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Multiple roller bearing-shaft systems
47
Wear, 81 (1982) 47 - 58
MULTIPLE ROLLER BEARING-SHAFT
G. IOANNOU,
SYSTEMS
A. WOOWAT and R. GOHAR
Department of Mechanical Engineering, Imperial College of Science and Technology, Exhibition Road, London SW7 2BX fGt. Britain} (Received
December
21,1981)
Summary
A design method for shaft and roller bearing systems is described, in which shaft flexure causes the loads on the rollers to be no longer purely radial. The resulting pressure distributions on the rollers are predicted without recourse to any assumptions relating to plain strain. Some experimental results are also presented, confirming the accuracy of the theory.
1. Introduction
The load which a rolling element bearing supports is usually transmitted to it through the shaft on which it is mounted and thence to the housing which encompasses the outer ring. In most engineering applications, the standard design procedure for the shaft and bearing systems treats the shaft and bearings separately, making the simplifying assumption that the reaction between them is only radial [ 1 - 31. However, even a relatively small misalignment of the bearing, in addition to any deflections that may be present, can cause significant changes in the stress distribution within its elements, resulting in a considerable decrease of the life expectancy. The classical Lundberg-P~rn~en 143 stress-life relationship for roller bearings shows that the life expectancy varies inversely with the eighth power of the contact stress. For non-self-aligning bearings, shaft flexure as well as misalignment of the bearings caused by tolerance errors, settling of the machine foundations and thermal distortions of the frames all contribute to induce moment loads on the bearings in addition to the radial loads. Thus, if the shaft deflections and bearing loadings are to be accurately predicted, the bearings and their supports must be treated as integral parts of the loaded system. Apart from the. expedient of selecting bearings which are obviously overlarge for the given loading, more subtle design procedures based on bearing location, variable shaft diameter and the proper choice of the roller axial profile can ~tematively be employed. This design method can only be followed with any confidence if comprehensive computer programs are 0043-1648J82/0000-0000/$02.75
@ Elsevier Sequoia/Printed
in The Netherlands
48
available to predict accurately bearing stresses for a given loading situation, In ref. 5 an example of such a design method was given for a simple system comprising a shaft supported by two roller bearings and radially loaded at midspan. Because of its simplicity, a method by Harris [ 6 J , which assumes a linear roller to race deflection relationship and plain strain of the rollers, could be used to find an explicit expression for the misalignment 6,,, of the on the roller worst-loaded roller. Knowing O,,,, the pressure distribution could be found by using a numerical method employed in ref. 7. Furtherby more, the expression for 8,,, in ref. 5 suggested that the contribution the roller bearing to the angular stiffness of the shaft supports was relatively slight. This state of affairs becomes less obvious in more complex shaft and bearing systems. Thus the design procedure described below does not make any ~sumptions of linear load deflection relationships, or plain strain, The method can be used for shafts subjected to several loads and moments and supported by two or more roller bearings. Some experimental results are also presented.
2. Main assumptions
used
(I) The roller axial clearances in the rings and the edge-pinching effect that can be produced under misaligned conditions on rollers carrying no radial load do not significantly affect the resulting deflections and angular misalignments. (2) In accordance with normal design practice, boundary lubrication is assumed throughout. At present, three-dimensional elastohydrodynamic lubrication theory for rollers is not sufficiently comprehensive to be included, although the pressure distributions obtained from prelim~~y results are found to be similar to those for dry contact.
3. Multiple-bearing
systems
Figure 1 shows an n bearing system subject to a completely generalized radial loading. It follows from the mathematical analysis that the reaction at any bearing location h is a function of the loading existing at and in between the supports located at positions h - 1 and h + 1 and is given by
Fh- ?
Fh
Fig. 1. Multiple-bearing system.
Fh+l
49
f
-
$f,ftfh+l+
+
2 @,h -
r
lh +l
b,h+l)
(1)
h+l
and
l+lt’%,h--%h+l) II
(2)
where 1 CA@)=
k=n(k)
kxl
2
~h,kah,k2(3~k
--+b,k)
I
1 C,(h)
=
k=n(h) 2
2
Ph,k(lh
--k,k)2(ih
+ 2%k)
k=l 6
k=n(h) Tk.,kuk,k(zh
2
-%,k)
k=l 1 CD(h)=
T
CE@)=
2
1
C,(h) = -it2
&z(h) =
2’
k=n(h)
2 km1
Ph,k%,k2(dh
(3)
-%,k)
k=n(h)
,zl
Ph,kah,kflh
-%k)2
kj$h)%,k%,k(zb Ti
kj$k)Th,k(lh
-3ah,k)
-ah,k)(bt
-3ah.k)
e
Equations (1) - (3) apply directly to all in~~edia~ support locations of an n bearing system but present a problem when applied to evaluate the reactions at support locations 1 and n because they include CIO,6r,D, and 0 ni.1, &,.+I, respectively. One way out of this is to make all terms that
50
include them zero, implying that there are no loads on shaft segments I, and 1n + 1 : thus eqns. (3) are zero for h = 1 and h = n + 1 and the second moment of area and the length of the overhang shaft segments (i.e. II, I, + 1, and I,, l,, + 1 ) are set to zero and one respectively in order to give a zero value for the
terms relating to the non-existent support locations 0 and n + 1 and to avoid the indeterminate case that would arise of l1 and/or l,,+r = 0. Using eqns. (1) and (2) for every bearing location in an IZ bearing system, 2n equations in 4n unknowns are obtained. In order to solve the system, two more equations per bearing are required. These are obtained by considering the relation between 13,, , 6,, ,, , Mh and F/,.
