Tiêu chuẩn iso tr 01281 1 2008

Microsoft Word C043028e doc Reference number ISO/TR 1281 1 2008(E) © ISO 2008 TECHNICAL REPORT ISO/TR 1281 1 First edition 2008 12 01 Rolling bearings — Explanatory notes on ISO 281 — Part 1 Basic dyn[.]

Basic dynamic radial load rating, C r , for radial ball bearings

According to Hertzian theory, the maximum orthogonal subsurface shear stress (τ₀) and the corresponding depth (z₀) can be described in relation to the applied radial load (Fᵣ), such as the maximum rolling element load (Qₘₐₓ) or the maximum contact stress (σₘₐₓ) These relationships also incorporate the dimensions of the contact area between the rolling element and raceways Specifically, the maximum shear stress and depth are expressed as τ₀ = ζb and z₀ = ζb, linking contact mechanics to load conditions in bearing design.

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⎝ ∑ ⎠ where σ max is the maximum contact stress; t is the auxiliary parameter; a is the semimajor axis of the projected contact ellipse; b is the semiminor axis of the projected contact ellipse;

Q is the normal force between a rolling element and the raceways;

E o is the modulus of elasticity; Σρ is the curvature sum; à, v are factors introduced by Hertz

For a given rolling bearing, parameters such as τ_o, a, l, and z_o can be expressed based on bearing geometry, applied load, and revolutions By incorporating a constant of proportionality into the correlation, the original equation is modified When specifying a certain number of revolutions (e.g., 1,000,000) and a desired reliability level (e.g., 90%), the equation can be solved to determine the rolling element load corresponding to the basic dynamic load rating for point contact rolling bearings, introducing the constant of proportionality, A_1.

Q C is the rolling element load for the basic dynamic load rating of the bearing;

D w is the ball diameter; γ is D w cos α/D pw ; in which

D pw is the pitch diameter of the ball set, α is the nominal contact angle;

Z is the number of balls per row

The basic dynamic radial load rating, C 1 , of a rotating ring is given by:

The basic dynamic radial load rating, C 2 , of a stationary ring is given by:

1 is the rolling element load for the basic dynamic load rating of a ring rotating relative to the applied load;

2 is the rolling element load for the basic dynamic load rating of a ring stationary relative to the applied load;

J r = J r (0,5) is the radial load integral (see Table 3);

J 1 = J 1 (0,5) is the factor relating mean equivalent load on a rotating ring to Q max (see Table 3);

J 2 = J 2 (0,5) is the factor relating mean equivalent load on a stationary ring to Q max (see Table 3)

The relationship between C r for an entire radial ball bearing, and C 1 and C 2 , is expressed in terms of the product law of probability as: c h c h

Substituting Equations (6), (7) and (8) into Equation (9), the basic dynamic radial load rating, C r , for an entire ball bearing is expressed as:

The constant "A 1" is an experimentally determined proportionality factor that plays a crucial role in analyzing bearing specifications The parameter "r_i" refers to the cross-sectional groove radius of the inner ring's raceway, while "r_e" denotes the groove radius of the outer ring's raceway Additionally, "i" indicates the number of ball rows within the bearing, which influences load distribution and performance Understanding these key parameters is essential for precise bearing design and optimization in mechanical applications.

The contact angle, α, along with the number of rolling elements (Z) and the diameter (Dₙ), are key factors influenced by bearing design Additionally, the ratios of the raceway groove radii, rᵢ and rₑ, to the half-diameter of the rolling element are critical parameters that affect bearing performance and durability Proper selection and optimization of these design elements ensure efficient load distribution and smooth operation of rolling-element bearings.

In practice, it is beneficial to designate the initial terms on the right-hand side of Equation (10) as a factor, f_c, since the variables (ball), D_w/2, and γ = D_w cosα / D_pw are non-dimensional This simplifies analysis and calculation, enhancing clarity and efficiency in engineering applications Using non-dimensional parameters like f_c aligns with best SEO practices by targeting relevant keywords such as "non-dimensional parameters," "engineering analysis," and "efficient calculation."

Radial ball bearings require careful consideration of manufacturing faults, which are addressed by introducing a reduction factor, λ, to adjust their basic dynamic radial load rating from its theoretical value This reduction factor, λ, is conveniently incorporated into the factor f c, simplifying calculations The value of λ is determined through experimental methods to ensure accurate assessment of bearing performance.

