Rolling Element Bearings

Transcription

1 28 Rolling Element Beaings Xiaolan Ai The Timken Company Chales A. Moye The Timken Company (etied) 28.1 Intoduction 28.2 Rolling Element Beaing Types Ball Beaings Rolle Beaings Special Beaings 28.3 Beaing Mateials 28.4 Contact Mechanics Theoy of Hetzian Contact Beaing Intenal Contact Geomety Non-Hetzian Contact 28.5 Beaing Intenal Load Distibution Beaings unde Geneal Load Conditions Beaings unde Pue Eccentic Thust Load 28.6 Beaing Lubication Elastohydodynamic Lubication Effect of Spin Motion Effect of Lubicant Stavation Lubication Methods 28.7 Beaing Kinematics Oute Raceway Contol Inne Raceway Contol Minimum Diffeential Spin 28.8 Beaing Load Ratings and Life Pediction Basic Load Ratings Beaing Life Pediction 28.9 Beaing Toque Calculation Stating Toque Running Toque Beaing Tempeatue Analysis Heat Geneation Heat Tansfe Beaing Enduance Testing Testing Pocedue and Data Analysis Beaing Failue Analysis Contact Fatigue Suface Depession and Factue Mechanical Wea Coosion Electic Ac Damage Discoloing and Oveheating 28.1 Intoduction Rolling element beaings ae typical tibological components. They utilize olling contacts between the olling elements and aceways to suppot load while pemitting constained motion of one body elative to anothe. The standad configuation of a olling element beaing compises inne and oute ings, a set of olling elements aanged in a ow between the inne and oute ings, and a etaine o cage to maintain a pope annula spacing between the olling elements (Figue 28.1). Some beaings also have seals as integated components. Due to thei wide availability and vesatility, olling element beaings ae, pehaps, the most widely used beaing type. Rolling element beaings ae chaacteized by little o

2 FIGURE 28.1 A single-ow, deep-goove, adial ball beaing. (Coutesy of SKF Beaing Industies Co.) no sliding motion. They usually geneate less fiction and have low stating toque compaed to hydodynamic beaings. Unlike hydodynamic beaings, the pefomance of olling element beaings is less susceptible to changes in load, speed, and tempeatue. Most olling element beaings ae capable of caying both adial and thust loads. Because of olling contact, the dependency on lubicant is not as citical. This makes olling element beaings easie to maintain. Well-designed and well-built olling element beaings can opeate ove wide anges of load and speed. Such featues often put olling element beaings on the top of the beaing selection list Rolling Element Beaing Types Vaious means have been used to categoize olling element beaings such as by the geomety of the olling elements, by the manne in which a beaing is used o by cetain technical featues that the beaing possesses. Howeve, the most common classification is by the olling element geomety. Thee ae two majo categoies: ball beaings and olle beaings. Within each categoy, beaings can be futhe divided into sub-categoies. Each type of beaing has chaacteistics o popeties that make it paticulaly suitable fo cetain applications. The main factos to be consideed when selecting the optimum beaing type ae: Available space Load condition Speed Tempeatue Misalignment Mounting and dismounting pocedues Dynamic stiffness Motion eo Noise facto This section outlines the most popula beaing types used in vaious automotive and industial applications. Fo compehensive listings on beaing types and usage, the eade is efeed to vaious beaing catalogs povided by beaing manufactues Ball Beaings Deep-Goove Ball Beaings A deep-goove ball beaing consists of a set of balls olling between an inne and oute aceway (Figue 28.1). Both inne and oute aceways have high shouldes on each side. A deep-goove beaing can cay significant adial load and, due to the high degee of confomity between the balls and aceways, modeate thust loads. When the elastic deflection of beaing aces is used to intoduce the balls into beaing aceways, beaings can have uninteupted aceway gooves and ae capable of caying substantial

3 load in eithe diection, even at high speeds. This type of beaing is often efeed to as Conad type. To incopoate the geatest possible numbe and size of balls, a deep-goove ball beaing may ely on a filling slot to intoduce balls into the beaing. This type of beaing is efeed to as Max type o filling slot type. Max-type beaings ae capable of caying highe adial loads Angula-Contact Ball Beaings Angula-contact ball beaings can be egaded as a vaiation of deep-goove ball beaings. Unlike deepgoove beaings, an angula-contact beaing has at least one ace ing that has only one side shoulde. Angula-contact ball beaings ae capable of caying an appeciable thust load in one diection with o without a adial load. Because an angula-contact beaing must have a thust load acting on it, no endplay (lateal movement) exists within the beaing. The line that connects the nominal contact between the inne aceway and a ball and the contact between the oute aceway and the ball lies at an angle with the adial plane of the beaing. This angle is called the contact angle. It anges fom 15 to 40 fo commecially available angula-contact ball beaings. Angula-contact ball beaings ae commonly used in pais to accommodate heavie adial loads, twodiectional thust loads, o vaious combinations of adial and thust loads. Tandem mounting is used fo heavy unidiectional thust loads and duplex mounting of eithe face-to-face o back-to-back is used fo two-diectional thust loads Double-Row Ball Beaings Double-ow ball beaings ae simila in design to single-ow ball beaings. They ae designed with onepiece inne and oute ings. Most double-ow ball beaings ae made with opposite contact angles in ows like two face-to-face o back-to-back angula-contact ball beaings in duplex mounting. They ae also made as two deep-goove ows. Like two single-ow ball beaings in paied mounting, a double-ow ball beaing is often used to cay heavy adial loads. It is also capable of caying thust loads in eithe diection. Thee ae advantages to using a double-ow ball beaing ove two single-ow ball beaings in paied mounting. Double-ow ball beaings take less axial space and the intenal elationship between two ows is pedetemined and is not, o less, affected by mounting pactice Thust Ball Beaings Thust ball beaings geneally have a 90 contact angle. Howeve, ball beaings with contact angle geate than 45 ae also classified as thust ball beaings. A thust ball beaing whose contact angle is 90 can only be used to suppot thust loads Rolle Beaings Cylindical Rolle Beaings In cylindical olle beaings, the olles ae axially guided between integal flanges on at least one of the beaing ings. The most popula type is the single-ow design, which offes vaious flange aangements. Figue 28.2 shows a typical single-ow cylindical beaing with two flanges on the oute ing. Cylindical FIGURE 28.2 A single-ow, adial cylindical olle beaing. (Coutesy of SKF Beaing Industies Co.)

4 a b FIGURE 28.3 Tapeed olle beaings. (a) A single-ow, tapeed olle beaing. (Coutesy of the Timken Company). (b) On-apex design esults in tue olling motion. olle beaings have exceptionally high adial load-caying capacity. They ae also capable of caying a small amount of thust load when two flanges ae positioned on opposing ings at opposite sides. In theoy, tue olling motion exists in cylindical beaings unde adial load. Theefoe, cylindical beaings have lowe stating toque and lowe opeating tempeatue than othe olle beaings. This makes them suitable fo high-speed applications. To achieve geate load capacity without inceasing the tendency of olle skewing on the aceways, cylindical beaings ae fequently constucted in two o moe ows athe than with one ow of longe olles Tapeed Rolle Beaings Tapeed olle beaings have tapeed inne and oute aceways, with tapeed olles guided between them by a vey accuately positioned flange known as the ib (Figue 28.3a). The inne and oute aceways have diffeent contact angles. The extensions of the inne and oute aceways and the olles ae designed to convege at a common apex point on the axis of otation (Figue 28.3b). The on-apex design esults in tue olling motion of the olles on the aceways along the line of contact. At the contacts between the ib and olle-ends, howeve, sliding and spinning exist. The tapeed aceways allow a tapeed olle beaing to cay combined adial and thust loads o thust loads only. The atio of thust to adial load capacity is detemined by the contact angle between the

5 oute aceway and the axis of otation. The long olle-aceway contact gives the tapeed olle beaing a high load-caying capacity. Tapeed olle beaings ae used in pais. One beaing is adjusted against the othe to achieve the desied endplay o pe-load. To acquie geate adial load-caying capacity and eliminate poblems of axial adjustment o themal gowth, tapeed olle beaings ae designed in two ows. Most double-ow, tapeed olle beaings have a single inne ace and double oute aces; o a single oute ace and double inne aces. Tapeed olle beaings ae also available in fou-ow aangements Spheical Rolle Beaings Most spheical olle beaings have an oute aceway that is a potion of a sphee. The beaings ae intenally self-aligning and pemit angula displacement of the shaft elative to the housing. The most popula design has two ows of olles that lie at an angle elative to the axis of the beaing. The olles ae in eithe symmetical o asymmetical bael shape. The cuvatue of olles in the diection tansvese to the otation confoms closely to the inne and oute aceways. The high degee of confomity between the olles and aceways makes spheical olle beaings suitable fo heavy-duty applications. Because of the non-zeo contact angle, spheical beaings ae also capable of caying a cetain thust load in eithe diection along with the adial load. Unlike cylindical olle beaings o tapeed olle beaings, tue olling motion at contact between the olles and aceways cannot be achieved with spheical beaings. Theefoe, spheical olle beaings have inheently highe fictional toque than cylindical beaings and ae not suitable fo high-speed applications Rolle Thust Beaings Cylindical olle thust beaings ae the simplest of this type. A typical cylindical olle thust beaing is compised of a pai of paallel thust plates (washes): a ow of cylindical olles sits between the thust plates and a cage etaining the olles. Cylindical olle thust beaings inheently expeience a lage amount of sliding as a esult of the spin motion between the olles and aceways. Fo this eason, cylindical olle thust beaings ae limited to slow-speed applications. To educe the magnitude of olle spinning, seveal olles can be used in each cage pocket athe than a single long olle. The spin motion of olles on aceways can be eliminated by using tapeed olles with an on-apex design, as illustated in Figue 28.3b. Beaings of such design ae efeed to as tapeed olle thust beaings. Because at least one beaing aceway is tapeed, an outboad flange is equied to confine olles fom being expelled in the adial diection. Sliding contact exists between the outboad flange and olleends. Fiction foces geneated at the sliding contact limit the beaings to elatively slow-speed applications Special Beaings Clutch Beaings A clutch beaing combines the functionality of a beaing and a clutch togethe. It tansmits toque between the inne and oute ings in one diection and allows fee oveun in the opposite diection. When tansmitting toque, eithe the inne ing o the oute ing can be used as the input membe. The tansition fom the oveun to locked opeation nomally occus with small backlash. Clutch beaings ae geneally used in indexing, backstopping, o oveunning Smat Beaings The functionality of a beaing can be extended by integating electonic systems. The speed-sensing beaing used in automotive wheel application is a typical example of smat beaings. A typical speedsensing beaing is a self-contained, double-ow tapeed olle beaing featuing an integal sensing system designed to povide speed infomation fo anti-lock bake systems (ABS) and taction contol systems (TCS). The sensing system contains a taget wheel within the beaing and a micochip speed-sensing element that detects the otational speed of the taget wheel. The senso s speed infomation can also be used fo the vehicle dynamics contol system, navigation system, speed contol, speedomete, odomete, tip compute, and othe puposes.

