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The contact angle of various rolling bearings and its effect on bearing selection
The contact angle effect on bearing
of various selection
rolling
bearings
and its
H-H. Schreiber*
The trend for lightweight construction means that the designer must look for new solutions to his design problems. Formerly, a component found to be too weak to cope with increased machine capacities was simply replaced by a stronger one, Now, the demand is for a better one. But what is better? The part is expected to take more load without taking up more space. One approach would be a material of higher strength but that is not the subject of this article. An aIternative is an even greater exploitation of the strength of the material by a more appropriate design of the component. This means that all forces resulting from deviations of the lines of force must be reduced to a minimum.
12
P7
% 10
Two simple examples will elucidate the problem. If, for instance, a pull is transmitted through a simple cylindrical rod the stress is evenly distributed across the whole section. If the rod is replaced by circular chain links higher stresses
&
0
i-
F
p6 A
p5
6
-
\
Fig I Contact angle LYand load angle /3 in an angular ball bearing
contact
60=’ Load
will develop.in them: in other words, the chain link does not employ the material strength as well as the cylindrical rod. All kinds of notches also give rise to stress concentrations, If the design can be modified to avoid sharp recesses or notches the part can accommodate higher loads. The flow of force through a component should therefore always follow a path that keeps the additional stresses as low as possible. If the flow of force is pictured as the path along which the force flows it can be recognized that any deviation from the straight line has an adverse effect on the flow of force. A rectilinear flow of force is in any case the best: however, it is not always possible. So, as always happens in design, a compromise must be made. *Technical Director, Kugelfisher Schweinfurt, West Germany 214
TRIBOLOGY
November
Georg Schafer and Company,
1968
angle
900 P
Fig 2 Magnitude of ball loads in an angular contact ball bearing of (Y = 30” at various load angles fi
THE FLOW OF FORCE IN A ROLLING
BEARING
In a rolling bearing the loads transmitted from the shaft to the housing are taken up by a number of rolling elements. The ball loads in Fig 1 follow the direction of the contact lines: these are the lines connecting the points of contact between balls and raceways. The angle formed by these lines and the bearing radial plane is called contact angle LY. All contact lines intersect at a common point on the bearing
0369 0 =900
0369 0 =750
0369 0=60”
369 R=450
The graph reveals a basic difference between the load curves and the friction curves. Whereas the curves for maximum ball load and dynamic equivalent load drop with increasing load ahgle the friction curves are at a minimum when the load angle is small. With pure axial load ie with 6 = 90” the load is evenly distributed across all the balls (Figs 2 and 3). According to Fig 4 the flow of force is, in this case, an optimum one, with. regard to fatigue life, static load and yield. Friction values are, on the other hand, lower when just a few balls are loaded and if low friction is of particular importance the higher loading on these balls is usually acceptable. To select a suitable actual performance be decided first.
B =400
R=35
0~32~
r3=300
HEARING
bearing for a particular characteristics required
application the must therefore
SELECTION
axis, called the pressure centre. To maintain the equilibrium between ball loads and external load F the latter must also pass through the pressure centre.
In practice the designer cannot select a load angle for a bearing but must select the bearing for a load angle. In other words, a bearing has to be chosen with properties that will give an optimum performance under the given operating conditions. To this effect, the bearing properties mentioned have to be considered in their inter-relation with contact angle (Y, which is the controlling design criterion for the bearing.
The upper part angular contact The ball loads directed at the
The various standard types of rolling contact bearing cover a wide range of contact angles as shown in Fig 5. As a guide to bearing selection it is useful to examine the effect of contact angle on the various performance characteristics of bearings.
Fig 3 The number of load carrying balls and the magnitude of ball loads at various load angles /l in the angular contact ball bearing shown ln Fig 1
of Fig 2 shows the ball complement of an ball bearing with a contact angle (Y = 30”. shown are the reactions to an external load F pressure centre at an angle ,9 = 35”.
The chart in the lower part of Fig 2 shows the ball loads as a function of load angle (between 30” and 90”) at a constant external load. With 6 = 90”, that is under pure axial load, the load is evenly distributed across all the balls. As angle 6 decreases the load on the lower balls initially increases and then decreases (except for the apex ball) while the load on the upper balls becomes steadily smaller. As can be seen a decreasing angle 6 reduces the number of load-carrying balls. Thus down to 6 = 45” all 12 balls are loaded, at /3 = 40” only 7 and at 6 = 32” only 3 balls.