Consequently, the factor f c is given by:

Based on References [1] and [2], the following values were assigned to the experimental constants in the load rating equations: e = 10/9 c = 31/3 h = 7/3

Substituting numerical values into Equation (11) indicates that the load rating is proportional to D_w^1.8 for small balls up to about 25 mm in diameter, based on limited test results For larger balls exceeding 25 mm in diameter, the load rating increases at a slower rate, and an approximate relationship of D_w^1.4 can be assumed, reflecting the different scaling behavior for bigger ball sizes.

Values of f c in ISO 281:2007, Table 2, are calculated by substituting raceway groove radii and reduction factors given in Table 1 into Equation (15)

The value for 0,089A 1 is 98,066 5 to calculate C r in newtons.

Basic dynamic axial load rating, C a , for single row thrust ball bearings

4.2.1 Thrust ball bearings with contact angle α ≠ 90°

As in 4.1, for thrust ball bearings with contact angle α ≠ 90°:

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In thrust ball bearings, the basic dynamic axial load rating must be adjusted to account for load distribution among the rolling elements This adjustment includes applying a reduction factor, η, which accounts for the unequal load sharing Additionally, a reduction factor, λ, originally used for radial ball bearings, is also considered to refine the load rating for thrust bearings, ensuring accurate performance assessments.

Consequently, the factor f c is given by:

Similarly, to take the effect of ball size into account, substitute experimental constants e = 10/9, c = 31/3, and h = 7/3 into Equations (16) and 17) to give:

The value for 0,089A 1 is 98,066 5 to calculate C a in newtons Values of f c in ISO 281:2007, Table 4, rightmost column, are calculated by substituting raceway groove radii and reduction factors given in Table 1 into

4.2.2 Thrust ball bearings with contact angle α = 90°

As in 4.1, for thrust ball bearings with contact angle α = 90°:

Similarly, to take the effect of ball size into account, substitute experimental constants e = 10/9, c = 31/3, and h = 7/3 into Equations (21) and (22), to give:

To calculate the value for 0.089A 1, use 98.0665 N to determine C a in newtons According to ISO 281:2007, Table 4, second column from the left, the values of f c are obtained by substituting raceway groove radii and reduction factors, which are provided within the standard.

Basic dynamic axial load rating, C a , for thrust ball bearings with two or more rows of

According to the product law of probability, the relationships between the basic axial load rating of an entire thrust ball bearing and those of both the rotating and stationary rings are established This fundamental principle helps in understanding how load capacities are distributed within the bearing components Proper comprehension of these relationships is essential for selecting the right thrust ball bearing for specific applications, ensuring optimal performance and longevity.

C ak is the basic dynamic axial load rating as a row k of an entire thrust ball bearing;

C a1k is the basic dynamic axial load rating as a row k of the rotating ring of an entire thrust ball bearing;

C a2k is the basic dynamic axial load rating as a row k of the stationary ring of an entire thrust ball bearing;

C a is the basic dynamic axial load rating of an entire thrust ball bearing;

C a1 is the basic dynamic axial load rating of the rotating ring of an entire thrust ball bearing;

C a2 is the basic dynamic axial load rating of the stationary ring of an entire thrust ball bearing;

Z k is the number of balls per row k

Substituting Equations (26), (27), and (29) into Equation (28), and rearranging, gives:

Substituting experimental constants c = 31/3 and h = 7/3 gives:

The load ratings C a1 , C a2 , C a3 … C an for the rows with Z 1 , Z 2 , Z 3 … Z n balls are calculated from the appropriate single row thrust ball bearing equation in 4.2.

Basic dynamic radial load rating, C r , for radial roller bearings

Using a procedure similar to that for deriving Equation (10) for point contact in section 4.1, but incorporating Equations (4) and (5), the basic dynamic radial load rating of radial roller bearings with line contact can be accurately determined.

B 1 is an experimentally determined proportionality constant; γ is

D we cos α /D pw in which D pw is the pitch diameter of roller set;

D we is the mean roller diameter; α is the nominal contact angle;

L we is the effective contact length of roller; i is the number of rows of rollers;

Z is the number of rollers per row

Here, the contact angle, α, the number of rollers, Z, the mean diameter, D we , and the effective contact length,

The design of bearing systems relies heavily on precise calculations, with the dimensionless term γ = D_we cos α / D_pw playing a crucial role in this process Since γ is non-dimensional, it is practical to categorize the components up to “i L_we …” in Equation (31) as a specific factor, denoted as f_c This approach simplifies the analysis and enhances the applicability of the formula in real-world bearing design scenarios.

