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Formulations of Calculating Load Forces for Universally Loaded Bearings

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Formulations of Calculating Load Forces for Universally Loaded Bearings

  1. 1. Formulations of Calculating Load Forces for Universally Loaded Bearings Zhan Wang September 25, 2014 This document is based on the [1], [2, chap.7] and [3, chap.1]. Most of the formulas are reformations of the formulas given in [1]. Most of the figures come from [2, chap.7] and [3, chap.1]. The detail deriving process can be found in [3, chap.1]. 1 Input Parameters All symbols used in this document is list in Table 1. They are the same as symbols defined in [1, p. 2,3,4]. Table 1: Symbols (o: output value; i: given internally, depend on the bearing type.) Name Input Cal. Const. Description A x x distance, in millimeters, between raceway groove curvature centers of ball bearing having no clearance and having an initial contact angle Aj x distance, in millimeters, between raceway groove curvature centers of ball bearing at the position where element j locates, when the load is applied cP x x spring constant, in newtons per millimeter to the power of 3/2, of a rolling element with point contact cs x x spring constant, in newtons per millimeter to the power of 8/9, of a roller lamina Dpw x x pitch diameter, in millimeters, of ball or roller set Dw x x nominal ball diameter or roller diameter (maximum or in the mid- dle), in millimeters Dwe x x roller diameter at the contact point, in millimeters, applicable in the calculation of load ratings E x x modulus of elasticity, in megapascals E(χ) x x complete elliptic integral of the second kind e subscript for outer ring or housing washer 1
  2. 2. fe x outer-race conformity fi x inner-race conformity fp x conformity of roller crown radius F(ρ) x x relative curvature difference Fa o bearing axial load (axial component of actual bearing load), in newtons Fr o bearing radial load (radial component of actual bearing load), in newtons i subscript for inner ring or shaft washer K(χ) x x complete elliptic integral of the first kind Lwe x x effective roller length, in millimeters, applicable in the calculation of load ratings MX o moment, in newton millimeters, acting on tilted bearing ns x x number of laminae P(x) i x profile function, in millimeters Ri x x distance, in millimeters, between the center of curvature of the inner race groove and the axis of rotation Rp x x roller crown radius re x x cross-sectional raceway groove radius, in millimeters, of outer ring or housing washer ri x x cross-sectional raceway groove radius, in millimeters, of inner ring or shaft washer s x x radial(diametrical) operating clearance, in millimeters, of bearing yk x x distance, in millimeters, between center of lamina k and roller center Z x x number of rolling elements α x x nominal contact angle, in degrees, of a bearing αj x operating contact angle, in degrees, of the rolling element j α0 x x initial contact angle, in degrees γ x x auxiliary parameter δ x total elastic deflection, in millimeters, of both contacts of a rolling element δj x elastic deflection, in millimeters, of the rolling element j δji x elastic deflection due to the inner ring contact, in millimeters, of the rolling element j δje x elastic deflection due to the outer ring contact, in millimeters, of the rolling element j δj,k x elastic deflection, in millimeters, of the lamina k of the roller j δa x relative axial displacement, in millimeters, of both bearing rings δr x relative radial displacement, in millimeters, of both bearing rings νE x x Poisson’s ratio ρ x x curvature, in reciprocal millimeters, of the contact surface ρ x x curvature sum, in reciprocal millimeters ϕj x angular position, in degrees, of rolling element j χ x x ratio of semi-major to semi-minor axis of the contact ellipse ψ x relative angular displacement(total misalignment), in degrees, be- tween inner raceway and outer raceway 2
