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Cage slip characteristics of a cylindrical roller bearing with a trilobe-raceway
CJA 879 12 July 2017 Chinese Journal of Aeronautics, (2017), xxx(xx): xxx–xxx
No. of Pages 12
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Chinese Society of Aeronautics and Astronautics & Beihang University
Chinese Journal of Aeronautics [email protected] www.sciencedirect.com
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Cage slip characteristics of a cylindrical roller bearing with a trilobe-raceway
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Sier DENG a,b, Yujia LU c,*, Wenhu ZHANG d, Xue SUN d, Zhenwei LU e
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a
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School of Mechanical Engineering, Henan University of Science and Technology, Luoyang 471003, China Collaborative Innovation Center of Major Machine Manufacturing in Liaoning, Dalian 116024, China c Patent Examination Cooperation Center of the Patent Office, SIPO, Henan, Zhengzhou 450002, China d School of Mechanical Engineering, Northwestern Polytechnical University, Xi’an 710071, China e Technology Center of LUOYANG LYC BEARING CO., LTD, Luoyang 471003, China
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Received 19 September 2016; revised 5 July 2017; accepted 5 July 2017
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b
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KEYWORDS
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Bearing rating life; Cage slip ratio; Dynamics; Raceway contour; Trilobe-raceway
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Abstract Based on dynamic analysis of rolling bearings, the nonlinear dynamic differential equations of a cylindrical roller bearing with a trilobe-raceway were established and solved by the GSTIFF (gear stiff) integer algorithm with a variable step. The influences of structural parameters and the tolerance of the trilobe-raceway, working conditions of the bearing, and the outer ring installation method on cage slip characteristics were investigated. The results show that: (i) The cage slip ratio and bearing rating life of a cylindrical roller bearing with a trilobe-raceway would reduce when the low-radius (radius of the outer raceway contour at the lowest point) and D-value (difference value between the high and low points of the outer raceway contour) decrease, and the former (low-radius) contributes more significantly. (ii) The cage slip ratio of a cylindrical roller bearing with a trilobe-raceway rises with the increase of the bearing speed, and decreases with the increase of the radial force; the variation range increases with the increase of the low-radius. (iii) When the installation angle of the outer ring increases in a period, the cage slip ratio remains unchanged while the bearing rating life rises up a little. Therefore, when installing a cylindrical roller bearing with a trilobe-raceway, the location of the maximum radius shall be under that of the radial force to improve the bearing rating life. (iv) With the increase of the roundness of the base circle where the radius of the lowest points of the trilobe-raceway contour locates, the cage slip ratio rises gradually and the bearing rating life decreases. Ó 2017 Production and hosting by Elsevier Ltd. on behalf of Chinese Society of Aeronautics and Astronautics. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/ licenses/by-nc-nd/4.0/).
* Corresponding author. E-mail address: [email protected] (Y. LU). Peer review under responsibility of Editorial Committee of CJA.