The basic dynamic radial load rating for radial roller bearings is adjusted to account for stress concentrations such as edge loading Additionally, a constant life formula exponent is used instead of a varying one to ensure accurate life estimation.

Adjustment for stress concentration is represented by a reduction factor, λ, while exponent variation is accounted for by a factor, ν Both factors are experimentally determined and incorporated into the combined factor, f_c, which simplifies the calculation process The overall factor, f_c, effectively captures the influence of stress concentration and exponent variation on the material's behavior, ensuring accurate and reliable results in analysis and design.

The Weibull slope, e, and the constants, c and h, are determined experimentally Based on References [1] and

[2] and subsequent verification tests with spherical, cylindrical, and tapered roller bearings, the following values were assigned to the experimental constants in the rating equations:

7 h= 3 Substituting experimental constants e = 9/8, c = 31/3, and h = 7/3 into Equations (32) and (33),

The value for 0,483B 1 is 551,133 73 to calculate C r in newtons Values of f c in ISO 281:2007, Table 7, are calculated by substituting the reduction factor given in Table 2 into Equation (35)

Basic dynamic axial load rating, C a , for single row thrust roller bearings

4.5.1 Thrust roller bearings with contact α ≠ 90°

In thrust roller bearings, the basic dynamic axial load rating must be adjusted downward to account for uneven load distribution among the rolling elements Additionally, a reduction factor, η, similar to the λ factor used in radial roller bearing load ratings, is introduced to accurately reflect the bearing's load-carrying capacity.

Consequently, the factor f c is given by: c h c h c h c h c h e c h e c h c h c h c h e c h f B

The value for 0,483B 1 is 551,133 73 to calculate C a in newtons Values for f c in ISO 281:2007, Table 10, second column from left, are calculated by substituting reduction factors given in Table 2 into Equation (39)

4.5.2 Thrust roller bearings with contact angle α = 90°

The value for 0,41B 1 is 472,453 88 to calculate C a in newtons Values of f c in ISO 281:2007, Table 10, second column from left, are calculated by substituting reduction factors given in Table 2 into Equation (43)

Basic dynamic axial load rating, C a , for thrust roller bearings with two or more rows of

According to the probability product law, the relationship between the fundamental dynamic axial load rating of a thrust roller bearing and those of its rotating and stationary rings is delineated, providing essential insights into bearing performance and design.

C ak is the basic dynamic axial load rating as a row k of an entire thrust roller bearing;

C a1k is the basic dynamic axial load rating as a row k of the rotating ring of an entire thrust roller bearing;

C a2k is the basic dynamic axial load rating as a row k of the stationary ring of an entire thrust roller bearing;

C a is the basic dynamic axial load rating of an entire thrust roller bearing;

C a1 is the basic dynamic axial load rating of the rotating ring of an entire thrust roller bearing;

C a2 is the basic dynamic axial load rating of the stationary ring of an entire thrust roller bearing;

Z k is the number of rollers per row k

Substituting Equations (44), (45), and (47) into Equation (46), and rearranging, gives:

( 1)/2 ( 1) we we sin sin sin sin

The load ratings, C a1 , C a2 , C a3 … C an for the rows with Z 1 , Z 2 , Z 3 … Z n rollers of lengths L we1 , L we2 ,

L we3 … L wen , are calculated from the appropriate single row thrust roller bearing equation in 4.2

Table 1 — Raceway groove radius and reduction factor for ball bearings

Raceway groove radius Reduction factor

Single row radial contact groove ball bearings

Single and double row angular contact groove ball bearings

Double row radial contact groove ball bearings 0,52 D w 0,90 —

Single and double row self- aligning ball bearings 0,53 D w 0 5, 1 1 D w γ

Single row radial contact separable ball bearings (magneto bearings)

NOTE Values of f c in ISO 281:2007, Tables 2 and 4, are calculated by substituting raceway groove radii and reduction factors in this table into Equations (15), (20), and (25), respectively.

Table 2 — Reduction factor for roller bearings

Table No in ISO 281:2007 Bearing type λν η

NOTE Values of f c in ISO 281:2007, Tables 7 and 10, are calculated by substituting reduction factors in this table into Equations (35), (39), and (51), respectively.

Expressions for dynamic equivalent load

5.1.1 Theoretical dynamic equivalent radial load, P r , for single row radial bearings

In a single row radial bearing, the life of the bearing ring is primarily influenced by the mean values of the rolling element loads When the indices 1 and 2 are assigned to the rotating ring and stationary ring respectively, these load values are crucial for determining the bearing's durability Understanding the relationship between ring rotation and load distribution helps optimize bearing performance and longevity Proper assignment of ring indices ensures accurate assessment of load conditions, which is essential for maintaining reliable operation and extending the service life of radial bearings.

