American J. of Engineering and Applied Sciences 4 (): 94-300, 0 ISSN 94-700 0 Science Publicaions Experimenal Sudy of Load Sharing in Roller-Bearing Conac by Causics and Phooelasiciy Konsandinos G. Rapis, George A. Papadopoulos, Theodore N. Cosopoulos and Andonios D. Tsolakis Deparmen of Mechanics, Naional Technical Universiy of Ahens 5 Heroes of Polyechnion Av, GR-57 73, Zografou, Ahens, Greece Absrac: Problem saemen: In his sudy a comprehensive mehodology for calculaing load sharing in Roller-bearing conac was presened based on he experimenal sress-opical mehod of causics and Phooelasiciy. Approach: The heoreical equaions describing he geomery of ransmied causic in relaion o he lengh of he conac zone were derived and a simple mahemaical se of equaions correlaing he maximum diameer of ransmied causic wih he magniude of load was given.for his conac problem, he basic heory of phooelasiciy was given. Resuls: The echniques of causics and phooelasiciy were applied on a se of PMMA (Plexiglas) Roller-bearings and on a se of PCBA (Lexan) Roller-bearings, respecively. Conclusion: The proposed mehod of he causics is a reliable alernaive for measuring load disribuion in Roller-bearings compared o phooelasiciy echnique. Key words: Causics measuremens, phooelasiciy mehod, roller-bearings, elasic consans, herzian conac, conac surface, complex sress funcion, plane srain, poisson s raios, ligh beam INTRODUCTION Imporan problems wih many pracical applicaions are he conac problems (Timoshenko and Goodier, 970). Various conac problems of surfaces under fricionless or fricional conac have been sudied by (Ciavarella and Decuzzi, 00; Yu and Bhushan, 996; Michailidis e al., 00), including lubricaed conacs, Durany e al. (00) and conac beween coaed bodies, Schwarzer e al., (999), wih eiher analyical or numerical mehods. For he soluion of such problems, excep he mechanical analysis, he experimenal mehod of causics (Theocaris, 970; Kalhoff, 993; Papadopoulos, 993) and he phooelasiciy mehod (Froch, 96) can also be menioned. The opical mehod of causics is suiable for he experimenal sudy of singulariies in sress fields creaed eiher by geomeric disconinuiies or by loading. In his sudy, he sress-opical mehod of causics was used o measure he conac load of bearing rollers. The sress disribuion prevailing a a classical Herzian conac, Johnson (987), is used o derive a suiable complex sress funcion, Muskhelishvili. Then, he equaions of he ransmied causics were derived in a comprehensive final form using Lauren series causic is mahemaically linked o he lengh of he bearing conac and hence he bearing load. MATERIALS AND METHODS A special es rig has been designed and consruced a he Laboraory of Machine Elemens of he Naional Technical Universiy of Ahens (NTUA) for esing Roller-bearing models and he infrasrucure of he Laboraory of Srengh of Maerials of NTUA has been used o perform he causics measuremens. The conac problem of wo cylinders: According o Herz heory (Durany e al., 00) when wo solid spheres are pressed ogeher wih a force, a circular surface of conac of radius α is obained, while, when wo solid cylinders are pressed ogeher wih a force, a narrow recangular area of conac of widh b and lengh l, is obained. The maximum pressure occurs a he cener of he conac area. The conac problem of wo cylinders can be solved by Mushkelishvili sress funcion Φ(z) (Schwarzer e al., 999). In he case of conac beween a cylinder of radius r and a spherical sea of radius R (Fig. ), if he bodies are pressed ogeher along he normal a O by a force P, here will be a local deformaion near he poin of conac producing conac expansion. The maximum diameer of he ransmied over a small surface, called surface of conac. Corresponding Auhor: George A. Papadopoulos, Deparmen of Mechanics, Naional Technical Universiy of Ahens, 5 Heroes of Polyechnion Av., GR-57 73, Zografou, Ahens, Greece 94
