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Stiffness of deep-groove ball bearings
Wear, 63 (1980) 89 - 94 0 Elsevier Sequoia S.A., Lausanne -Printed
89 in the Netherlands
STIFFNESS OF DEEP-GROOVE BALL BEARINGS H. R. EL-SAYED Production Alexandria
Engineering (Egypt)
Department,
(Received January 26.1979;
Faculty
of Engineering,
Alexandria
University,
in final form October 9,1979)
Summary Ball bearing stiffness is an import parameter in the design of machine tool spindles because of its effect on the performance of the spindle system. As bearing stiffness is given in manufacturers’ catalogues and handbooks it would be useful to derive an equation to predict this parameter. Such an equation for predicting the stiffness of deep-groove ball bearings is derived and expressed in terms of the available bearing dimensions. Experimental verification showed that the predicted results are satisfactory.
1. Introduction Evaluation of the performance of machine tool spindles depends largely on the stiffness of the bearings used in the spindle head. The prediction of bearing stiffness is one of the principal steps in the design of such machine elements. No information pertinent to the evaluation of bearing stiffness is provided in ball bearing catalogues. However, info~ation on many problems inherent to ball bearing performance is available in the literature [l - 8) although no information is available concerning bearing stiffness. This paper presents a solution to the problem.
2. Bearing deflection The inner and outer rings of the deep-groove ball bearing are secured to the shaft and the housing respectively. During running they remain almost circular but some elastic contact deformation take place between them and the rolling balls. If the bearing initial clearance is neglected the centre line of the shaft will therefore traverse a distance 6, given by 6, =6i +6o Hertz [ 91 evaluated the elastic contact deformation curvature in contact under a load P:
(11 6 of two bodies of any
By applying eqn. (2) to the configuration shown in Fig. 1, the elastic contact deformation between the rolling element (ball) and the bearing raceway is found to be
(3)
(a)
(b)
Fig. 1. Dimensions in (a) the rolling and (b) the perpendicular planes.
In ball bearings the balls contact the inner and outer races simultaneously. Hence the total deflection 6 t caused by an applied load W is
(4) where
(-&=2_1_1 r
ri =
cos-l
r. = cos-l
R,
l/Ri
( !
r.
+ l/ri
) URo co 1
ci
l/r0 -
Mi/mi and M,/mo are determined and r. respectively.
from Fig. 2 according
to the values of ri
91
Fig. 2. Value of M/m.
3. Evaluation
of the bearing deflection
(4) shows that the bearing deflection 6, depends on r, Ri, ri, cannot be determined from the data provided in manufacturers’ catalogues and handbooks. To facilitate the evaluation of the magnitude of the bearing deflection the parameters should be expressed in terms of the available bearing dimensions, i.e. D, and di. To satisfy this objective samples of deep-groove ball bearings (series 62 and 63 FAG and SKF) were selected, Numerous trials were made to determine relations between the unknown parameters and the available bearing dimensions. The following equations were found to provide satisfactory results: Equation
R,, r. and PO. These parameters
r = 0.64(0,
- di)
Ri = 0.09Do
(5)
+ 0.41di
R, = 0.410,
+ 0.09di
r, = r. = 0.17(0,
- di)
(6)
(7) (8)
P, = 5 W/r
= 3.5w
Do -di
2.40,
+ di
(9)
4. Bearing stiffness The bearing deflection 6t arises from the application fore bearing stiffness X can be expressed as
x = W/6,
of a load W. There-
(10)
92
which on substituting &z
from eqns. (4) and (9) becomes
LW’i3
(11)
where L is the stiffness factor which depends mensions and is given by
on the bearing type and di-
-1
M. __.?c1'3 1
+"OC1,3
6
mi
o
Bearing stiffnesses predicted from eqn. (11) are shown in Figs. 3 and 4 where the stiffnesses of some bearings selected from series 62 and 63 FAG and SKF are shown as a function of the applied load.
4 i
Series
62
FAG
Series
Series
62
SKF
Series 63 SKF
W 2 0 0
LOO
,.’
800
1200
Radial
W /
Load
2000
1600
W
2400
0
400
Kg
800
1200
63
FAG
Radial
bad
1600
2000 W
2400
Kg
Fig. 3. Stiffnesses of series 62 FAG and SKF ball bearings. Fig. 4. Stiffnesses of series 63 FAG and SKF ball bearings.
5. Experimental
verification
The test rig shown in Fig. 5 was constructed and used to verify the theoretically predicted bearing stiffness equation. A shaft 80 mm long is attached to the bearing bore and the bearing outer ring is attached to the base block. The load was applied as shown in Fig. 5 using a universal testing machine. The load and the reading on the dial gauge were recorded. The bearing deflection was determined by subtracting the corresponding shaft deflection from the dial gauge reading. The results obtained are compared with the predicted results in Fig. 6.
93
_TheoretKd
0
LOO
800
1200
1600
2000 W
2LOM)
Kg
Fig. 5. Experimental test rig. Fig. 6. Comparison of theoretical and experimental results.
