The paper presents the results of theoretical analysis of the effect of supply pressure
and rotational speed on the radial stiffness of the integrated angular cylindrical–face
hydrostatic bearings designed for precise technological devices. The obtained theoretical results were verified experimentally using the spindle system of the grinding wheel of cylindrical grinder

Sikora, Małgorzata

English

Lodz University of Technology Repository

Mechanics and Mechanical EngineeringVol. 20, No. 3 (2016) 321{330c Lodz University of TechnologyRadial Stiness of Combined Journal and Thrust Hydrostatic BearingMa lgorzata SikoraInstitute of Machine Tools and Production EngineeringLodz University of TechnologyStefanowskiego 1/15, 90-924  Lodz, Polandnorbertkepczak@dokt.p.lodz.plwitold.pawlowski@p.lodz.plReceived (20 June 2016)Revised (26 July 2016)Accepted (18 August 2016)The paper presents the results of theoretical analysis of the eect of supply pressureand rotational speed on the radial stiness of the integrated angular cylindrical{facehydrostatic bearings designed for precise technological devices. The obtained theoreticalresults were veried experimentally using the spindle system of the grinding wheel ofcylindrical grinder.Keywords: hydrostatic bearing, combined journal, integrated angular, stiness.1. IntroductionThe continuous search for a more perfect way of the realization of rotational oradvance motion of assemblies of technological devices has determined the devel-opment of technology of hydrostatic lubrication. This is particularly important inthe machine tool industry, where the machining accuracy depends on the spindleunit. The advantages such as high accuracy of motion, high load capacity andrigidity, damping of vibration and a very low frictional resistance even at start-up,resulted in that the hydrostatic bearings are currently used in machine tools, in suchcases, where other varieties of bearings can not meet the extreme requirements [1].They are indispensable when demands are toward particularly high precision spin-dle running and excellent vibration damping. Compared with the hydrodynamicand rolling bearings they allow for repeated reduction in run out of the spindle [2{4]as well as the obtaining the mirror smoothness of the machined surface [5]. Theyare suitable for very heavy static and dynamic loads. Additionally, they have a highstiness in the range of low loads, typical for the operating conditions of precisionmachine tools. These bearings can operate at very low (including zero) and veryhigh rotational speeds. In a properly designed and operated hydrostatic bearing it322 Sikora, M.never comes to the metallic contact of journal and sleeve during operation, whichprovides practically unlimited service life (durability) of such bearings.In general, to support the machine spindle with hydrostatic bearings there arerequired two radial bearings and one two{sided axial (thrust) bearing. Sometimesthe integrated (combined) radial{axial bearings, which are adapted to simultane-ously, transfer of radial and axial forces. The application of such bearings allowssimplifying the design of the spindle unit. Power losses in these bearings are smallerthan in the independent axial and radial bearings. In addition, the integrated bear-ings (combined) enable a reduction in oil ow, which reduces the output of pumpsupplied the system. The result is a simpler design, easier to make, cheaper andmore favourable in terms of energy [6-9].Among the hydrostatic axial-radial bearings it is distinguished cylindrical-facebearing (Fig. 1), which is described in the literature for a long time - the mostdetailed in the books that can be considered as a classic in this eld [7{9]. Rowecalled them Yates bearings. Although neither mentioned books nor the presenttime books or manuals [10{13] does not take into account adequately an eect ofrotational speed on the properties of these bearings, in particular on the stiness,which is the objective function subjecting