GB/Z 36517-2018Rolling bearings -- Methods for calculating the modified reference rating life for universally loaded bearings (English PDF)

4.1 General

This chapter describes radial and thrust ball bearings that are subjected to radial and axial loads, taking into account radial clearance and tilt.

Analysis of internal load distribution. For the calculation of different geometrical parameters or under more complex load conditions, this technical text can be

The formula given in the article is introduced.

The internal load distribution of the bearing is only for static equilibrium calculation; it is assumed that dynamic effects such as centripetal force and gyroscopic force are not significant, this assumption

Generally effective for low and medium speeds. At high speeds, the effects of centripetal force and gyroscopic force may become prominent and may significantly change the internal load of the bearing

distributed.

4.2 Bearing internal load distribution

4.2.1 Elastic deformation of point contact

The elastic deformation of the point contact can be calculated by Hertz theory. The elastic deformation delta of a single point contact is.

delta=

4.5

1-nuE2

piE

K chi( )

sum rho

chi2E (chi)

Q2/3 (1)

The ratio of the elliptical long semi-axis to the short semi-axis is the root of equation (2).

Chi2-1

K chi( )

Echi( ) -1

Êê

úú-Frho( )=0 (2)

Among them, the first type of complete elliptic integral K (chi).

K chi( )=integral

pi/2

1- 1-

Chi2

÷ sinphi( ) 2

Êê

Úú

-1/2

Dphi (3)

The second type of complete elliptic integral E(chi).

Echi( )=integral

pi/2

1- 1-

Chi2

÷ sinphi( ) 2

Êê

Úú

Dphi (4)

Curvature at the inner ring contact and sum rhoi.

sum rhoi=

Dw

1-gamma-

Dw

2ri

÷ (5)

Curvature at the contact of the outer ring and sum rhoe.

sum rhoe=

Dw

1 gamma-

Dw

2re

÷ (6)

The relative curvature difference Fi(rho) at the inner ring contact.

Firho( )=

1-gamma

Dw

2ri

÷/2

1-gamma-

Dw

2ri

÷ (7)

The relative curvature difference Fe(rho) at the contact of the outer ring.

Ferho( )=

1 gamma

Dw

2re

÷/2-

1 gamma-

Dw

2re

÷ (8)

The total elastic deformation delta in contact with the inner and outer rings is.

delta=

4.5

1-nuE2

piE

K chii( )

sum rhoi

chii2Echii( )

K chie( )

sum rhoe

chie2Echie( )

úQ

2/3 (9)

This derives the load-deformation relationship formula (10).

Q=cPdelta3/2 (10)

Wherein, the elastic constant cP is.

cP=1.48

1-nuE2 K

Chii( )

sum rhoi

chii2Echii( )

K chie( )

sum rhoe

chie2Echie( )

-3/2

(11)

4.2.2 Static balance

For radial ball bearings with radial working clearance s measured in the diameter direction, original contact angle alpha0=arccos[1 - (s/2A)]

In terms of the total elastic deformation deltaj of the rolling elements.

deltaj= \u003cAcosalpha0 deltarcosphij( ) 2 (Asinalpha0 deltaa Risinpsicosphij)2 -A\u003e

(12)

If the right side of equation (12) is a negative value, it is set to zero.

Note. The original contact angle alpha0 is generally different from the nominal contact angle alpha in ISO 281.

In equation (12), A is the center-of-gravity of the channel radius ri and re, see Figure 1.

A=ri re-Dw (13)

Figure 1 Auxiliary geometry parameters

The distance Ri between the center of curvature of the inner ring channel and the axis of rotation is.

Ri=

Dpw

4.2.2.1 The sum of all forces

Fr-cPsum

j=1

deltaj3/2cosalphajcosphij=0 (16)

Fa-cPsum

j=1

deltaj3/2sinalphaj=0 (17)

4.2.2.2 sum of all moments

Mz-

Dpw

÷cPsum

j=1

deltaj3/2sinalphajcosphij=0 (18)

4.3 rated life

4.3.1 Rolling element load corresponding to the basic dynamic load rating

4.3.1.1 General

The rolling element loads Qci and Qce corresponding to the basic dynamic load ratings of the inner and outer rings are derived from ISO /T R1281-1 [1].