4. Multiple-bearing
systems with loaded overhangs
The simplest way to introduce loaded overhangs, although still utilizing eqns. (1) - (3), is to consider dummy supports at the tips of the overhangs and to equate the dummy reactions at those positions to zero.
5. Roller bearings under radial and moment
loads
The theory and numerical computation scheme for estimating the dry contact pressure distribution between a roller and its race when under radial and moment loads has been fully described in refs. 5 and 7. Using the resulting computer program developed by Johns and Gohar (called herein the J-program), the pressure distribution on a roller under a given load and misalignment, as well as the resulting deflection of the roller and the moment on it, can be obtained. To simplify the computer program implementation of the bearing-shaft system solution, the results of the J-program were used to determine, by linear regression, the constants of two equations describing the behaviour of a roller. The equations, which were chosen to model the behaviour of a roller as implied by the J-program, are as follows: M, = K1 W,*8,
P
(4)
and QJI = K,W,r(B,
+ l)f
(5)
where cr, /3, y, c, K1 and K2 are constants for a given roller, which depend on its material properties, geometry and axial blend profile (the reduction in diameter towards its ends) [ 51. Equations (4) and (5) imply that if there is no central deflection at the interfaces of a roller there cannot be a load Q+ or moment M, present. Similarly, for no roller misalignment 8, there is no moment M, but there can be a load Q$ .
51
Considering a roller bearing acted on by a radial load Fh and a misaligning moment Mlt, as shown in Fig. 2, the compression of a roller at any position J/, for a positive or zero diametral clearance, will be given by 6, = L,X cos I// -+Q(l - cos $)
(6)
and the extent of the load zone (the extent of the load zone is the angle at which 6, = 0) will be given by
cos-l( 26,p,d+p,)
+I=
(7)
while, if the bearing is preloaded, i.e. it has a negative diametral clearance, the total compression of a roller at any position $J will be given by 6, = L,X’ cos i/J-fp&
(3)
where S,,, ’ is the additional compression of the bottom roller due to the external load or, seen another way, the deflection of the shaft due to the load. The extent of the load zone will then be given by
forlpdl G 2L,,’
J/l
(9)
and +, = 180”
for iA 2 2L,,’
since, for lpdl > 26,,,‘, 6, does not become zero for any value of J/ and thus & = 180”. Noting that 6 corresponds to the total roller compression at the roller centre-line due to loading and since W denotes the roller and race compression at one interface, we can write s = 2W
IlO)
Further, the misalignment 8, of a roller at any position varies approximately according to cos $ so that 0, = e,,,kos
$1
(11)
For static equilibrium,
(13) Finally, there exists a relationship between the angle 8, of misalignment of the bearing inner ring and shaft and the angle 8,,, of mis~i~ment of the bottom roller, as illustrated in Fig. 3:
52
INNER
RING i 2wmar
UNOER
OUTER
Fig. 2. Roller bearing Fig. 3. Relationship
subjected between
RING
to radial and moment
/ loading.