,Maximum
ball
Fig 3 again shows the distribution of the ball loads for 8 different load angles 6. The figures on the abscissa identify the position of the balls as shown in Fig 2; the length of the perpendiculars drawn above the figures indicates the magnitude of the ball load. The distribution of the ball loads as shown for angular contact ball bearings has also been calculated for deep groove ball bearings and roller bearings. With this the flow of force in a rolling bearing is known. Now, what is the optimum flow of force under given operating conditions? The question is: optimum is which respect? EFFECTS
Rolling
2 3 4
friction
OF THE FLOW OF FORCE
Rolling bearings are characterized by properties such as fatigue behaviour, yield and rigidity, friction and wear, and high-speed behaviour. These properties must therefore be kept in mind when considering whether, under the given external conditions, the flow of force is optimum: in other words whether the capabilities of the bearing are fully exploited. The four curves plotted in Fig 4 against load angle P refer to an angular contact ball bearing with a contact angle cr = 30”. They show: 1
Load
Maximum ball load. This is indicative of the static capacity and the yield Dynamic equivalent load, which is used in calculating the fatigue life Rolling friction Hysteresis. 3 and 4 determine friction and wear
The ordinate of Fig 4 was left unscaled since the graph is only intended to show the trend of the variables. The curves apply to a constant external load.
Dynamic
0"
300 -Load
angle
equivalent
load
90”
60’ p--c
Fig 4 Maximum ball load, dynamic equivalent load, hysteresis and rolling friction at various load angles for angular contact ball bearing where (Y = 30” TRIHOLOGY
November
1968
215
,I=!
(ala=O”
:. R
I~ R
(c)a=35”
(b)a=15”
(dla=35”
”
R
(fl a=60°
(e)a =40”
(g)a=90”
r7
F7
(k)a=lS’
(j)a=lO”
(hja-0”
a=20
75”
60”
(i)a=O” -6 5j
lti
(Ua=50"
h)a=so”
Fig 5 Standardized rolling bearing types with their contact angles a Deep groove ball bearing b Self-aligning ball bearing c Four-point-contact ball bearing d Double row angular-contact ball bearing e Single row angular-contact ball bearing f Angular contact thrust ball bearing g Thrust ball bearing h Cylindrical roller bearing i Barrel type roller bearing j Spherical roller bearing k Tapered roller bearing 1 Spherical roller thrust bearing m Cylindrical roller thrust bearing
I
I
30”
60” Load
angle
9
(3
Fig ‘7 Maximum ball load versus load 6for angular contact ball bearing with different contact angles LY
a=20°
30°
40°
50°
6o”
75O
Effect on fatigue Fig 6 shows the dynamic equivalent load versus load angle for ball bearings with different contact angles. The graph indicates which contact angle and which bearing will provide, for a given direction of outside load, the lowest equivalent load and thus the longest fatigue life. According to the graph the deep groove ball bearing with cy = 0” should be used for a 0 -42” load angle range, the angular contact ball bearing with (Y = 40” for load angles between 42” and 70” the angular contact thrust ball bearing with LY = 60” for load angles from 70” to 90” and the thrust ball bearing with o! = 90” for pure thrust load. Effect on static capacity and yield Similarly, information can be obtained from the curves in Fig 7 on the best contact angle/load angle relationship for angular contact ball bearing to achieve the lowest maximum ball load. This maximizes the static capacity and minimizes yield.
I
00
300 Lood
angle
60° 13
900
F’ig 6 Equivalent load versus load angle 6 for ball bearings with different contact angles (Y 216
TFtIBOLOGY
November
1968
Effect on friction As shown in Fig 4 the pattern of the friction curves is entiraly different from that of the load curves. Similarly to Figs 6 and 7, Fig 8 shows rolling friction versus load angle 6 for angular contact ball bearings with different contact angles. It appears that the lowest rolling friction values are obtained at a load angle p about 5” larger than the pertinent contact angle (Y. This favourable flow of force makes for low wear in addition to minimum friction.
HIGH-SPEED
BEHAVIOUR
When consider&z the oatimum flow of force the behaviour of the various bearing &es at higher speeds must not be overlooked. For each rolling bearing a so-called limiting speed is given in the catalogues which should not be exceeded for usual operating conditions. This limit is primarily a function of operating
temperature which, in turn, depends largely on friction and heat dissipation; it is however also influenced by rolling element kinematics. At higher speeds the effect of gyroscopic moments of the balls becomes noticeable. These moments tend to spin the balls as shown in Fig 9. This moment is countered by a frictional moment. Since the gyroscopic moment increases with the square of the speed, whereas friction hardly changes with
Fig 10 Skid marks
O0
30’
60” Load
angle
90’
on the raceway
of a thrust ball bearing
3.01
f3 2.0-
Fig 8 Rolling friction versus load angle 8for angular contact ball bearings with different contact angles LY
1.5 1.0 o-7,0.5 -
0.3 0.2 -
Fig 9 Gyroscopic moment in an angular contact ball bearing. This moment tends to spin the ball. 3o”
Fig 11 Limiting bearings.
