Subsurface Stresses in Rolling & Sliding Machine Components [Sadeghi, Sui; Int.comp.Eng.conf., 1988]

8/13/2019 Subsurface Stresses in Rolling & Sliding Machine Components [Sadeghi, Sui; Int.comp.Eng.conf., 1988]

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Purdue University

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International Compressor Engineering Conference School of Mechanical Engineering

1988

Subsurface Stresses in Rolling/Sliding MachineComponentsFarshid SadeghiPurdue University

Ping C. SuiPurdue University

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Sadeghi, Farshid and Sui, Ping C., "Subsurface Stresses in Rolling/Sliding Machine Components" (1988).International CompressorEngineering Conference. Paper 680.hp://docs.lib.purdue.edu/icec/680

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SUBSURFACE STRESSES IN ROLLING/SLIDINGMACHINE COMPONENTSFauhid SadeghiAssistant Profes..or

Ping C. SuiGraduate Assistant

School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907ABSTRACT

The internal stress distribution in elastohydrodynamic lubrication of rolling/sliding line contactwas obtained. The technique involves the full EHD solution and the use of Lagrangian quadrature toobtain the internal stress distributions in the x, y, z-directions and the shear stress distribution as afunction of the normal pressure and the friction force. The principal stresses and the maximum shearstrese were calculated for dimeru.ionless loads ranging from (2.0452 ; ; 10 5) to (2.3 x 10-


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E1 Equivalent Young's modulus l/E1 1/2 ((1-v;)/E1 + (1-v;)/Eb), PafT Traction coefficientG Material parameter, aE1H Dimeruiionlees film thickness, hRfb2H. Dimensionless film thickness where dP Pressure viscosity exponent, m2/NI ; Dimensionless viscosity of the lubricant Ambient viscosity of the lubricant, Ns/m 2p Relative densityP. Relative density where H =H.p Dimensionless density Norm . stress in the rolling direction, Pa.

ux Dimensionless normal stresa in the rolling direction, ax/PHa1 Normal stress in the solid, Pa1:11 Dimensionless normal stress in the solid, a7 /PHmu: Dimensionless maximum shear stresei'.,, Shear stress, Par. Dimensionless shear stress, r.,,/Pa

THE GOVERNING EQU TIONSThe Reynolda Equation with the appropriate asaumptions {13] could he written in thedimensionless form as:

H (jf_) _ . ir2U ] -. H _ P H ) =0dX; 2 P1 1)where the boundary conditions are P1 O for X1 ""X,.10 and P dP dX = O for X X-.o- Theconstant load condition is:

. ...2The film shape in the dimensionless form is given by [13];

~ '


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The viscosity /pressure relationhip used in thi analysis was proposed by Roelands [14]. TheRoelands equation in dimensionless form is;(5)

The exponent z can be expressed in terms of and as; 5.1 X 10 9, ln( 0 ) + 9.61 (6)

where G = aE1The density pressure relationship used by Dowson and Higginson [15] in the dimeneionlese form isemployed in this analysis.

(7)

STRESS EQUATIONSA triaxial state of stress exists at the conjuction region of rolling/sliding contacts; and the solidsin contact are in a condition of plane strain. The plane strain condition implies that the shear stresses

TY ,- ' = O. Rolling/sliding contacts transmit normal pressure and tangential traction due to friction.Figure 1 depicts an elastic baI space loaded over Xmin < X


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X-i 2- .1.. --2 .L = - 1- f y21x - X'l P(X1dX' YIX - x1 Q(X')dX'r.,. b2 axay II x... Aj x""' A'Wbere [ 1

/2A = (X-X')2 +Y2with the boundary conditions given as [16];

O",(X,O) -P(X)x..,.

. ~ d Xf x,. (X - X1

cry(X,o) - P(X). u f )3/2 u.r X O )= - - -Y HW '\132 H ( 2W )1/2 Af ..11 dX

(12)

(13)

(14)(15)16)

Using equations (10) through (16) the subsurface normal otresses and shearing stress at any point in anll:HD iubrioa.tion of rolling/sliding eontact ca.n be obtained. These stresses were used to obtain thestress invariants and principal stresses. The principal stresses are employed to obtain the maximumshear stress in the rolling/sliding elements. The maximum shear stres$ is given by:

Tmax t(um.,. - O minl (17}RESULTS AND DISCUSSION

Figures 2 through 5 iUUBtrate the pressure and film thickness for dimensionless loads ofW = 2.0452xl0-6 to 2.3xl0-< and dimensionleiiS velocity ranging from U = 10-10 to io-12 at materialparameter G 5007. The fast approach Newton-Raphson technique [13] was employed to solve thesimultaneous system of Reynolds and elasticity equations. Figure 2 depicts that the pressure spikeoccurs at X = .44 from the center of the contact for W = 2.0452 x 10- and U = 10-10 However, asthe speed is decreased (Figure 2), the magnitude of the pressure spike decre"""" and the point ofmaximum preSAure, where dP/rfX 0 occurs near the center of the contact. This condition also occursa.s the load is increased for a constant velocity. A comparison of pressure profiles in Figures 2 and 3 fora constant velocity of U = 10-11 reveals this fact, The points of maximum pressure and pressme spikeare important since they significantly effect the surface ahear stress. Figures 6 through 15 illustrate thecontour plots of maximum ahear stresli distributions for the dimensionless velocity varying fromU 10-10 to 10-12 and loads from W 2.0452JC10-li to 2.3xlo- for pure rolling a.nd slip conditions,Figure 6 iUUBtrates t h ~ maximum shear streSB distribution for the loa.d W = 2.0452x10- a.nd speedU = 10-lO under pure rolling condition. Figure 6 depicts that the maximum shear stress reaches itsmaximum of 0.491 at the surface of the element and a point of stress concentration appears at thepressure spike. FigID"e 7a illustrates the maximum shear atreoo distributions at the same load and slip

as in Figure 6 at the lower velocity of U = 10-11 In this case the maximum shear stress reaches itsmaximum of 0.290 at 0.680 below the surface. However, as the slip is increa.Bed, the maximum shearstress moves towards the surface. An examination of Figure 7h indicates that at this load and speedthe maximum shear stress of 0.343 occurs on the surface for 20 percent slip. Figure 8 elucidates the1naxi1num shear stress contour for the load W = 2.0452xl0-5 and speed U = 10-n under pure rollingand 100 percent slip conditions. Figure Sb indicates that for this load a.nd velocity at 100 percent slipthe 1na.x mum shear stress has moved towards the surface. ffowever, it fails Lo reach the surface. Themaximum shear stress reaches its 1nuimum of 0.295 at 0.780 below the surfaee for pure rolling (FigureBa) and at 0.660 below the surface for 100 percent slip eondition (Figure 8b ). Figure 9 depicts themaximum shear stre"" distribution at the pressure spike and the point of maximum pressure wheredP /dX = 0. These points are of interest since they significantly elfe


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the surface (Figrnce 10). However, at this load and speed a small amount of slip significantly effects thedepth at which the maximum shear stress oecurs (Figme lOh). Figure lOb illustrates that at 0.6percent slip the maximum shear stress of 0.323 occurs at the surface. An examination of Figure lObindicates that a. region of maximum shear stress appears to be at a depth of 0.40 below the surface nearthe pressure spike region. However, closer e:ramination of Figure lOb reveals that the maximum shearstre$S occurs on the surface near the point of maximum pre8Sure. Similar ob,,.,rvations can be made forthe same load but at lower velocities. Figures 11 and 12 indicate that, at the lower speeds the slip hasto be significantly increased for the maximum shear strel I to move towards the surface.

Figures 13 through 15 are the contours for 2.3:i:lo- load at speeds ranging from U = 10-10 toio-12 under pure rolling a11d slip conditions. For this high load under pure rolling condition (Figure13a), the shear stress reaches its maximum of 0.2114 a.t 0.780 below the surface. For 3 x 10- percentslip the maximum shear tress is 0.314 and occurs on the surface. For the same load at lower speed ofU-= 10-11 (Figure 14), the maximum shear is 0.2 1 1 and occurs at 0.780 below the surface under purerolling condition (Figure 14a) and 0.313 on the surface for 4 x 10...; percent slip. Figure 15 illustratesthe shear stress contour for the 10-12 speed case. The figure illustrates that the maximum shear stressoccurs at 0.780 below the surface under pure rolling and move to the surface for 6x 10-' percent slip.

,.JGTt

E6 035

- -1 . .1 : : : :10 '41__ _. . . . . U.::: 10' JI l l= 10 11

4- I 11 ol I 41:1X O . X l f d m . l : I E ( X ~ ll./bl

095

'

u 1 1___ . , . ,U 10

'

056.,,028

0 0 0 1 . . . . . i . ~ ~ . . . : ~ ' - O : . : ; , . = ~ ~ ' - ' - - . . . J a o o36 -:JD l4 1 11 48 15 14ilCorm1i l4lte {JI. .. 1/ bl

X - C ( m r d l n . : : i t ~ ( l < ~ 1l"d"1;.lr-\X 1


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F i ~ 1 ~ r c 7. D mQ8killl-et.nJ U l x 10