or and emax,
emax = pr
(14)
Therefore, using eqns. (4) - (7) and (10) - (14), the equations roller bearing with clearance become =
+
c
K,
-d”f,h
x
(10
for a
h
+h, Fh
NO LOAO
h{iWmax,hi
cos
+
-$&l,h(l
--cos
$)jYh
x
’
max,h
cos
$1
+
l)‘h
cos
(15)
$
where the sign of F,, is the same as that of Wmax,h,
x
10 max,h
COS
d’IPh
ICOS
where the sign of Mh is opposite
ti1.h
=
cos-l
4,w,.,p:;h+p
(16)
$1
to that of emax,h, and
-d,h
)
(17)
By using eqns. (4), (5) and (8) - (14), the equations with preload become h
+‘h, Fh
=
for a roller bearing
f
1 cos
&,h(lWt&U,h
I+!’ -
+pd,
h)‘h
x
- F I,h
x
where the
w
Sign
max,
h cos
$1
+
l)QJ cos $
of Fh is the same as that of Wmax,h,
(18)
53
+h,
Mh = *
h
c K1 h(lWmax,hl cos$ -+Pd,h)Qh x ’
_\Ll,h
x 10max,h COS $‘IphICOS$‘I where the sign of M,, is opposite +l,h
= cos-l
$l,h
=
( 4,;;;x.,,)
(19)
to that of Omax,h, and for
ibd,hl
G
41wrnax,hl
for @d,h! 2
4iWmax,hl
cw
180”
’ is the additional compression at one roller and race interface due to ( wmax external loading. It should be noted that, in the way in which the equations are assembled, W,,, E IV,,,‘, but if the actual loading on the bottom roller is to be calculated, w,,, = 1w,,,‘i - +pd.) Equations (1) - (3) and (15) - (20), together with the J-program, enable a generalized shaft and bearing system and the pressure distribution on the worst-loaded roller in each bearing to be obtained.
6. Design procedure The method assumes that the given shaft-bearing has been roughly designed by some standard procedure based on formulae or graphs [ 61. Thus first estimates of the radial deflection and shaft slope at every support are available. The J-program is then run for a range of loads and misalignments for each type of roller used in such a way that its results will produce regression equations that will not require extrapolation for the final solution. The sets of data produced by the J-program are fed into another program which estimates the regression constants and solves the shaft-bearing equilibrium equations. The output of this program specifies system equilibrium conditions with respect to the worst-loaded roller of each bearing and gives rollerto-race deflection and misalignment. The J-program is then used again to obtain the final pressure distributions. The maximum pressures on each of these worst-loaded rollers will determine whether the bearing chosen is suitable or not. If a maximum pressure is not acceptable by exceeding some design stress level, a bearing with rollers of different profile or a larger bearing is used at conspicuous support locations and the procedure is repeated until a satisfactory design is obtained.
7. Design examples 7.1. A shaft supported by two roller bearings with an offset radial load The bearings (Fig. 4(a)) had the specifications given in Table 1. As some experiments were to follow, the load P was varied from 490.5 to 3433.5 N. The corresponding inputted misalignments of the worst-loaded
20
(d)
Em
-LO
of uaLy -3Q
J
-2 0
AWL
-10
PamON
00 Im43
IQ
20
30
L-Q
EtaoFROtLER
Fig. 4. (a) Two-bearing system; (b) theoretical shaft deflection contours and experimental. points for various loads (x, 490.5 N; +, 981.0 N;A, 1471.5 N; 0, 1962.5 N; 0, 2452.5 N; 0, 2943.0 N; 0, 3433.5 N); (c) pressure distribution (bearing 1); (d) pressure distribution (bearing 2).
(b)
(cl
lGNlm2I
PREURE
55 TABLE 1 Nominal outside diameter (outer ring} ~ominu~ bore diameter (inner ring) Number of rollers Diametral clearance Roller length Roller diameter Flat region Dab~ff radias Shaft material
0.052
m 0.025 m 12 ZO-30pm 0.007144 m 0.007144 m 0.006078 m 0.000533 m En24 S steel
roller ranged from 0” to 0.2”. The computer solution gives the load and moment on the worst-loaded roller of each bearing, together with its combined deflection and misalignment with the race. Furthermore, relevant radial pressure distributions are supplied for the design load P as well as shaft centre-line deflections for the range of P chosen. The full lines in Fig. 4(b) give the theoretical shaft deflections as given by the computer solution for various loads, while Figs. 4(c) and 4(d) give the pressure distributions on the worst-loaded roller of bearing 1 and bearing 2 respectively for a radial load on the shaft of 3433.5 N. 7.2. Experiments
Experimental confirmation of the results was obtained from a specially designed universal static deflection apparatus, which is fully described in ref. 8. Up to three bearings could be accommodated to support a single shaft. Approximate point loading was supplied using Bowden cable loops tensioned through lever arms at up to two chosen locations. Rails over the shaft enabled a capacitance transducer to scan its top surface to within 0.027 m of each bearing vertical centre-line (accuracy, +0.65 I.trn (the maximum departure from an ideal straight line calibration); resolution, 10T2 ym). A comparison of the theoretical and experimental shaft deflection profiles, for the first example, is shown in Fig. 4(b). To compare with theory, the bearings were arranged with a roller at $ = 0. The results include the deflection of part of the unloaded overhang beyond bearing 2. For the highest experimental load applied, the computed pressure distributions on the bottom rollers of bearings 1 and 2 are shown in Figs. 4(c) and 4(d) respectively. The co~esponding pressure distribution based on the plain strain theory of Harris [6] is shown for comparison for bearing 1. It should be noted that there is a considerable underestimation of the maximum pressure on the basis of Harris’ theory. 7.3. A shaft supported by three roller bearings with two radial loads As another example we take the case of a shaft supported by three widely separated roller bearings (Fig. 5(a)). This is unusual in engineering practice, as normally in such arrangements two of the bearings are close to
56
57
each other and are often of different types, a typical example being the supports for machine tool spindles. Nevertheless, this example serves to test the accuracy and flexibility of the computer programs. The bearings used were the same as those of the previous example. The load PI was kept constant at 981.0 N and Pz was varied from 0 to 2943.0 N. Figure 5(b) shows the theoretical and experimental shaft deflection profiles. There is clearly a poorer correlation between theory and experiment, especially between bearings 1 and 2. Nevertheless, the general deformed shape of the shaft is obvious from the results. In both the examples given, the unloaded overhang shape was linear as would be expected from a deflection due to slope only. The pressure distributions of the worst-loaded rollers of bearings 2 and 3 are also shown in Figs. 5(c) and 5(d) for Pz = 2943 N. Because bearing 2 is loaded on both sides, it suffers relatively little mis~i~ment, the pressure concentrations being predominantly due to its reaction load Q. The slight aberrations before the pressure peaks are caused by computing errors and can be reduced if more pressure elements are taken.