2 a r 1 1 max 1 r a a r 2 2 max 2 r a cos sin cos sin

Q max is the maximum rolling element load;

J r is the radial load integral;

J a is the axial load integral;

Z is the number of rolling elements; α is the nominal contact angle

Radial and axial load integrals are given by:

This article discusses key parameters influencing bearing contact mechanics, including the contact type and zone Specifically, for point contact, the contact exponent t equals 3/2, while for line contact, t is 1.1 The parameter ϕ₀ represents half of the loaded arc, providing insights into the extent of load distribution along the contact surface Additionally, ε is introduced as a parameter that indicates the width of the loaded zone within the bearing, which is crucial for analyzing load capacity and wear behavior Understanding these parameters helps optimize bearing design and performance under various operating conditions.

(52) for point and line contact respectively

If P r1 and P r2 are the dynamic equivalent radial loads for the respective rings, then with radial displacement of the rings (ε=0 5),

= = (53) where the values J 1 (0,5), J 2 (0,5) and J r (0,5) are given in Table 3

C r is the basic dynamic radial load rating;

C 1 is the basic dynamic radial load rating of a rotating ring;

C 2 is the basic dynamic radial load rating of a stationary ring; w is equal to pe, where p is the exponent on life formula and e is the Weibull slope

Point contact Line contact Point and line contact

Single Double Single Double Single Double

Quantity row bearing row bearing row bearing

For radial displacement of the bearing rings with ε = 0.5 and a fixed outer ring load, the basic dynamic load ratings are denoted as C₁ = Cᵢ for the inner ring and C₂ = Cₑ for the outer ring Equation (54) is used to calculate the displacement, incorporating the parameters rᵣ, aᵣ, and their respective values This analysis helps in assessing bearing performance under specified load conditions, ensuring optimal design and reliability of rotating machinery.

(55) for point and line contact respectively

For ε = 0,5 and fixed inner ring load (C 1 = V f C e ; C 2 = C i /V f ), is found r f r

P =V F (56) where V f is the rotation factor

The factor V f varies between 1 ± 0,044 and 1 ± 0,038 for point and line contact respectively In ISO 281:2007, the rotation factor V f has been deleted

NOTE The value of 1,2 for the rotation factor V f was given in ISO/R281 for radial bearings, except self-aligning ball bearings, as safety factor

For axial displacement of the bearing rings (ε = ∞) and fixed outer ring load (C 1 = C i , C 2 = C e ), r a i r

The factor f 1 (C i /C e ) varies between 1 and 1/V f = J 1 (0,5)/J 2 (0,5) Introducing as a good approximation the geometric mean value 1/ V f between these two values (see Table 3),

For non-self-aligning bearings, consideration has to be given to the effect of the manufacturing precision on the factor, Y

The value of Y given in Equation (58) is corrected by the reduction factor η

For combined loads, Equation (54) gives related values of F r /P r and F r cot α/P r corresponding to the curves given in Figure 1 for the limiting cases C 1 /C 2 ≈ 0 and C 2 /C 1 ≈ 0

The points A represent ε = 0,5, i.e radial displacement of the bearing rings For these points, a r a r

= ⎬⎭ (60) for point and line contact, respectively a) Point contact b) Line contact

C 1 basic dynamic radial load rating of a rotating ring

C 2 basic dynamic radial load rating of a stationary ring

P r dynamic equivalent radial load for radial bearing α nominal contact angle

Figure 1 — Dynamic equivalent radial load, P r , for single row radial bearings with constant contact angle, α

5.1.2 Theoretical dynamic equivalent radial load, P r , for double row radial bearings

In double row radial bearings, indices I and II are assigned to each respective row to facilitate identification and analysis The lifespan of both the rotating and stationary rings primarily depends on their mean operational values, which serve as crucial determinants for bearing durability and performance Ensuring accurate assessment of these mean values is essential for optimizing bearing longevity and maintaining reliable machinery operation.

For a bearing without internal clearances,

For double row bearings, the equivalent bearing load can be calculated using Equation (54), similar to single row bearings, by incorporating the values of J r, J a, J 1, and J 2 Specifically, J r (0.5), J a (0.5), and J 1 (0.5) are used, which correspond to the parameters when ε I = ε II = 0.5, as detailed in Table 3.