Which, was obained from he equilibrium of displacemens and: κ + κ + K = + 4µ 4µ (4) µ, µ = The shear modulii of he wo bodies and k, k = The elasic consans which are given as: Fig. : Geomery of conac beween a cylinder and a spherical sea For plane srain : κ = 3 4ν For plane sress :,, 3 ν, (5) κ, = + ν, where, v, v are he Poisson s raios of he maerial of which were made he wo bodies. For he sudy of his problem, he maerial of he wo bodies was Plexiglas (PMMA) and hus he elasic consans are he same: µ = µ = µ, ν = ν = ν, κ = κ = κ (6) For he plane sress sae he K and κ are given as: Fig. : Geomery of he conac line of he wo bodies where P is a poin of he iniial curve of he causic Assuming ha he radii of curvaure r, R are very big in comparison wih he dimensions of conac surface, he conac surface is assumed as a line of lengh l (Fig. ). In his case he Mushkelishvili complex sress funcion is: κ + K = µ (7) 3 ν κ = (8) + ν By inegraing he sress funcion () and he Rel. (), we obain: R r Φ (z) = i z z l (9) KrR Φ (z) = l z + f ()d πk l l ( z) l Wih z being he complex variable. From he equilibrium of forces, we obain: () And: l rr = KP π (R r) (0) + l l f ()d l = KP f () ( ) r R () = (3) The force P is calculaed by he Rel.(0), if he lengh l of conac line is known and inversely, he conac lengh can be calculaed if he force is known. Boh, force and conac lengh can be calculaed by he experimenal mehod of causics. If he wo cylinders are on he eiher side each oher, wih a conac line of lengh l, he Rels (3), (9) and (0) becomes: 95
f () ( ) r R = + () R + r Φ (z) = i z z l () KrR l rr = KP π (R + r) (3) Applicaion of he sress-opical mehod of causics for measuremens of conac load: In every conac beween wo elasic solids a sress singulariy appears because of he high srain gradiens a he conac region. The opical mehod of causics is able o ransform he sress singulariy ino an opical singulariy, using he reflecion laws of geomeric opics (Theocaris, 970; Kalhoff, 993; Papadopoulos, 993) hus providing all he informaion needed for he evaluaion of he sress singulariy. According o he heory of causics for opical isoropic maerials, he equaions of he iniial curve and he causic for he ransmied ligh rays of he convergen ligh beam (indicaed by subscrip ) are (Papadopoulos, 993; 005; 004): 4C Φ "(z) = (4) * * W = λ m z + 4C Φ (z) Wih: z c z + z C = andλ = * 0 0 i m λm zi (5) (6) Φ(z) = complex he sress funcion, z = The complex variable, z = x+iy λ m = The opical seup magnificaion facor, z 0 = The disance beween specimen and reference plane z i = The disance beween specimen and ligh beam focus, is he specimen hickness c = The sress opical consan of he specimen maerial The complex sress funcion for he case of he conac of wo elasic bodies is given by Muskhelishvili (977): ( l z ) z Φ ( z) = + i kr kr Subsiuing Eq. 7 ino Eq. 4 and 5 he iniial curve radius equaion and he causic parameric equaions are obained: ** C 3 r = Re( z) = l + l C * ** o kr k mr And: (7) and Goodier, 970): 96 (8) C z c = = (9) λ ** Cl X = rλm cosθ ± cos θ 3 r ** Cl Y = rλm sin θ ± sin θ 3 r (0) () The heoreical ransmied causic curves calculaed for wo differen conac lenghs are illusraed in Fig. 3. Solid lines correspond o l = 0.00m, while doed lines correspond o l =.6mm. The inner causics ( ε -shape) correspond o focus poin lying in fron of he specimen and he ouer causics ( C -shape) correspond o a focus poin lying behind he specimen. For he causics ploed in ** Fig. 3 i was assumed ha r = 0.00m, C = 0.00m and λ m =. In Fig. 3 i becomes obvious ha he maximum diameer of he ransmied causic is parallel o he x- axis. Therefore by measuring he maximum diameer D x = X max of he causic which is parallel o he horizonal axis, he conac lengh lcan be calculaed by solving numerically he following sysem of equ.: g Dx = rλm cosθ + sin θ θ = sin C l g = r ** 3 ( ) + 8g 4g () (3) (4) Concenraed force a a poin of a sraigh boundary: A verical concenraed force P acs on a horizonal sraigh boundary of an infiniely large plae (Fig. 4). The sress funcion is given as (Timoshenko