  3. 3. ψj x relative angular displacement(total misalignment), in degrees, be- tween inner raceway and outer raceway in the plane of rolling element j 2 Ball Bearings 2.1 General Description To describe the kinematics of the inner- and outer-rings, we define a From marker Mi in the center of the outer ring and a To marker in the center of the inner ring. Both markers are left hand ruled coordinates system. We assume that the Y axis of the From marker Mi is defined pointing to the direction of the unit axis direction ea of the outer ring. The direction ea is also the axial load direction (or axial displacement direction). The unit radial load direction, denoted by er, is also defined w.r.t. the From marker Mi. And the Y axis of the To marker Mj is defined pointing to the direction of the axis direction of the inner ring. In the initial state, no load is applied to the bearing. Due to a diametrical clearance in the no load state, a nonzero initial contact angle α0 will occur. The formula for calculating this angle is α0 = arccos [1 − (s/2A)] (1) and the angle is described in figure 1(a). After the load is applied, the contact angle will change from α0 to αj, which is given by equation (24) and described in figure 1(b). When a ball is compressed by load Q, since the centers of curvature of the raceway grooves are fixed with respect to the corresponding raceways (that is because the contact happens on individual points on the raceway grooves, the global curvatures and radii of the raceway grooves do not change), the distance between the centers is increased by the amount of the normal approach between the raceways [3]. From figure 1(b), it can be seen that Aj = A + δji + δje (2) δj = A + δji + δje − A = Aj − A, (3) where all the symbols are given in Table 1. 2.2 Relative Displacement After the load is applied, the relative displacement of the inner- and outer-ring may be defined [3]: • δa: relative axial displacement of both bearing rings • δr: relative radial displacement of both bearing rings 3
  4. 4. re α0 (a) Ball–raceway contact before applying load. δje δji Aj αj (b) Ball–raceway contact after load is applied. Figure 1: Ball–raceway contact. • ψ: relative angular displacement of both bearing rings These relative displacement are shown in figure 2. 2.2.1 Relative Axial Displacement deltaa The relative axial displacement, in millimeters, of both bearing rings, deltaa is state dependent. It is defined as: δa = rMiMj · ea (4) where ea is unit axis direction of the outer ring. 2.2.2 Radial Displacement δr The radial displacement, δr, is state dependent, and can be calculated by projecting the dis- placement of two markers into radial load direction, er: δr = rMiMj · er , (5) where er is er = ea × (rMiMj × ea) ea × (rMiMj × ea) . (6) 4
  5. 5. ψZ Y Figure 2: Displacements of an inner ring (outer ring fixed) due to application of combined radial, axial, and moment loadings. [3] If the ea is set to be parallel to Y axis of the From marker Mi, then δr can be obtained by just setting the y component of the vector rMiMj to be zero. 2.2.3 Relative Angular Displacement ψ The relative angular displacement(total misalignment) ψ, in degrees, state dependent, is the angle through which the axis of the inner ring rotates w.r.t. the axis of the outer ring in the y − z plane (around X axis)[2, 67], if we follows the orientation assumption in the begin of this document. It is defined as ψ = αMiMj , (7) where αMiMj is the relative angle of two markers around X axis. 2.3 Contact Geometry When two solids are pressed together by a force, the contact area is elliptical. To describe this contact area, two geometry definitions are needed: curvature sum (ρ) and curvature difference F(ρ). Their concepts are well explained in [2, p. 68]. Figure 3 shows the geometry of two contact solids. For the inner contact and outer contact, as the curvatures are different, the constant ρi, ρe, Fi(ρ) and Fe(ρ) need to be calculated according to equation 8, 9, 10 and 11, respectively. ρi = 2 Dw 2 + γ 1 − γ − Dw 2ri (8) 5
  6. 6. ρe = 2 Dw 2 − γ 1 + γ − Dw 2re (9) Fi(ρ) = γ 1 − γ + Dw 2ri / 2 + γ 1 − γ − Dw 2ri (10) Fe(ρ) = −γ 1 + γ + Dw 2re / 2 − γ 1 + γ − Dw 2re (11) where • Dw is the nominal ball diameter, in millimeters; • γ = Dw cos α/Dpw – α is nominal contact angle, i.e. the contact angle of the load-free bearing. For deep- groove ball bearings having a nominal contact angle zero degree. – Dpw is the pitch diameter in millimeters, of ball set. • ri is the cross-sectional raceway groove radius, in millimeters, of the inner ring. • re is the cross-sectional raceway groove radius, in millimeters, of the outer ring. This contact area ellipse can be described by the ratio χ of the semi-major to semi-minor. The ratio χ depends only on the geometry of two contact solid. The ratio χi (or χe), of inner contact (or outer contact) is the root of equation (12). They are constant and need to be calculated only once in the pre-prosseing. 1 − 2 χ2 − 1 K(χ) E(χ) − 1 − F(ρ) = 0 (12) • K(χ) is the complete elliptic integral of the first kind, defined by equation (3) in [1]. • E(χ) is the complete elliptic integral of the second kind, defined by equation (4) in [1]. • F(ρ) is calculated according equation (10) and equation (11). Brewe and Hamrock [4], using a least squares method of linear regression, obtained simplified approximations for χi, χe, K(χ) and E(χ) [2, p.127]. These equations are: χ ≈ 1.0339 Ry Rx 0.636 (13) K ≈ 1.0003 + 0.5968 Ry Rx (14) 6