)(CoordITTlllr, x ~ a / t i )02

(s} pure rolling conditioo: (b) 100 ellp e.onditivuD i r n ~ 1 1 i e . h : i n l e H m..xinnun heJlr t.rieH .ctantvure t r W = 2,0(52 10and U = l x 10 1 1:,

o O O r - ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ - , 60525" 450

' 375300.225150

40 Slip

20'% 511 .1Slip

~ 7.0O_.......-60% Shp

40 40-t. Shp20'"/., SlipI ?1 S tp

1000 ozs 050 07'5 100 t25 150 175 200 125 150 ?5. ::?00

1 m 4 n ~ l e s s o.,ip1ti into he Solid ~ / b )

(a) .. the miu;imum p r ~ o U r eFigUTll I. Dimtnni.onlr,111ii: maxhmun l'.lhenr titr ' e. 1:1i11t. "ibut.lun in t b ~ OliJ at.

diff' rtmt lip rondltionn, W 2.0452 10-t nd U = l x 10 16

634


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(b) 0.1 ftlip ctmdttianF i g u r ~ 1(1, Dbnentionlieu ~ I m u m hut ' trctH w11 t t l r I f()l' w _, 1.0 % ur

and U 1 io- 10-.

X - C 0 0 4 ' d l n J 1 ~ (X'"l/bl02

X COOtd1nQ11t 00: 1/tl-04 o ;; oe

(b) 0.8 alip oeondillon

Flgurll'I 11, Dime:niU.J\)H ll rn0


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(a) ptJn rolllrtg comUtionFlgurit j 'L Dim;i:1n11oinnlr1U1 m.axinmm 1hei1.l" t.l" il:H t i o l l t . ( ) U ~ for W = '2.3 x i o ~

a n d U = lx lO. rn .

XMC.o


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Tablea l and 2 contain the maximum shear stress and its location for different load and velocitiesunder pure rolling and slip conditions. Table l indicates that under pure rolling conditions themaximum shear stress for high loads occurs at 0.78 below the surface and near the center of thecontact. However, at low loads and high speeds the maximum shear stress occurs closer to the llllfaceand moves towards the edt zone. Table 2 contains the maximum shear streSB under slip conditionsobtained from the present method (Lagrange Quadrature Method, LQM) and the result of Smith [7].The coefficient of friction for each loading, speed, and slip condition obtained from the LQM ispresented in Table 2. The coefficient of friction for each condition was used to obtain the maximumshear streaa for Smith's [7] analysis. A comparison of the maximum shear stresses in Table 2 indicatesthat, Tmax obtained from the LQM is significantly higher (111%) at low loads and high speeds. However,the error is reduced as the load is increased. The LQM indicates that, r"'01 occurs on the surface for allloads, speeds, and slip conditions examined except for the lowest load and speed. However, Smith'sMethod [7] indicates that, r. . always occurs below the surface for the enmined eonditiollll. Themethod developed represents a more accurate description of rolling/sliding contacts, since itsimultaneously solves the Reynolds and elasticity equa.tions to obtain the pressure profile and the stressdistributions in the elements.

Table 1. Maximum shear stress and its location under pure rolling conditions.x y ~Wx 1 5 u x 1012 LQM LQM LQM

2.0,52 100 0.520 0.000 o.49110 O.ll60 0.680 0.291

0.295 0.780 0.29S7.0 100 o.uo 0.780 0.281

10 0.080 0.780 o.2g51 0.000 0.780 0.299

13.0 100 0.120 11.780 0.29010 0.020 0.780 0.2981 OJlOO 0.780 0.300

23.0 100 O.o IS 0.780 0.29410 0.000 0.780 0.2911l 0.000 0.780 0.300

T11-ble 1. Maximum shear 11tres11 11.nd it11 location under slip conditions.x y x y ,_ rw 10 U x 1012 % lip fT LQM LQM [ J 111 LQM (7]

2.'&2 100 0 32x10-< 0.52 0.0 o.oo 0.78 0.4gl o.aoo10 20 4 I l t - i l 0.74 o.o 0.1-i 0.78 0.3 3 0.3illI 100 11 I 1 2 0.50 0 66 0.36 0 12 0.310 0.3il67.0 100 6X 10-1 12 x lo-"' 0.18 0.0 o.o 0.72 0.323 0.30

10 x 10 -1 1S x 10- 0.18 o.o MO 0.72 0.362 Q.3091 l 14 x 10- 0.1 o.o 0.40 0.72 0.357 0.30913.0 100 14 10 .. 95 10 .. O.lft o.o Q.M 0.74 0.318 0.30410 15x10"""' 87x10 ... 0.14 0.0 0.28 0.74 0.310 0.304x 10_. 10x10 . . O.H 0.0 0.34 0.72 0.333 0.30ii23.0 100 3 10- 82 10-> 0.14 0.0 0.28 0.76 0.314 0.30310 xur 81x10- o.u 0.0 0.28 0.76 0.313 0.303

l x 10- 7g 10- o.H 0.0 0.22 0.7ft 0.312 0.303CONCLUSIONS

A solution to the problem of internal stress in elatohydrodynamic lubrication of r o l l i n g s l ~ d i n gcontacts is presented using the Newton-Ra.phson and Lagrangian quadrature method. For the highlyidealized lubricant considered, the following conclusions were drawn:

1. Under pure rolling condition the maximum shear stress in general occurs at 0.780b below thesurface.

2. Slip has significant effect on the depth at which the maximum shear stress occurs.s. For fow loads, the higher the speed, the closer to the surface the maximum shear stress occurs.4. For high loads small amounts of slip draws the maximum shear stress to the surface.5. The point of stre concentration occurs near the exit zone under low load conditions, however, asthe load is increased this point moves toward the center of con.ta.et.

637


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ACKNOWLEDGEMENTThe authors would like to express their appreciation to the National Science Foundation,Tribology Program for their 511pport of this project and to the program director Dr. S. Jahanmir for his

amstance and encouragement.REFERENCES

1 Shigley, J.E., "Mechanical Engineering Design," Third Edition, McGraw Hill 1977.2. Morton, W.B., and Close, L.J., "Note on Hertz Theory of the Contact of Elastic Bodies,"Philosophical Magazine Series 6, Vol. 43, p. 320, 1922.3 Thoma, H.R. and Hoeraoh, V.A. "Stresses Due to the Pressure of Oue Elastic Solid UponAnother," Univ. Ill., Engr. Expt. Sta., Bull. 212, 1930.4. Foeppl, L., "Der Spannungszustand und die AUBtrengung des WerkStoffe, bei der Bernehrung Zweier

Korper," Forschung a.uf dcm Gebiek des Ingeniennvesens Ausgabe, A. Vol, 7 pp. 209-221, 1936.5. Lundberg, G. and Palmgren, A. "Dynamic Capacity of Rolling Bearings," Ing. Veta.nakap, Akad Hand ., No. 196, 1947.i: Poritsky, H., "Stresses and Deflections of Cylindrical Bodies in Contact with Application to Contactof Gears and Locomotive Wheels," Journol of Applied Milcha.nics Trarui. ASME, Vol. 72, pp. 191-201, 1950.7. Smith, J.O., and Liu, C.K., "Stresses Due to Tangential and Normal Loa& on an Elastic Solid WithApplication to Some Contact Stress Problem;i," Journal of Applied :Mechanics 1952.8. Dowson, D., Higginson, G.R. and Whitaker, RA., "Stress Distribution in Lubricated Rolling

Contacts, Iruit. Mech. En;;rs., 1963.Y Hamilton, G.H., and Goodman, C.E., "The StreliS Field Created by Circular Sliding Contact,

SME Journal of Applied Mechanics 1966.10. Bryant, M.D., and Keer, L.J., "Rough Contact Between Elastically and Geometrically IdenticalCurved Bodi..,.," SME Journal of p p l i ~ d M:chanics Vol. 49, June 1982, pp. 345-359.11. Kannel, J.W., Tevaarwerk, J.L., "Subsurface Stress Evaluations Under Rolling/Sliding Contacts,"Trans of tiu: ASME ASLE Vol. 106, pp. 96-103, 1983.12. Okamura, H., "A CoJJtribution to the Nurnerical Analysis of Isothermal ElaetohydrodynamicLubrication," Tribology of Reciprocating Engines: r o c e ~ d i n g s of the 9th Leeds-Lyon Symposium onTribology B u t ~ r w o r t h s Guilford, England, pp. 313-320, 1982.13. Houpert, L.G., and Hamrock, B.J., Fast Approach for Calculating Film Thiekness and Pressures inElaetohydrodynamic Lubricated Contaets at High Loads," SME Journal of Tribology Vol. 108, pp.411-420, 1118614. Roelands, C.J.A., Vlugtcr, J.C., Watermann, H.l., ..The Viscosity Temperature PressureRelationship of Lubrieating Oils and its Correlation with Chemical Constitution," Journal of Ba.si.t;Engineering pp. 601-606, 1963.15. Dowson, D., Higginson, G.R. "Elastohydrodynarnic Lubrication," Pergarnon Press, 1966.16. Kannel, J.W., Walowit, J.A., Bell, J.C., Allen, C.M., "The Determination of Strescs in RollingContact Elernents," SME Journal of Lubrication TechMlogy Vol. 89, pp. 483-467, 1967,

Documents

Subsurface Stresses in Rolling & Sliding Machine Components [Sadeghi, Sui; Int.comp.Eng.conf., 1988]