6 TABLE 28.1 Gade Chemical Composition of High-Cabon Beaing Steels Chemical Composition (%) C Si Mn C Mo P S Application ASTM-A295(50100) Not commonly ASTM-A295 (51100) used JIS-G4805 (SUJ1) ASTM-A (52100) Typical steels fo ASTM-A (52100) small and BS-535A medium-size DIN-100C beaings JIS-G4805 (SUJ2) NF-100C ASTM-A gade Used fo lage-size ASTM-A gade beaings JIS-G4805 (SUJ3) ASTM-A gade Used fo ultalage-size ASTM-A gade JIS-G4805 (SUJ5) beaings Futue smat beaings will featue sensing elements capable of detecting speed, tempeatue, load, vibation, and othe opeating paametes that can be used not only fo system contol, but also fo beaing and beaing system health monitoing Beaing Mateials A beaing s enduance life is lagely detemined by the stength of beaing mateial in elation to the contact stesses the beaing expeiences duing opeation. While a well-designed intenal geomety and good suface finish effectively educe the actual contact stess and thus polong the beaing s sevice life, high-quality beaing mateial is essential. Beaing mateials ae selected on the basis of stength, fatigue esistance, wea esistance, elevated tempeatue esistance, coosion esistance, toughness, hadenability and dimensional stability. A geat majoity of olling element beaings ae fabicated fom vacuum-efined, high-quality lowalloy o cabon steels. Beaing steels can be categoized as high-cabon steels, which contain 0.8% o moe cabon by weight; and low-cabon steels, which contain less than appoximately 0.2% cabon. High-cabon steels, with chomium-alloy additions, ae usually though-hadened to ensue a suface Rockwell hadness of 58 to 64 HRc. Fo lage beaings, paticulaly those with thick coss-sections, inceased amounts of manganese, chomium, silicon, molybdenum, and/o nickel ae intoduced to enhance the hadenability. Though-hadened steels ae used extensively in ball beaings while casehadened steels ae pedominately used in olle beaings. Table 28.1 lists common gade and espective chemical compositions fo high-cabon steels. Low-cabon steels ae alloyed with nickel, chomium, molybdenum, and manganese to incease hadenability. Cabon is diffused into the steel unde contolled tempeatue and atmospheic composition to incease cabon content at suface and subsuface layes to appoximately 0.56 to 1.10%. The cabuized steel is heat-teated to povide a hadened suface case fo high contact load-caying capability and a tough, ductile coe fo heavy shock load absoption. The softe coe also helps to etad suface cack popagation. Low-cabon steels ae histoically used in tapeed olle beaings, although they ae wellsuited fo othe types of beaings. Table 28.2 lists the most commonly used cabuizing steels. The hadness of beaing steels dops shaply when the tempeing tempeatue is exceeded. Because most beaing steels ae tempeed between 160 and 280 C, beaings should not be used at tempeatues

7 TABLE 28.2 Gade Chemical Composition of Cabuizing Beaing Steels Chemical Composition (%) C Si Mn Ni C Mo P S Application ASTM-A (5120H) ASTM-A Used fo small-size beaings (4118H) ASTM-A (8620H) DIN 20NiCMo JIS-G4052/ (SC420H) JIS-G4052/ (SCM420H) JIS-G4052/4103 (SNCM220H) ASTM-A (4320H) JIS-G4052/4103 (SNCM420) ASTM-A (9310H) JIS-G4052/4103 (SNCM815) Used fo mediumsize beaings Used fo lage-size beaings TABLE 28.3 Chemical Composition of Special Beaing Steels Gade Chemical Composition (%) C Si Mn Ni C Mo W V Co Tempeatue Limits ( C) Applications M Used fo hightempeatue M-50-NiL beaings CBS CBS TBS VASCO X M M M SKH C Used fo hightempeatue BG and ASM coosion-esistant beaings above 120 C, if tempeed at 160 C; o above 200 C if tempeed at 280 C. When beaings ae used at elevated tempeatues, steels with high-tempeatue hadness ae equied. Table 28.3 lists some of the widely used high-tempeatue beaing steels along with thei chemical compositions. Coosion-esistant alloys should be used fo beaings in applications whee they ae exposed to coosive envionments. Steels with high chomium content possess good coosion esistance. Of the alloys listed in Table 28.3, 440C, BG-42, and ASM 5749 ae good candidates fo coosion-esistant beaings. Non-feous beaing mateials such as ceamic also povide excellent coosion esistance and can be consideed.

8 TABLE 28.4 Popeties of Engineeing Ceamics and Beaing Steels Mateial Density (g/cm 3 ) Hadness (HV) Young s Modulus (GPa) Flexual Stength (MPa) Factue Toughness (MPa-m 1/2 ) Linea Themal Expansion Coefficient 10 6 / C Themal Shock Resistance ( C) Themal Conductivity (W/m-K) Electic Resistance (Ω-cm) Silicon nitide (Si 3 N 4 ) Silicon cabonate (SiC) Alumina (AL 2 O 3 ) Patly stabilized ziconia (ZO 2 ) Beaing steel Ceamics ae supeio to feous alloys in coosion, heat, and wea esistance, but ae limited in application because they have lowe toughness and factue esistance and ae not able to sustain shock loads. Futhemoe, ceamic beaing mateials ae moe expensive than feous alloys and vey expensive to machine. New engineeing ceamics, to a cetain degee, have impoved some of the mateial popety limitations and have inceased use in beaing applications. Today, a fai numbe of beaings ae made totally, o patially, fom a vaiety of ceamics, including alumina, silicon cabide, titanium cabide, and silicon nitide. With the advances in mateial science and manufactuing technology, the demands fo ceamic beaings will continue to gow. Table 28.4 povides the popeties of engineeing ceamics fo beaing applications Contact Mechanics Rolling element beaings ae typical mechanical components that opeate unde concentated-contact conditions. Loads caied by olling element beaings ae tansmitted though the discete contacts between the olling elements and the two aceways. Even unde modeate beaing load, the stesses at the contact ae quite high, being on the ode of 1 to 4 GPa. Well-designed beaings ae, howeve, capable of caying an appeciable amount of load. This is attibuted to the fact that contact stess inceases slowly with the applied load (to one thid powe fo point contact and one half powe fo line contact) and that the mateial is in geneal compession. In olling element beaings, point contact efes to the conjunction of two sufaces such that unde no load, the initial contact is a single point. As load is applied, the contact develops into a finite aea of a geneally elliptical shape. Point contact exists between the olling elements and aceways of ball beaings, and of olle beaings with high-cown on olles and aceways. It also exists in olle beaings between the olle-ends and flanges. Line contact is the conjunction of two sufaces such that the initial contact unde no load is a staight line. When loaded, the line speads to fom an elongated ectangle. A concept of modified line contact is fequently used (Lundbeg et al., 1947). The modified line contact is the majo fom of contact between the olling elements and aceways fo olle beaings. This chapte section outlines the classical Hetzian theoy of contact that allows quick calculations of contact stess and defomation to be made fom the applied load, mateial popeties, and the intenal geomety. Issues of non-hetzian contact ae also discussed Theoy of Hetzian Contact Point Contact When two elastic solids contact unde load W, a contact aea develops. Fo a point contact, the aea, in geneal, assumes an elliptical shape and has a semi-majo, a, in one diection and a semi-mino, b, in the pependicula diection. Fo puposes of discussion, assume a lies in the x-diection; b lies in the

9 y-diection; and R x and R y ae pincipal elative adii (composite adii) of cuvatue at oigin of the contact. The atio of the semi-mino axis b to the semi-majo axis a is defined as ellipticity k = b/a (0 k 1). The ellipticity is elated to the atio of suface cuvatue adii, k = R y /R x, though a tanscendental equation, which can be appoximated as follows (Johnson, 1985): k k 2 3 (28.1) Denote R e = (R x R y ) 1/2 as the geometical aveage adius of suface cuvatues, the geometical aveage contact size, denoted as c = (ab) 1/2, can be obtained as: 13 () 3 WR e c = Fe () 2E (28.2) whee E is efeed to as the effective Young s modulus and is detemined fom the Young s moduli E 1 and E 2, and Poisson atios ν 1 and ν 2 of the contacting bodies, = + E ν1 ν2 E E (28.3) F(e) in Equation 28.2 is an auxiliay function, vaying only with the eccenticity of the contact ellipse, e = (1 k 2 ) 1/2. Fo e > 0 (k < 1), F(e) can be appoximated in tems of k by the following equation: Fe () k 16 1 k lnk 0< k < k 13 (28.4) Fo the cicula contact whee e = 0, F(e) is equal to unity, F(e) = F(0) = 1. Having defined the shape k and size c of the contact ellipse, the semi-majo and semi-mino axes, a and b, maximum Hetzian pessue p k, and maximum defomation at cente of the contact δ can be detemined a= k c W b= k c W W p = 3 h 2 π c W 2 12 k K() e 2 9W δ= 2 Fe 2E R () e W 2 3 (28.5) (28.6) (28.7) (28.8) whee K(e) is the complete elliptical integal of the fist kind. The value can be detemined by the following appoximation (Hamock et al., 1983): K() e π π k < k ln 0 1 (28.9)

10 TABLE 28.5 Maximum Shea Stess and Its Depth k z/b τ max /p h The stesses in x- and y-diections at the cente of the contact ae: ( ) + k σx = ph 2ν+ 1 2ν 1 k ( ) + 1 σy = ph 2ν+ 1 2ν 1 k (28.10) (28.11) The maximum shea stess occus at a cetain depth along the z-axis below the suface. The maximum shea stess has been used as the citical stess fo contact fatigue analysis. Table 28.5 gives the maximum shea stess along with its depth as functions of ellipticity. Detailed infomation egading stess calculations at abitay points below the contact suface is given in Jones (1964) and Sackfield (1983a,b) Line Contact In line contact, one of the pincipal adii, R x, appoaches infinity (R x ). Accodingly, the atio of suface cuvatue adii k educes to zeo (k = R y /R x = 0), and R e is edefined as R e = R y. The contact aea becomes a long stip of width 2b. Assume W l is the contact load pe unit length. The half-contact width b and Hetzian pessue p h can be expessed as: 8WR l e b = π E 12 (28.12) p h Wl = W π b 2 12 (28.13) Assume the defomation is zeo at a cetain peset depth (z = R e ) fo both contacting sufaces. The maximum composite defomation δ at the cente of contact can be appoximated by: ( ) ( + ) π ν ν ν ν δ = π + Wl 2 E E W R ln e 2 E E l 1 2 (28.14) Altenative empiical equations fo calculating suface defomation have been developed fo modified line contact (Lundbeg, 1939; Kunet, 1961). Assume two cylindes with paallel axes ae pessed togethe unde load W. The effective contact length is l eff. The total defomation of the contact sufaces can be appoximated by a powe-law function δ= C W l t t eff (28.15)

11 It is suggested that fo steel, C = , t = 0.925, and t = 0.85 when W is in Newtons and l eff and δ ae in millimetes. Fo simplicity concens, t = 9/10 has been widely adopted. The stesses at a geneal point (y, z) within the contact solids ae given by McEwen (1949) σ y p h b m 2 2 z + n = + z m + n σ z p = m τ yz 2 2 h b m z + n n ph b n m 2 = z m + n (28.16) (28.17) (28.18) whee m = sign( z) ( b y + z ) y z + b y + z ( ) n = sign( y) ( b y + z ) y z b y + z ( ) (28.19) (28.20) Vaious citical stesses (othogonal shea stess, maximum shea stess, von Mises stess, etc.) fo contact fatigue life calculations can be deived fom the stess components Beaing Intenal Contact Geomety The pevious section povided useful equations fo contact analysis. To use these equations, beaing intenal geomety must be defined. As can be seen, the shape and dimension of the contact ellipse o stip, the maximum Hetzian pessue, and suface defomation all elate to two impotant paametes: (1) the pincipal effective cuvatue atio, k = R y /R x, of the contact sufaces and (2) the geometical aveage adius, R e = (R x R y ) 1/2 (fo line contact, k = 0 and R e is defined as R e = R y ). As will be seen, these two paametes also play impotant oles in lubicant film thickness and beaing toque calculations. This section defines the two geomety paametes fo commonly used beaing types Ball Beaings The intenal geomety of a ball beaing is detemined by the aceway-goove adii i and o in the beaing s axial plane, ball diamete d b, beaing pitch diamete D p, and the inne- and oute-aceway contact angles β i and β o. The adii of inne- and oute-aceway gooves and the pitch diamete ae expessed in tems of ball diamete d b as i = g i (d b /2), o = g o (d b /2) and D p = Gd b, espectively (Figue 28.4). Fo the inne-aceway contact, it can be shown that: k cos β i 1 = 1 G 1 g i (28.21)

12 FIGURE 28.4 Intenal geomety of a ball beaing. R e cosβ i 1 = 1 G 1 g i d b 2 (28.22) Simila equations ae obtained fo the oute-aceway contact: k cos β o 1 = 1+ G 1 g o (28.23) R e cos β o 1 = 1+ G 1 g o d b 2 (28.24) whee D o di G = d = d b b (28.25) and d i and D o ae the inne- and oute-aceway diametes measued at the cental adial plane. The theoetical adial cleaance δ 0 is negligibly small, and this is not included in the calculation. In the absence of centifugal foces, β i and β o ae identical. Fo beaings unde pue adial loads, β i = β o = 0.

13 FIGURE 28.5 Intenal geomety of a tapeed olle beaing Rolle Beaings A olle beaing s intenal geomety is defined by the olle cental o mean diamete d, effective pitch diamete D p *, inne-aceway contact angle β i, and oute-aceway contact angle β ο (Figue 28.5). The effective pitch diamete D p * is elated to the inne-aceway mean diamete d m and oute-aceway mean diamete D m though the following equation: ( ) * 1 βi+ β o βo β i Dp = Dp+ Dm dm tan tan (28.26) whee D p = (D m + d m )/2 is the beaing s pitch diamete. Fo spheical olle beaings, D m is defined by the diamete of the oute-aceway sphee D o and the contact angle of the oute aceway β o ; fo example, D m = D o cosβ o. Fo most olle beaings, including spheical olle beaings, line contact exists between the aceways and olles. Theefoe, k = 0. At the inne-ing aceway and olle contact, the geometical aveage adius is: R e dm = 1 2G * cos βo βi 2 ( ) (28.27) At the oute-ing aceway and olle contact, the geometic aveage adius is expessed as: R e Dm = 1 2G * cos βo βi 2 ( ) (28.28) whee G* is the atio of effective pitch diamete to olle mean diamete, G* = D p */d.