LO’ 50” 60’ Contact angle a
speed and minimum
TRIBOLOGY
70”
80”
90”
thrust load for ball
November
1968
217
speed, the gyroscopic moment will dominate at higher speeds. This induces ball skidding leading to the formation of skid marks on the raceways (Fig 10). If the bearing is still to function at high speed the friction moment must be increased. However, since the friction coefficient is fairly constant the only way to increase the friction moment is by adding more thrust load. Limiting speed and thrust load are therefore related. In Fig 11 the upper curve shows how at constant thrust load the limiting speed, relative to the limiting speed of a thrust ball bearing, increases with a decreasing contact angle. According to the chart an angular contact thrust ball bearing with o = 60” has a 30% higher limiting speed than a thrust ball bearing with (Y = 90” For a radial angular contact ball bearing with a contact angle (Y = 40” the increase is 80%. The lower curve of the same graph reveals that, fn order to obtain a given constant speed, the required thrust load decreases with a decreasing contact angle o. The diagram explains the fact that, contrary to all that has been said about the flow of force, deep groove ball bearings and angular contact ball bearings are frequently installed for pure thrust loads in those applications where speeds are high at low loads.
218
THIBGLGGY
November
1968
THE OPTIMUM
BEARING
Based on what has been said above about the relationship between the contact angle and the flow of force and with the knowledge of their effect on bearing performance, a specific rolling bearing might be developed for each individual application. Its design would meet exactly the operating conditions prevalent in this particular application. Such a bearing would however lack a wide application range and would be impractical as a standardized machine component. Economic large-series production would hardly be possible and subsequent replacement problematic. There is available a wide range of standardized bearing types with various contact angles which allows a bearing of optimum design to be selected, that is a bearing, which by its type, contact angle, speed and other characteristics matches the requirements of the application. ACKNOWLEDGEMENT The material in this article was originally published in ‘Ball and Roller Bearing Engineering’ and we are grateful to the FAG Bearing Company Ltd for permission to reproduce it.
of various selection
rolling
bearings
and its
H-H. Schreiber*
The trend for lightweight construction means that the designer must look for new solutions to his design problems. Formerly, a component found to be too weak to cope with increased machine capacities was simply replaced by a stronger one, Now, the demand is for a better one. But what is better? The part is expected to take more load without taking up more space. One approach would be a material of higher strength but that is not the subject of this article. An aIternative is an even greater exploitation of the strength of the material by a more appropriate design of the component. This means that all forces resulting from deviations of the lines of force must be reduced to a minimum.
12
P7
% 10
Two simple examples will elucidate the problem. If, for instance, a pull is transmitted through a simple cylindrical rod the stress is evenly distributed across the whole section. If the rod is replaced by circular chain links higher stresses
&
0
i-
F
p6 A
p5
6
-
\
Fig I Contact angle LYand load angle /3 in an angular ball bearing
contact
60=’ Load
will develop.in them: in other words, the chain link does not employ the material strength as well as the cylindrical rod. All kinds of notches also give rise to stress concentrations, If the design can be modified to avoid sharp recesses or notches the part can accommodate higher loads. The flow of force through a component should therefore always follow a path that keeps the additional stresses as low as possible. If the flow of force is pictured as the path along which the force flows it can be recognized that any deviation from the straight line has an adverse effect on the flow of force. A rectilinear flow of force is in any case the best: however, it is not always possible. So, as always happens in design, a compromise must be made. *Technical Director, Kugelfisher Schweinfurt, West Germany 214
TRIBOLOGY
November
Georg Schafer and Company,
1968
angle
900 P
Fig 2 Magnitude of ball loads in an angular contact ball bearing of (Y = 30” at various load angles fi
THE FLOW OF FORCE IN A ROLLING
BEARING
In a rolling bearing the loads transmitted from the shaft to the housing are taken up by a number of rolling elements. The ball loads in Fig 1 follow the direction of the contact lines: these are the lines connecting the points of contact between balls and raceways. The angle formed by these lines and the bearing radial plane is called contact angle LY. All contact lines intersect at a common point on the bearing
0369 0 =900
0369 0 =750
0369 0=60”
369 R=450
The graph reveals a basic difference between the load curves and the friction curves. Whereas the curves for maximum ball load and dynamic equivalent load drop with increasing load ahgle the friction curves are at a minimum when the load angle is small. With pure axial load ie with 6 = 90” the load is evenly distributed across all the balls (Figs 2 and 3). According to Fig 4 the flow of force is, in this case, an optimum one, with. regard to fatigue life, static load and yield. Friction values are, on the other hand, lower when just a few balls are loaded and if low friction is of particular importance the higher loading on these balls is usually acceptable. To select a suitable actual performance be decided first.