the maximization.This paper considers in theoretical and experimental analysis the of two vari-ables, i.e. the supply pressure ps and rotational speed n on the stiness of axial{radial, cylindrical{frontal hydrostatic bearing. Theoretical dependencies were ex-perimentally veried with the application of grinding spindle system of cut in centercylindrical grinder, which was equipped in a front node with analyzed axial-radialbearing.2. Object of analysisThe design and the main characteristic dimensions of the analysed hydrostaticaxial{radial bearing are shown in Fig. 1 while the numeric values are summarizedin Tab. 1.Ll lhDDDDDlk lkh0z h0z0syxFigure 1 Principal dimensions of the hydrostatic bearingsRadial Stiness of Combined Journal and Thrust Hydrostatic ... 323Table 1 Characteristic dimensions of the bearingMagnitude ValueRadial bearing Diameter D [mm]Length L [mm]Radial gap h0 [m]Land width:- leakage ow l [mm]- inter-recess s [mm]Number of packets k90134544124Axial bearing Collar diameter Dl [mm]Pockets diameter Dk [mm]Axial gap h0z [m]11611029Analysed bearing arrangement was fed with oil ISO 11158 - HL (L-HL 46) atconstant pressure ps, which was adjustable by overow valve. Before each of pocketsof radial bearing it was placed a slot restrictor with a slot face providing a laminarow of [14], whose design with characteristic dimensions are shown in Fig. 2.Figure 2 Principal dimensions of the hydrostatic bearings3. Static stiness of the bearingStatic stiness is determined by the ratio of the increase in the applied externalload to the displacement caused as a result of load actions. Loading force shouldbe applied in such a way that its point of application and the force direction wasconsistent with the actual layout at the grinding [15].324 Sikora, M.Due to the fact that the verication investigation are performed using the spindlesystem of the grinding wheel of claw of cut in grinder, the present discussion focusesonly on its radial properties. Under the terms of the cut in transverse cylindricalgrinder the radial stiness of analysed axial{radial bearing was determined for twoplanes Oxz (tangential force) and Oyz (thrust force).Radial displacement of the spindle journal versus the load radial force, in accor-dance with [6] is described by the following equations of motion:md2ydt2+Kdydt+Kpy +K!x =Wy (1)md2xdt2+Kdxdt+Kpx K!y =Wx (2)where:W { loading force of bearing,K { coecient of damping resistance [16, 17],Kp { pressurized component of stiness [16, 17],K! { motional component of stiness [16, 17].In the case where the journal is only statically loaded then the displacementderivatives in equations (1) and (2) are equal to zero, and dependences concerningthe displacement in the bearing take the following form [16]:y =KpWy  K!WxK2p +K2!x =KpWx +K!WyK2p +K2!(3)Equations (3) are accurate enough for the following assumptions and limitations: oil is incompressible and Newtonian uid, and its viscosity is constant withinthe bearing, ow in the restrictors and lands of bearing is laminar, depth of the chamber is substantially greater than the height of slots at thesteps, pressure drops at the lands width leakage ow and inter{recess are linear, relative displacements x and y are in the range [16]: 0,30  x/h0  0,30;-0,30  y/h0  0; 30, width of the lands satisfy the conditions: 0 < s  0; 075D ; 0 < l  0; 15 L.On the basis of equation (3) the dependencies of the radial stiness of bearing inthe assumed planes Oxz and Oyz take the form [18]:cy =Wyy=K2p +K2!Kp   WxWyK!cx =Wxx=K2p +K2!Kp +WyWxK!(4)Radial Stiness of Combined Journal and Thrust Hydrostatic ... 