14 Fo spheical olle beaings with symmetic olles, β i = β o. Fo cylindical olle beaings, β i = β o = 0. Fo high-cowned olles and aceways, olles may not fully engage with aceways along the olle length when the applied load is low elative to the load unde which the cowns wee designed. In such cases, elliptical contact exists. Assume the cown cuvatue adius of the olles is R w, and the inne and oute aceways have cown cuvatue of adii of i = g i R w and o = g o R w, espectively. The following elationships exist when β o β i is small. At the inne-ing aceway, k i d = 1 cos β G 1+ 1 g 2R i w (28.29) At the oute-ing aceway, R e cos β i = G g k o d = + 1 cos β G 1+ 1 g 2R i o Rd w 2 w (28.30) (28.31) R e cos β o = + G g o Rd w 2 (28.32) Equations though ae applicable to spheical beaings in elliptical contact. In this case, beaing aceways ae consideed as being concavely cowned. Accodingly, negative values ae used fo g i and g o Rolle-End and Flange Contact Fo tapeed olle beaings, olles ae etained by flanges, also known as ibs, at both ends of the inne ing. The ib located at the lage end of the inne ing sees a setting foce fom each olle. The lage end of a olle is a potion of a sphee with the adius R s being a faction of the apex length L a, R s = ζl a (0 < ζ < 1). Elliptical contact exists between the ib and each olle-end. The ib and olle-end geomety ae designed such that the cente of the contact is located at a height H, oughly half the ib height, and the majo axis of contact ellipse is oiented along the sliding and olling diection of olles. Assume the ib angle is θ. As seen fom Figue 28.6, the following elationships exist: k H h = = hr s cosθ g g ( ) ( γ θ) sin γ θ cos D = 2 R sin βi + θ = 1 h cos( βi+ θ)+ g s ( ) i (28.33) (28.34) (28.35) (28.36)

15 FIGURE 28.6 Rolle-end and ib contact geomety. Re = k 12 Rs Ri g i = R s (28.37) (28.38) γ = (β o β i )/2 is the olle half-included angle. R i is the base-adius of the inne aceway at the lage end, and D is the base-diamete of olles at the lage end. The above equations also apply to cylindical beaings whee γ = β i = β o = Non-Hetzian Contact Hetzian contact exists only in the idealized situation. Fo pactical applications, Hetzian assumptions may not always be satisfied. While this, in geneal, does not undemine the usefulness of the Hetzian theoy, it gives ise to the necessity of non-hetzian contact. Fo finite line contact as seen in olle beaings, plane stain assumption is not applicable at the end potions of the contact whee edge loading occus. Moeove, contact sufaces ae fa fom pefectly smooth. Suface impefections o damage, whethe esulting fom manufactuing pocesses o fom mishandling duing tanspotation and assembling o fom opeation, ae pactically inevitable. A suface impefection seves as a stess-aise. It can cause peceptible stess deviation fom the Hetzian theoy. Unde thin-film lubication conditions, suface oughness also plays an impotant ole in alteing contact stesses. Life eduction due to suface oughness becomes an impotant issue. In this section, issues egading non-hetzian contact ae discussed Edge Stess and Rolle Cowning When a finite olle contacts a aceway of geate length, the mateial in the aceway is in geate tension along the contact stip at olle ends than at the cental potion of the olle. Coespondingly, the olle eceives a high compessive stess at its ends. This condition of edge-loading is shown in Figue 28.7 whee a spheical olle is in contact with the aceway unde heavy load conditions. To pevent edgeloading, olles and aceways ae usually patially o fully cowned to elieve the edge stesses. Cowning the contact membes of a beaing also gives the beaing potection against edge stess esulting fom

16 Pessue, P = p/ph Y coodinate (mm) X coodinate (mm) FIGURE 28.7 Contact pessue in a spheical olle and aceway contact unde heavy load conditions. FIGURE 28.8 Contact pessue and inteio von Mises stess fo a olle in contact with a scatched suface. (Fom Ai, X. and Sawamiphakdi, K. (1999), Solving elastic contact between ough sufaces as an unconstained stain enegy minimization by using CGM and FFT techniques, ASME J. Tibol., 121, With pemission.) misalignment (Hatnett, 1984). The amount of cowning is detemined by the maximum load the beaing was designed fo and by the degee of misalignment anticipated in the application. Substantial studies have been conducted ove the yeas to povide methods of designing pofiles based on a unifom contact pessue o contact stesses along the length of contact. Fo moe detailed discussion on olle pofiling, the eade is efeed to Lundbeg (1939), Poon et al. (1978), Rahnejat et al. (1979), Hatnett (1984), and Reusne (1987) Effect of Suface Scatch and Indentation Suface scatches o indentations ae common suface defects. They esult fom mishandling duing the tanspotation and assembly pocesses. Lubicant contamination is anothe majo souce of suface damage. The pesence of a suface scatch o indentation can esult in a noticeable stess concentation at the edges of the defect. It bings the maximum stess close to the suface, causing suface-oiginated spall initiation and subsequent popagation. Figue 28.8 shows the contact pessue and inteio von Mises stess distibutions fom the efeence (Ai et al., 1999) fo a scatched olle in contact with its aceway. The scatch has slightly aised shouldes on both edges. Pessue spikes and high stess concentations at the edges of the scatch ae clealy demonstated. Studies have indicated that the magnitudes of the pessue spike and stess concentation ae lagely dependent on the atio of depth to half-width of the scatch o indentation. The atio epesents the aveage slope of the indentation pofile. A deep slope gives ise to high contact pessue and stesses.

17 Effect of Roughness With inceasing demand on powe density, beaings ae often equied to opeate unde thin-lubicantfilm conditions. When the lubicant film is penetated by the suface oughness, suface aspeity causes peceptible stess concentation on the suface o in the nea-suface laye. Consequently, suface distesselated contact failue, mostly in the fom of suface pitting and spalling, becomes an impotant issue. The contact poblem unde this condition is known as ough contact. Unlike smooth Hetzian contact, the contact aea is now a myiad of mico-contacts of highly iegula shapes within the maco-contact. The contact pessue at these mico-contacts can be high enough to cause localized popety changes both in contact bodies and in lubicant. Because the contact sufaces in olling element beaings ae, in geneal, smoothe than othe machine components such as geas, the mutual dependency between aspeities is damatic. Mico-contact aeas ae often connected and the atio of actual to nominal contact aeas is close to unity. Figue 28.9 depicts a typical suface oughness pofile poduced by ginding pocesses. The cente-line-aveage oughness is 0.11 µm and aveage aspeity slope is about 1 degee. When the suface engages in contact, mico-contact aeas develop. Accodingly, contact stesses of diffeent magnitudes ae geneated. Figue shows contact pessue and footpint in a bounday-lubicated contact. As demonstated in Figue 28.10, suface oughness causes peceptible pessue fluctuation. While educing suface oughness deceases pessue ippling, loweing the suface slope has a dominating impact on educing pessue fluctuation and peak pessue. In life-sensitive applications, beaing aceways must be honed o popely machined to povide good suface qualities Beaing Intenal Load Distibution Beaing load is distibuted among olling elements. The load each olling element eceives is dependent, in geneal, on the azimuth position of the olling element. The load distibution is impotant fo pedicting beaing contact stess, toque, tempeatue, and fatigue life. A beaing s intenal load distibution can be obtained by analyzing contact deflections and beaing geomety constaints. As a load is applied to the beaing, defomations of vaious magnitudes occu at diffeent azimuth locations. The defomation must comply with cetain geometical constaints so that the beaing s geometical integity is not violated. Assume the oute ing of the beaing is fimly seated in a housing and the inne ing is well-fitted on a solid shaft o a thick wall hollow shaft. In the absence of centifugal foces, the load each olle eceives can be detemined fom the defomation/load elationship outlined in Section In this section, beaing load distibution is deived Beaings unde Geneal Load Conditions When a beaing is subjected to a adial and an axial load F and F a, espectively, the centes of the inne ing and oute ing ae displaced elative to each othe in both adial and axial diections. Assume the movement is δ in the adial diection and δ a in the axial diection. The defomations at any olling element position x is expessed by δψ δ β ψ δ β δ e ( )= cos cos + a sin cosβ 2 (28.39) whee β is the contact angle and δ e is the diametical effective cleaance. The equation can be eaanged in tems of δ max = δ(0): 1 δψ ( )= δmax 1 ( cosψ) 2ε 1 (28.40)

18 FIGURE 28.9 Suface oughness pofile: gound suface.

19 FIGURE Contact pessue distibution and footpint fo contact with gound sufaces.

20 whee ε is the load zone paamete and is defined as: 1 δ δ ε = + β 2 1 a e tan δ 2δ (28.41) Fo pue thust load, ε appoaches infinity (ε = ). The deflection at azimuth position ψ is the sum of defomations between the olling element and each aceway: δ= δ + δ ( o) ( i) (28.42) Fom the load and defomation elationship defined in Section , the load can be expessed in tems of the total defomation as t W = k t δ (28.43) whee t kt = k + k t o 1 1 t ( ) () t i t (28.44) k t(i) and k t(o) ae defined by Equation 28.8 fo point contact and by Equation fo line contact. Accodingly, t = 3/2 fo point contact and t = 40/37 o 10/9 fo line contact. Assume W max as the maximum olle load. It can be easily shown fom Equation that: W( ψ) δ ψ = ( ) W δ max max t (28.45) Thus, the load distibution can be witten as: 1 W( ψ)= Wmax 1 ( cosψ) 2ε 1 t (28.46) To achieve the state of static equilibium, the following conditions must be satisfied: F ZW J = max () ε F ZW J a = max a() ε cosβ sinβ (28.47) (28.48) whee J (ε) and J a (ε) ae Sjövall integals (Sjövall 1933) and ae defined as: xl 1 1 J ()= ε 1 1 cosψ cosψdψ 2 π 2ε xl ( ) t (28.49)

21 TABLE 28.6 J (ε) and J a (ε) fo Single-Row Beaings ε Point Contact (t = 3/2) Line Contact (t = 10/9) J (ε) J a (ε) J R (ε) J (ε) J a (ε) J R (ε) J a (0.31) (ε) 0 0/0 a 0/0 a 0 0/0 a 0/0 a 0 0/0 a a 0/0 denotes a special numbe that any poduct o division by this numbe is zeo. xl 1 1 Ja()= ε 1 ψ dψ 2 π 2ε 1 cos xl ( ) t (28.50) ψ l is the half load-zone angle and is a function of the load-zone paamete ε. ( ) cos ε ε ψl( ε)= π ε > 1 (28.51) The values of Sjövall integals ae listed in Table Fo an angula-contact beaing, with a given contact angle β, unde the combined adial and thust load condition, ε can be found fom the following equation with the aid of Table J () ε F J()= ε β J F a() ε = tan a (28.52) The maximum load of a olling element is given by: W max = 1 Z F Fa J + J 2 2 a (28.53) Fo a adial beaing, with zeo contact angle β = 0, unde pue adial load, ε can be found fom the following equation with the aid of Table F = Zk J () ε δ t R e t (28.54)

22 When δ e 0, and When δ e < 0, and J J ε 0 ε 1 /2 J ε ε 1 2ε R()= () ε 1 /2 < ε < J ε ε 2 1 R()= () ε t t (28.55a) (28.55b) The values of J R (ε) fo vaious ε ae listed in Table Fo angula-contact ball beaings unde pue thust load, the contact angle β is usually geate than the initial contact angle β 00 that exists unde zeo load. The actual contact angle can be detemined fom the following equation (Eschmann, 1964). sinβ = l cos 2 2 l sin β 00 + δ ( o) β + 00 l sin β + δ 00 ( o) 2 (28.56) whee δ (o) is the defomation at the contact between the ball and oute ace. It can be calculated fom Equation l is the distance between the centes of cuvatues of the inne- and oute-aceway gooves. ( ) db l= gi+ go 2 2 (28.57) whee d b is the ball diamete, g i and g o ae the atios of the inne and oute aceway goove adii to ball diamete Beaings unde Pue Eccentic Thust Load When a pue thust load F a is applied on a thust o angula-contact beaing, each olling element shaes an equal amount of load; that is, Fa W( ψ)= Z sin β (28.58) Howeve, if the load is applied eccentically, the adial planes of the two aceways ae no longe paallel and they tilt at an angle φ to each othe. The load is distibuted unevenly among the olling elements. The equations fo δ(ψ), W(ψ), and F a deived in the pevious section ae applicable povided that the load-zone paamete ε is edefined as: 1 δ ε = a φdp (28.59)

