NTN TECHNICAL REVIEW No.75（2007）
[ Technical Paper ]
Dynamic Analysis for Needle Roller Bearings
Under Planetary Motion
Tomoya SAKAGUCHI*
A dynamic analysis tool for needle roller bearings in planetary gear
systems has been developed. This tool uses two-dimensional analysis,
considering three degrees of freedom for of both rollers and cage and two
translational degrees of freedom for the planet gears on the radial plane. In
addition, elastic deformation of the cage can be simulated using a
Component-Mode-Synthesis method in order to evaluate cage stress.
Numerical results on the maximum principal stress of the cage using this
dynamic analysis tool indicates that the effect of the planet gear orbital
rotation speed on the cage stress is higher than that of the planet gear
rotation speed and the maximum cage stress is in nearly proportional to the
square of the orbital rotation speed. The reason is that stress on the base of
the cage bar increases in the unloaded zone due to supporting the roller
centrifugal force induced by the orbital motion of the planet gear. In addition, relatively high stresses of the cage
were observed in the two following cases: the roller just passing the load zone and accelerates due to centrifugal
force and collides with the front bar, and cage bar contact force due to a roller in the load zone opposes the
cage moments generated by rollers in the non-load zone.
author developed, on general-purpose mechanism
analysis software, a dynamic analysis tool for a
tapered roller bearing that is capable of considerations
about six degrees of freedom of motion of the cage
and rollers as well as elastic deformation of the cage2).
The author previously reported that the trajectory and
stress output from this tool coincide well with the test
result 2).
The above-mentioned dynamic analysis tool for
tapered roller bearings has been applied to needle
roller bearings under planetary motion. The stress
acting on bearing cages was analyzed; wherein, the
dynamic analysis was limited to three degrees of
freedom on the radial plane of bearing in order to
attain better analysis efficiency.
1. Introduction
A rolling bearing consists of a pair of bearing rings,
a number of rolling elements situated between the
bearing rings, and a cage that separates the rolling
elements at equal intervals. A cage will not fail in an
ordinary application; however, a cage can fail in an
application that accompanies a varying load or
planetary motion. To prevent a failure, a cage must be
provided with greater mechanical strength;
unfortunately, determination of forces acting on a cage
through a test is difficult. Therefore, a dynamic
analysis technique for a rolling bearing which
considers cage behavior will offer a useful solution to
this challenge.
ADORE 1) by Gupta is a known dynamic analysis
technique for rolling bearings that positively reflects
cage motions but, the cage assumed in this technique
needs to be a simple-shaped solid body. Furthermore,
this technique is not capable of directly determining a
cage stress. If we attempt to determine a cage stress
through FEM-based structural analysis, it will be
difficult to rationally define the load and supporting
conditions for the cage. To address this issue, the
2. Dynamic analysis model
The planetary gear mechanism to be analyzed is
schematically illustrated in Fig. 1. There are seven
assumptions about his model, wherein, the model can
be simplified as shown in Fig. 2 due to these
assumptions.
*Elemental Technological R&D Center
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Dynamic Analysis for Needle Roller Bearings Under Planetary Motion
transmission torque (Fig. 2).
6. The cage is assumed to be an elastic body based
on the Component-Mode-Synthesis method 4) in
order to take into account its elastic deformation. In
other words, the elastic deformation of the cage is
expressed as the sum of the elastic deformation in
inherent deformation mode and that in restrained
deformation mode. All other members are each
assumed to be a solid member; and local elastic
contact is considered for regions where interference
can occur owing to their geometrical shapes.
7. Generally, three or more planet gears are situated
on the carrier. Assuming that all the planet gears
are equivalent with each other, one planet gear only
is analyzed. For convenience of analysis, the
weight of planet gears is ignored.
Fig. 1 Actual forces and motions on a planet gear
Based on these assumptions, it will be possible, like
with a conventional dynamic analysis procedure for
bearings 2), to analyze needle bearings in a dynamic
planet gear. This analysis is done by describing the
contact forces, frictional forces and rolling viscosity
resistances occurring between the rollers and raceway
and the rollers and cage on the general-purpose
mechanism analysis software “MSC.Adams” 5).
To be able to assess these interference forces, the
author has again used a model for the 2D dynamic
analysis for cylindrical roller bearings 3). Despite
change in the number of dimensions, the basic
concept for tapered roller bearings 2) is same as that
for the interference model in the present study.