3254. Experimental studiesFor experimental verication of equation (4) the investigation of spindle system ofgrinding wheel of cut in center grinder equipped with hydrostatic bearings: front(from the grinding wheel side) and integrated axial-radial, cylindrical- face and therear - a typical radial bearing.Radial stiness of the spindle system was determined for two axes Ox and Oyangulary displaced in relation to the force vector Fr. The angle between the axisOx and the direction of the force Fr is equal to  = 45 (Fig. 3). Therefore, itcan be assumed that the spindle was loaded with two components acting along theaxis Ox (tangential force) and Oy (thrust force). The values of these forces are asfollows:Frx = Fr  sin Fry = Fr  cos (5)Lay{out of a measuring system for determination of the relationship betweenthe spindle displacement and loads in the radial direction is shown in Fig. 3.b a.1KomputerPCSpider1Fzx y23B A MaMrFry FrxFrFigure 3 Lay{out of the measuring system: 1 { dierential non{contact displacementmeasurement sensors, 2 { amplier, 3 { computerTo determine the transverse stiness of the axial{radial bearing the model wasassumed in which the external radial force Fr is applied to the tip of the spindle ata distance a = 170 mm from its centre, while the non-contact sensors 1 are locatedat the distance equal to aM = 125 mm. Displacements in the radial directiongenerated by force Fr were measured in two mutually perpendicular planes, Oyzand Oxz. The measurement system consists of two sets of dierential non{contactdisplacement sensors 1, wherein the signal amplied by the amplier 2 Spider wasrecorded on a PC 3 with a resolution of 0,01m. The value of the applied force isdetermined by a set consisting of a strain gauge force sensor (measuring range of0 to 5 kN, accuracy class 0.05%) and the microprocessor meter (non{linearity of< 0.0015%).The displacement of the spindle in the plane Oyz caused by the force of Fry areshown in Fig. 4. In the following part of this paper due to the analogy of load anddisplacements the detailed analysis was carried{out only the stiness in the planeOyz. The dependences given for the plane Oyz also apply in the plane Oxz after326 Sikora, M.the appropriate modication of recording, e.g. WAx, WBx, Frx; x, cx instead WAy,WBy, Fry, y, cy, and so on.ByyRAyW AyRByW ByAzF ryNMb ..aMByA yM³y MuyMwaFigure 4 The displacements of the spindle in the plane Oyz caused by force FryAccording to Fig. 4, as the result of the experimental investigation it was obtaineda total displacement of the spindle tip at the measuring point (point M), and it isequal in the plane of action of force Fry to the sum of displacements resulting fromthe exibility of the bearings and the spindle:yMu = yMl + yMw (6)where: yMw { displacement of a point M due to the exibility of the spindle equalto [6]:yMw = M FryyMl { displacement of a point Mdue to the exibility of the bearings is deter-mined from the relationship:yMl = yA1 +aMb  yB aMb(7)Factor M formulates the displacement of the pointM of spindle which is supportedon a perfectly rigid supports and which is caused by concentrated force, applied toa point determined by the coordinate a. Coecient M ratio is an indicator of theexibility of the spindle in place of the sensors mounting. Considering the actualdimensions of the test spindle the value M was determined by means of analyticalmethod. For the considered system it was obtained: M = 5,8510 9 m/N.Knowing yMl calculated from the formula (6) the searched (unknown) journaldisplacement in the front radial bearing the forward movement of the pivot journalbearing is yA:yA =yMl + yBaMb1 + aMb(8)Radial Stiness of Combined Journal and Thrust Hydrostatic ... 327wherein the value yB is considered as a known and determined by the set of theoreti-cal dependences (3). Thus, the dependence on the radial stiness of the axial{radialbearing obtained by experimental way is:cAd =WAyA=WA 1 + aMbyMl + yBaMb(9)Experimental study of the spindle unit was carried out for three rotational speeds n:0, 1000, 1500 rpm and three pressures ps values: 1.0, 1.5 and 2.0 MPa. During theinvestigations the viscosity of the oil owing through the bearing was 0,0263 Pas.0,02,04,06,08,010,012,01,0 1,5 2,0StiffnesscA·107[N/m]Supply pressure ps [MPa]cayd cayt cazd caztAyd Ayt Axd Axta)-10,0-8,0-6,0-4,0-2,00,02,01,0 1 ,5 2,0StiffnesscA·108[N/m]Supply pressure ps [MPa]cayd cayt cazd caztb)Ayd