B =400
R=35
0~32~
r3=300
HEARING
bearing for a particular characteristics required
application the must therefore
SELECTION
axis, called the pressure centre. To maintain the equilibrium between ball loads and external load F the latter must also pass through the pressure centre.
In practice the designer cannot select a load angle for a bearing but must select the bearing for a load angle. In other words, a bearing has to be chosen with properties that will give an optimum performance under the given operating conditions. To this effect, the bearing properties mentioned have to be considered in their inter-relation with contact angle (Y, which is the controlling design criterion for the bearing.
The upper part angular contact The ball loads directed at the
The various standard types of rolling contact bearing cover a wide range of contact angles as shown in Fig 5. As a guide to bearing selection it is useful to examine the effect of contact angle on the various performance characteristics of bearings.
Fig 3 The number of load carrying balls and the magnitude of ball loads at various load angles /l in the angular contact ball bearing shown ln Fig 1
of Fig 2 shows the ball complement of an ball bearing with a contact angle (Y = 30”. shown are the reactions to an external load F pressure centre at an angle ,9 = 35”.
The chart in the lower part of Fig 2 shows the ball loads as a function of load angle (between 30” and 90”) at a constant external load. With 6 = 90”, that is under pure axial load, the load is evenly distributed across all the balls. As angle 6 decreases the load on the lower balls initially increases and then decreases (except for the apex ball) while the load on the upper balls becomes steadily smaller. As can be seen a decreasing angle 6 reduces the number of load-carrying balls. Thus down to 6 = 45” all 12 balls are loaded, at /3 = 40” only 7 and at 6 = 32” only 3 balls.
,Maximum
ball
Fig 3 again shows the distribution of the ball loads for 8 different load angles 6. The figures on the abscissa identify the position of the balls as shown in Fig 2; the length of the perpendiculars drawn above the figures indicates the magnitude of the ball load. The distribution of the ball loads as shown for angular contact ball bearings has also been calculated for deep groove ball bearings and roller bearings. With this the flow of force in a rolling bearing is known. Now, what is the optimum flow of force under given operating conditions? The question is: optimum is which respect? EFFECTS
Rolling
2 3 4
friction
OF THE FLOW OF FORCE
Rolling bearings are characterized by properties such as fatigue behaviour, yield and rigidity, friction and wear, and high-speed behaviour. These properties must therefore be kept in mind when considering whether, under the given external conditions, the flow of force is optimum: in other words whether the capabilities of the bearing are fully exploited. The four curves plotted in Fig 4 against load angle P refer to an angular contact ball bearing with a contact angle cr = 30”. They show: 1
Load
Maximum ball load. This is indicative of the static capacity and the yield Dynamic equivalent load, which is used in calculating the fatigue life Rolling friction Hysteresis. 3 and 4 determine friction and wear
The ordinate of Fig 4 was left unscaled since the graph is only intended to show the trend of the variables. The curves apply to a constant external load.
Dynamic
0"
300 -Load
angle
equivalent
load
90”
60’ p--c
Fig 4 Maximum ball load, dynamic equivalent load, hysteresis and rolling friction at various load angles for angular contact ball bearing where (Y = 30” TRIHOLOGY
November
1968
215
,I=!