The bent curves given in Figure 2 are found for the limiting cases C 1 /C 2 ≈ 0 and C 2 /C 1 ≈ 0

Both rows are loaded if ε I < 1, i.e if a r a r

< ⎬⎭ (64) for point and line contact, respectively

When the value of F_a exceeds a certain threshold, only one row of the bearing is loaded In such cases, the life of double row bearings can be accurately calculated using the principles of single row bearing theories, ensuring precise lifecycle predictions under specific loading conditions.

If P rI is the equivalent radial load for the loaded row considered as a single row bearing and P r is the equivalent load for the double row bearing,

Figures 1 and 2 are calculated on the assumption of a constant contact angle Figures 1 a) and 2 a) are also approximately applicable to angular contact groove ball bearings, if cot α′ is determined from Equation (66):

D w is the ball diameter a) Point contact b) Line contact

P rI dynamic equivalent radial load for the loaded row considered as a single row bearing α nominal contact angle

Figure 2 — Dynamic equivalent radial load, P rI , for double row bearings with constant contact angle, α

5.1.3 Theoretical dynamic equivalent radial load, P r , for radial contact groove ball bearings

Figure 3 is applicable to radial contact groove ball bearings The curve AC has been determined from

Equation (54) and the approximate equation

′≈⎢⎣ − ⎥⎦ ⎜⎝ − ⎟⎠ ⎜⎜⎝ ⎟⎟⎠ (67) and gives the functional relationship between F r /P r and F a cot a′/P r where α′ is the contact angle calculated from Equation (68) (Reference [1])

Equation (68) is obtained from Equation (67) for a centric axial load F a = F r = 0, i.e ε = ∞ and J a = 1

P r dynamic equivalent radial load for radial bearing α′ contact angle calculated from Equation (68)

Figure 3 — Dynamic equivalent radial load, P r , for radial contact groove ball bearings

5.1.4 Practical expressions for dynamic equivalent radial load, P r , for radial bearings with constant contact angle

From a practical standpoint, it is preferable to replace the theoretical curves in Figures 1 and 2 by broken lines

A 1 BC for single row bearings and ABC for double row bearings, as in Figure 4

A, A 1 , B, C points a intercept of line BC on abscissa (x-co-ordinate of point C) b intercept of line AB on ordinate (y-co-ordinate of point A)

P rI dynamic equivalent radial load for radial bearing, row I

Y I axial load factor for radial bearing, row I α nominal contact angle ξ value of F a cot α/P rI at the x-co-ordinate of point B

Figure 4 — Dynamic equivalent radial load, P rI , for radial bearings with constant contact angle, α The equation for the straight line A 1 B in Figure 4 is r rI

P Therefore, for F a /F r u ξ tan α, we have

P rI = F r (69) and the straight line passing through the points B (ξ, 1) and C (a, 0) is given by r rI a rl

From this equation, for F a /F r > ξ tan α, it follows that rI r a 1 r 1 a

For the double row bearings, the equation for the straight line AB is

Therefore, for F a /F r u ξ tan α, it follows that

Further, from Equation (70), which represents straight line BC, we find for F a /F r > ξ tan α

Integrating the above, Table 4 shows expressions of dynamic equivalent radial load, P r , and of factors X and Y, for radial bearings with constant contact angle, α

Table 4 — Expressions for dynamic equivalent radial load, P r , and factors X and Y for radial bearings with constant contact angle, α

Single row bearings Double row bearings a r

5.1.5 Practical expressions for dynamic equivalent radial load, P r , for radial ball bearings

The contact angle of radial ball bearings typically varies with the applied load However, Table 4 can approximately be used for angular contact groove ball bearings by substituting the contact angle α with the axial load-specific contact angle α′, as described in Equation (66) This adjustment allows for accurate analysis of bearing behavior under axial loads, ensuring proper selection and performance evaluation.

For single row and double row radial contact groove ball bearings, the theoretical curve in Figure 3 is replaced by the broken line A 1 BC in Figure 5

A 1 , B, C points a intercept of line BC on abscissa (x-co-ordinate of point C)

P r dynamic equivalent radial load for radial bearing α′ contact angle calculated from Equation (68) ξ value of F a cot α/P rI at point B (and its x-co-ordinate)

Figure 5 — Dynamic equivalent radial load, P r , for radial contact groove ball bearings

For this type of bearing,

For self-aligning ball bearings, the contact angle can be considered as independent of the load (α′ = α); also η can be assumed to be unity

5.1.6 Practical expressions for dynamic equivalent axial load, P a , for thrust bearings

Radial (X a) and axial (Y a) load factors for single and double direction bearings with an inclination angle α not equal to 90° are derived based on the dynamic equivalent radial load (P r) These factors are essential for accurately assessing bearing performance and are determined from specific expressions tailored to single-row and double-row radial bearings Incorporating these factors enhances the precision of load analysis and supports optimized bearing design.