Ayt Axd Axt-10,0-8,0-6,0-4,0-2,00,02,01,0 1 ,5 2,0StiffnesscA·108[N/m]Supply pressure ps [MPa]cayd cayt cazd caztc)Ayd Ayt Axd AxtStiffness10[N/m]cA78Stiffness10[N/m]cA8..Stiffness10[N/m]cA.Figure 5 Radial stiness of the axial{radial cylindrical{frontal bearing for: a) n = 0 rpm,b) n = 1000 rpm, c) n = 1500 rpmIn order to determine the radial stiness of the axial{radial bearing the system wasloaded by radial force Fr equal 200, 400 and 600 N. The determined characteristics328 Sikora, M.of the transverse stiness of the axial-radial bearing hydrostatic bearing are shownin Fig. 5 in the form of bar graphs. The results concerning the experimentalinvestigation were denoted by full ll and the results obtained by theoretical wayby pattern of diagonal stripes. It should be noted that in the case of Fig. 5a it wasapplied the multiplier, which is less than an order than for Figs. 5b and 5c.On the basis of the results of theoretical analysis and experimental tests it canbe concluded that for a stationary spindle the stiness is linear as a function ofthe supply pressure (Fig. 5a). This is conrmed by correlation coecients R2 tregression line that for all obtained characteristics were not less than 0.995.If the spindle rotates then the signicant eect on its stiness have the hydro-dynamic phenomena in the bearings, which are included in the theoretical stinessof the motion component of stiness K!.To assess the compatibility of the results obtained experimentally with the re-sults of theoretical calculations the relative dierences cx and cy stiness weredetermined:c =cd   ctct 100% (10)For axes Oy the most cases was characterized by the relative dierence maximumof up to 20%, the largest dierence was for the supply pressure of 1.0 MPa androtational speed of 1,500 rpm and it was 40%. However, it should be noted thatsuch a large, relative dierence eects at the load, e.g. 200 N in the dierence ofdisplacement not greater than 1.0 m. But for the axis Ox almost all test cases arecharacterized by maximum relative dierence of 15%, only for the supply pressure of2.0 MPa and for a stationary spindle the relative dierence was larger and amountedto 29%, which resulted in a maximum dierence for a load to 200 N and is equal to0.5 m.-8-7-6-5-4-3-2-1012340 500 1000 1 500 2 000 2 500 3 000StiffnesscA·108[N/m]Rotation speed n [rpm]1,0 oœ OY 1,0 oœ OZ1,5 oœ OY 1,5 oœ OZ2,0 oœ OY 2,0 oœ OZp s 1,0 a xle Oyp s 1,5 a xle Oyp s 2,0 a xle Oyp s 1,0 axle Oxp s 1,5 axle Oxp s 2,0 axle OxStiffness10[N/m]cA8----Figure 6 Dependence of the stiness of the axial{radial bearing on the rotational speed fordierent supply pressureRadial Stiness of Combined Journal and Thrust Hydrostatic ... 329Fig. 6 shows the characteristics of the eect of rotational speed on the stiness of thedierent supply pressures in the extended speed range with respect to experimentalresearch.Fig. 5b, Fig. 5c and Fig. 6 show the inuence of the supply pressure ps andthe rotational motion on the radial stiness of the bearing. Attention is drawn toa much greater eect of rotational motion on the stiness in direction of axis Oythan the axis Ox.Particular emphasis is given the fact that in the case of the spindle rotatingat 1000 rpm for a pressure of 2.0 MPa signicantly higher stiness was obtainedin the direction Oy than for 1500 rpm. This situation is due to the correlation ofthe hydrodynamic and ow phenomena occurring between the chambers of bearinghaving a direct eect on the angle between the displacement and load vector [16].It should also be noted that the studied spindle system also has signicantlyhigher stiness in Oyz than in Oxz plane. This feature is extremely benecialbecause it is a plane of the action of grinding thrust force, and thus is decisive forthe accuracy of the machine.5. ConclusionsThe results obtained according to the experimental investigation have conrmedthe theoretical study that for the stationary spindle the variation in the stiness isa linear function of supply pressure. Supply pressure