23 whee D p is the beaing pitch diamete. Because of eccentic loading, a tilting moment M develops, whee Dp M = ef l a= ZW max J () ε sin β 2 The eccenticity e l is detemined fom the following equation: e l D J p () ε = 2 J ε a () (28.60) (28.61) 28.6 Beaing Lubication Elastohydodynamic Lubication Unde healthy opeating conditions, contact sufaces in a olling element beaing ae sepaated and, theefoe, potected by a thin laye of self-pessuized lubicant film. The applied load is tansmitted fom one suface to the othe suface though this pessuized lubicant film. The fomation of the lubicant film is the subject of the Elastohydodynamic Lubication (EHL) theoy and has been descibed elsewhee (Dowson et al., 1977; Goha, 1988). The lubicant film between the contact sufaces has an almost unifom thickness h c at the cental egion. At the outlet, a distinct constiction occus and the film eaches its minimum thickness h min. The thickness of the lubicant film in elation to suface oughness plays impotant oles in the detemination of fictional toque, heat geneation, wea, and fatigue failue. Equations deived fom the numeical egession of EHL simulations o expeimental measuements ae available fo film-thickness calculations. Based on numeical simulation esults, Dowson and Higginson (1961) established an empiical fomula to detemine the minimum film thickness at the outlet of the EHL conjunction fo line contact poblems unde fully flooded, isothemal conditions. Dowson (1968) evised the equation to make it compatible with the law of dimensionless analysis. In dimensional fom, the equation eads: ( ) ( ) ( ) hmin = 265. α η E Re L ue w Lube & mateial popeties Geomety Opeation conditions (28.62) whee η and α ae viscosity and pessue-viscosity coefficient of the lubicant, espectively; E is the effective Young s modulus; R e is the geometical aveage adius given in Section ; L is the contact length; u e is the entainment speed and is given in the following section as a function of beaing geomety and otational speed fo vaious beaings; and w is the applied contact load. The cental film thickness is appoximately 4/3 of the minimum film thickness fo most of the EHL egime. Howeve, the atio of cental film thickness to minimum film thickness educes asymptotically to unity as load inceases and speed deceases. The atio also educes unde conditions of lubicant stavation. Fo point contact poblems, Hamock and Dowson (1977a) pesented a minimum-film-thickness equation. The equation was esticted to cicula contact o tansvese elliptical contact whee the majo axis of the ellipse lies pependicula to the diection of entaining motion. The estiction on the diection of entaining motion was lifted and genealized solutions fo entainment along eithe axis, o along any diection inclined to the pincipal axes, wee pesented by Chittenden et al. (1985a,b). A evised minimum film thickness equation was expessed in tems of the diectional suface cuvatue atio k* (0 < k* < ); that is:

24 ( ) ( ) *.. k* h E ue w Re k e min = α η Lube & mateial popeties Opeating conditions Geomety configuations (28.63) When the entaining motion is in the diection of the majo axis of ellipse, k* >1 and k* = 1/k. When the entainment is along the diection of the mino axis of ellipse, k* <1 and k* = k. The atio of film thickness to the composite ms value of suface oughness, σ = (σ σ 2 2 ) 1/2, is called the lambda (λ) atio. It is the most used paamete among othe quantities associated with EHL. It indicates the condition of lubicant film fomation and somewhat eflects the seveity of aspeity contact between contact sufaces. Thee lubication egimes fom full-film lubication to mixed lubication and to bounday lubication can be distinguished based on the value of the lambda atio. When the lubicant film thickness exceeds thee times the composite ms oughness, a full sepaation of the contact sufaces is achieved. The contact load is caied almost entiely by the lubicant film. In full film egime, contact stesses ae less affected by suface oughness. The oveall pefomance can be pedicted by the classical EHL theoy consideing smooth sufaces. When the lambda value becomes less than thee but geate than one, a peceptible amount of aspeity contact occus. Local lubicant film can be inteupted at the tip of tall aspeities. This egime is called patial EHL o mixed lubication egime. The contact load is shaed between lubicant film and contacting aspeities. The load shaing atio and contact fiction foces ae stongly dependent on the lambda value (Tallian, 1972). A majoity of olling element beaings opeate in this egime. As the lambda atio becomes less than one, the contact falls in the bounday lubication egime whee aspeity contact pedominates. Sevee suface distess is expected. The oveall pefomance can be well-pedicted by the dy contact analysis Effect of Spin Motion In addition to olling motion, spinning a otation of a olling element about an axis nomal to the contacting sufaces often exists at the contacts between the olling elements and aceways. A study on point EHL contact (Taniguchi et al., 1996) shows that spin motion educes the minimum film thickness. The effect on cente film thickness is, howeve, insignificant. The eduction in the minimum film thickness depends on load and speed paametes. An adjustment facto fo the minimum film thickness was given by Taniguchi et al. (1996) as: ϕ spin and W is the non-dimensional load defined by: hmin 047. = = W Ω h min0 whee Ω is the spin-to-oll atio defined as: Ω= 2ω sre u k e (28.64) (28.65) whee ω s is spin angula velocity. W kw = ER 2 e (28.66) Effect of Lubicant Stavation When lubicant supply to the inlet of the contact is insufficient, eduction in film thickness occus. This phenomenon is efeed to as lubicant stavation. Unde such conditions, the thickness of the lubicant

25 film is lagely detemined by the availability of lubicant supplied at the inlet to the contact. The usage of lubicant becomes moe efficient. Lubicant at the inlet zone is entained into the contact with little o no side flows. The eduction in film thickness has been studied by many eseaches, including Wolveidge et al. (1971), Wedeven et al. (1971), Kingsbuy (1973), Chiu et al. (1974), and Elod et al. (1974). A concept of inlet lubicant-ai-meniscus distance to the cente of contact was adopted to quantify the seveity of lubicant stavation. Empiical equations wee subsequently developed to coelate the eduction in lubicant film thickness to inlet meniscus distance (Castle et al., 1972; Hamock et al., 1977b). The pimay limitation of these meniscus-distance-based equations is that they only apply to the stavation egime whee the meniscus is clealy obseved outside the Hetzian contact. As the inlet meniscus distance appoaches the half Hetzian contact width, the appoach beaks down. In a ecent wok, inspied by the wok of Kingsbuy (1973), Chevalie et al. (1998) used the available lubicant film thickness at inlet to define lubicant supply conditions. An equation based on the inlet available film thickness was developed to estimate the eduction in cental film thickness fo cicula contact poblems: ϕ h c stv = = q hcf γ 1+ γ q (28.67) whee h c is the cental film thickness; h cf is the cental film thickness unde fully flooded conditions; and γ epesents the atio of available film thickness at the inlet to the modified fully flooded film thickness at the contact cente by consideing lubicant compessibility. γ = h ρh (28.68) ρ = ρ/ρ 0 is the non-dimensional lubicant density; ρ is the lubicant density; ρ 0 is the density unde ambient pessue; h spl is the thickness of the supplied lubicant at the inlet; and q is an exponent and it inceases as load inceases o the entainment speed deceases. Fo most of EHL egime, q vaies fom 2 to 5. As q, Equation educes to spl cf γ ϕ = stv 1+ γ = min ( γ,1) (28.69) Equation undelines the physics. When lubicant supply at the inlet is abundant, the thickness of lubicant film is govened solely by load, speed, and contact geomety. Inceasing lubicant supply at the inlet has no effect on film thickness. Howeve, when the lubicant supply at the inlet is insufficient, the fomation of lubicant film elies heavily on the availability of lubicant at the inlet. The film thickness at the contact cente cannot exceed what can be geneated at the inlet fom the limited lubicant supply. As γ deceases, the cental film thickness h c deceases faste than the minimum film thickness h min. The atio of h c to h min appoaches unity. When γ < 0.5, Equation can be used quite accuately fo calculating eductions in minimum film thickness. The concept of inlet lubicant availability also applies to elliptical contact as well as line contact. Fo tansvese elliptic contact (k * < 1) and line contact poblems, side flows at the inlet to the contact encounte geate esistance. Consequently, q assumes a geate value when Equation is used fo a tansvese elliptic contact o line contact poblem Lubication Methods In the poceeding equations and text, oil lubication is consideed and the EHL film between contacting sufaces is implied. Thus, fo fluid lubication, the detemination of films and opeating expectations

26 ae staightfowad. In selecting a lubicant and method of lubication, it is ecommended that the fist choice be as simple a selection as possible. Fo a simple system, staight mineal oil usually is the fist choice. With easonable tempeatue geneated (tempeatue anging fom 50 C to 300 C), this system could be a simple sump Oil Bath The oil bath o closed sump is a popula means to lubicate beaings. A pope oil level should be about at the middle to almost the top of the bottom olling element. Fo many systems, this povides fo film fomation and emoves sufficient heat fom the beaings. If load, tempeatue, o system speed leads to poblems, then the lubicant viscosity o type of lubicant may need to be changed. When a wide ange of tempeatue in the system is anticipated, a lubicant with highe viscosity index may be used. Fo beaings used at elevated tempeatue, a highe viscosity oil, additives o, in special cases, a synthetic lubicant such as polyalphaolefin should be consideed Ciculating Oil Systems If the above suggestions do not solve the poblems, then a system may equie ciculating the lubicant though the beaings with an oil ciculation system. The oil ciculation system usually includes an extenal esevoi and pump and, in some cases, a heat exchange that will maintain lubicant and beaing at a equied tempeatue ange. Vey often, the appeaance of too much wea debis o contamination may also indicate the need fo filtation to be added to the system. Consideing the envionmental pessues to maximize lubicant life and educe fluid contamination, the addition of a filtation system including a monitoing pocedue may be equied so that the quality of the oil going though the beaings and pehaps geas of the oveall system can be continually assessed Oil Mist and Oil Jet Oil mist lubication can be used fo lage beaing systems o high speed. Fo this type of lubication, small amounts of oil doplets ae caied by ai onto beaings with an atomize. This povides enough lubicant fo fluid film fomation and some cooling. It uses vey little oil but can be limited in use, depending how well the mist can be contained within the system. An oil jet may be equied fo ultahigh speed beaings because at such speeds, the lubicant would be diven off the beaing components. Thus, a jet velocity on the ode of 10 to 20 m/s allows the lubicant spay to each the intenal beaing sufaces and povides the necessay fluid contact Geases Gease is a lubicant made up of about 10% soap thickene and 80% base oil, with the addition of ust and oxidation inhibitos and specific additives to povide anticoosion, viscosity index impovement, o high film stength, depending on the need fo extended life unde equied tempeatue, load, speed, and specific dy to wet conditions. It is ecognized that moe olling beaings ae lubicated with gease athe than oil because gease lubication is simple, moe economical, and can povide sealing to beaings and thus keep out contaminants and esist wate ingess moe than oils. Conventional geases ae usually limited to lowe speeds anging up to 3500 pm (Boehinge, 1992). Lithium-based geases ae commonly used because they can be pumped, have esistance to small paticle ingess, and can be stoed. Fo a olling element beaing, gease that fills about 30 to 50% of the beaing inside cavity is ecommended. Geate fill can cause highe tempeatues o the gease may be foced out fom the beaings. As to its shotcomings, gease does not flush out wea paticles geneated within a beaing as well and maintaining the pope amount of lubicant within a beaing is not as easily contolled as is possible with an oil. Finally, gease will not be the choice of lubicant when heat geneated in a system must be dissipated quickly. Consideing the EHL film fomation with gease, tests on six lithium geases (Dalmaz and Nantuo, 1987) indicated that beaing fatigue life elated to the viscosity of the base oil in the gease and theefoe to the EHL film thickness. It has been ecommended (Chetta et al., 1992) that an EHL film eduction facto of 0.5 to 0.7 be used with gease in a olling element beaing. This was also suggested by Aihaa