P Φ = rϑsin ϑ π (5) The sresses σ r, σ ϑ, σ ϑr are given by he relaions: Φ Φ P cosϑ σ r = + = r r r ϑ π r (6) Φ 0 σ ϑ = = r Φ τ rϑ = ( ) = 0 r r ϑ (7) (8) Fig. 3: Theoreical causics for elasic conac beween wo bodies For a circle (or disk) of diameer d (Fig. 4), a any poin of he circle, he sress σ r is: P σ r = (9) π d From Eq. (4-5) he iniial curve and he parameric equaions of causic are: r o = r = (C ) /3 (30) 3 X = λm(c ) cosϑ + cos ϑ (3) Fig. 4: Geomery of he applied force P a semi-infinie plae 3 Y = λm(c ) sin ϑ + sin ϑ (3) Wih: z dc P 0 C = (33) λ m π For values of ϑ beween π/ and +π/, we obain he iniial curve and he respecive causic around he poin of he concenraed force P. Figure 5 presens he forms of he iniial curve and he respecive causic. The maximum longiudinal diameer of he causic along he Oy axis (Fig. 5), Can be derived from he condiion: Y = 0 ϑ (34) Fig. 5: Geomery of he iniial curve and he respecive causic a he poin of he applied force max ϑ = 60 (35) Relaion (3), for ϑ=ϑ max =60, gives: Which gives? 97 max 3 3 3 max m D = Y = λ C 3 (36)
By subsiuing Rel.(33) ino Rel.(36), we obain: πd P = 8 3z d c 3 max 0 λm (37) From Rel. (37) he load P can be experimenally calculaed by he diameer of he causic which is formed a he poin of is applicaion. The phooelasic mehod: According o he sressopical law, he difference of he principal sresses is (Froch, 96): Nfσ σ σ = (38) N = The isochromaic fringe order, is he hickness of he specimens = The maerial fringe value or sress-opical consan f σ The sress-opical consan of he specimen maerial can be calculaed by he disribuion of he difference, (σ - σ ), of he principal sresses a he cener of a disk, according o relaion (Timoshenko and Goodier, 970): Fig. 6: Geomery of he Roller-bearing (a) 8P σ σ = (39) π D Then, he Rel. (38) becomes: NπfσD P = (40) 8 Relaion (40) gives he compressive load P a he cener of he disk, or he sress-opical consan f σ, if he compressive load is given. (b) Fig. 7: Causics a he conac poins (a) for wo rollers and (b) for hree rollers RESULTS AND DISCUSSION The specimens were loaded by a saic concenraed load P hrough he inner ring of he Roller In his sudy he load disribuion in Roller bearing bearing (Fig. 6). Thus, he load P is ransmied by he is sudied. For his sudy, a simulaion of Roller bearing inner ring o roller and hen o he ouer ring. The was made (Fig. 6). Plexiglas (PMMA) specimens were conac surfaces beween rings and rollers are assumed used for he sudy by he mehod of causics and Lexan o be poins. The disribuion of he sresses is given by (PCBA) specimens for he sudy by he mehod of he Herz heory. phooelasiciy. The hickness of he Plexiglas Figure 7 presens he causics a he conac poins specimens was 0.0093 m and ha of he Lexan beween rings and rollers (a) for wo rollers and (b) for specimens was 0.0 m. The radii of inner ring were R i, = 0.0055 m, R i, = 0.075 m, he radii of ouer ring hree rollers configuraion. The applied concenraed were R o, = 0.0305 m, R o, = 0.045 m and he diameer loads were P=7.39 and P=94.990 KNm for wo of rollers was D = 0.03 m. loaded rollers and P=68.904 and P=98.470 KN m for hree loaded rollers. 98
(a) (b) Fig. 8: Isochromaic fringe paerns a he conac poins (a) for wo rollers and (b) for hree rollers Table : Calculaed loads for each roller Load of` Number of each Sum of P (KN loaded roller loads a/a m ) rollers (KN m ) (KN m ) P (%) LSF By causics 7.39 30.70 73.74.86 0.4 43.00 0.59 94.990 44.00 97.4.37 0.46 53.30 0.56 3 68.904 3 7.40 73.55 6.74 0.0 57.570 0.84 8.840 0.3 4 98.470 3.00 03.45 5.0 0. 74.50 0.76 6.90 0.7 By phooelasiciy 33.50 5.40 30.84 7.8 0.46 5.40 0.46 38.575 8.500 37.00 4.6 0.48 8.500 0.48 3 45.70 3 6.67 43.64 4.65 0.4 30.830 0.68 6.67 0.4 The load for each roller was calculaed by he Rel.(37), wih z 0 =.90 m, d = 0.0093 m, λ m =.93 and c =. 0 0 m N. The calculaed loads for each roller and he percen divergence of loads are presened in Table. The small divergence of he loads are caused by he lack of separaors (reainers) in he model, leading o non-symmery in he rollers. This can be seen in phoographs of figures. Also, he causics a he conac poins are no symmerical (are roaed) because of he shear loads which are creaed a he conac poins. From phoographs of Fig. 7(a) i is appeared ha he angle of roaion of he causic o be one degree. So, he shear loads were calculaed abou Q = Panφ=0.536 and 0.75 KN m. The Fig. 8 presens he isochromaic fringe paerns a he conac poins beween rings and rollers and ono he rollers (a) for wo rollers and (b) for hree rollers configuraion. 