8. conclusion The pressure distribution on the worst-loaded rollers in shaft-roller bearing systems has been calculated. The theory used has avoided the usual design assumptions of plain strain and linear load-deflection relationships in the rollers. The maximum static pressures incurred generally exceed those forecast by simpler theories and will therefore enable more confident predictions to be made for the bearing fatigue life.
Nomenclature
cA,
E
CB, CC, CD,
Fh
h
Ih
k K1, K,,tz, ih
max k’fh
K2 g2,h
CE, CF,
CG
distance between the point of application of a load and the bearing centre to its left for the kth applied load on shaft segment h loading constants for shaft segment h Young’s modulus for the shaft material radial force at bearing location h subscript denoting either the hth bearing or the hth shaft segment in a bearing-shaft system second moment of area of the cross section of shaft segment h summation index regression coefficients (constants for a given roller) K1, K2 referring to the rollers of bearing h shaft length between bearing centres for segment h subscript denoting a variable relating to the most highly loaded roller in a roller bearing moment at bearing location h
58
pd pd, h ph, k
QIL
Th,k Wmax w max’ Wmax,h We
Q, P,Y,
5
6 max
6 max
eh 8
max
moment on a roller at angular location $ in a roller bearing number of applied loads on a shaft between two bearings or the total number of bearings in a shaft-bearing system bearing diametral clearance the diametral clearance of bearing h the kth applied radial load on shaft segment h radial load on a roller at angular position $J in a roller bearing the kth applied moment load on shaft segment h W for the worst-loaded roller in a roller bearing Wmax in a roller bearing with preload excluding the compression due to the preload Wmax for the hth bearing roller compression at one contact interface with the race at angular position $ regression coefficients (constants for a given roller); when they carry the subscript h, they refer to the rollers of bearing h total compression of the worst-loaded roller in a roller bearing additional total compression of the worst-loaded roller of a roller bearing with preload excluding the total compression of the roller due to the preload shaft deflection at bearing location h total roller compression at angular position $ in a roller bearing shaft rotation at bearing location h angular misalignment of the worst-loaded roller in a roller bearing angular alignment of the worst-loaded roller of bearing h angular rotation of the inner ring of a bearing angular misalignment of a roller at angular position $ in a roller bearing angular extent of the load zone; when carrying the subscript h it refers to bearing h
References 1 A. D. Deutschman, W. J. Michels and C. E. Wilson, Machine Design (Theory and Practice), Macmillan, London, 1975. 2 SKF Hauptkatalog, Katalog 2800T/Dd 6000, April 1970. 3 M. F. Spotts, Design ofMachine Elements, Prentice-Hall, New York, 4th edn., 1971. 4 G. Lundberg and A. Palmgren, Dynamic capacity of rolling bearings, Acta Polytech., Mech. Eng. Ser., 1 (3) (1947) 7. 5 P. M. Johns and R. Gohar, Roller bearings under radial and eccentric loads, Tribal. Int., (June 1981) 131 - 136. 6 T. A. Harris, Rolling Bearing Analysis, Wiley, Chichester, 1966. 7 M. Heydari and R. Gohar, The influence of the axial profile on pressure distribution in radially loaded rollers, J. Mech. Eng. Sci., 21 (6) (1979) 381 - 388. 8 A. Woowat, Roller bearing-shaft systems, 3rd Year Design Project Rep., 1980 (Department of Mechanical Engineering, Imperial College, London).
