a coaxial 6 EWEIS,M,VDI-Forscn.Heft 371 (1935) piston operating in a cylind~r,and are cer 7 HORGEN,H, Diss, ETH Zurich'1942 tainly suited to represent the influence of 8 CASTLEMAN 1R,A. Physics 7 (1~36) the p~ramet~rs under'discussion on the lu~ 9 FURUHAMA S. Bullet.Jap, Soc ,Mech ,Eng, 2 bricatiori conditions. 1 10 WAKURI 1 Y, Bullst .J~p.Soc;Mech.Eng.13 Trunk pistons run eccentrically and inclined 11 EILON-SAUNDERS,Inst,Mech,Eng.Proc,171 in the cylirider,which leads to the question, 12 KRUSE H, 1 Motortechn;Z,10+12(1965)~ how the lubrication conditions ch~nge 1 and t 1 . 5+ 9(1966), what value the calculation described has in this case, 13\ RAYLEIGH,O,M,Phil.Mag, 35 (1918) 14 ARCHIBALD 1F.R. ,ASME Trans, 72 (1950) 4, APPLICAT,ION OF THE CALCULATION TO TRUNJ.< 15. ~~RMEISTER.J,, Diss. TU Hannover 1973 PISTONS ~ Burmeister (1S),in expanding this calcu~~tion recently solved the Reynolds Differntial· Equation by approximation on a digital com~ puter,for various eccentricities and angl~~ of inclination of a piston of the 'same siz~ as the one t~k~n as an example here, In Reynolds Diffe~ential Equation,the second term of the l~ft hand side can'no'Ionger b~ ignored, that' is, thq.t '-whicfi d~~cribes the circumferential'pressure gerier~tion,The se- 79