27 and Dowson (1978), especially unde longe opeation when base oil moves fom the gease and esults in beaing stavation Solid Lubication Othe lubication methods ae solid lubication and gas lubication. Gas lubication is usually not consideed fo olling element beaings, but solid lubication is used when sevee conditions must be addessed. When tempeatues ae exteme o specific contaminants must be avoided, solid lubication can be used although actual usage is much less than oil o gease, consideing the costs and that most solid lubicants ae not off-the-shelf poducts. Solid lubicant advantages include: thee may be little o no movement of the lubicant, low volatility, and functionality at vey low o high tempeatues. The disadvantages ae poo themal conductivity, themal expansion diffeent than metals, and wea without means to eplace the solid mateial won away. Because solid lubicants ae vey special, thei selection is moe complex than the selection of oils and geases. The fist solid lubicant was pobably gaphite, although molybdenum disulfide has been ecognized and used fo quite some time. Above 350 C, gaphite povides low fiction and may adhee fimly to metal sufaces but the puity and specific fom of gaphite must be known to povide wothwhile popeties as a solid lubicant. Unfotunately, the same may be tue fo othe solid lubicants, both natual and those developed in the laboatoy. Because solid lubicants have diffeent chaacteistics, elevant infomation must be eviewed when consideing the use of solid lubicants. Infomation on solid lubicants is available (see, e.g., Landsdown (1998) and Sliney (1992)). Whateve the lubicant used in olling element beaings, it should be selected based on the undestanding of the oveall system the beaings opeate in and the anges of speed, load (two most impotant), and tempeatue, along with the mateials of the beaings and othe components such as cage mateial o seals so that expected beaing life can be achieved Beaing Kinematics Kinematic analysis is vey impotant fo evaluating a beaing s pefomance. It detemines the entainment speed u e as well as the sliding and spinning velocities equied fo lubicant film thickness calculation and subsequent beaing toque and tempeatue analyses. In this section, olling speed (entainment speed), sliding speed, and spin angula velocity at both inne- and oute-aceway contacts ae discussed. Conside a efeence coodinate system in a beaing s axial plane as shown in Figue To facilitate the discussion, assume fo the moment that the otating axes of olling elements ae fixed in space and no obital movement exists. Denote ω, ω i, and ω o as the angula velocities of the olling elements, the inne aceway, and the oute aceway, espectively, in elation to the efeence coodinate system. To simplify the analysis, futhe assume that ω lies in the same plane as ω i and ω o. The assumption is valid fo olle beaings with minimum o no olle skewing. The assumption is also valid fo ball beaings unde low and modeate speed applications whee the gyoscopic moment of balls is negligible. In addition, this assumption may hold tue fo ball beaings unde heavy loads whee the gyoscopic moment is esisted by the fictional moment at the inne- and oute-aceway contacts. In geneal cases, a olling element contacts the inne aceway at a contact angle β i and the oute aceway at a contact angle β o. As the olling element evolves about the beaing s axis (in geneal cases), the olling element also otates about its own axis that lies at an angle β with espect to the beaing axis. At the inne-aceway contact, as shown in Figue 28.11, the angula velocity of the olling element ω esolves into two components, one paallel with the contact suface, ω cos(β β i ), one pependicula to the contact suface, ω sin(β β i ). Similaly, the angula velocity ω i also esolves into two components, ω i cosβ i and ω i sinβ i in elation to the contact suface. The olling speed, also known as the entainment speed, at contact is: u ei 1 = γ ω d cos( β β ) γ cosβ + 1 D ω 4 ωi i d i p i (28.70)

28 FIGURE Rolling element and aceway contact. whee γ d = D e /D p. D e is the diamete of olling elements measued at the cental adial plane of each olling element, and D p is the beaing pitch diamete. Should goss slip occu at the inne-aceway contact, the sliding speed is: v i 1 ω = ( )+ 1 D 2 ω γ cos β β γ cosβ i ω d i d i p i (28.71) The spin angula velocity of a olling element nomal to the contacting sufaces at the inne-aceway contact can be expessed as: ω si ω = sin( β β )+ sin β ω ωi i i i (28.72) Similaly, at the oute-aceway contact, the olling speed, sliding speed, and spin angula velocity can be obtained as: u v eo o 1 ω = ( )+ + 1 D 4 ω γ cos β β γ cosβ o ω d o d o p o 1 ω = ( ) 1 D 2 ω γ cos β β γ cosβ o ω ω so d o d o p o ω = sin( β β ) sin β ω ωo o o o (28.73) (28.74) (28.75)

29 In applications, beaing otational speeds ae usually pescibed by the absolute angula velocities of the inne and oute ings, ω i and ω o. The following elationships exist between the elative and absolute angula velocities: ωi = ωi ωc (28.76) ωo = ωo ωc (28.77) whee ω c is the angula velocity of the cage o the olling element set. Now one has eight equations (Equations to 28.77) and eleven unknown vaiables. Eight of the eleven unknown vaiables ae placed on the left-hand side of the above equations. These unknown vaiables ae not attainable unless the thee emainde unknowns, ω, β, and ω c, ae solved. Obviously, the thee unknowns ae govened by a beaing s intenal geomety design and suface fictional conditions at the inne- and oute-aceway contacts. Rathe sophisticated numeical analysis of foces and moments is equied to detemine these unknown quantities (Hais, 1966, 1971a,b; Gupta, 1975, 1979a,b,c,d). Seveal compute pogams ae publicly available. The eade is efeed to COSMIC, Univesity of Geogia (Athens), fo detailed pogam listings. As a pactical matte, appoximations can be made to simplify the analysis. Fo olle beaings, the oientation angle β of the otation axis fo each olle is constained by beaing geomety. Only two unknown vaiables ω and ω c need to be detemined. The two unknowns ae usually obtained by assuming non-goss slip conditions at both the inne- and oute-aceway contacts (v i = v o = 0). Fom the non-goss slip conditions, it can be shown that ω γdcosβo+ 1 = γ cos β β ( ) d o ω o 1 ω m c i (28.78) ω c m ω ω = m 1 c o i c (28.79) whee cos( β β ) 1+ γ 1 γ i m c = cos( βo β) d d cosβ o cosβ i (28.80) The entainment speed and spin angula velocity at the olle and inne aceway contact ae detemined as: ω si u ei m = ( ) 1 γ cosβ D o i 1 m ω ω 2 c d i c 1 mc = cos βi βi β βi ωo ωi γ tan( )+ sin d 1 m c p (28.81) (28.82)

30 The entainment speed and spin angula velocity at the olle and oute aceway contact ae: ω so u eo 1 D d o = + γ cosβ o i 1 m ω ω 2 c 1 1 = + cosβo βo β βo ωo ωi γ tan( ) sin d 1 m c p (28.83) (28.84) Fo ball beaings, the motion of balls is moe complex. Each ball has an additional degee of feedom. The oientation angle β of the otation axis fo each ball is unknown. Thee spin contol hypotheses, known as oute aceway contol, inne aceway contol, and minimum diffeential spin, ae available Oute Raceway Contol Expeiments have evealed that balls of an angula-contact ball beaing nomally opeate with minimal spinning motion at one aceway, with nealy all the spinning motion occuing at the othe aceway (Shevchenko and Bolan, 1957; Jones, 1959). Whee spinning motion pevails is mainly dictated by fiction and contact conditions and by contact confomability o between the ball and aceway. Oute aceway contol theoy assumes, in addition to non-goss slip at both inne- and oute aceway contacts, that no spin exists at the oute aceway contact, ω so = 0, and all spinning motion occus at the inne aceway. The otating axis of each ball intesects the extended tangent of the contact between the ball and oute aceway at beaing axis. Fo elatively high-speed, lightly loaded and oil-film-lubicated beaings, the oute aceway contol theoy is a ational appoximation. The load and the spin-esisting fictional moment ae geate at the oute aceway contact than at the inne aceway contact due to centifugal effects. With the additional constaint ω so = 0, given by the oute aceway contol theoy, the oientation angle β of the otation axis fo each ball can be detemined. tan β = γ d sin β o + cos β o (28.85) The entainment speed at both the inne- and oute-aceway contacts and the spin angula velocity at the inne-aceway contact ae given in Equations 28.81, 28.83, and 28.82, espectively Inne Raceway Contol The inne aceway contol theoy assumes, in addition to the non-goss slip conditions at both inne and oute contacts, that thee is no spin at the inne contact, ω si = 0. The otating axis of each ball intesects the contact tangent of ball and inne aceway at beaing axis. Unde this constaint, it can be shown that: tan β sin βi γd sin 2 βi = cos β γ cos 2 β i d i (28.86) Equations fo u ei and u eo have the same foms as those fo oute aceway contol. The spin angula velocity at the oute-aceway contact is given in Equation Inne aceway contol pobably occus only in maginal lubicated beaings opeated at low speeds. The fomability of lubicant film is lowe at the inne aceway than at the oute aceway. Consequently, the spin-esisting moment is highe at the inne aceway than at the oute aceway. Inne aceway contol

31 may pevail when the inne-aceway goove is designed to give a geate contact confomability with balls in the beaing s axial plane than the oute-aceway goove. Thus, the inne-aceway contact has geate ellipticity and consequently a highe spin-esisting moment than the oute-aceway contact if the centifugal foces ae negligible Minimum Diffeential Spin Fo oil-lubicated beaings, in which elastohydodynamic films exist at both inne- and oute-aceway contacts, spin occus at both contacts. It has been shown that even unde less favoable lubication conditions, spinning and olling occu simultaneously at both inne- and oute-aceway contacts. The spin moment at the inne aceway counteacts the spin moment at the oute aceway to achieve equilibium. The minimum diffeential spin hypothesis assumes that no goss sliding exists at inne- and oute-aceway contacts, and that spin angula velocities at the inne and oute aceways have the same magnitude. The aithmetical summation of spin angula speed is zeo, ω si + ω so = 0. The minimum diffeential spin theoy tends to appoximate the minimum fiction enegy pinciple. Should the spinesisting moments at both the inne and oute aceways be the same, minimum diffeential spin theoy eflects exactly the minimization of fictional enegy. The minimum diffeential spin theoy esults in the following appoximation: tan β sin βo+ sin βi cos β + cos β o i (28.87) Rolling speeds and spin angula velocities fo the inne- and oute-aceway contacts can be obtained fom Equations and and fom Equations and 28.84, espectively. Because of the on-apex design of tapeed olle beaings, no spin exists at eithe the inne- o outeaceway contacts if olles ae not skewed. Howeve, the same cannot be said of spheical beaings. Peceptible spin occus at both inne- and oute-aceway contacts Beaing Load Ratings and Life Pediction Basic Load Ratings Beaing load ating is an impotant basis fo beaing selections. Beaing manufactues povide two basic load atings: the basic static load ating and the basic dynamic load ating. The concept of dynamic load ating is intoduced to eflect the fatigue natue of beaing failue. Rolling contact fatigue is a unique fom of mateial failue that occus at the suface o sub-suface laye due to epeated stesses. Because of the mico-scale inhomogeneities of beaing mateial, the stength o esistance to fatigue of beaing mateial vaies fom point-to-point. Even unde identical stess conditions, thee will be a wide dispesion in beaing lives. Fo this eason, beaing life is associated with a failue pobability o a suvival pobability. The concept of pecentage life is used. Fo example, L 10 life efes to the numbe of hous at a cetain speed o the numbe of evolutions that 10% of a goup of supposedly identical beaings ae expected to fail, o 90% of the beaings ae expected to suvive unde a specific load. L 10 life can also be intepeted as the numbe of hous at a cetain speed o the numbe of evolutions that an individual beaing will suvive with 90% eliability unde the specified load conditions. Reseach ove the yeas has shown that fo a given failue pobability, beaing life deceases as load inceases. ISO adopts L 10 = 10 6 evolutions as the standad ating life to establish the basic dynamic load ating. The basic dynamic load ating is defined as a constant load applied to a beaing with a stationay oute ing that will endue the standad ating life of 1 million evolutions, L 10 = The basic dynamic load ating fo adial beaings is a cental adial load of constant diection, denoted as C 1, while the basic dynamic load ating fo thust beaings is an axial load in the same diection as the cental axial and is

32 TABLE 28.7 Basic Dynamic Load Rating Ball Beaing a Rolle Beaing Radial beaing b m f c (i cos β) 7/10 Z 2/3 D9/5 w b m f c (il we cos β) 7/9 Z 3/4 D 29/27 we Angula contact beaing b m f c (cos β) 7/10 tanβz 2/3 D9/5 w b m f c (L we cos β) 7/9 tanβz 3/4 D 29/27 we Thust beaing β = 90 b m f c Z 2/3 D9/5 w b m f c L 7/9 we Z 3/4 D 29/27 we a When D w > 25.4 mm, use 3.645D 7/5 w fo D 9/5 w. Note: b m : Rating facto fo contempoay beaing mateial and manufactuing quality, the value of which vaies fom 1.0 to 1.3, depending on beaing type and design. f c : Facto that depends on the geomety of the beaing components, the accuacy to which the beaing components ae made, and the mateial. The values of f c fo vaious beaings ae given in ISO i: Numbe of ows of olling elements in a beaing. β: Nomal contact angle of a beaing. Ζ: Numbe of elements pe ow. D w, D we : Diamete at the cental plane of a olling element (mm). L we : Effective contact length of olles (mm). TABLE 28.8 Beaing Type Maximum Pemissible Contact Stess Maximum Stess (MPa) Self-aligning ball beaings 4600 Non-self-aligning ball beaings 4200 Rolle beaings 4000 denoted as C 1a. The basic dynamic load atings fo vaious beaings ae calculated by the equations shown in Table The basic static load ating is used to detemine the maximum pemissible load that can be applied to a non-otating beaing. The static adial load ating C 0, and the static thust load ating C 0a, ae based on a maximum contact stess between olling element and beaing aceway at the cente of the contact in a non-otating beaing with a 180 and 360 load zone, espectively. The maximum stess vaies slightly fo diffeent beaing types. Table 28.8 lists the maximum contact stesses. The maximum stess level listed in Table 28.8 may cause visible light Binell maks on beaing aceways. These maks will not have a measuable effect on fatigue life when the beaing is subsequently otating unde lowe application loads Beaing Life Pediction Beaing Life Theoy In studying the failue of bittle engineeing mateials, Weibull (1939) concluded that the stength of a mateial was a statistical vaiable, and assumed that the fist initial cack led to the beak of the entie stuctue unde stess. By applying the calculus of pobability, Weibull stated that the suvival pobability S could be expessed by: ln 1 S dv Γσ = ( ) V (28.88) whee Γ(σ) is a mateial chaacteistic and is a function of stess conditions σ. Lundbeg et al. (1947) extended Weibull s weakest link theoy to ductile beaing mateials by aguing that a beaing s life consists of cack initiation life and cack popagation life, and that initiation life pedominates the beaing s life. Based on extensive beaing tests, Lundbeg and Palmgen poposed an empiical elationship:

33 ln 1 c e 0 N = ( ) dv V h S Γσ τ z V 0 (28.89) whee τ 0 is the maximum othogonal shea stess, z 0 is the depth below the load-caying suface at which the maximum shea stess occus, V is the volume of stessed mateial, and N is the numbe of stess cycles befoe failue with suvival pobability of S. c, e, and h ae exponents. Assume Hetzian contact exists between the olling elements and aceways. Fo an assembly of one o moe contacts, Equation can be expessed in tems of beaing load P by substituting beaing geomety and by applying Hetzian contact theoy: L n C = Lc P (28.90) whee n is an exponent (n = 3 fo ball beaings and n = 10/3 fo olle beaings). L c10 is the efeence ating life unde which the dynamic load ating C is established. In ISO standads, L c10 = 10 6 evolutions and, accodingly, C = C 1. As shown in Table 28.7, the influences of beaing geomety and mateial quality ae lumped into the dynamic load ating C 1. P is the equivalent load and can be calculated fom the following equation. P= XF + YF a (28.91) F and F a ae beaing loads in adial and axial diections, espectively. Factos X and Y ae detemined based on the atio of F a /F. The values of X and Y ae given in ISO and beaing catalogs fo specific beaings Beaing Life Adjustment Factos Actual beaing opeating conditions may vay peceptibly fom the efeence conditions unde which the basic dynamic load atings ae established. To pemit the quantitative evaluation of envionment diffeences, such as eliability equiement, beaing mateial quality and pocessing, opeation conditions, and failue citeia, vaious life adjustment factos have been intoduced to the basic life Equation by beaing manufactues. ISO combines thee factos into the following life equation to give the adjusted life: L aa a L na = (28.92) L na is the adjusted beaing life, in evolutions, and a 1, a 2, and a 3 ae life adjustment factos. The definition of life adjustment factos and the vaiables that have been incopoated in each facto ae discussed as follows. Life adjustment facto fo eliability (a 1 ): Fo applications that equie beaings to have eliability othe than 90% (S = 0.9), the life adjustment facto can be obtained fom: a 1 = 1 S ln ln e (28.93) whee e is the Weibull slope; e = 1.5 is often ecommended.

34 Life adjustment facto fo special beaing popeties (a 2 ): This facto allows impovements in beaing mateial and pocess to be incopoated in the beaing life estimate. Fo beaings made fom standad beaing quality alloy and cabon steels, a value of a 2 = 1 is ecommended. Beaings ae also made fom highe gade steels with bette contolled efining pocesses. These pemium steels contain fewe and smalle inclusion impuities than standad steels and povide the benefit of extending beaing fatigue life whee fatigue life is limited by nonmetallic inclusions. When beaings made fom pemium steels ae used, a value of a 2 geate than 1 can be adopted, povided that the maximum contact stess is less than 3 GPa and that favoable lubication conditions ae maintained. Howeve, if a eduction in life is expected as the esult of a special heat teatment, a educed a 2 value may be consideed. Values of a 2 ange fom 0.6 to 6.0 fo diffeent chemisties, melting pactices, cleaness levels and metal woking, heat teatment, and suface modification pactices. Infomation on a 2 can be found in Zaetsky (1992). Life adjustment facto fo opeating conditions (a 3 ): Vaiables consideed in the life adjustment facto fo opeating conditions ae load distibution, tempeatue, and lubication. The inclusion of load distibution and tempeatue equies extensive analytical and expeimental wok. Thee is no geneal consent among beaing manufactues egading these effects on beaing life. ANSI/ABMA and ISO decided not to give geneal ecommendations. Adequacy of lubication is cental to a 3. The ANSI/ABMA and ISO standads ae based on nominal lubication conditions whee the lambda atio λ is equal to o slightly geate than 1. A value of a 3 = 1 is ecommended fo such conditions. a 3 inceases as λ inceases and educes as λ deceases. Life adjustment factos wee intoduced to account fo vaiances not included in the basic life model (Equation 28.90). Howeve, choosing appopiate values of life adjustment factos is not as easy as it might appea. Envionmental and design factos ae often mutually dependent. Changing one facto can affect othes. Some factos may not even be multiplicative. Not to estict vaious means of combining the appopiate life factos, the ISO beaing life ating technical committee is cuently woking on a poposal in which a 2 and a 3 will be eplaced by a single facto a xyz. This facto is deived fom a stessbased life calculation model that incopoates all possible factos affecting the genealized stess field. Thee ae still effects on beaing life that cannot be popely accounted fo by life adjustment factos. To incopoate these effects, extensive analytical wok and beaing life tests ae equied. The ISO and ANSI/ABMA may futhe evise thei life adjustment factos and/o life calculation method to eflect the advance of moden beaing technology. It is woth noticing that altenative beaing-life models ae also available fo beaing fatigue life pediction. These models can geneally be classified into two categoies: the engineeing model and the eseach model. Tallian (1992) povided a compehensive eview of some of the published models. Discussions of vaious fatigue-life pediction models ae beyond the scope of this handbook and thus ae not pesented hee. The ISO and vaious beaing manufactue associations may have thei own pactice fo the selection of life calculation methods and life adjustment factos. Fo a specific application, seek a beaing manufactue fo advice Beaing Toque Calculation When efficiency and enegy consumption of a mechanical system ae of concen, a beaing s fictional toque is often analyzed to detemine its acceptance. Beaing toque calculations ae also pefomed to assess beaing heat geneation. When a beaing s toque is obtained unde nomal unning conditions, it is efeed to as unning toque. Fo angula-contact ball beaings and tapeed olle beaings, toque is often used to detemine the appopiate amount of pe-load. Beaing toque obtained unde slow speed conditions o setup conditions is efeed to as stating toque o setup toque. This section

35 addesses beaings stating toque as well as unning toque and povides empiical and analytical equations fo toque pedictions of angula-contact ball beaings and tapeed olle beaings. The majo souce of beaing toque deives fom the olling esistance and, if applicable, the sliding esistance at the contacts between the olling elements and aceways. Beaing toque also esults fom the sliding esistance at the contacts between the olling elements and cage, and between the olling elements and flange, if a flange exists. In modeate and high-speed applications, lubicant chuning is an additional souce of fictional esistance. The olling esistance is caused by intenal fiction in the lubicant and beaing mateial and by the mico-slip. Sliding esistance esults pimaily fom suface aspeity inteactions intoduced by the sliding and/o spinning motion. Unde high-speed and light-load conditions, goss sliding occus at aceway contacts, which futhe complicates the toque analysis. A complete analytical appoach fo beaing toque pedictions is vey difficult, if not impossible. Toque models used by beaing manufactues ae mostly established though expeimentation. Howeve, thee ae cases whee beaing toque models can be deived analytically o semi-analytically Stating Toque Stating Toque fo Angula-Contact Ball Beaings Fo angula-contact ball beaings, the stating toque esults pimaily fom the slippage associated with spin motion between the balls and oute aceway. At slow speed, as suggested by inne-aceway contol theoy, the spin-esisting moment at the inne aceway pevails. Thus, the stating toque M is sought by only consideing the spin moment at contacts between the balls and oute aceway. Z j = 1 M = M sin β sj j (28.94) whee M sj is the toque caused by Coulomb fiction foces fo the contact at the j th azimuth location. Poitsky et al. (1947) integated the fiction foce ove a contact ellipse and showed that: 3 M W a E e 8 = µ ( ) sj a j j j (28.95) whee W j, a j, and β j ae the element load, the half majo axis of contact ellipse, and the contact angle at the j th azimuth location, espectively (see Section 28.4). µ a is sliding fiction coefficient fo the dy o bounday lubicated contact. E(e j ) is the elliptic integal of the second kind, the value of which can be obtained fom mathematical handbooks o fom the following empiical equation (Hamock et al.,1983): E() e + π 1 k k (28.96) whee k is defined in Section If the beaing is unde pue thust load, as in the beaing setup situation, beaing toque can be expessed in tems of the axial load F a as: M = µ Fe e 13 E a R e 4 3 F 13 a E k Z sin β { Lubicationand Mateial () () Intenal Geomety Axial Load (28.97) whee k is the ellipticity atio at the oute-aceway contact, R e is the geometic aveage adius at the oute aceway contact, and F(e) can be calculated fom equations given in Section 28.4.

36 Stating Toque of Tapeed Rolle Beaings Although the on-apex design of tapeed olle beaings enables olles to achieve pimaily tue olling motions on the aceways at evey point along the line of contact, sliding and spinning inheently occu at the ib and olle-end contact fo evey olle. Unde slow-speed o staved lubication conditions, and when the load on a tapeed olle beaing exceeds a cetain level, usually about the level of pe-load, the sliding fiction at the ib and olle-end contact becomes a decisive facto. Beaing toque caused by olling esistance at olle and aceway contacts is compaatively negligible. This is due, in lage pat, to the inability of foming a full lubicant film at the ib and olle-end contact to sepaate the mating sufaces unde these conditions. Sevee aspeity-to-aspeity contact occus. The fiction toque fom the ib and olle-end contact is peceptibly high. Theefoe, the stating toque of a tapeed olle beaing is sought by consideing only the sliding and spinning motion at the ib and olle-end contact. The fictional toque at the ib and olle-end contact M ib consists of two components: a sliding toque M sld and a spinning toque M spn : Mib = Msld + Mspn (28.98) M M =µ HF cosγ sld a a = χµ HF cosγ spn a a (28.99) (28.100) whee γ is the half included angle of olles; H is the ib contact height and is detemined by the beaing intenal geomety fom Equation 28.33; and χ is a facto depending pimaily on ib and olle-end geomety and load distibution. Fo most applications, χ 0.5 is a good appoximation. The stating toque can be calculated fom the following appoximation: ( ) µ M =µ 1+ χ HF cos γ 1. 5 HF cosγ a a a a (28.101) Running Toque Running Toque of Ball Beaings The pediction of unning toque fo ball beaings is lagely dependent on empiical equations. Based on expeimental studies, Palmgen (1959) esolved the total toque into a load-elated moment M l and a lubicant viscous moment M v. The load-elated moment eflects elastic olling esistance, including mico-slip and hysteesis loss, while the viscous-elated moment accounts mostly fo the losses of lubicant film sheaing and bulk lubicant chuning. M = M + M l v (28.102) This geneal appoach has been adopted by vaious beaing manufactues. Howeve, the empiical equations established by vaious beaing manufactues fo M l and M v calculations may vay to eflect the diffeences in intenal geomety design. Catalogs of manufactues must be consulted fo appopiate toque calculation equations. NSK uses the following equations fo load- and viscous-elated moments calculations fo ball beaings unde high-speed applications: M = D F N mm l p a ( ) a b M = D 3 n η Q N mm v p i ( ) (28.103) (28.104)

37 whee D p = Beaing pitch diamete (mm) F a = Axial load (N) n i = Inne ing speed (pm) η = Lubicant viscosity at oute ing tempeatue (MPa.s) Q = Lubicant flow ate (kg/min) The exponents a and b ae functions of beaing angula speed only and ae given, espectively, by: a= n i (28.105) b= n i (28.106) Running Toque of Tapeed Rolle Beaings Running toque of tapeed olle beaings esults pimaily fom the hysteesis and viscous-elated olling esistance at the contacts between the olles and aceways, and fom the slide-and-spin-elated sliding esistance at the contacts between the olle-ends and ib face. As the speed of the beaing inceases, and when ample lubicant is available at the inlet to the contact, contact sufaces between the olle-ends and ib may be fully sepaated by a laye of lubicant film. The thickness of the lubicant film at the olleend and ib contact is appoximately twice the film thickness at the olle and aceway contact. Consequently, the fiction loss fom the olle-end and ib contact educes substantially. Figue shows schematically the vaiation of fictional toque as a function of beaing speed fo tapeed olle beaings. Fo a tapeed olle beaing, the unning toque is composed of thee components: the inne aceway toque M ace(i), the oute aceway toque M ace(o), and the ib toque M ib. M = M + M + M () ( ) ace i ace o ib (28.107) The aceway toque fo a combined adial and axial load is obtained by advancing the wok of Aihaa (1987). M 031. = φ ace i TH φsp ρd ζds () ( ) 2 Re R o Facos2γ α RlE sin β e o 031. (28.108) FIGURE Beaing toque as a function of speed.