Wear, 81 (1982) 47 - 58
MULTIPLE ROLLER BEARING-SHAFT
G. IOANNOU,
SYSTEMS
A. WOOWAT and R. GOHAR
Department of Mechanical Engineering, Imperial College of Science and Technology, Exhibition Road, London SW7 2BX fGt. Britain} (Received
December
21,1981)
Summary
A design method for shaft and roller bearing systems is described, in which shaft flexure causes the loads on the rollers to be no longer purely radial. The resulting pressure distributions on the rollers are predicted without recourse to any assumptions relating to plain strain. Some experimental results are also presented, confirming the accuracy of the theory.
1. Introduction
The load which a rolling element bearing supports is usually transmitted to it through the shaft on which it is mounted and thence to the housing which encompasses the outer ring. In most engineering applications, the standard design procedure for the shaft and bearing systems treats the shaft and bearings separately, making the simplifying assumption that the reaction between them is only radial [ 1 - 31. However, even a relatively small misalignment of the bearing, in addition to any deflections that may be present, can cause significant changes in the stress distribution within its elements, resulting in a considerable decrease of the life expectancy. The classical Lundberg-P~rn~en 143 stress-life relationship for roller bearings shows that the life expectancy varies inversely with the eighth power of the contact stress. For non-self-aligning bearings, shaft flexure as well as misalignment of the bearings caused by tolerance errors, settling of the machine foundations and thermal distortions of the frames all contribute to induce moment loads on the bearings in addition to the radial loads. Thus, if the shaft deflections and bearing loadings are to be accurately predicted, the bearings and their supports must be treated as integral parts of the loaded system. Apart from the. expedient of selecting bearings which are obviously overlarge for the given loading, more subtle design procedures based on bearing location, variable shaft diameter and the proper choice of the roller axial profile can ~tematively be employed. This design method can only be followed with any confidence if comprehensive computer programs are 0043-1648J82/0000-0000/$02.75
@ Elsevier Sequoia/Printed
in The Netherlands
48
available to predict accurately bearing stresses for a given loading situation, In ref. 5 an example of such a design method was given for a simple system comprising a shaft supported by two roller bearings and radially loaded at midspan. Because of its simplicity, a method by Harris [ 6 J , which assumes a linear roller to race deflection relationship and plain strain of the rollers, could be used to find an explicit expression for the misalignment 6,,, of the on the roller worst-loaded roller. Knowing O,,,, the pressure distribution could be found by using a numerical method employed in ref. 7. Furtherby more, the expression for 8,,, in ref. 5 suggested that the contribution the roller bearing to the angular stiffness of the shaft supports was relatively slight. This state of affairs becomes less obvious in more complex shaft and bearing systems. Thus the design procedure described below does not make any ~sumptions of linear load deflection relationships, or plain strain, The method can be used for shafts subjected to several loads and moments and supported by two or more roller bearings. Some experimental results are also presented.
2. Main assumptions
used
(I) The roller axial clearances in the rings and the edge-pinching effect that can be produced under misaligned conditions on rollers carrying no radial load do not significantly affect the resulting deflections and angular misalignments. (2) In accordance with normal design practice, boundary lubrication is assumed throughout. At present, three-dimensional elastohydrodynamic lubrication theory for rollers is not sufficiently comprehensive to be included, although the pressure distributions obtained from prelim~~y results are found to be similar to those for dry contact.
3. Multiple-bearing
systems
Figure 1 shows an n bearing system subject to a completely generalized radial loading. It follows from the mathematical analysis that the reaction at any bearing location h is a function of the loading existing at and in between the supports located at positions h - 1 and h + 1 and is given by
Fh- ?
Fh
Fig. 1. Multiple-bearing system.
Fh+l
49
f
-
$f,ftfh+l+
+
2 @,h -
r
lh +l
b,h+l)
(1)
h+l
and
l+lt’%,h--%h+l) II
(2)
where 1 CA@)=
k=n(k)
kxl
2
~h,kah,k2(3~k
--+b,k)
I
1 C,(h)
=
k=n(h) 2
2
Ph,k(lh
--k,k)2(ih
+ 2%k)
k=l 6
k=n(h) Tk.,kuk,k(zh
2
-%,k)
k=l 1 CD(h)=
T
CE@)=
2
1
C,(h) = -it2
&z(h) =
2’
k=n(h)
2 km1
Ph,k%,k2(dh
(3)
-%,k)
k=n(h)
,zl
Ph,kah,kflh
-%k)2
kj$h)%,k%,k(zb Ti
kj$k)Th,k(lh
-3ah,k)
-ah,k)(bt
-3ah.k)
e
Equations (1) - (3) apply directly to all in~~edia~ support locations of an n bearing system but present a problem when applied to evaluate the reactions at support locations 1 and n because they include CIO,6r,D, and 0 ni.1, &,.+I, respectively. One way out of this is to make all terms that
50
include them zero, implying that there are no loads on shaft segments I, and 1n + 1 : thus eqns. (3) are zero for h = 1 and h = n + 1 and the second moment of area and the length of the overhang shaft segments (i.e. II, I, + 1, and I,, l,, + 1 ) are set to zero and one respectively in order to give a zero value for the
terms relating to the non-existent support locations 0 and n + 1 and to avoid the indeterminate case that would arise of l1 and/or l,,+r = 0. Using eqns. (1) and (2) for every bearing location in an IZ bearing system, 2n equations in 4n unknowns are obtained. In order to solve the system, two more equations per bearing are required. These are obtained by considering the relation between 13,, , 6,, ,, , Mh and F/,.