6. Discussion and conclusions The theoretical results are shown in Figs. 3 and 4 and compared with the experimental results in Fig. 6. The difference between the theoretical and experimental results may be due to neglect of the effect of initial clearance and the approximation made in expressing the parameters determining stiffness in terms of available bearing dimensions D, and di. However, this difference is reasonable. The following conclusions can be drawn from these results. (1) It is possible to express the bearing stiffness of non-filing slot-type ball bearings in terms of the dimensions given in the manufacturers’ catalogues. (2) The equation derived predicts the stiffness of deep-groove ball bearings with reasonable accuracy.
Nomenclature
c di DO E i M/m PO
sum of the curvatures of the ball and raceway in the rolling and the perpendicular planes, mm-l inside diameter of the bearing, mm outside diameter of the bearing, mm modulus of elasticity of the bearing material, kgf mrne2 Poisson’s ratio constant load on the most heavily loaded ball, kgf
94
r fi ;Pi ii0
ball radius, mm inside radius of the raceway in the perpendicular plane, mm outside radius of the raceway in the perpendicular plane, mm inside radius of the raceway in the rolling plane, mm outside radius of the raceway in the rolling plane, mm elastic contact deformation between the most heavily loaded ball and the inner bearing ring, mm elastic contact deformation between the most heavily loaded ball and the outer bearing ring, mm sum of the curvatures of the two bodies in the two principal planes, mm-l
References 1 T. A. Harris, Prediction of temperature in a rolling contact bearing assembly, Lubr. Eng., 20 (4) (April 1964) 145 - 150. 2 T. A. Harris, Optimizing the fatigue life of flexibly mounted rolling bearings, Lubr. Eng., 21 (10) (Oct. 1965) 421 - 428. 3 B. G. Brothers and G. R. Bremble, The effect of geometric conformity between a ball and its track on the free rolling resistance, Wear, 20 (2) (1972) 137. 4 N. G, Popinceanu and M. D. Gafitanu, A study of rolling bearing fatigue life with mineral oil lubrication, Wear, 22 (1) (1972) 21. 5 T. A. Harris and M. H. Mindel, Rolling element bearing dynamics, Wear, 23 (3) (1973) 331. 6 R. A. Goodelle, W. J. Derner and L. C. Root, Determination of static load distributions from elastic contacts in rolling element bearings, Lubr. Eng., 14 (4) (1971) 275. 7 R. K. Allan, Rolling Bearings, Pitman, London, 1964. 8 T. A. Harris, Rolling Bearing Analysis, Wiley, New York, 1966. 9 D. F. Wilcock, Bearing Design and Application, McGraw Hill, New York, 1957.
89 in the Netherlands
STIFFNESS OF DEEP-GROOVE BALL BEARINGS H. R. EL-SAYED Production Alexandria
Engineering (Egypt)
Department,
(Received January 26.1979;
Faculty
of Engineering,
Alexandria
University,
in final form October 9,1979)
Summary Ball bearing stiffness is an import parameter in the design of machine tool spindles because of its effect on the performance of the spindle system. As bearing stiffness is given in manufacturers’ catalogues and handbooks it would be useful to derive an equation to predict this parameter. Such an equation for predicting the stiffness of deep-groove ball bearings is derived and expressed in terms of the available bearing dimensions. Experimental verification showed that the predicted results are satisfactory.
1. Introduction Evaluation of the performance of machine tool spindles depends largely on the stiffness of the bearings used in the spindle head. The prediction of bearing stiffness is one of the principal steps in the design of such machine elements. No information pertinent to the evaluation of bearing stiffness is provided in ball bearing catalogues. However, info~ation on many problems inherent to ball bearing performance is available in the literature [l - 8) although no information is available concerning bearing stiffness. This paper presents a solution to the problem.
2. Bearing deflection The inner and outer rings of the deep-groove ball bearing are secured to the shaft and the housing respectively. During running they remain almost circular but some elastic contact deformation take place between them and the rolling balls. If the bearing initial clearance is neglected the centre line of the shaft will therefore traverse a distance 6, given by 6, =6i +6o Hertz [ 91 evaluated the elastic contact deformation curvature in contact under a load P:
(11 6 of two bodies of any
By applying eqn. (2) to the configuration shown in Fig. 1, the elastic contact deformation between the rolling element (ball) and the bearing raceway is found to be
(3)
(a)
(b)
Fig. 1. Dimensions in (a) the rolling and (b) the perpendicular planes.
In ball bearings the balls contact the inner and outer races simultaneously. Hence the total deflection 6 t caused by an applied load W is
(4) where
(-&=2_1_1 r
ri =
cos-l
r. = cos-l
R,
l/Ri
( !
r.
+ l/ri
) URo co 1
ci
l/r0 -
Mi/mi and M,/mo are determined and r. respectively.
from Fig. 2 according
to the values of ri
91
Fig. 2. Value of M/m.