  7. 7. r1x r1y r2y r2x 2 Y Y X X Figure 3: Geometry of contact bodies. E ≈ 1.5277 + 0.6023 ln Ry Rx (15) For 1 ≤ χ ≤ 10, the errors in the calculation of χ are less than 3%, errors on K are essentially nil except at χ = 1 and vicinity where they are less than 2%, and errors on E are essentially nil except at χ = 1 and vicinity, where they are less than 2.6%. For the inner contact, the directional equivalent radii Rix and Riy are defined by 1 Rix = ρx ball + ρx innerRing = 2 Dw + 2 Dw ( γ 1 − γ ) = 2 Dw ( 1 1 − γ ), (16) 1 Riy = ρy ball + ρy innerRing = 2 Dw − 1 ri (17) For the outer contact, the directional equivalent radii Rex and Rey are defined by 1 Rex = ρx ball + ρx outerRing = 2 Dw − 2 Dw ( γ 1 + γ ) = 2 Dw ( 1 1 + γ ), (18) 1 Rey = ρy ball + ρy outerRing = 2 Dw − 1 re . (19) In equation (16) and equation (18), we assume that the curvature is positive for convex surfaces and negative for concave surfaces. 7
  8. 8. 2.4 Basic Formulas 2.4.1 The Spring Constant cP After we obtain χi and χe, the spring constant cP can be calculated using: cP = 1.48 E 1 − νe 2 K(χi) 3 ρi χi 2E(χi) + K(χe) 3 ρe χe 2E(χe) − 3 2 (20) In equation (20), • E and νe are modulus of elasticity and Poisson’s ratio, given in the input parameter list. • The ratio χi and χe are constant, and are given by solving equation (12) or approximated using equation (13). • K(χ) is constant, and is defined by equation (3) in [1] or approximated using equation (14). • E(χ) is constant, and is defined by equation (4) in [1] or approximated using equation (15). • The curvature sum ρi and ρe are constant, and can be calculated according to equa- tion (8) and equation (9). 2.4.2 Angular Position of rolling element j Assuming the Y axis of the From marker is parallel to the axis direction, then the angular position ϕj can be defined as ϕj = ϕ1 + 2π Z (j − 1) j = 1, 2, ...Z (21a) ϕ1 = 0.5 ∗ mod (βMiMj , 2π) (21b) where we assuming that the initial position of the first ball is on the Y axis of the To marker Mj, and all the balls rotate in a speed, which is half of the inner ring rotation speed. 2.4.3 The total elastic deflection of rolling element j The loci of raceway groove curvature radii centers in the initial state is shown in figure 4. If the outer ring of the bearing is considered fixed in space as the load is applied to the bearing, then the inner ring will be displaced and the locus of inner-ring raceway groove radii centers will also be displaced as shown in figure 5. From figure 5 it can be determined that Aj, the distance between the centers of curvature of the inner- and outer-ring raceway grooves at the position where rolling element j locates, is given by Aj = (A cos α0 + δr cos ϕj)2 + (A sin α0 + δa + Ri sin ψ cos ϕj)2 , (22) In equation (23), 8
  9. 9. • A, constant, is the distance, in millimeters, between raceway groove curvature centers of ball bearing having no clearance and having an initial contact angle, calculated using A = ri + re − Dw. The concept is well explained in [2, 64]. See figure 1(a). • α0, constant, is the initial contact angle, in degrees, α0 = arccos [1 − (s/2A)]. See fig- ure 1(a). • δr, state dependent, is the relative radial displacement of both bearing rings, calculated using equation (5). • ϕj, state dependent, is the angular position of rolling element j, given in equation (21). • δa, state dependent, given in equation (4). • Ri is the distance, in millimeters, between the center of curvature of the inner race groove and the axis of rotation. Ri = Dpw 2 + ri − Dw 2 cos α0. • ψ, state dependent, is the relative angular displacement of both bearing rings given in equation (7). The total elastic deflection of the rolling element j, δj, is given by δj = max 0, Aj − A . (23) 2.4.4 Operating Contact Angle of Rolling elementj The operating contact angle of rolling element j, αj is αj = arctan A sin α0 + δa + Ri sin ψ cos ϕj A cos α0 + δr cos ϕj , (24) where δa, ψ, ϕj, δr are state dependent, and calculated using equation (4), equation (7), equa- tion (21), equation (5). 2.5 Load Force 2.5.1 Radial Load Force The formula of the radial load force is Fr = cP Z j=1 δ 3 2 j cos αj cos ϕjer. (25) In equation (25), • cP is constant w.r.t the type of bearing, and can be calculated using equation (20). 9