99 The applied concenraed loads were P = 33.50 and P = 38.575 KN m for wo loaded rollers and P =47.70 KN m for hree loaded rollers. Lexan (PCBA) specimen wih a sress-opical consan f σ = 7. KN m was used. The load for each roller was calculaed by he Rel.(40), wih D = 3 0 3 m (D is he diameer of he rollers). The calculaed loads for each roller are presened in Table. The calculaed Load Sharing Facor (LSF = P i /P) for he experimenal configuraions are presened in Table. From he resuls i is concluded ha boh mehods give accurae experimenal resuls bu a small divergence was observed which is indebed o he accurae calculaions and o he fac ha here was no used separaors (reainers) beween rollers, hus he rollers were no been in symmeric places, as i appears in phoographs. CONCLUSION In his sudy he experimenal sress-opical mehod of causics was applied on roller-bearing specimens o deermine he load sharing during muliple bearing rollers conac. The equaions of he causics were derived and used o calculae he magniude of he applied load a each roller conac poin. The obained resuls of he load sharing by causics were compared o ha by phooelasic mehod wih which good agreemen was verified. Furhermore, i was demonsraed ha he proposed mehod has advanages over phooelasiciy, because resoluion of he measuremens is no compromised by he small dimensions of he load bearing area and he exreme sress gradiens observed in i, bu insead i is reasonably accurae, highly repeaable and much faser involving a single measuremen of he diameer of he projeced causic. Apar from bearings, his echnique (causics) can be expanded in oher applicaions such as gears, splines, chains. REFERENCES Ciavarella, M. and P. Decuzzi, 00. The sae of sress induced by he plane fricionless cylindrical conac II: The The general case (elasic dissimilariy). In. J. Solids Srucures, 38: 455-4533. DOI: 0.06/S000-7683(00)0090-0 Durany, J., G. Garcìa and C. Vàzquez, 00. Numerical simulaion of a lubricaed Herzian conac problem under imposed load. Finie Elemen Anal. Design, 38: 645-658. DOI: 0.06/S068-874x(0)00097-X
Froch, M.M., 96. Phooelasiciy. Wiley, New York. hp://books.google.com.pk/books?id=svjgaac AAJ&dq=Phooelasiciy&hl=en Johnson, K.L., 987. Conac Mechanics. Cambridge Universiy Press, Unied Kingdom., ISBN: 05347963, pp: 45. Kalhoff, J.F., 993. Shadow Opical Mehod of Causics. In: Handbook on Experimenal Mechanics, A.S. Kobayashi (Ed.). VCH, pp: 430-500. ISBN: 56086406 Michailidis, A., V. Bakolas and N. Drivakos, 00. Subsurface sress field of a dry line conac. Wear, 49: 546-556. DOI: 0.06/S0043-648(0)0054-7 Muskhelishvili, N., 977. Some basic Problems of he Mahemaical Theory of Elasiciy. s Edn., Springer, Moscow, ISBN: 0: 9006070, pp: 768. Papadopoulos, G.A., 993. Fracure mechanics: The experimenal mehod of Causics and he De.- Crierion of Fracure. s Edn., Springer-Verlag, London, ISBN: 354097680, pp: 85. Papadopoulos, G.A., 004. Experimenal esimaion of he load disribuion in Bearings by he Mehod of Causics. Exp. Mech., 44: 440-443. DOI: 0.007/BF048098 Papadopoulos, G.A., 005. Experimenal sudy of he load disribuion in bearing by he mehod of causics and phooelasiciy, J. Srain Anal., 40: 357-365. DOI: 0.43/03093405X5963 Schwarzer, N., F. Richer and G. Hech, 999. The elasic field in a coaed half-space under Herzian pressure disribuion. Surface Coaings Technol., 4: 9-303. DOI: 0.06/S057-897(99)00057- Theocaris, P.S., 970. Local yeilding around a crack ip in Plexiglas. J. Appl. Mech., 9: 409-45. DOI: 0.5/.34085 Timoshenko, S.P. and J.N. Goodier, 970. Theory of Elasiciy. 3rd Edn., McGraw-Hill, ISBN: 0: 007064708, pp: 608. Yu, M.M.-H. and B. Bhushan, 996. Conac analysis of hree-dimensional rough surfaces under fricionless and fricional conac. Wear, 00 : 65-80. DOI: 0.06/S0043-648(96)0733-9 300