That is, for single direction bearings, when F a /F r > ξ tan α

(75) and for double direction bearings, when F a /F r > ξ tan α, a 2 r a a2 r a2 a

Integrating the above, Table 5 shows expressions for dynamic equivalent axial load, P a , for thrust bearings and factors X a and Y a

Table 5 — Expressions for dynamic equivalent axial load, P a , and factors X a and Y a for thrust bearings

Single direction bearings Double direction bearings a r

Factors X, Y, and e

Based on initial test results, single row radial contact groove ball bearings have a ξ value of 1.2, while other bearings show a ξ value of 1.5, aligning closely with theoretical curves However, subsequent testing has provided updated insights into these values.

ISO/R281 took values of ξ = 1,05 for radial contact groove ball bearings and single row angular contact groove types with α = 5°; ξ = 1,25 for other angular contact groove types; and ξ = 1,5 for self-aligning types

The reduction factor, η, depends on the contact angle, α, and is given by

Based on experience and preliminary tests, Reference [1] gives k = 0,4 and Reference [2] k = 0,15 to 0,33 In

ISO/R281, k = 0,4 (= 1/2,5) was used for radial contact groove bearings (α = 5°) and angular contact groove bearings with α = 5°, 10° and 15° and k = (1/2,75) is used for angular contact groove bearings with α = 20° to

NOTE ISO/R281 did not include factors for bearings with α = 45° Factors for this angle are specified in

Radial contact groove ball bearings and angular contact groove bearings with a nominal contact angle of 15° experience significant variations in the actual contact angle depending on the load According to ISO 281:2007, Table 3, all relevant factors are provided as functions of the relative axial load, ensuring accurate performance assessment under different loading conditions.

The values of contact angle, α′, under an axial load, F a , can be calculated from

⎝ ⎠ ⎣ ⎦ (79) for radial contact groove ball bearings (considering them as angular contact groove bearings with a nominal contact angle, α = 5°), and from Equation (66) for angular contact groove bearings with a nominal contact angle, α

For 2r/D w = 1,035, c = 0,000 438 71 is given, with units in newtons and millimetres

Table 6 shows the values of contact angle α′ calculated from Equations (66) and (79) for 2r/D w = 1,035

For angular contact groove ball bearings with a 20° contact angle (αW 20°), the effect of axial load on the contact angle is relatively minimal Consequently, ISO 281:2007, Table 5, provides a single set of X, Y, and e factors for each contact angle, simplifying load calculations and enhancing reliability.

With regard to the calculation rules applied to these bearings, see 5.2.2.3

Table 6 — Values of contact angle α′ for radial and angular contact groove ball bearings

1 000 6,894 8 23,263° 24,444° 26,360° a For radial contact groove bearings F i Z D a / 2 w b 1 MPa = 1 N/mm 2

5.2.2 Values of X , Y , and e for each type of radial ball bearing

Integrating the above, methods of calculating values of X, Y, and e are as follows (see Tables 10 and 11)

5.2.2.1 Radial contact groove ball bearings

The Y1 value, initially calculated as 0.964 (approximately 0.96) at F i Z D a / w² = 6.89 MPa, is adjusted to 1.00 based on its relationship with the Y1 value for an angular contact groove type with αW 20°, as shown in Figure 6 The derived contact angle, α' = 23.262°, is refined to 22.512° (α' = tan⁻¹ 0.41445) Consequently, the initial e value of 0.45142 (approximately 0.45) is modified to 0.4352 (approximately 0.44), using the formula e = 1.05 tan 22.512° or 0.41 × 1.05 / (1 - 0.41 sin 5°).