The cage as an elastic body was treated in a
manner essentially same as that for a cage in a
tapered roller bearing 2). However, for the current
analysis, a 2D cage model was developed: as
illustrated in Fig. 3. A 3D geometrical cage model was
cut at the middle cross-sectional plane of the bearing
and constaints for the 2D plane were set up on this
plane. Furthermore, the boundary point in the
constraint deformation mode per the ComponentMode-Synthesis method 4) was situated at the
midpoint of each bar of the cage on the above
mentioned cross sectional plane. The cage mass on
the general-purpose mechanism analysis software
package is half as that of the actual cage. To cope
with this situation, the other constituting elements
were cut at their center in the axial direction so that
the mass and inertial moment of each constituting
element are half those on the actual cage.
Accordingly, the magnitude of the interference force,
such as a contact force or frictional force, acting
between these elements was assumed to be a half the
actual forces.
It was assumed that the interference force from a
Fig. 2 Analyzed forces and motions on a planet gear
1. The analysis technique adopted is a twodimensional dynamic analysis where degrees of
freedom are limited only within a radial plane.
2. Apparent forces including the centrifugal force are
taken into account.
3. The carrier rotates at a constant speed about a
fixed axis.
4. The outer ring raceway (planet gears) has two
translational degrees of freedom, provided that the
rotation speed of the planet gears is known.
5. It is assumed that the radial internal clearance on
the bearings is smaller than the mesh clearance on
the gearing and all the centrifugal force on the
planet gears acts on the rolling bearings. More
specifically, it is assumed that the sum of
interference forces from the sun gear and that from
the ring gear each acting on the planet gears is
equivalent only to the reaction force of the
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NTN TECHNICAL REVIEW No.75（2007）
roller acts on a corresponding pocket on the contact
surface at the center of the rib. The maximum major
cage stress often occurs at the base of a bar of cage
pocket, and as a result, the above-mentioned
assumption tends to slightly overestimate the stress.
Since the cage in question is guided by the outer
ring raceway, the contact force and frictional force
acting between the cage and the outer ring were
evaluated. This contact force was calculated at three
particular points along the cage outer surface, for
each cage pocket, as shown in Fig. 4. Each contact
force value was calculated based on the amount of
geometrical indentation into the outer ring raceway.
The elastic deformation characteristics of the cage
were calculated using, the “I-deas NX Series*1” (UGS)6),
which is an FEM software package. For dynamic
calculation, a mechanism analysis software package,
“MSC.Adams” 5), as well as an optional package,
“ADAMS/Flex” 5), which is an optional variant of
MSC.Adams, were used. For stress evaluation,
“ADAMS/Durability” 5), which is another optional
variant of MSC.Adams, was used.
3. Examples of analysis
The technical data for the planetary gear system
and needle roller bearings used for the present study
are summarized in Table 1. The cage used was an
outer ring guided steel cage (model KMJ-S) fabricated
through a press-forming and welding process.
Operating
conditions
Bearing
Planetary
technical data gear system
Table 1 Specifications of needle roller bearing and
planetary gear system
Pitch circle diameter of sun gear, mm
Pitch circle diameter of planet gears, mm
Pitch circle diameter of ring gear, mm
Mass of planet gears, kg
184.2
57.4
299.0
0.045
Outer ring raceway dia., mm
Inner ring raceway dia., mm
Roller dia., mm
Roller length, mm
Number of rollers
19.85
13.85
2.997
13.8
11
Orbital rotation speed of planet gears Nc, min-1
Rotation speed of planet gears Np, min-1
Torque transmitted per planet gear T, N･m
Lubricating oil
Typical temperature of lubricating oil, ˚C
5940
25000
16, 32
ISO VG100
120
4. Stress generating mechanism of the cage
The analysis result shown in Fig. 5 was obtained at
the instant where the orbital rotation speed of the
planet gears was at 5940 min-1 and the major cage
stress was at the maximum, wherein the direction of
motion of the related components is same as that
shown in Fig. 2. In Fig. 5, the centrifugal force is in
the upward direction. The maximum major stress
occurred at the base of pocket bar when a roller in the
non-load zone came into contact with this bar. The
contact force triggering this stress was induced by the
centrifugal force on the roller.
To be able to review the state of contact between
the rollers and cage on a running bearing, the
maximum stress history is illustrated in Fig. 6. The
stress values presented are non-dimensional values
obtained by dividing by the fatigue strength of the
cage material. To be able to locate the load zone, the
displacements on the adjacent rollers along the radial
direction of the planetary gear system are shown in
this graphical representation. The above-mentioned
maximum stress occurs because a cage bar exiting
the load zone will be subjected to the centrifugal force
of a roller in the non-load zone, thereby the a tensile
stress and a compression stress sequentially occur in
the right and left non-load zones as shown by 1 and
2 in Fig. 7. This maximum stress occurs at the .0173
point in Fig. 6. In the load zone, the frictional force
Fig. 3 Cage elastic model introduced into twodimensional dynamic analysis
Fig. 4 Locations of interaction forces on cage outside
*1: The I-deas NX Series is a trademark or registered
trademark of UGS Corp. or its subsidiary in USA and
other nations.