4.3.1.2 radial ball bearings

For the inner ring, the Qci of single and multi-row bearings can be calculated using the radial basic dynamic load rating Cr.

Qci=

Cr

0.407Zcosalpha( )i0.7

1 1.044

1-gamma

1 gamma

1.72 ri

Re

2re-Dw

2ri-Dw

Êê

Úú

0.41

{ }

10/3æ

(19)

For the outer ring, the Qce of single and multi-row bearings can be calculated using the radial basic dynamic load rating Cr.

Qce=

Cr

0.389Zcosalpha( )i0.7

1 1.044

1-gamma

1 gamma

1.72 ri

Re

2re-Dw

2ri-Dw

Êê

Úú

0.41

{ }

-10/3æ

(20)

4.3.1.3 Thrust ball bearing with nominal contact angle alpha!=90°

For inner rings or shaft rings, Qci can be calculated using the axial basic dynamic load rating Ca.

Qci=

Ca

Zsinalpha 1

1-gamma

1 gamma

1.72 ri

Re

2re-Dw

2ri-Dw

Êê

Úú

0.41

{ }

10/3æ

(twenty one)

For the outer ring or race, the Qce can be calculated using the axial basic dynamic load rating Ca.

Qce=

Ca

Zsinalpha 1

1-gamma

1 gamma

1.72 ri

Re

2re-Dw

2ri-Dw

Êê

Úú

0.41

{ }

-10/3æ

(twenty two)

4.3.1.4 Thrust ball bearing with nominal contact angle alpha=90°

For the shaft collar, Qci can be calculated using the axial basic dynamic load rating Ca.

Qci=

Ca

Z 1

Ri

Re

2re-Dw

2ri-Dw

Êê

Úú

0.41

{ }

10/3æ

(twenty three)

For the race, Qce can be calculated using the axial basic dynamic load Ca.

Qce=

Ca

Z 1

Ri

Re

2re-Dw

2ri-Dw

Êê

Úú

0.41

{ }

-10/3æ

(twenty four)

4.3.2 Rolling body equivalent dynamic load

The rolling element equivalent dynamic load Qei of the inner ring or the shaft ring that rotates relative to the bearing load is.

Qei=

Zsum

j=1

Qj3

(25)

The rolling element equivalent dynamic load Qei of the inner ring or the shaft ring which is stationary with respect to the bearing load is.

Qei=

Zsum

j=1

Qj10/3

(26)

The equivalent dynamic load Qee of the rolling element of the outer ring or race relative to the bearing load is.

Qee=

Zsum

j=1

Qj10/3

(27)

The rolling element equivalent dynamic load Qee of the outer ring or race rotating relative to the bearing load is.

Qee=

Zsum

j=1

Qj3

(28)

For a normal load distribution, the difference between the equivalent dynamic load of the rolling element of the rotating inner ring and the stationary inner ring is less than 2%. The difference

The difference is generally negligible, especially when the equivalent dynamic load deviations of the rolling elements on the inner and outer rings can partially compensate each other.

When calculating, it is generally considered that the inner ring is rotating and the outer ring is stationary.

4.3.3 Basic reference rating life

The basic reference rated life L10r can be calculated using the rolling element load and the rolling element equivalent dynamic load corresponding to the basic dynamic load rating.

L10r=

Qci

Qei

-10/3

Qce

Qee

-10/3é

Êê

Úú

-9/10

(29)

4.3.4 Reference equivalent dynamic load

The reference equivalent dynamic load Pref,r of the radial ball bearing is.

Pref, r=

Cr

L10r1/3

(30)

The reference equivalent dynamic load Pref, a of the thrust (axial) ball bearing is.

Pref, a=

Ca

L10r1/3

(31)

4.3.5 Correcting the reference rating life

The corrected reference life of the radial ball bearing Lnmr can be calculated using the life correction factor aISO , aISO can use ISO 281.2007

Equation (31) ~ formula (33) to calculate.