5.2.2.2 Angular contact groove ball bearings with αu 15°

For single row bearings with α = 5°, the values of X 1 , Y 1 , and e are the same as those for the radial contact groove type above

For double row bearings with α = 5°

= = ′ For F ZD a / w 2 = 6,89 MPa, the contact angle, α′, of 22,512° is used Therefore,

F a axial load i number of rows of balls or rollers

Z number of balls or rollers per row α nominal contact angle a Calculated values b Adjusted values c Values given in TC 4 N36 d Include factor i for radial contact groove bearings

Figure 6 — Adjustment of Y 1 values for radial and angular contact groove ball bearings

Namely, for α = 10°, X 1 = 0,462 7 ≈ 0,46, Y 1 = 0,429 86 cot α′, and for α = 15°, X 1 = 0,442 3 ≈ 0,44,

For the above-stated reason, in the case of the calculated value of Y 1 being less than 1, Y 1 has to be set equal to 1,00 (see Figure 6) Therefore, we have for F i Z D a / w 2 = 6,89 MPa

− and also for F ZD a / w 2 = 5,17 MPaand F ZD a / w 2 = 6,89 MPa

5.2.2.3 Angular contact groove ball bearings with α = 20° to α = 45°

Table 7 — Values of X 1 for bearings with α = 20° to α = 45° α X 1

For values of Y 1 , in principle, the values in Table 8, taken from document ISO/TC 4 N36

(= TC 4 N56 = TC 4 N110), are used (see Note), where the first and second values are adjusted, in consideration of the relationship with the values of Y 1 for αu 15° (see Figure 6)

Then values of Y 2 , e, and Y 3 are calculated from Equations (80) which are obtained from Equations (73):

NOTE The values of Y 1 are obtained from Equation (81):

1 (1/3) sin (81) where the values of contact angle α′ are determined by the equation cosα′ = cos 0,972 402α

The relation between α and α′ is obtained from Equation (66) for e i w w w c

= + = + and the rolling body load, in megapascals, is given by a w 2

Z D α′= This is equivalent to 0,5 kgf/mm 2

For bearings with an inclination angle of α = 45°, not covered in ISO/R281, a Y₁ value of approximately 0.50 is determined using Equation (81) This value is specified in ISO 281-1:1977 alongside other factors such as Y₂, Y₃, and e, which are calculated from Equation (80).

5.2.3 Tabulation of factors X, Y , and e for radial ball bearings

Table 10 summarizes the basic equations for calculating the factors X, Y, and e as well as α′, ξ, and η values for each type of radial ball bearing

Table 10 — Summary of factors X, Y and e for radial ball bearings Single row bearings Double row bearings a r

F e F>Bearing type X 1 Y 1 X 3 Y 3 X 2 Y 2 e α′ξη Radial contact groove bearings 0 04 1,ξ η−04 cot100, ,α η′WDetermined from Equation (1)a 1,05 α 5°

1,05 (for single row) 1,25 (for double row) sin5° 1 25,− 10° 15°

′Wtan04,ξ ξα η′u Determined from Equation (2)b sin 1 25,α − 20° 1,00c 25° 0,87c 30° 0,76c 35° 0,66c 40° 0,57c

Angular contact gro ove bear ings

The value of Y α ′′ is determined by the equation Y α ′′ = −1.04 cot(π/1.3 s), where α ′ is derived from the relation cos α ′ = cos α₀ Notably, the Y₁ values of 1.04 and 0.89, obtained for α = 20° and 25°, are adjusted to 1.00 and 0.87, respectively, to ensure a smooth fit of the data for angles less than 20°.

5.2.4 Calculated values Y and e different from standard

Table 11 shows the Y and e values calculated using values of contact angle α  ′ given in Table 6, and which differ from the values given in ISO 281:2007, Table 2 The maximum discrepancy is within ± 0,02

The slight variation observed is attributed to the values of factors Y and e, which are associated with the contact angle α′ However, direct calculation of α′ is not possible using the provided values of F ZD a / w 2 or F i ZD a / w 2, as outlined in Equations (66) and (79).

Therefore, the discrepancies are thought to arise from the inaccuracy of the calculated values of contact angles α′

Table 11 — Calculated values of Y and e different from ISO 281:2007, Table 2 (αu 15°) a/ w 2

Angular contact groove bearings α = 15° e 0,45 0,54 b b a For radial contact groove bearings, F iZD a / w 2 b Adjusted values

From Tables 3, 4, and 5, fundamental equations that determine factors X, Y, and e are obtained, as shown in

Table 12 — Fundamental equations for factors X , Y , and e

Variable ξ is taken to have the same value of 1,25 as for angular contact groove ball bearings, and η is taken as 1 − (1/3)sin α

As variation of contact angle with the axial load, F a , is very small in thrust ball bearings with large contact angle, the nominal contact angle α can be used

Integrating the above, values of X a , Y a , and e are calculated as follows: a1 a2 a3 a1 a2 a3

The values of factors X, Y, and e for α = 45° to α = 85° in ISO 281:2007, Table 5, are calculated from

Equation (82) ISO/R281 prescribed the values only for α = 45°, α = 60°, and α = 75°; a Y a3 value of 0,54 for α = 60° was found to be incorrect and the amended value of 0,55 appears in ISO 281:2007

For radial roller bearings, ξ = 1,5 and η = 1 − 0,15 sin α are used (Reference[2])

Radial roller bearings feature three distinct types of contact between the rollers and bearing rings: point contact against both rings, line contact against both rings, and a combination of line contact on one ring with point contact on the other Understanding these contact types is essential for optimizing bearing performance and longevity Proper selection of bearing type based on contact characteristics can enhance operational efficiency in various mechanical applications.