4. Multiple-bearing
systems with loaded overhangs
The simplest way to introduce loaded overhangs, although still utilizing eqns. (1) - (3), is to consider dummy supports at the tips of the overhangs and to equate the dummy reactions at those positions to zero.
5. Roller bearings under radial and moment
loads
The theory and numerical computation scheme for estimating the dry contact pressure distribution between a roller and its race when under radial and moment loads has been fully described in refs. 5 and 7. Using the resulting computer program developed by Johns and Gohar (called herein the J-program), the pressure distribution on a roller under a given load and misalignment, as well as the resulting deflection of the roller and the moment on it, can be obtained. To simplify the computer program implementation of the bearing-shaft system solution, the results of the J-program were used to determine, by linear regression, the constants of two equations describing the behaviour of a roller. The equations, which were chosen to model the behaviour of a roller as implied by the J-program, are as follows: M, = K1 W,*8,
P
(4)
and QJI = K,W,r(B,
+ l)f
(5)
where cr, /3, y, c, K1 and K2 are constants for a given roller, which depend on its material properties, geometry and axial blend profile (the reduction in diameter towards its ends) [ 51. Equations (4) and (5) imply that if there is no central deflection at the interfaces of a roller there cannot be a load Q+ or moment M, present. Similarly, for no roller misalignment 8, there is no moment M, but there can be a load Q$ .
51
Considering a roller bearing acted on by a radial load Fh and a misaligning moment Mlt, as shown in Fig. 2, the compression of a roller at any position J/, for a positive or zero diametral clearance, will be given by 6, = L,X cos I// -+Q(l - cos $)
(6)
and the extent of the load zone (the extent of the load zone is the angle at which 6, = 0) will be given by
cos-l( 26,p,d+p,)
+I=
(7)
while, if the bearing is preloaded, i.e. it has a negative diametral clearance, the total compression of a roller at any position $J will be given by 6, = L,X’ cos i/J-fp&
(3)
where S,,, ’ is the additional compression of the bottom roller due to the external load or, seen another way, the deflection of the shaft due to the load. The extent of the load zone will then be given by
forlpdl G 2L,,’
J/l
(9)
and +, = 180”
for iA 2 2L,,’
since, for lpdl > 26,,,‘, 6, does not become zero for any value of J/ and thus & = 180”. Noting that 6 corresponds to the total roller compression at the roller centre-line due to loading and since W denotes the roller and race compression at one interface, we can write s = 2W
IlO)
Further, the misalignment 8, of a roller at any position varies approximately according to cos $ so that 0, = e,,,kos
$1
(11)
For static equilibrium,
(13) Finally, there exists a relationship between the angle 8, of misalignment of the bearing inner ring and shaft and the angle 8,,, of mis~i~ment of the bottom roller, as illustrated in Fig. 3:
52
INNER
RING i 2wmar
UNOER
OUTER
Fig. 2. Roller bearing Fig. 3. Relationship
subjected between
RING
to radial and moment
/ loading.