Lnmr=a1aISO

Cr

Pref,r

(32)

For thrust ball bearings, the corrected reference rating life is.

Lnmr=a1aISO

Ca

Pref,a

(33)

Among them, the life correction coefficient aISO can be calculated using the formula (37) to the formula (39) of ISO 281.2007.

5.1 General

This chapter describes the internal load distribution of radial roller bearings subjected to radial loads under consideration of radial clearance and tilt.

Analysis. For different geometrical parameters of bearings or analytical calculation methods under more complex load conditions, the formula given in this technical document can be used.

roll out.

The internal load distribution of the bearing is only for static equilibrium calculation; it is assumed that dynamic effects such as centripetal force and gyroscopic force are not significant, this assumption

Generally effective for low and medium speeds. At high speeds, the effects of centripetal force and gyroscopic force may become prominent and may significantly change the internal load of the bearing

distributed.

5.2 Bearing internal load distribution

5.2.1 Elastic deformation of line contact

According to reference [4], the elastic deformation of the line contact rolling body can be described as.

Q=cLdelta10/9 (34)

Among them, the elastic constant cL of the steel contact parts is.

cL=35948Lwe8/9 (35)

Figure 2 Total deformation of the roller contact

5.2.2 Slice model

For the case where the raceway is cylindrical, the elastic deformation of the skewed rolling body can be described by a slicing model.

To calculate the elastic deformation, the rollers are divided into nS identical slices, as shown in Figure 3. The number of slices nS should be at least 30.

Calculate the load-deformation formula of the load qj,k on the kth slice of the roller j.

Qj,k=cSdeltaj,k10/9 (36)

Where the elastic constant cS is.

cS=

35948Lwe8/9

nS

(37)

For the radial displacement deltar of the inner ring, the elastic deformation deltaj of the rolling element j is.

deltaj=deltarcosphij-

(38)

In the plane of the rolling element j, the total skew angle psij (shown in Figure 4) between the raceways is.

Psij=arctantanpsicosphij( ) (39)

This derives the elastic deformation deltaj,k of the kth slice of the rolling element j.

deltaj,k=< deltaj-xktanpsij> (40)

If the right side of equation (40) is a negative value, it is set to zero.

Note. The assumptions in equation (40) are not completely correct when there is the effect of the rib load and the difference between the inner and outer ring contours.

Further subtract the contour depth from the deformation.

deltaj,k=< deltaj-xktanpsij-2P xk( ) > (41)

If the right side of equation (41) is a negative value, it is set to zero.

Figure 3 slice model

Figure 4 Skewed roller bearing

5.2.3 Roller profile

If a purely cylindrical roller is loaded, edge stresses may occur which may greatly exceed the calculated Hertzian pressure. Therefore, usually

Reshape the roller.

For steel rollers and general application conditions, the contour function P(xk) of equations (42) to (44) is introduced.

For rollers of length Lwe <= 2.5Dwe.

P xk( )=0.00035Dweln

1- 2xk/Lwe( ) 2

Êê

Úú (42)

For rollers of length Lwe > 2.5 Dwe, the contour function defined by the segmentation should be used.

When xk <=

Lwe-2.5Dwe

Time.

P xk( )=0 (43)

When xk >

Lwe-2.5Dwe

Time.

P xk( )=0.00050Dweln

1- 2xk - Lwe-2.5Dwe( )[ ]/2.5Dwe{ }2

÷ (44)

The contour functions in equations (42) to (44) give approximations. Actual roller design based on manufacturer's expertise may deviate significantly

These reference geometry parameters.

5.2.4 Static balance

The equations can be obtained according to the static equilibrium conditions of the external force and moment acting on the bearing ring and the reaction force of the rolling element, see 5.2.4.1 and

5.2.4.2, which can be solved by iteration.

5.2.4.1 The sum of all forces

Fr-

cL

nSsum

j=1

Cosphijsum

nS

k=1

deltaj,k10/9( )=0 (45)

5.2.4.2 sum of all moments

MZ-

cL

nSsum

j=1

Cosphijsum

nS

k=1

Xkdeltaj,k10/9( )=0 (46)

5.3 rated life

5.3.1 Rolling element load corresponding to the basic dynamic load rating

5.3.1.1 General

The rolling element loads Qci and Qce corresponding to the basic dynamic load ratings of the inner and outer rings are derived from ISO /T R1281-1 [1].