  10. 10. ϕj y xe xi ze zi Figure 4: Loci of raceway groove curvature radii centers before applying load. (From Jones, A., Analysis of Stresses and Deflections, New Departure Engineering Data, Bristol, CT, 1946.) Aj ϕj YiY Xi Xi Xe ψ ZiZiZe Figure 5: Loci of raceway groove curvature radii centers after displacement. (From Jones, A., Analysis of Stresses and Deflections, New Departure Engineering Data, Bristol, CT, 1946.) 10
  11. 11. • Z is number of rolling elements, given in the input parameter list. • δj, states dependent, is the elastic defection, in millimeters, of the rolling element j, and can be calculated using equation (23) • αj, states dependent, is the operating contact angle, in degrees, of the rolling element j, and can be calculated using equation (24) • ϕj, states dependent, is the angular position, in degrees, of rolling element j, and can be calculated using equation (21) • er, states dependent, is the unit direction vector of the radial displacement. 2.5.2 Axial Load Force The formula of the axial load force is Fa = cP Z j=1 δ 3 2 j sin αjea. (26) In equation (26), ea is the unit direction vector of the axial displacement, and all other components are the same as described in section 2.5.1. 2.5.3 Load Moment The load moment is in the unit of newton millimeter, and the formula is MX = Dpw 2 cP Z j=1 δ 3 2 j sin αj cos ϕj. (27) The direction of the load moment is pointing to the X axis of the From marker Mi. In equa- tion (26), Dpw is the pitch diameter in millimeters of ball set and given by the parameter list. All other components are the same as described in section 2.5.1. All the forces and moments applied on the outer ring are opposite to the load forces and moment. 3 Roller Bearings 3.1 Lamina Model To commence the analysis, it is assumed that any roller–raceway contact can be divided into (ns, ns ≥ 30) slices or laminae situated in planes parallel to the radial plane of the bearing. It is 11
  12. 12. also assumed that shear effects between these laminae can be neglected owing to the small magni- tudes of the contact deformations that develop. (Only contact deformations are considered.) [3, p.26]. For lamina k of roller j, the load qj,k is qj,k = csδ 10 9 j,k (28) where the spring constant of lamina, cs, for contacting parts made of steel cs = 35948L 8 9 we ns . (29) 3.2 The Elastic Deflection of the Lamina k of the Roller j The elastic deflection, in millimeters, of the lamina k of the roller j may be considered to be composed of three components: • elastic deflection due to radial load at roller azimuth location j: δj ; • displacement due to the profile depth: −2P(yk); • deflection due to bearing misalignment (angular displacement). 3.2.1 Elastic Deflection due to Radial Load: δj For a radial displacement δr, of the inner ring, the elastic deflection of the rolling element j, δj is δj = δr cos ϕj − s 2 (30) where δr and ϕj are state dependent, and s, constant, is the radial operating clearance of bearing. 3.2.2 Profile Function: P(x) The profile function depends on the expertise of the manufacturer. Some standard profile func- tions for different types of rollers are given in [1, p.12,17,18]. 3.2.3 Deflection due to Bearing Misalignment The bearing misalignment angle (angular displacement) ψ is described in figure 6. The angle ψ is positive when rotates clockwise. In figure 7, yk is the distance between center of lamina k and roller center. The red dot in figure 7 denotes the original point of yk. Then the deflection due to bearing misalignment is (−yk tan ψj), where ψj = arctan (tan ψ cos ϕj) (31) 12
  13. 13. ψ ψ ye + − yi Figure 6: Misalignment of cylindrical roller bearing rings. [3] X X Y yk Original point of yk Z Figure 7: Misaligned roller bearing. [3] 13