3. Evaluation
of the bearing deflection
(4) shows that the bearing deflection 6, depends on r, Ri, ri, cannot be determined from the data provided in manufacturers’ catalogues and handbooks. To facilitate the evaluation of the magnitude of the bearing deflection the parameters should be expressed in terms of the available bearing dimensions, i.e. D, and di. To satisfy this objective samples of deep-groove ball bearings (series 62 and 63 FAG and SKF) were selected, Numerous trials were made to determine relations between the unknown parameters and the available bearing dimensions. The following equations were found to provide satisfactory results: Equation
R,, r. and PO. These parameters
r = 0.64(0,
- di)
Ri = 0.09Do
(5)
+ 0.41di
R, = 0.410,
+ 0.09di
r, = r. = 0.17(0,
- di)
(6)
(7) (8)
P, = 5 W/r
= 3.5w
Do -di
2.40,
+ di
(9)
4. Bearing stiffness The bearing deflection 6t arises from the application fore bearing stiffness X can be expressed as
x = W/6,
of a load W. There-
(10)
92
which on substituting &z
from eqns. (4) and (9) becomes
LW’i3
(11)
where L is the stiffness factor which depends mensions and is given by
on the bearing type and di-
-1
M. __.?c1'3 1
+"OC1,3
6
mi
o
Bearing stiffnesses predicted from eqn. (11) are shown in Figs. 3 and 4 where the stiffnesses of some bearings selected from series 62 and 63 FAG and SKF are shown as a function of the applied load.
4 i
Series
62
FAG
Series
Series
62
SKF
Series 63 SKF
W 2 0 0
LOO
,.’
800
1200
Radial
W /
Load
2000
1600
W
2400
0
400
Kg
800
1200
63
FAG
Radial
bad
1600
2000 W
2400
Kg
Fig. 3. Stiffnesses of series 62 FAG and SKF ball bearings. Fig. 4. Stiffnesses of series 63 FAG and SKF ball bearings.
5. Experimental
verification
The test rig shown in Fig. 5 was constructed and used to verify the theoretically predicted bearing stiffness equation. A shaft 80 mm long is attached to the bearing bore and the bearing outer ring is attached to the base block. The load was applied as shown in Fig. 5 using a universal testing machine. The load and the reading on the dial gauge were recorded. The bearing deflection was determined by subtracting the corresponding shaft deflection from the dial gauge reading. The results obtained are compared with the predicted results in Fig. 6.
93
_TheoretKd
0
LOO
800
1200
1600
2000 W
2LOM)
Kg
Fig. 5. Experimental test rig. Fig. 6. Comparison of theoretical and experimental results.
6. Discussion and conclusions The theoretical results are shown in Figs. 3 and 4 and compared with the experimental results in Fig. 6. The difference between the theoretical and experimental results may be due to neglect of the effect of initial clearance and the approximation made in expressing the parameters determining stiffness in terms of available bearing dimensions D, and di. However, this difference is reasonable. The following conclusions can be drawn from these results. (1) It is possible to express the bearing stiffness of non-filing slot-type ball bearings in terms of the dimensions given in the manufacturers’ catalogues. (2) The equation derived predicts the stiffness of deep-groove ball bearings with reasonable accuracy.
Nomenclature
c di DO E i M/m PO
sum of the curvatures of the ball and raceway in the rolling and the perpendicular planes, mm-l inside diameter of the bearing, mm outside diameter of the bearing, mm modulus of elasticity of the bearing material, kgf mrne2 Poisson’s ratio constant load on the most heavily loaded ball, kgf
94
r fi ;Pi ii0
ball radius, mm inside radius of the raceway in the perpendicular plane, mm outside radius of the raceway in the perpendicular plane, mm inside radius of the raceway in the rolling plane, mm outside radius of the raceway in the rolling plane, mm elastic contact deformation between the most heavily loaded ball and the inner bearing ring, mm elastic contact deformation between the most heavily loaded ball and the outer bearing ring, mm sum of the curvatures of the two bodies in the two principal planes, mm-l
References 1 T. A. Harris, Prediction of temperature in a rolling contact bearing assembly, Lubr. Eng., 20 (4) (April 1964) 145 - 150. 2 T. A. Harris, Optimizing the fatigue life of flexibly mounted rolling bearings, Lubr. Eng., 21 (10) (Oct. 1965) 421 - 428. 3 B. G. Brothers and G. R. Bremble, The effect of geometric conformity between a ball and its track on the free rolling resistance, Wear, 20 (2) (1972) 137. 4 N. G, Popinceanu and M. D. Gafitanu, A study of rolling bearing fatigue life with mineral oil lubrication, Wear, 22 (1) (1972) 21. 5 T. A. Harris and M. H. Mindel, Rolling element bearing dynamics, Wear, 23 (3) (1973) 331. 6 R. A. Goodelle, W. J. Derner and L. C. Root, Determination of static load distributions from elastic contacts in rolling element bearings, Lubr. Eng., 14 (4) (1971) 275. 7 R. K. Allan, Rolling Bearings, Pitman, London, 1964. 8 T. A. Harris, Rolling Bearing Analysis, Wiley, New York, 1966. 9 D. F. Wilcock, Bearing Design and Application, McGraw Hill, New York, 1957.