directly aects the stinessof the spindle system, which increases with pressure. In both investigated radialplanes the system showed a similar stiness.In the case of the rotating spindle due to the hydrodynamic phenomena theradial stiness of the hydrostatic bearings depends on the supply pressure and therotational speed of the spindle. This speed has a clear eect on the value and direc-tion of displacement. Rotational motion of spindle results in having several timesthe increase in the system stiness of the axis Oy (reaction), which is particularlyrequired in high-precision spindle grinding systems. Underestimation of such eectmay lead to errors in the design of spindle systems with hydrostatic bearings.It should be noted that such a signicant variation in eect of rotational speedand the supply pressure on the stiness in the direction of axis Oy may contribute,especially in the case of precision machine tools, the diculty in obtaining therequired, reproducible preciseness of parts which are machined in the case of thefailure of constant parameters, i.e. the supply pressure and rotational speed.References[1] Hale, L. C., Wul, T. A. and Sedgewick, J. C.: Testing a low{inuence spindlemotor, Precision Engineering, 29, 1{10, 2005.[2] Frank, W., Neugebauer, R. and Voll, H.: Nutzung des Leistungsmogens vonSpindellagerungen, Konstruktion, 3, 53{58, 1995.[3] Kane N. R., Sihler J. and Slocum A. H.: A hydrostatic rotary bearing withangled surface self{compensation, Precision Engineering, 27, 125-139, 2003.[4] Wardle, F.: Ultra{precision Bearings, Woodhead Publishing, 2015.[5] Zhou, L., Tachibana, T. Syoji, K., Kuriyagaw,a T., Haga, T., Unno, K. andOhshita, H.: Form mirror grinding of inner race of ceramic bearing, Transaction of330 Sikora, M.the Japan Society of Mechanical Engineers, Part C Vol. 63 No. 612 (August), 2905{2910, 1997.[6] Przyby l, R.: Podstawowe problemy projektowania zespo low wrzecionowychwyposa_zonych w  lo_zyska hydrostatyczne, II. 13, 115{122, Dabrowski L. (ed.):Obrobka scierna w technikach wytwarzania. Badania i aplikacje. Warszawa: OcynaWydawnicza Politechniki Warszawskiej, 2005.[7] Rowe, W.B.: Hydrostatic and Hybrid Bearing Design, London, Butterworths, 1983.[8] Rowe, W. B. and O'Donoghue, J. P.: Design Procedures for Hydrostatic Bear-ings, Brighton: Machinery Publishing Co. Ltd., 1971.[9] Stanseld, F. M.: Hydrostatic bearings for machine tools, Brighton: MachineryPublishing Co. Ltd. 1970.[10] van Beek, A.: Advanced engineering design, Delft: University of Technology, 2006.[11] Shigley, J. E., Mischke, C. R. and Brown, T. H. Jr.: Standard Handbook ofMachine Design, New York: McGraw Hill, 2004.[12] ISO 12168{1, Plain bearings - Hydrostatic plain journal bearings without drainagegrooves under steady{state conditions, Part 1: Calculation of oil-lubricated plain jour-nal bearings without drainage grooves, 2001.[13] ISO 12168{2, Plain bearings { Hydrostatic plain journal bearings without drainagegrooves under steady{state conditions, Part 2: Characteristic values for the calculationof oil{lubricated plain journal bearings without drainage grooves, 2001.[14] Sikora, M.: Metody wyznaczania charakterystyk przep lywowych d lawikow hy-draulicznych, Mechanik, nr 8/9, 349-354/958, 2014.[15] Kwapisz, L. and Rafa lowicz, J.: Szlierki. WNT Warszawa, 1976.[16] Przyby l, R.: Poprzeczne  lo_zyska hydrostatyczne w zespo lach wrzecionowych obra-biarek, Zeszyty Naukowe Politechniki  Lodzkiej, nr 921, Wydawnictwo Politechniki Lodzkiej,  Lodz 2003 .[17] Sikora, M.: W lasciwosci poprzeczno{wzd lu_znego  lo_zyska hydrostatycznego, Tribolo-gia teoria i praktyka, nr 2, 69{82, 2012.[18] Lewandowski, D. and Przyby l, R.: Sztywnosc zespo lu wrzecionowego  lo_zysko-wanego hydrostatycznie, Mechanik, 5, 273{277, 1985.

Radial Stiffness of Combined Journal and Thrust Hydrostatic Bearing.

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Radial Stiffness of Combined Journal and Thrust Hydrostatic Bearing.

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