  14. 14. In summation, the elastic deflection of the lamina k of the roller j is δj,k = max(0, δj − yk tan ψj − 2P(yk)) (32) where δj and ψj are state dependent. 3.3 Load Force Fr = cs Z j=1 cos ϕj ns k=1 δ 10 9 j,k er . (33) where cs, Z and ns are constant, and all other components are state dependent. MX = cs Z j=1 cos ϕj ns k=1 yk δ 10 9 j,k , (34) where cs, Z, ns and yk are constant, and all other components are state dependent. The direction of the load moment is pointing to the X axis of the From marker Mi. All the forces and moments applied on the outer ring are opposite to the load forces and moment. 4 Reference Geometries 4.1 Pitch Diameter Dpw The pitch diameter Dpw is usually approximately equal to the mean of the bore diameter d and the outer diameter of the outer ring D, or to the mean of the inner- and outer-ring raceway contact diameter. Therefore, Dpw ≈ 1 2 (d + D) (35) Dpw ≈ 1 2 (di + de). (36) In most cases, d and D are provided by the manufacturer. If the value of di and de can also be obtained from the manufacturer, equation (36) is preferred to be used to approximate the pitch diameter since it is more precise. 4.2 Point Contact Bearing For the point contact bearings: 14
  15. 15. • deep groove ball bearings, angular contact ball bearings and separable ball bearings, • spherical roller bearings, • self-aligning ball bearings, • thrust ball bearings and thrust angular contact ball bearings, • thrust spherical roller bearings, the calculation method are similar to that present in section 2. In this case, the spring constant cP is calculate via equation (20), and then the cross-sectional raceway groove radii of inner- and outer-ring, ri and re, are two important constant values for the calculation. Usually, ri and re are dependent on the diameter of the rolling element. Two conformity values are used for defining the radii: ri = fiDw (37) re = feDw, (38) where fi and fe are the inner- and outer-race conformity. 4.2.1 Deep Groove Ball Bearings Figure 8: The geometry of deep groove ball bearings. [5] The nominal contact angle, α, of deep groove ball bearings is 0◦ . The input parameters are • (d, D, E, Z, νe, Dw), • s, α = 0◦ , fe = 0.53 and fi = 0.52, where the parameters inside the bracket are the common parameters needed for every type of bearings. The conformities fe and fi are better to be set as input parameters with default 15
  16. 16. values, instead of internal values. Because different standards have suggested slightly some different values. And the initial contact angle α0 can be calculated using equation (1). The bearing diameter clearance s can be given according ISO 5753, 2009 or from the man- ufacturer. 4.2.2 Angular Contact Ball Bearings Figure 9: The geometry of angular contact ball bearings. [5] Angular contact ball bearings are specifically designed to operate under thrust loads, and the clearance built into the unloaded bearing along with the raceway groove curvatures determines the bearing initial (free) contact angle [3, p.63]. That is the initial contact angle of angular contact ball bearings is equal to the nominal contact angle: α0 = α, and should not be calculated using equation (1). Thus, the diametrical clearance s is not needed. The input parameters are • (d, D, E, Z, νe, Dw), • α, fe = 0.53 and fi = 0.52. And the standard nominal contact angle α0 are 15◦ , 25◦ and 40◦ [5, p.11]. 4.2.3 Spherical Roller Bearings (radial) In figure 10, Dw is the maximal roller diameter. For the radial spherical roller bearings, the roller diameter at the contact point, Dwe, is equal to Dw: Dwe = Dw . (39) 16
  17. 17. Figure 10: The geometry of spherical roller bearings. [5] For the axial spherical roller bearings, Dwe = Dw. Then in this case, Dwe should be deter- mined according the contact angle and other geometry constraints, but this is not included in [1]. The input parameter for spherical roller bearings (radial) are • (d, D, E, Z, νe, Dw), • α and fp = 0.97. The inner- and outer-race conformities are not input parameters, since for this type, re and ri are calculated using re = Dpw 2 cos α + Dwe 2 (40) ri = re . (41) For spherical roller bearings, in order to avoid edge stress, rollers may be crowned as shown in figure 10. The stress distribution is thereby made more uniform depending on the applied load [2, p.144]. The conformity of roller crown radius, fp, is needed for defining the rollers: Rp = fpre . (42) Then the equations for calculating curvature sum and relative curvature difference, 8, 9, 10 and 11, should be updated by replacing the radius in plane Y from 0.5Dwe to Rp: ρi = 2 Dwe 1 1 − γ + 1 Rp − 1 ri (43) 17