5.3.1.2 Radial roller bearings

For single-row and multi-row bearings, the rolling element loads Qci and Qce corresponding to the basic dynamic load ratings of the inner and outer rings are available in radial basis

The fixed load Cr is calculated.

Qci=

Lambdanu

Cr

0.378Zcosalpha( )i7/9

1 1.038

1-gamma

1 gamma

143/108é

Êê

Úú

{ }

(47)

Qce=

Lambdanu

Cr

0.364Zcosalpha( )i7/9

1 1.038

1-gamma

1 gamma

143/108é

Êê

Úú

-9/2

{ }

(48)

According to reference [1],

Lambdanu=0.83 (49)

The value of lambdanu needs to be carefully analyzed for contact stress as described in reference [5], [6] or [7], or in the application of equation (60).

The stress concentration approximation formula.

5.3.1.3 Thrust roller bearings with nominal contact angle alpha!=90°

The rolling element loads Qci and Qce corresponding to the basic dynamic load rating of the inner ring (shaft ring) and the outer ring (seat ring) can be used for the axial basic dynamic load rating.

Load Ca to calculate.

Qci=

Lambdanu

Ca

Zsinalpha 1

1-gamma

1 gamma

143/108é

Êê

Úú

{ }

(50)

Qce=

Lambdanu

Ca

Zsinalpha 1

1-gamma

1 gamma

143/108é

Êê

Úú

-9/2

{ }

(51)

In the formula.

Lambdanu=0.73 (52)

The value of lambdanu needs to be carefully analyzed for contact stress as described in reference [5], [6] or [7], or in the application of equation (60).

The stress concentration approximation formula.

5.3.1.4 Thrust roller bearing with nominal contact angle alpha=90°

The rolling element loads Qci and Qce corresponding to the basic dynamic load rating of the collar and the race can be calculated from the axial basic dynamic load Ca.

Qci=

Lambdanu

Ca

Z x 2

2/9 (53)

Qce=

Lambdanu

Ca

Z x 2

2/9 (54)

In the formula.

Lambdanu=0.73 (55)

The value of lambdanu needs to be carefully analyzed for contact stress as described in reference [5], [6] or [7], or in the application of equation (60).

The stress concentration approximation formula.

5.3.2 Basic dynamic load rating of bearing slicing

The basic dynamic load qci of a bearing slice of the inner ring is.

Qci=Qci(

nS

(56)

The basic dynamic load qce of a bearing slice of the outer ring is.

Qce=Qce(

nS

(57)

5.3.3 Edge stress concentration

In the case where the rolling elements are only slightly trimmed or severely deflected, edge stresses may occur, which should be considered in the calculation of the rated life. roll

The contact stress distribution over the length of the moving body can be calculated using references [5], [6] or [7]. Based on the calculated length of the roller

For the contact stress distribution, the stress concentration approximation function fi[j,k] on the inner ring raceway can be obtained from equation (58), and the outer ring raceway is obtained from equation (59).

Fe[j,k].

Fij,k[ ]=

pHij,k

Dwe1-gamma( )

Lwe

nS

Êê

Úú/qj,k (58)

Fej,k[ ]=

pHej,k

Dwe1 gamma( )

Lwe

nS

Êê

Úú/qj,k (59)

As the first approximation, the stress concentration function f[k] determined by the contact stress calculation can be used for the kth slice.

Fik[ ]=fe k[ ]=1- 0.01ln1.985

2k-nS-1

2nS-2

Êê

Úú (60)

This approximation function is only valid for approximate contours obtained using equations (42), (43), and (44), and is assumed to satisfy medium load and total bearing deflection.

The angle is less than 4'. For general calculations, the methods described in references [5], [6] and [7] are recommended.

5.3.4 Equivalent dynamic load on a slice

Equivalent dynamic load q on the kth slice of the inner ring rotated relative to the load.