38 M 031. = φ ace o TH φsp ρd ζds ( ) ( ) 2 Re R i F a α RlE sin β e o 031. (28.109) whee φ TH = L (28.110) is a themal facto that is intoduced to conside the inlet shea heating. φ TH is dependent only on u L = βη 0 k cd 2 e (28.111) whee β is the tempeatue coefficient of viscosity, η o is lubicant viscosity, and k cd is the themal conductivity of lubicant. φ SP = αη u e 0 R e (28.112) is a speed facto, and ρ D = D is the aspect atio of olles. ( ) 031. ζ DS ( ) 031. ZJa = 031. ( ZJa ) (28.113) is a load distibution facto. J a (0.31) is called the extended Sjövall integal and is defined as: J ( ) 031. a xl 1 1 = cosψ π ε xl ( ) 031. t dψ (28.114) (0.31) The values of J a ae listed in Table 28.6 fo line contact (t = 10/9). Othe paametes used in the above equations ae defined as follows: α = Pessue-viscosity coefficient of lubicant β 0 = Contact angle at the oute aceway contact l = Rolle length D = Rolle mean diamete Z = Numbe of olles R e = Effective contact adius R i, R o = Mean adii of the inne and oute aceways The ib toque is expessed in a simila fom as Equation : M ib = ( 1+ χ)µ HFa cosγ (28.115)

39 TABLE 28.9 at 20 C Opeating Tempeatue of Beaings in Diffeent Machines Opeating Beaing Application Appoximate Opeating Tempeatue ( C) Cutte shaft of planing machine 40 Bench dill spindle 40 Hoizontal boing spindle 40 Cicula saw shaft 40 Double-shaft cicula saw 40 Blooming and slabbing mill 45 Lathe spindle 50 Vetical tuet lathe 50 Wood cutte spindle 50 Calende oll of a pape machine 55 Backup olls of hot stip mills 55 Face ginding machine 55 Jaw cushe 60 Axle box beaing/locomotive o passenge coach 60 Hamme mill 60 Wie mill oll 65 Vibatoy moto 70 Rope standing machine 70 Vibating sceen 80 Impact mill 80 Ship s popelle thust block 80 Vibating oad olle 90 Unde nomal unning conditions, contacts between the olle-ends and ib face ae patially o fully sepaated by a lubicant film. Fiction coefficient µ at these contacts is educed by vitue of the low sheaing esistance of lubicant film. The eduction in µ is detemined by the load shaing atio ϑ, which in tun is a function of the lambda atio (λ): µ= ϑµ a+ ( 1 ϑ)µ ϑ= e λ f (28.116) (28.117) µ a and µ f ae fiction coefficients fo dy contact (aspeity contact) and lubicated contact, espectively. The empiical appoach as outlined fo unning toque calculations of ball beaings is also applicable fo vaious othe beaing types, including tapeed olle beaings, when a quick and simple toque estimate unde nomal opeating conditions is desied. The eade is efeed to Bandlein et al. (1999) o Hais (1991) fo details Beaing Tempeatue Analysis The expected beaing opeating tempeatue is impotant fo designing the beaing aangement, lubication and sealing, and fo detemining the pope beaing setting and fitting pactice. Excluding extaneous heat, the opeating tempeatue of a olling element beaing unde medium speed and load conditions is elatively low. Table 28.9 povides efeence values fo the aveage opeating tempeatue in vaious applications (Bandlein et al., 1999). When exposed to extaneous heat, the tempeatue that a beaing assumes can be vey high. Table lists the typical opeating tempeatue fo beaings exposed to extaneous heat (Bandlein et al., 1999). The opeating tempeatue of beaings in a mechanical system is detemined by the amount of heat geneated within the beaing and the amount of heat that is tansfeed to o away fom the beaing.

40 TABLE Opeating Tempeatue of Beaings Exposed to Extaneous Heat Beaing Application Extaneous Heat Appoximate Opeating Tempeatue ( C) Electic taction moto Electic heat fom amatue, housing cooled by ai Hot gas fans Heat tansmission to the beaing via the shaft fom the 90 impelle exposed to hot gas Wate pump in a vehicle engine Heat fom cooling wate and engine 120 Tubo-compesso Dissipation of compession heat though shaft 120 Intenal combustion engine cankshaft Dissipation of combustion heat though cankshaft; 120 cooled housing Dye olls of pape machine Heating steam of C though beaing jounal Calende fo plastic substances Supply of heat caie at C though beaing 180 jounal Wheel beaings of kiln tuck Heat adiation and tansmission fom kiln space The actual system in which beaings ae opeated can be vey complex. Howeve, the pinciple of tempeatue analysis emains the same. The pinciple is based on the law of enegy consevation, known as the fist law of themodynamics, and on the ate equations of heat tansfe. Conside a contol volume, a egion of space bounded by contol sufaces though which enegy and mass can pass. The contol volume can be a finite egion o a diffeential (infinitesimal) egion. Fo beaing tempeatue analysis, it is customay to use finite-contol volumes. These contol volumes ae bounded by sufaces that define a beaing s bounday dimensions. The law of enegy consevation states that the net ate at which themal and mechanical enegy entes a contol volume, plus the ate at which themal enegy is geneated within the contol volume, must equal the ate of incease in enegy stoed within the contol volume. Unde steady-state conditions, the amount of heat geneated in the beaing is caied away by the net dissipated heat flow. E = E E gen out in (28.118) To assess the tempeatue level at which a olling element beaing opeates, the heat geneated and the heat tansfeed though vaious modes must be detemined. This section intends to povide the basic elements being used by the beaing industy in tempeatue analysis Heat Geneation The powe loss in a olling element beaing pimaily comes fom the fictional loss duing opeation that manifests itself mainly in a fom of heat geneation. Thus, the ate of heat geneation in the beaing can be estimated fom the beaing s fictional toque though the following equation: Hf =ωm (28.119) whee ω is the otational speed and M is the beaing toque. Fo ball beaings and olle beaings, the fictional toque can be obtained by equations given in the pevious section. The fictional toque equations fo ball beaings ae based on empiical fomulae. The effect of sliding between the balls and cage pockets is included and so is lubicant chuning effect. Thus, Ė gen = H f (28.120) Fo olle beaings, howeve, the fiction between the olles and cage pockets is not consideed. In addition, lubicant chuning loss is not included in the toque calculation. Lubicant chuning effects

41 must be accounted fo when a beaing opeates at a elatively high speed and chuns a fai amount of lubicant. The ate of heat geneation due to viscous dag can be appoximated by: H c Z ξ ld D m pω 295. c l φvc v 32g = ( ) (28.121) whee Z = Numbe of olles ξ = Density of oil-ai mist; weight of lubicant in beaing cavity divided by the fee volume within the beaing bounday dimension l = Effective olle length D m = Rolle mean diamete D p = Beaing pitch diamete ω c = Obital angula speed of olles g = Gavitational acceleation c v = Dag coefficient φ VC = Effective volume coection facto, 0 < φ VC < 1 Theefoe, fo olle beaings, the ate of heat geneation is calculated as: Ėgen = Hf + Hl (28.122) Heat Tansfe Heat geneated in beaings is dissipated into the system via thee basic modes: conduction, convection, and adiation. It is possible to quantify the heat tansfe pocesses using ate equations. Because heat conduction within a solid and heat convection between the solid and fluid ae the pimay foms of heat tansfe in a beaing system, the ate equations fo these two modes ae addessed below. The ate equation fo adiation, if needed, can be found elsewhee (Pinkus, 1990) Conduction Heat conduction is caused by the diffusion of enegy fom the moe enegetic to the less enegetic molecules o paticles though andom motion. The ate of heat tansfe via conduction is given by Fouie law: q k A T cd = cd l (28.123) whee k cd is the themal conductivity; A is the aea nomal to the heat flux; and T/ l is the tempeatue gadient along the diection of heat flux Convection Heat convection is the esult of enegy diffusion due to andom molecula motion and enegy tansfe due to bulk motion. The actual physical events associated with heat conduction may be quite complex when latent heat exchange is involved. Regadless of the paticula natue of the convection heat tansfe pocess, the appopiate ate equation is, in simple fom, known as Newton s cooling law: ( ) q = h A T T cv cv s (28.124) whee T s is the suface tempeatue and T is the efeence tempeatue of the fluid. Fo extenal flow, the fee steam tempeatue T = T is used; fo intenal flow, the efeence tempeatue T efes to the mean tempeatue T = T m. The popotionality constant h cv is the heat convection coefficient o film conductance. It encompasses all the paametes that influence convection heat tansfe. In paticula, it

42 depends on conditions of the bounday laye, which is affected by suface geomety, the natue of fluid motion, and the themodynamic and tanspot popeties of the fluid. Any study of convection ultimately educes to a study of detemining h cv. The equation povided by Ecket (1950) can be used as an appoximation fo h cv when consideing the convection heat tansfe fom beaing to oil that contacts the beaing assuming a lamina flow field (Hais, 1991): h cv u 13 c = kcd P νdp 12 (28.125) whee k cd and ν ae the themal conductivity and kinematic viscosity of the oil; P is the Pandtl numbe of the oil; and u c is the cage speed. The equation is also suitable fo calculating the convection heat tansfe fom the housing insidesuface to oil, taking u c equal to 1/3 of the cage speed and D p equal to the housing diamete. The tempeatue obtained though heat flow analysis using a finite-contol volume outlined above gives the aveage beaing bulk tempeatue. The actual tempeatue at the contact between the olling elements and aceways, o between the olle-ends and flange, can be peceptibly highe than the bulk tempeatue. Fo flash tempeatue analysis at beaing contact, the eade is efeed to Gao et al. (2000). Thee ae cases whee compehensive numeical pogams must be used to accomplish beaing tempeatue analysis. When such need aises, seek a beaing manufactue fo advice Beaing Enduance Testing Beaing enduance testing is pefomed to establish life atings, conduct quality auditing, and evaluate mateial, heat teatment, intenal geomety, and suface finish impovements. Geat vaiation can be expected despite the fact that beaings ae un unde identical conditions. Thus, statistical methods in planning and intepetation of beaing enduance tests have become a necessity. Accodingly, testing pocedues have been developed to optimize the tests based on statistics Testing Pocedue and Data Analysis Extensive testing esults ove the yeas have indicated that beaing life is appoximately a powe law function of load unde a given failue pobability. On a log-log plot, the life and load elation falls on a staight line. A family of such staight lines can be obtained, with each coesponding to a cetain failue pobability. Fo a given load, the cumulative pecentage of failue assumes the Weibull distibution. On Weibull pobability pape, the gaphical pesentation of the elationship between the failue pobability and beaing life also assumes a staight line unde constant load conditions. Beaing enduance testing is a time-consuming and costly pocess. To educe testing time and cost, beaings ae usually tested unde acceleated conditions. Inceasing applied beaing load is pehaps the most commonly adopted means fo test acceleation. Cae must taken to ensue that the load incease does not alte the beaings failue mode. As a geneal guideline, the maximum applicable load should not cause any plastic defomation. Fo this consideation, the applied load should not poduce a maximum Hetzian stess in excess of 3.3 GPa. Even with acceleated tests, it is neithe economical no possible to test the entie beaing population. Instead, only a finite collection of beaings is tested. Statistical analysis is theefoe used to deive the undelying life distibution fo the geneal population fom the finite beaing samples. Seveal methods exist to design testing pocedues that educe testing time. Sudden death testing, sequential testing, and paallel testing ae exemplay test stategies. Among the vaious test methods, sudden death tests ae the most ecommended test pocedues in the beaing industy. In sudden death tests, beaing samples of size m ae divided into l goups with n beaings in each goup (m = nl). When the fist failue occus in a goup, the test is suspended fo that goup. Afte all goups ae tested, l failues ae obtained. Each failue epesents an estimate of L q life that is associated

43 with a definite pecentage q of the beaing population failing as if the entie population wee tested. This pecentage is called ank. Because the tue ank is not known, estimations ae made such that in a long un, the positive and negative eos of the estimate cancel each othe. A ank with such popety is called median ank and is denoted as q = 0.5. Fo a detailed discussion on the detemination of median anks, the eade is efeed to Johnson (1951). The l estimates of L q ae tabulated in ascending ode. The median ank fo the j th estimate in l can be obtained appoximately fom the following equation (Johnson 1974): j () l = l (28.126) Plotting L q as abscissa in logaithm scale and 1/(1 (l) 0.5) as odinate in log-log scale, the Weibull distibution of L q is established. The best estimate of L q is the median life coesponding to the 50% level. An α% confidence band can be constucted by seeking the (50 ± α/2)% ank values of the life estimate L q. The eade is efeed to Johnson (1974) fo detailed discussion. To ensue meaningful esults, test contols ae equied. Caution must be execised to make sue that the test beaings ae fee fom mateial and manufactuing defects, and that all pats confom to established dimensional and fom toleances. Moeove, test conditions must be adequate fo the vaiable that is unde evaluation. The failue mode is of fatigue natue and no othe vaiables o atifacts alte the outcome of test esults. Finally, each failed beaing should be caefully examined to exclude nonfatigue-elated failues in the life estimate and to detemine if the enduance test seies was adequately contolled befoe and duing the test Beaing Failue Analysis Pesent-day beaings seldom fail when popely installed and maintained. Pematue failues esult pimaily fom damage to beaings while handling befoe and duing installation, and damage caused by impope installation, setting, and opeation. When pematue failue occus, post-motem investigations ae often conducted to detemine the causes of failue so that simila failues can be pevented. In many cases, failues ae easily identified visually on beaings, but the oot cause of these failues is difficult, if not impossible, to detemine. This section intends to povide some possible cause-and-effect elationships to help the eade detemine the causes of the most common types of beaing failue so that peventive actions can be taken. A vaiety of means have been poposed to classify the types of beaing failue, some based on the mechanisms o causes of failue, and othes based on failue appeaances o failue locations, o a combination of all of the above. In this section, beaing failues ae categoized into a numbe of majo types accoding to the mechanisms. Each type may futhe be distinguished by specific modes based on the natue of the failue Contact Fatigue Contact fatigue can have diffeent appeaances, depending on the initial causes. It anges fom suface pitting, peeling, to spalling o flaking, and to section cacking in ae cases. Whateve the initial causes, cyclic stess is the common facto. Pitting. Pitting is indicated by the development of small cavities in the contact path. It is a suface-oigin fatigue failue associated pimaily with aspeity contact and small debis dents that act as stess aises. Pitting can also esult fom wate coosion. In such cases, pits usually have ough iegula bottoms (Figue 28.13). Peeling. Peeling is chaacteized by the fomation of cacks that oiginate at a vey shallow angle to the suface and popagate paallel to the suface at depths much smalle than the maximum nominal stess.