or and emax,
emax = pr
(14)
Therefore, using eqns. (4) - (7) and (10) - (14), the equations roller bearing with clearance become =
+
c
K,
-d”f,h
x
(10
for a
h
+h, Fh
NO LOAO
h{iWmax,hi
cos
+
-$&l,h(l
--cos
$)jYh
x
’
max,h
cos
$1
+
l)‘h
cos
(15)
$
where the sign of F,, is the same as that of Wmax,h,
x
10 max,h
COS
d’IPh
ICOS
where the sign of Mh is opposite
ti1.h
=
cos-l
4,w,.,p:;h+p
(16)
$1
to that of emax,h, and
-d,h
)
(17)
By using eqns. (4), (5) and (8) - (14), the equations with preload become h
+‘h, Fh
=
for a roller bearing
f
1 cos
&,h(lWt&U,h
I+!’ -
+pd,
h)‘h
x
- F I,h
x
where the
w
Sign
max,
h cos
$1
+
l)QJ cos $
of Fh is the same as that of Wmax,h,
(18)
53
+h,
Mh = *
h
c K1 h(lWmax,hl cos$ -+Pd,h)Qh x ’
_\Ll,h
x 10max,h COS $‘IphICOS$‘I where the sign of M,, is opposite +l,h
= cos-l
$l,h
=
( 4,;;;x.,,)
(19)
to that of Omax,h, and for
ibd,hl
G
41wrnax,hl
for @d,h! 2
4iWmax,hl
cw
180”
’ is the additional compression at one roller and race interface due to ( wmax external loading. It should be noted that, in the way in which the equations are assembled, W,,, E IV,,,‘, but if the actual loading on the bottom roller is to be calculated, w,,, = 1w,,,‘i - +pd.) Equations (1) - (3) and (15) - (20), together with the J-program, enable a generalized shaft and bearing system and the pressure distribution on the worst-loaded roller in each bearing to be obtained.
6. Design procedure The method assumes that the given shaft-bearing has been roughly designed by some standard procedure based on formulae or graphs [ 61. Thus first estimates of the radial deflection and shaft slope at every support are available. The J-program is then run for a range of loads and misalignments for each type of roller used in such a way that its results will produce regression equations that will not require extrapolation for the final solution. The sets of data produced by the J-program are fed into another program which estimates the regression constants and solves the shaft-bearing equilibrium equations. The output of this program specifies system equilibrium conditions with respect to the worst-loaded roller of each bearing and gives rollerto-race deflection and misalignment. The J-program is then used again to obtain the final pressure distributions. The maximum pressures on each of these worst-loaded rollers will determine whether the bearing chosen is suitable or not. If a maximum pressure is not acceptable by exceeding some design stress level, a bearing with rollers of different profile or a larger bearing is used at conspicuous support locations and the procedure is repeated until a satisfactory design is obtained.
7. Design examples 7.1. A shaft supported by two roller bearings with an offset radial load The bearings (Fig. 4(a)) had the specifications given in Table 1. As some experiments were to follow, the load P was varied from 490.5 to 3433.5 N. The corresponding inputted misalignments of the worst-loaded
20
(d)
Em
-LO
of uaLy -3Q
J
-2 0
AWL
-10
PamON
00 Im43
IQ
20
30
L-Q
EtaoFROtLER
Fig. 4. (a) Two-bearing system; (b) theoretical shaft deflection contours and experimental. points for various loads (x, 490.5 N; +, 981.0 N;A, 1471.5 N; 0, 1962.5 N; 0, 2452.5 N; 0, 2943.0 N; 0, 3433.5 N); (c) pressure distribution (bearing 1); (d) pressure distribution (bearing 2).
(b)
(cl
lGNlm2I
PREURE
55 TABLE 1 Nominal outside diameter (outer ring} ~ominu~ bore diameter (inner ring) Number of rollers Diametral clearance Roller length Roller diameter Flat region Dab~ff radias Shaft material
0.052
m 0.025 m 12 ZO-30pm 0.007144 m 0.007144 m 0.006078 m 0.000533 m En24 S steel
roller ranged from 0” to 0.2”. The computer solution gives the load and moment on the worst-loaded roller of each bearing, together with its combined deflection and misalignment with the race. Furthermore, relevant radial pressure distributions are supplied for the design load P as well as shaft centre-line deflections for the range of P chosen. The full lines in Fig. 4(b) give the theoretical shaft deflections as given by the computer solution for various loads, while Figs. 4(c) and 4(d) give the pressure distributions on the worst-loaded roller of bearing 1 and bearing 2 respectively for a radial load on the shaft of 3433.5 N. 7.2. Experiments
Experimental confirmation of the results was obtained from a specially designed universal static deflection apparatus, which is fully described in ref. 8. Up to three bearings could be accommodated to support a single shaft. Approximate point loading was supplied using Bowden cable loops tensioned through lever arms at up to two chosen locations. Rails over the shaft enabled a capacitance transducer to scan its top surface to within 0.027 m of each bearing vertical centre-line (accuracy, +0.65 I.trn (the maximum departure from an ideal straight line calibration); resolution, 10T2 ym). A comparison of the theoretical and experimental shaft deflection profiles, for the first example, is shown in Fig. 4(b). To compare with theory, the bearings were arranged with a roller at $ = 0. The results include the deflection of part of the unloaded overhang beyond bearing 2. For the highest experimental load applied, the computed pressure distributions on the bottom rollers of bearings 1 and 2 are shown in Figs. 4(c) and 4(d) respectively. The co~esponding pressure distribution based on the plain strain theory of Harris [6] is shown for comparison for bearing 1. It should be noted that there is a considerable underestimation of the maximum pressure on the basis of Harris’ theory. 7.3. A shaft supported by three roller bearings with two radial loads As another example we take the case of a shaft supported by three widely separated roller bearings (Fig. 5(a)). This is unusual in engineering practice, as normally in such arrangements two of the bearings are close to
56
57
each other and are often of different types, a typical example being the supports for machine tool spindles. Nevertheless, this example serves to test the accuracy and flexibility of the computer programs. The bearings used were the same as those of the previous example. The load PI was kept constant at 981.0 N and Pz was varied from 0 to 2943.0 N. Figure 5(b) shows the theoretical and experimental shaft deflection profiles. There is clearly a poorer correlation between theory and experiment, especially between bearings 1 and 2. Nevertheless, the general deformed shape of the shaft is obvious from the results. In both the examples given, the unloaded overhang shape was linear as would be expected from a deflection due to slope only. The pressure distributions of the worst-loaded rollers of bearings 2 and 3 are also shown in Figs. 5(c) and 5(d) for Pz = 2943 N. Because bearing 2 is loaded on both sides, it suffers relatively little mis~i~ment, the pressure concentrations being predominantly due to its reaction load Q. The slight aberrations before the pressure peaks are caused by computing errors and can be reduced if more pressure elements are taken.