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Dynamic Analysis for Needle Roller Bearings Under Planetary Motion
Dimensionless stress
2.0
2.0
1.0
1.0
0.0
0.0
-1.0
-1.0
-2.0
0.015
0.0175
0.02
0.0225
-2.0
0.025
Dimensionless displacement
on the cage is generated due to the frictional force
between the cage outer surface and the outer ring
raceway as well as the centrifugal force on the roller in
the non-load zone.
In Fig. 6, a relatively large peak in tensile stress is
present immediately after exiting the load zone. This is
because around the end of the load zone, the roller is
situated at the trailing end of the pocket and after
exiting the load zone, the roller accelerates due to the
centrifugal force acting on it and hits the bar in front of
it (4 in Fig. 7).
occurring from the raceway is greater than the
centrifugal force occurring on the roller, thereby the
centrifugal force on the roller does not act on the
corresponding cage bar. At the 0.0212 s point in Fig.
6, a compressive force appears to be present. This
means that in the latter half of the load zone, the bar is
in contact with the roller in front of it (3 in Fig. 7), and
as a result of this situation, a decelerating moment is
exerted onto the cage. This decelerating moment
occurs because an accelerating moment occurs in the
non-load zone on the cage. The accelerating moment
Time sec.
Fig. 6 Stress histories of a cage pocket bar and the
adjacent rollers displacements along the radial direction
of planetary gear system
a) all over the bearing
b) local view near the maximum principle stress point
Fig. 5 Analysis result of a needle roller bearing under
planetary motion (Nc: 5940 min-1, T = 32 N･m)
Fig. 7 Principal mechanisms to make cage stress rise under
planetary motion
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NTN TECHNICAL REVIEW No.75（2007）
torque T was 32 Nm and the orbital rotation speed
was 7000 min-1. This situation was true with the
phenomenon 4 in Fig. 7. However, when the orbital
rotation speed was 8000 min-1, a different result was
obtained as the collision shown in the phenomenon 4
in Fig. 7 was less significant partly because the load
zone position shifted due to an increase in the
centrifugal force acting on the planet gears.
Next, the outcome from the reversed transmission
torque direction was studied. Consequently, it has
been found that, at the rotation speed range of 5000
min-1 or lower, the stress at T = –32 Nm is greater
than that at T = –16 Nm. Additionally, at the rotation
speed range of 6000 min-1 or higher, the stress at T =
–32 Nm is smaller than that at T = –16 Nm. In a higher
rotation speed range, and when T= –16 Nm, a stress
of greater magnitude occurred from collision in the
situation 4 in Fig. 7. However, when T = –32 Nm, the
roller at the trailing end of the load zone has already
begun to come into contact with the cage bar in front
of it in the orbital rotation direction of the planet gears.
As a result, the stress was relatively smaller (show in
Fig. 8) due to a smaller difference in relative
speed between the roller and cage car.
5. Effects of the operating conditions
of the planetary gear system onto
the cage stress
Fig. 8 plots the interrelation between the maximum
cage stress and the orbital rotation speed of the planet
gears. Assuming that the magnitude of the
transmission torque T was 32 Nm or 16 Nm, the
calculation result has been plotted. For transmission
torques in the reverse direction, the calculation result
has also been plotted. Though the stress varies
depending on the magnitude and direction of
transmission torques, the cage stress generally
increases with a greater orbital rotation speed of the
planet gears. As discussed in the previous section,
there is a close interrelation between the centrifugal
force on a roller and the cage stress. In Fig. 8, the
stress is roughly in proportion with the square of an
orbital rotation speed of planet gears.
A greater stress occurred when the transmission
2.5
T : 32
Dimensionless stress
2.0
T : 16
T : -32
1.5
T : -16
1.0
0.5
0.0
0
2000
4000
6000
8000
10000
Planet gear orbital rotation speed Nc, min-1
Fig. 8 Cage stress with various speeds of planet gear orbital rotation
and transmitted torque
2.5
Np : +, T : 32
Dimensionless stress
2.0
Np : -, T : 32
Np : +, T : -32
1.5
Np : -, T : -32
1.0
0.5
0.0
0
20000
40000
60000
80000
Planet gear rotation speed Np, min-1
Fig. 9 Cage stress with various speeds of planet gear rotation and
transmitted torque
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As can be derived from Fig. 8, the factor
most apparently affecting the cage stress at
a given planet gear orbital rotation speed is a
collision phenomenon (4 in Fig. 7) between
the roller immediately after exiting the load
zone and the corresponding cage bar.
