Dynamic Analysis for Needle Roller Bearings Under Planetary Motion

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. 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 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 cage 2). 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. 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 -94-

Dynamic Analysis for Needle Roller Bearings Under Planetary Motion Fig. 1 Actual forces and motions on a planet gear 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 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. 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 Component- Mode-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 -95-

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. Planetary gear system Bearing technical data Operating conditions 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 Outer ring raceway dia., mm Inner ring raceway dia., mm Roller dia., mm Roller length, mm Number of rollers Orbital rotation speed of planet gears N c min -1 Rotation speed of planet gears N p min -1 Torque transmitted per planet gear T N m Lubricating oil Typical temperature of lubricating oil, C 184.2 57.4 299.0 45 19.85 13.85 2.997 13.8 11 5940 25000 16, 32 ISO VG100 120 4. Stress generating mechanism of the cage 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. 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 -96-

Dynamic Analysis for Needle Roller Bearings Under Planetary Motion 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 212 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 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). Dimensionless stress - - 15 175 2 Time sec. Fig. 6 Stress histories of a cage pocket bar and the adjacent rollers displacements along the radial direction of planetary gear system - - 225 25 Dimensionless displacement 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 (N c : 5940 min -1, T = 32 N m) Fig. 7 Principal mechanisms to make cage stress rise under planetary motion -97-

NTN TECHNICAL REVIEW No.75 2007 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 Dimensionless stress Dimensionless stress 2.5 1.5 0.5 0 2000 4000 6000 8000 10000 Planet gear orbital rotation speed N c, min -1 T : 32 T : 16 T : -32 T : -16 Fig. 8 Cage stress with various speeds of planet gear orbital rotation and transmitted torque 2.5 1.5 0.5 0 20000 40000 60000 80000 Planet gear rotation speed N p, min -1 Np : +, T : 32 Np : -, T : 32 Np : +, T : -32 Np : -, T : -32 Fig. 9 Cage stress with various speeds of planet gear rotation and transmitted torque 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. 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 -98-

Dynamic Analysis for Needle Roller Bearings Under Planetary Motion 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. 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 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. 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 Photo of author Tomoya SAKAGUCHI Elemental Technological R&D Center -99-