  18. 18. ρe = 2 Dwe 1 1 + γ + 1 Rp − 1 re (44) Fi(ρ) = 2 Dwe 1 1−γ − 1 Rp + 1 ri 2 Dwe 1 1−γ + 1 Rp − 1 ri (45) Fe(ρ) = 2 Dwe 1 1+γ − 1 Rp + 1 re 2 Dwe 1 1+γ + 1 Rp − 1 re (46) where γ = Dwe cos α/Dpw and α is the nominal contact angle, i.e. the contact angle of the load-free bearing. 4.2.4 Self-aligning Ball Bearings(single row) Figure 11: The geometry of self-aligning ball bearings. [5] The input parameters for self-aligning ball bearings(single row) are • (d, D, E, Z, νe, Dw), • α = 0◦ , fi = 0.53. The outer-race conformity is not input parameter, since for this type, re is calculated using re = 0.5 1 + 1 γ Dw . (47) 18
  19. 19. 4.2.5 Thrust Ball Bearings Figure 12: The geometry of thrust ball bearings. [5] The input parameters are • (d, D, E, Z, νe, Dw), • s, α = 90◦ , fe = 0.54 and fi = 0.54. 4.2.6 Thrust Angular Contact Ball Bearings The description of geometry for thrust angular contact bearings is the same as for angular contact ball bearings. The only difference in geometry is a value of 0.54 for the conformity, instead of the 0.52 and 0.53 used for angular contact ball bearings. The input parameters are • (d, D, E, Z, νe, Dw), • s, α, fe = 0.54 and fi = 0.54. Standard thrust angular contact ball bearings have a contact angle of 60◦ . 4.3 Line Contact Bearing For the line contact bearings: 19
  20. 20. Figure 13: The geometry of thrust angular contact ball bearings. [5] • cylindrical roller bearings and needle roller bearings, α = 0◦ • tapered roller bearings, • thrust cylindrical roller bearings and thrust needle roller bearings, α = 90◦ • thrust tapered roller bearings, the calculation method are similar to that present in section 3. In this case, the spring constant cs, given in equation (29), is not calculate pure theoretically, but also base on laboratory testing. But different crown profile functions are applied for each type. For the detail formulas, please refer to [1, p.17]. The input parameters are • (d, D, E, Z, νe, Dw), • s, α, Lwe, ns ≥ 30, P(x). In addition to the parameters used for ball bearings, the effective length of the roller Lwe is a required input parameter. The effective length is a little smaller than the length of the roller because of radii at the end of the roller. The roller diameter at contact point, Dwe should be determined from the geometry and kinematic constraints. The number of laminae ns should be at lest 30. 20
  21. 21. 4.3.1 Cylindrical Roller Bearings and Needle Roller Bearings Cylindrical roller bearings and needle roller bearings are similar, their nominal contact angle α = 0◦ . In figure 14(a) and figure 14(b), Dw is the (maximal) roller diameter and equals to the roller diameter at contact point Dwe. The standard profile functions, P(x), for these two types of bearings are defined by (eq.42 44) in [1], replacing xk by yk. (a) The geometry of cylindrical roller bearings. (b) The geometry of needle roller bearings. 4.3.2 Tapered Roller Bearings Figure 14: The geometry of tapered roller bearings. [5] 21
  22. 22. Tapered roller bearings use a conical roller instead of a cylindrical roller. The input roller diameter, Dw, is given for the middle of the roller and also the pitch diameter Dpw is defined for the middle of the rollers. The force is assumed to be at the medium height of the inner ring rim.[5, p.18] So the roller diameter at contact point, Dwe is equal to Dw. The standard profile functions, P(x), is defined by (eq. 77) in [1], replacing xk by yk. 4.3.3 Thrust Cylindrical Roller Bearings and Thrust Needle Roller Bearings Figure 15: The geometry of thrust cylindrical roller bearings. [5] Axial cylindrical roller bearings have a nominal contact angle of 90◦ . They only allow axial forces and bending moments. No radial forces can be applied. The clearance s in this case represents the axial clearance. 4.3.4 Thrust Tapered Roller Bearings The standard profile functions, P(x), is defined by (eq. 85) in [1], replacing xk by yk. References [1] ISO/TS 16281, Rolling bearings — Methods for calculating the modified reference rating life for universally loaded bearings. International Organization for Standardization: Switzerland, First Edition, 2008. [2] Harris, T.A. and Kotzalas, M.N., Essential Concepts of Bearing Technology. Taylor & Francis, Fifth Edition, 2006. 22
  23. 23. [3] Tedric A. Harris, Michael N. Kotzalas, Advanced Concepts of Bearing Technology,: Rolling Bearing Analysis. CRC Press, Fifth Edition, 2006. [4] Brewe, D. and Hamrock, B., Simplified solution for elliptical-contact deformation between two elastic solids. ASME Trans. J. Lub. Tech., 101(2), 231–239, 1977. [5] MESYS AG, MESYS Rolling Bearing Calculation. 2014. 23

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