Assuming that the magnitude of the
transmission torque onto the maximum cage
stress could be ignored, the result at
approximately 7000 min-1 was considered.
Consequently, it was learned that the stress
at T = 32 where collision occurred is 100%
greater compared with that at T = 16; and
that the stress at T = –16 is 70% greater
compared with that at T = –32. In other
words, it may be concluded that the
maximum cage stress when collision
phenomenon has occurred is approximately
85% greater compared with that when
collision phenomenon has not occurred.
The orbital rotational speed of the planet
gears was fixed at 5940 min-1 and the nondimensionalized maximum cage stress was
measured while varying the speed and
direction of rotation of the planet gears and
the direction of transmission torques. The
resultant maximum stress values are plotted
in Fig. 9. Under all the conditions, the cage
stress relative to the rotation speed of the
Dynamic Analysis for Needle Roller Bearings Under Planetary Motion
acting between the constituting elements of needle
roller bearings as well as the resultant cage stress.
The orbital rotation speed of the planet gears had
greater affect on the cage maximum stress compared
with the rotation speed of the planet gears and is
nearly proportional to the square of the orbital rotation
speed. This is because the orbital rotational motion of
the planet gears causes the centrifugal force to occur
on the rollers. This centrifugal force acts on the
corresponding cage bar, in the non-load zone, and the
stress increases at the base of this cage bar.
Furthermore, a relatively large cage stress was
observed if the roller at the trailing end of the load
zone is situated in the rear of the cage pocket
opposite the direction of rotation when the roller
exiting the load zone is accelerated by the centrifugal
force hitting the cage bar in front of it. Large stresses
also occur if the moment acting on the cage in the
non-load zone is balanced by the moment resulting
from the contact force between the roller and cage bar
in the load zone.
planet gears is proportional to the speed raised to the
0.34th power. The increase in the cage stress
resulting from the increase in the rotation speed stems
from the increase in the difference of the relative
speed between the rollers and the cage.
From Figs. 8 and 9, it would be understood that the
orbital rotation speed of the planet gears significantly
affects the cage stress. This trend, as discussed in the
previous section, stems from the centrifugal force on
the rollers. At the same time, the cage stress will vary
when the direction and magnitude of the transmission
torque as well as the speed and direction of rotation of
the planet gears are varied. The cage stress can
rapidly increase particularly when the roller at the
trailing end of the load zone is situated at the back of
the cage pocket and this roller exits the load zone and
is accelerated by the centrifugal force thereby hitting
the cage bar in front of it. If this situation takes place,
the cage stress can increase by approximately 85%
as compared with a case where no collision has
occurred.
The operating conditions under which the collision
phenomenon discussed above appears to be
determined by conditions including the angle of roller
orbital motion at the load zone exit as well as the
moment acting on the cage in the non-load zone.
However, the affects of these operating conditions
need to be detailed in the future.
References
1) Gupta, P. K.: Advanced Dynamics of Rolling
Elements, Springer-Verlag, New York (1984).
2) Sakaguchi, T., and Harada, K.: "Dynamic Analysis of
Cage Stress in Tapered Roller Bearings," Proc.
ASIATRIB 2006 Kanazawa, Japan, (2006)649-650.
3) Tomoya Sakaguchi, Kaoru Ueno: Dynamic Analysis of
Cage Behavior in a Cylindrical Roller Bearing, NTN
Technical Review, No.71 pp.8-17 (2003)
4) Craig, R. R., and Bampton, M. C. C.: "Coupling
substructures for dynamic analysis," AIAA J., 6 (7),
pp.1313-1319 (1986)
5) MSC.Adams (Registered trademark of MSC.Software
Corporation)
http://www.mscsoftware.co.jp/products/adams/
6) I-DEAS: http://www.ugs.jp/product/nx/I-deas.html
6. Conclusion
A 2D dynamic analysis tool for needle roller
bearings under a planetary motion has been
developed. This tool allows its users to take into
account the elastic deformation of a cage whose
mechanical strength tends to be low relative to the
constituting elements within a rolling bearing. The
users can calculate the resultant cage stress based on
the magnitude of this elastic deformation. Thus, this
tool has been used to study the interference force
Photo of author
Tomoya SAKAGUCHI
Elemental Technological
R&D Center
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