Table 13 shows the values of factors X, Y, and e for the three different cases of contacts and for contact angles α = 0°, α = 20°, and α = 40° They are calculated from Tables 3 and 4

Table 13 — Calculated values of X, Y , and e for bearings with a) point contact, b) line contact and c) line and points contacts α Bearing contact type 1

The average values of the factors are summarized in the penultimate row of Table 13, providing essential data for practical application To enhance usability, these mean values are rounded accordingly and listed below the table Notably, the value of X2 has been adjusted from 0.65 to 0.67 to reflect the relationship between Y1 and Y2.

As in the case of radial roller bearings, ξ = 1,5 and η = 1 − 0,15 sin α are used

According to Table 5 and Equations (83) a1 a1 a2 a2 a3 a3 tan 1 tan 1

The basic rating life of rolling bearings refers to the duration at which a bearing is expected to operate with 90% reliability, whether it is an individual bearing or a group of identical bearings working under the same conditions This metric is essential for predicting bearing performance and ensuring the reliability of machinery Understanding bearing life helps in optimal maintenance planning and reduces unexpected failures, ultimately enhancing equipment longevity.

Equations (85) and (86) are derived from Equations (4), (5), and (6):

The rolling element load in bearings is directly proportional to the overall bearing load, with QC and Q corresponding to the basic dynamic load ratings, C_r or C_a Additionally, the dynamic equivalent radial load (P_r) and the dynamic equivalent axial load (P_a) are key factors influencing bearing performance As derived from Equations (85) and (86), these relationships are fundamental for understanding bearing load capacity and ensuring accurate load calculations.

Substituting experimental constants e = 10/9 (point contact), e = 9/8 (line contact), c = 31/3 and h = 7/3 into

The contact between the rollers and raceways in roller bearings shifts from point to line contact as load increases, affecting the bearing’s lifespan The life exponent varies from 3 to 4 depending on different loading intervals within the same bearing, highlighting the need for a consistent calculation method To ensure accuracy across all roller bearing types and loading conditions, a uniform approach to life prediction is essential Therefore, applying the same life equations universally to all roller bearings simplifies analysis and improves reliability in lifespan estimation.

Differences between actual and calculated life values, caused by the use of this single exponent, are reduced by the use of a compensatory adjustment of the load rating (see 4.4)

7 Life adjustment factor for reliability

A relationship between the bearing life and its probability of survival is expressed as Equation (92) which is derived from Correlation (1): ln1 AL e

S is the probability of survival;

L is the bearing life; e is the Weibull slope

Inserting the basic rating life, L 10 , as L for S = 0,9 into Equation (92) gives a constant of proportionality, A, as:

From Equations (92) and (93), Equation (94) can be derived:

L n =a L 1 10 (94) where a 1 is the life adjustment factor for reliability, given by:

The failure distribution curve below 10 % failure probability is bent down towards higher fatigue lives (Reference [4])

The failure distribution curve below 10 % of the failure probability has been approached by a Weibull distribution with the Weibull slope e = 1,5

Values for the life modification factor for reliability, a 1 , shown in ISO 281:2007, Table 12, can be calculated from Equation (95) with e = 1,5

[1] L UNDBERG , G., P ALMGREN , A Dynamic capacity of rolling bearings Acta Polytechn.: Mech Eng Ser

[2] LUNDBERG, G., PALMGREN, A Dynamic capacity of roller bearings Acta Polytechn., Mech Eng Ser

[3] A OKI , Y On the evaluating formulae for the dynamic equivalent load of ball bearings J Jpn Soc

[4] TALLIAN, T Weibull distribution of rolling contact fatigue life and deviations therefrom ASLE Trans

Tiêu đề	Rolling Bearings — Explanatory Notes On Iso 281 — Part 1: Basic Dynamic Load Rating And Basic Rating Life
Trường học	International Organization for Standardization
Chuyên ngành	Standardization
Thể loại	technical report
Năm xuất bản	2008
Thành phố	Geneva

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