(ala=O”
:. R
I~ R
(c)a=35”
(b)a=15”
(dla=35”
”
R
(fl a=60°
(e)a =40”
(g)a=90”
r7
F7
(k)a=lS’
(j)a=lO”
(hja-0”
a=20
75”
60”
(i)a=O” -6 5j
lti
(Ua=50"
h)a=so”
Fig 5 Standardized rolling bearing types with their contact angles a Deep groove ball bearing b Self-aligning ball bearing c Four-point-contact ball bearing d Double row angular-contact ball bearing e Single row angular-contact ball bearing f Angular contact thrust ball bearing g Thrust ball bearing h Cylindrical roller bearing i Barrel type roller bearing j Spherical roller bearing k Tapered roller bearing 1 Spherical roller thrust bearing m Cylindrical roller thrust bearing
I
I
30”
60” Load
angle
9
(3
Fig ‘7 Maximum ball load versus load 6for angular contact ball bearing with different contact angles LY
a=20°
30°
40°
50°
6o”
75O
Effect on fatigue Fig 6 shows the dynamic equivalent load versus load angle for ball bearings with different contact angles. The graph indicates which contact angle and which bearing will provide, for a given direction of outside load, the lowest equivalent load and thus the longest fatigue life. According to the graph the deep groove ball bearing with cy = 0” should be used for a 0 -42” load angle range, the angular contact ball bearing with (Y = 40” for load angles between 42” and 70” the angular contact thrust ball bearing with LY = 60” for load angles from 70” to 90” and the thrust ball bearing with o! = 90” for pure thrust load. Effect on static capacity and yield Similarly, information can be obtained from the curves in Fig 7 on the best contact angle/load angle relationship for angular contact ball bearing to achieve the lowest maximum ball load. This maximizes the static capacity and minimizes yield.
I
00
300 Lood
angle
60° 13
900
F’ig 6 Equivalent load versus load angle 6 for ball bearings with different contact angles (Y 216
TFtIBOLOGY
November
1968
Effect on friction As shown in Fig 4 the pattern of the friction curves is entiraly different from that of the load curves. Similarly to Figs 6 and 7, Fig 8 shows rolling friction versus load angle 6 for angular contact ball bearings with different contact angles. It appears that the lowest rolling friction values are obtained at a load angle p about 5” larger than the pertinent contact angle (Y. This favourable flow of force makes for low wear in addition to minimum friction.
HIGH-SPEED
BEHAVIOUR
When consider&z the oatimum flow of force the behaviour of the various bearing &es at higher speeds must not be overlooked. For each rolling bearing a so-called limiting speed is given in the catalogues which should not be exceeded for usual operating conditions. This limit is primarily a function of operating
temperature which, in turn, depends largely on friction and heat dissipation; it is however also influenced by rolling element kinematics. At higher speeds the effect of gyroscopic moments of the balls becomes noticeable. These moments tend to spin the balls as shown in Fig 9. This moment is countered by a frictional moment. Since the gyroscopic moment increases with the square of the speed, whereas friction hardly changes with
Fig 10 Skid marks
O0
30’
60” Load
angle
90’
on the raceway
of a thrust ball bearing
3.01
f3 2.0-
Fig 8 Rolling friction versus load angle 8for angular contact ball bearings with different contact angles LY
1.5 1.0 o-7,0.5 -
0.3 0.2 -
Fig 9 Gyroscopic moment in an angular contact ball bearing. This moment tends to spin the ball. 3o”
Fig 11 Limiting bearings.
LO’ 50” 60’ Contact angle a
speed and minimum
TRIBOLOGY
70”
80”
90”
thrust load for ball
November
1968
217
speed, the gyroscopic moment will dominate at higher speeds. This induces ball skidding leading to the formation of skid marks on the raceways (Fig 10). If the bearing is still to function at high speed the friction moment must be increased. However, since the friction coefficient is fairly constant the only way to increase the friction moment is by adding more thrust load. Limiting speed and thrust load are therefore related. In Fig 11 the upper curve shows how at constant thrust load the limiting speed, relative to the limiting speed of a thrust ball bearing, increases with a decreasing contact angle. According to the chart an angular contact thrust ball bearing with o = 60” has a 30% higher limiting speed than a thrust ball bearing with (Y = 90” For a radial angular contact ball bearing with a contact angle (Y = 40” the increase is 80%. The lower curve of the same graph reveals that, fn order to obtain a given constant speed, the required thrust load decreases with a decreasing contact angle o. The diagram explains the fact that, contrary to all that has been said about the flow of force, deep groove ball bearings and angular contact ball bearings are frequently installed for pure thrust loads in those applications where speeds are high at low loads.
218
THIBGLGGY
November
1968
THE OPTIMUM
BEARING
Based on what has been said above about the relationship between the contact angle and the flow of force and with the knowledge of their effect on bearing performance, a specific rolling bearing might be developed for each individual application. Its design would meet exactly the operating conditions prevalent in this particular application. Such a bearing would however lack a wide application range and would be impractical as a standardized machine component. Economic large-series production would hardly be possible and subsequent replacement problematic. There is available a wide range of standardized bearing types with various contact angles which allows a bearing of optimum design to be selected, that is a bearing, which by its type, contact angle, speed and other characteristics matches the requirements of the application. ACKNOWLEDGEMENT The material in this article was originally published in ‘Ball and Roller Bearing Engineering’ and we are grateful to the FAG Bearing Company Ltd for permission to reproduce it.