44 (a) (b) FIGURE Pitting (a) poceeded with spalling, and (b) caused by acid coosion. (Fom Walp, H.O. (1971), Intepeting Sevice Damage in Rolling Type Beaings A Manual on Ball & Rolle Beaing Damage, ASLE, Pak Ridge, IL. With pemission.) (a) ROLLING (b) FIGURE (a) Spall initiated at suface damage. (Fom Hais, T.A. (1991), Rolling Beaing Analysis, 3d ed., John Wiley & Sons, New Yok. With pemission.) (b) Spall nucleated at subsuface on cylindical beaing inne-ing aceway. (Fom Dene, W.J. and Pfaffenbege, E.E. (1984), CRC Handbook of Lubication, Vol. II, Boose, E.R. (Ed.), CRC Pess, Boca Raton, FL. With pemission.) Peeling may appea as a fosted aea. Thus, it is sometimes called fosting. Peeling occus when a elatively lage pecentage of the suface aspeities ae in diect contact with the mating suface. Aspeity-scale plastic defomation is inevitable. The fact that depth of peeling often coelates with suface oughness suppots the oughness aspeity theoy. Pogessive supeficial pitting can also lead to peeling. Spalling. Spalling is identified by the fomation of lage and deep cavities in the loaded aea (Figue 28.14). In sevee cases, a spall can extend along o acoss the beaing aceway, esulting in the emoval of a lage volume of mateial fom the suface. The pogessive state of spalling is sometimes called flaking. Spalling is consideed the most common fom of fatigue failue. It can be eithe suface o subsuface initiated. A suface-oiginated spall is usually associated with suface defects that act as stess aises. Debis dents, nicks, and ginding gooves ae common sites fo suface-oiginated spalling. Suface-oiginated spalls often show an aowhead gowth patten (Figue 28.14a). A multitude of small suface defects esults in mico-spalling o pitting, which could potentially culminate in mico-peeling o fosting. Lage suface defects tend to

45 FIGURE (a) Flaking with conchoidal patten extending acoss the loaded pat of the ace. (Fom Neale, M.J. (1973), Tibology Handbook, John Wiley & Sons, New Yok. With pemission.) (b) A geatly advanced flaking on inne aceway of a cylindical olle beaing. (Fom Walp, H.O. (1971), Intepeting Sevice Damage in Rolling Type Beaings A Manual on Ball & Rolle Beaing Damage, ASLE, Pak Ridge, IL. With pemission.) cause deep and lage spalls that penetate to the depth of the maximum Hetzian shea stess. Subsufaceoiginated spalling is associated with stess concentations caused by constituents in the matix, paticulaly oxide-type inclusions (Figue 28.14b). When the Hetzian stesses ae dominant, cacks popagate tansganulaly to the maximum Hetzian stess depth, causing mateial to beak off fom the suface. Flaking. Flaking is identified by mateial beak-off with conchoidal and ipple pattens extending evenly acoss o along the loaded pat of the ace (Figue 28.15). Although thee is no clea distinction between flaking and spalling, some conside flaking as a sevee state of spalling. Flaking can be suface o subsuface oiginated. Suface-oiginated flaking is usually caused by a multitude of debis dents. Subsufaceoiginated flaking most likely esults fom an inclusion inside the mateial. Edge loading tends to esult in flaking moe sevee on one side of the aceway. Tansvese cacking. Tansvese cacking seldom occus fo case-cabuized beaing components. It mainly affects beaings made fom though-hadened steels. Tansvese cacking is the esult of cack popagation unde cyclic tensile stess due to pess fit o flexing of the section. This type of failue is usually peceded by anothe mode of fatigue Suface Depession and Factue Unlike contact fatigue, suface depessions and factues of beaing components ae usually inflicted by static load o impact load. The load fo inflicting such damage is geate than those that cause fatigue failue. Unde some cicumstances, suface depessions themselves may not cause significant concen. Howeve, the consequences ae usually detimental. Wheneve possible, this type of damage should be avoided. Binelling. Binelling is chaacteized by the plastic defomation of beaing sufaces due to exteme static o epeated shock loads. Binelling is identified as dents o gooves on beaing aceways confoming to

46 FIGURE Binelling maks on the inne aceway of a tapeed olle beaing. (Coutesy of the Timken Company.) the shape of olling elements (Figue 28.16). Indentations by foeign objects do not fall into this categoy. Binelling may not immediately teminate beaing opeation if vibations caused by binelling maks do not aouse significant concen. Howeve, the post ove-olling of binelling maks will lead to cack initiation and subsequent popagation. Beaings with binelling maks ae eventually failed by spalling. Gouges o nicks. Impope handling and installation often inflict beaings with gouges o nicks. Goss ginding opeation can also leave nicks on beaing aceways. These suface defects seve as stess aises and ae moe likely to cause suface-oiginated pitting o spalling unde cyclic loads. Debis buises. Debis buises ae suface indentations caused by foeign paticles that ae entained into the contacts between the olling elements and beaing aceways duing opeation. Unless the sufaces ae seveely indented, the poceeding failue modes, mostly pitting o spalling, ae the ultimate modes of failue. Factue. On ae occasions, beaings suffe fom component factue, a sudden failue caused pimaily by high stesses that exceed the mateial s ultimate stength limits. Factue occus in beaings with impope tight fit and beaings with high intenal tensile stesses that esulted fom impope heat teatment. Cage damage o beakage. Cage damage and cage beakage ae occasionally seen in beaing applications. Cage damage is usually a esult of mishandling o impope installation. Cage beakage is pimaily associated with tosional vibation o excessive acceleation and deceleation. Cage beakage can also esult fom excessive olle skewing due to misalignment o loss of intenal geomety Mechanical Wea Mechanical wea is a gadual suface deteioation by elative motion. Compaed to sliding beaings, mechanical wea in olling element beaings is fa less sevee. This is paticulaly tue fo cylindical and tapeed olle beaings whose sufaces ae subjected pimaily to pue olling contact. Thee ae fundamentally thee mechanical wea mechanisms existing in olling element beaings: adhesive wea, abasive wea, and fetting wea.

47 FIGURE (a) Scuffing at a olle end; suface showing spial scuffing maks; (b) scuffing steaks in aceway of ball thust beaing caused by light load and high speed. (Fom Walp, H.O. (1971), Intepeting Sevice Damage in Rolling Type Beaings A Manual on Ball & Rolle Beaing Damage, ASLE, Pak Ridge, IL. With pemission.) Adhesive wea. Adhesive wea is chaacteized by mateial tansfe fom one suface to the mating suface. It is often found on olle-ends and the coesponding guide face of the flanges, as well as at the intefaces between shafts and inne ings and between housings and oute ings. Adhesive wea can also occu at contacts between the olling elements and aceways when substantial sliding exists. Adhesive wea appeas in vaious foms, depending on its seveity. 1. Scoing. Scoing occus when lubicant film thickness is inadequate to sepaate the contact sufaces. Aspeity-to-aspeity contact occus. Unde high nomal and tangential stesses, heat geneation at the aspeity becomes excessive. It gives ise to stong adhesive o welded junctions at isolated spots. With elative motion between the contacting aspeities, junctions ae ton apat, esulting in noticeable mateial tansfe fom suface to suface. 2. Scuffing. Scuffing is pogessive scoing that occus on a geate scale when the moe welded junctions appea and the isolated scoing spots ae joined togethe (Figue 28.17). Scuffing is identified by sevee suface oughening, and is accompanied by peceptibly high fiction and tempeatue. 3. Smeaing. Smeaing is sometimes called galling. It is consideed scoing on a gand scale. It involves sevee plastic defomation and massive mateial tansfe fom suface to suface. Although the final smeaing mode may epesent the goss beakdown of vaious suface films and nea suface egion, it may be tiggeed by the deteioation in suface topogaphy as a esult of pogessive scoing o scuffing. 4. Seizing. Seizing is the final stage of adhesive wea. Welding between contacting aspeities is so sevee that it seizes the beaing, esulting in catastophic failue. Seizing is, howeve, aely seen in olling element beaings. Abasive wea. Abasive wea occus when aspeity-scale plastic defomation leads to mateial emoval and wea debis. It involves abasive foeign paticles hade than beaing mateials, often nonmetallic. Abasive wea is identified by micoscopic fuows and the dulling of contact sufaces (Figue 28.18). Excessive wea esults in a loss of beaing dimension o intenal geomety, which in tun affects beaing pefomance. Fetting. Fetting esults fom mico-movement between mating sufaces. Fetting mostly occus duing tanspotation o machine idling. It is ecognizable by the gooves won into the aceway by vibational movement between the olling elements and aceways (Figue 28.19a). Fetting also occus between beaing ings and the sufaces that they contact (Figue 28.19b). It usually indicates an inadequate fit. Fetting fatigue is now ecognized as one of the outcomes of continuous fetting.

48 FIGURE Advanced stage of abasive wea and coosion in inne aceway and olles of a spheical olle beaing. (Fom Walp, H.O. (1971), Intepeting Sevice Damage in Rolling Type Beaings A Manual on Ball & Rolle Beaing Damage, ASLE, Pak Ridge, IL. With pemission.) FIGURE (a) Fetting gooves won into the aceways by axial movement of the olles duing tanspotation. (Coutesy of the Timken Company.) (b) Fetting on the oute diamete of an oute ace ing caused by nonunifom seat in housing. (Fom Walp, H.O. (1971), Intepeting Sevice Damage in Rolling Type Beaings A Manual on Ball & Rolle Beaing Damage, ASLE, Pak Ridge, IL. With pemission.)

49 FIGURE Electic ac damage on beaing aceways showing fluted pattens. (Fom Hais, T.A. (1991), Rolling Beaing Analysis, 3d ed., John Wiley & Sons, New Yok. With pemission; Widne, R.L. and Littmann, W.E. (1976), Beaing Damage Analysis, NBS 423, National Bueau of Standads, Washington, D.C.) Coosion Coosion is a chemical wea. The common foms of coosive wea seen in beaing applications ae etching and hydolysis. Acid etching. Acid etching occus when a beaing opeates in an acid envionment. Acid substances eact with and dissolve steel sufaces to fom salts. As a esult, numeous iegula and dak-bottomed pits ae ceated on beaing sufaces, a condition that subsequently leads to suface-oiginated spalling. Hydolysis. Beaings exposed to mist conditions ae susceptible to hydolysis, a coosion by oxidation. Hydolysis is identified by localized iegula pits. The affected aea may appea ed to dak bown, depending on the state of oxidation Electic Ac Damage Impopely gounded electical equipment can cause beaings to suffe fom electic acing damage by localized mateial melting and metallugical popety alteation. Appaent visual effects ange fom andom isolated pits to fluted pattens composed of numeous pits (Figue 28.20) Discoloing and Oveheating Discoloing is associated with heat geneation at the contact whee goss sliding occus. The heat oxidizes contact sufaces, poducing oxidation colos fom dak bown to blue. Excessive heat can cause all pats of a beaing to show tempeing colos. Oveheating causes the lubicant to decompose, diminishing its ability to lubicate. The majo impact is, howeve, the loss of mateial stength to contact fatigue. Table lists the most common failue modes and the possible causes.