Production and hosting by Elsevier
1. Introduction
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For high-speed cylindrical roller bearings in the application of aircraft engines, cage slip always happens due to high-speed and light-load conditions, and serious cage slip will lead to early failure of such bearings, which has a profound effect
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http://dx.doi.org/10.1016/j.cja.2017.07.001 1000-9361 Ó 2017 Production and hosting by Elsevier Ltd. on behalf of Chinese Society of Aeronautics and Astronautics. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: DENG S et al. Cage slip characteristics of a cylindrical roller bearing with a trilobe-raceway, Chin J Aeronaut (2017), http://dx.doi. org/10.1016/j.cja.2017.07.001
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on the aircraft safety. One of the effective solutions to this light–load slip problem is to make the outer raceway of a cylindrical roller bearing to be a trilobe-raceway.1 There are three man-made preload locations circumferentially equally-spaced on the outer ring raceway of a cylindrical bearing to increase preload onto more rollers, decrease the rotational speed difference at different positions, reduce bearing skidding, and improve its service life. The skidding problem of bearings at high speed with insufficient load has always been a hot concern by scholars both at home and abroad. Harris2 developed a skid prediction model for a high-speed roller bearing by means of a quasi static analytical method, and investigated the effects of bearing load, rotating speed, and number of rolling elements on bearing skidding. Poplawski3 established a quasi dynamics analysis model for a high-speed roller bearing taking into consideration the guide surface between the cage and rings as well as the friction between the rolling element and the cage pocket, and analyzed the cage and roller slip as well as the film thickness and forces in the cage. Cavallaro and Nelias4 presented an analytical model for high-speed cylindrical roller bearings with flexible rings, and investigated the relationship between ring deformation and bearing load distribution as well as roller slipping speed and load distribution. Takafumi et al.5 proposed a theoretical model for cage slip of cylindrical roller bearings with the consideration of non-Newtonian fluid properties and temperature rise of lube oil, and analyzed the cage slip ratio under different speeds and loads. Arthanari and Marappan6 experimentally analyzed the effects of rotating speed, radial load, and the number of rollers on cage slip of a cylindrical roller bearing. Takabi and Khonsari7 built a dynamic model of a high-speed cylindrical roller bearing, and investigated the influence of different traction models on the sliding velocities and cage wear rate between rollers and races. Chinese scholars also did lots of research on bearing skidding. Li and Wu8 established a dynamic analytical model for highspeed cylindrical roller bearings, and dynamically simulated the roller skew, axial movement, roller and cage slip characteristics, etc. Hu et al.9,10 established a quasi static calculation model for inter-shaft cylindrical roller bearings, and analyzed the relationship between the roller slip ratio and radial load when bearing rings are under different working conditions. Tu and Shao11 considered the acceleration phase of roller bearings, and presented an analytical model to study bearings’ skidding characteristics under different forces and inner ring rotational accelerations. Chen et al.12 developed a bearing dynamic model under a whirling condition taking into consideration the bearing skidding of a high-speed rolling bearing under the whirling condition, and analyzed the effects of various outer loadings, whirling frequencies, and whirling radii on the bearing skidding. Deng et al.13 established a dynamic differential equation for a high-speed cylindrical roller bearing, and analyzed the effects of the clearance ratio of the cage, guiding type, etc. on the cage slip ratio and centroid trajectory. All the studies mentioned above have been focused on the performance analysis of cylindrical roller bearings under highspeed and light-load conditions, whereas studies on a cylindrical roller bearing with a trilobe-raceway have been focused on its raceway process technology,1,14,15 and little research has been done on the theoretical research of the dynamics of a cylindrical roller bearing with a trilobe-raceway.
S. DENG et al. Therefore, this paper analyzes the dynamics of a cylindrical roller bearing with a trilobe-raceway based on the dynamic analysis of the roller bearing, establishes nonlinear dynamics differential equations for a cylindrical roller bearing with a trilobe-raceway, and then adopts the GSTIFF (gear stiff) integer algorithm with a variable step to solve these equations. The focus of this paper is to study the influences of structural parameters of the raceway and parameters of working conditions as well as the relationship between the outer ring installation method and cage slip. The present paper provides some theoretical basis for the structure design of a cylindrical roller bearing with a trilobe-raceway.
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2. Calculation model of the trilobe-raceway
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An outer ring with a trilobe-raceway can be obtained by a predeformation machining method,1 and the principle of the method is shown in Fig. 1. As shown in Fig. 1(a), a uniformly-distributed load F is applied to the 1/3 symmetrical outer surface of the roughcast of the bearing ring along the circumferential direction, which leads to an elastic pre-deformation shown in Fig. 1(b). Keep the applied load constant while grinding the raceway to its theoretical base circle dimension as shown in Fig. 1(c), and the required trilobe-raceway contour can be obtained as a result of the deformation resilience after releasing the load, as shown in Fig. 1(d). The radius of the trilobe-raceway at different azimuth angles after machining can be expressed as:
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Re ðuÞ ¼ Regm Reg ðuÞ þ Reo
ð1Þ
where Reo is the original radius before machining; Reg ðuÞ is the radius of the raceway at different azimuth angles u after the elastic deformation of the outer ring is generated because of the applied force; Regm is the grinding radius, i.e., the theoretical base circle radius.