8. conclusion The pressure distribution on the worst-loaded rollers in shaft-roller bearing systems has been calculated. The theory used has avoided the usual design assumptions of plain strain and linear load-deflection relationships in the rollers. The maximum static pressures incurred generally exceed those forecast by simpler theories and will therefore enable more confident predictions to be made for the bearing fatigue life.
Nomenclature
cA,
E
CB, CC, CD,
Fh
h
Ih
k K1, K,,tz, ih
max k’fh
K2 g2,h
CE, CF,
CG
distance between the point of application of a load and the bearing centre to its left for the kth applied load on shaft segment h loading constants for shaft segment h Young’s modulus for the shaft material radial force at bearing location h subscript denoting either the hth bearing or the hth shaft segment in a bearing-shaft system second moment of area of the cross section of shaft segment h summation index regression coefficients (constants for a given roller) K1, K2 referring to the rollers of bearing h shaft length between bearing centres for segment h subscript denoting a variable relating to the most highly loaded roller in a roller bearing moment at bearing location h
58
pd pd, h ph, k
QIL
Th,k Wmax w max’ Wmax,h We
Q, P,Y,
5
6 max
6 max
eh 8
max
moment on a roller at angular location $ in a roller bearing number of applied loads on a shaft between two bearings or the total number of bearings in a shaft-bearing system bearing diametral clearance the diametral clearance of bearing h the kth applied radial load on shaft segment h radial load on a roller at angular position $J in a roller bearing the kth applied moment load on shaft segment h W for the worst-loaded roller in a roller bearing Wmax in a roller bearing with preload excluding the compression due to the preload Wmax for the hth bearing roller compression at one contact interface with the race at angular position $ regression coefficients (constants for a given roller); when they carry the subscript h, they refer to the rollers of bearing h total compression of the worst-loaded roller in a roller bearing additional total compression of the worst-loaded roller of a roller bearing with preload excluding the total compression of the roller due to the preload shaft deflection at bearing location h total roller compression at angular position $ in a roller bearing shaft rotation at bearing location h angular misalignment of the worst-loaded roller in a roller bearing angular alignment of the worst-loaded roller of bearing h angular rotation of the inner ring of a bearing angular misalignment of a roller at angular position $ in a roller bearing angular extent of the load zone; when carrying the subscript h it refers to bearing h
References 1 A. D. Deutschman, W. J. Michels and C. E. Wilson, Machine Design (Theory and Practice), Macmillan, London, 1975. 2 SKF Hauptkatalog, Katalog 2800T/Dd 6000, April 1970. 3 M. F. Spotts, Design ofMachine Elements, Prentice-Hall, New York, 4th edn., 1971. 4 G. Lundberg and A. Palmgren, Dynamic capacity of rolling bearings, Acta Polytech., Mech. Eng. Ser., 1 (3) (1947) 7. 5 P. M. Johns and R. Gohar, Roller bearings under radial and eccentric loads, Tribal. Int., (June 1981) 131 - 136. 6 T. A. Harris, Rolling Bearing Analysis, Wiley, Chichester, 1966. 7 M. Heydari and R. Gohar, The influence of the axial profile on pressure distribution in radially loaded rollers, J. Mech. Eng. Sci., 21 (6) (1979) 381 - 388. 8 A. Woowat, Roller bearing-shaft systems, 3rd Year Design Project Rep., 1980 (Department of Mechanical Engineering, Imperial College, London).