Fig. 1
Sketch map of the pre-deformation machining principle.
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2.1. Finite element analytical model of the outer ring with the trilobe-raceway Take a certain type of bearing as an example to study the relationship between the trilobe-raceway’s structural parameters and the application of loads. The finite element analytical model of the outer ring with the trilobe-raceway is established by adopting 20-node hexahedron units. The model established in Ref. 1 was simplified based on the geometrical relationship analyzed above, and the deformation caused by the grinding force in unloaded areas of the outer ring as well as the elastic deformation of the mold can be neglected. Based on the loads distribution needed for forming a trilobe-raceway, when under ideal processing conditions, gradient loads are applied to the 1/3 symmetrical outer surface of the roughcast of the bearing ring along the circumferential direction, and the included angles between loaded areas are 60°; one of the loaded areas is rotated at a slight degree uf to simulate non-ideal processing conditions, and by this way, the included angles between loaded areas become 60°, 60° + uf, and 60° uf. The finite element analytical model of the outer ring with the triloberaceway is shown in Fig. 2. Fig. 3 shows the impact of the loads rotating degree uf on the trilobe-raceway’s structural parameters when the original radius before machining is 34.86 mm and the applied loads are 2.0 MPa. As shown in Fig. 3, with an increase of the loads rotating degree, the high point of the trilobe-raceway counter increases and the low point decreases, i.e., the roundness of the base circle at low and high points rises, and the D-value between the highest and lowest points is also increased. This is because when a loaded area is rotated, the magnitudes of loads also take suitable changes to achieve the balance of applied loads; in the area where loads are increased, the deformation of the outer ring under applied loads increases, and when the grinding radius remains unchanged, the radius at the low point of the raceway rises while the radius at the high point reduces after molding; likewise, in the area where loads are decreased, the radius of the low point reduces and the radius of the high point rises after molding.
Fig. 2 Finite element analytical model of the outer ring with the trilobe-raceway.
Fig. 3 loads.
Wave shape of the raceway imposed with asymmetrical
2.2. Mathematical model of the outer ring with the triloberaceway
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As shown in Fig. 3, with the changes of the azimuth angle, the raceway radius is in accordance with a sinusoidal curve. In order to facilitate the expression of relationship between the trilobe-raceway radius and the azimuth angle, for an ideal trilobe-raceway, the obtained data can be fitted into a sinusoidal function in which the azimuth angle u is regarded as the independent variable and the radius of the raceway Re is regarded as the dependent variable as follows:
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Re ðuÞ ¼ A þ B cosðCu þ DÞ
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ð2Þ
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Meanwhile, as for a non-ideal trilobe-raceway, the obtained data can be fitted into six sinusoidal functions in the same form as Eq. (2), where the azimuth angle u is regarded as the independent variable, the radius of the raceway Re is regarded as the dependent variable, and A, B, C, and D are real constants.
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3. Dynamic analytical model of a cylindrical roller bearing with a Trilobe-raceway
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The schematic diagram of a cylindrical roller bearing with a trilobe-raceway is shown in Fig. 4. The outer ring is fixed, and the inner ring rotates at a constant speed x and bears a radial force Fr. The cage is guided by the outer ring, and the bearing face is lubricated by oil jets. Assume that the surfaces of bearing components are absolutely smooth with an ideal geometric shape, and the center of mass is coincident with the center of form. In Fig. 4, Oi and O0i are the inner ring centers of the bearing before and after loading, respectively; P0d is the radial clearance of the theoretical base circle of the outer ring; uj is the azimuth angle of the jth roller; Pdj is the radial clearance between the theoretical base circle and the outer raceway at the azimuth angle uj , and Pdj ¼ Re ðuj Þ Regm ; Re can be obtained from Eq. (2); dr is the radial displacement of the inner ring. As can be seen from Fig. 4, contact deformation between the raceway and part of rollers occurs before applying the radial force, where the radial clearance between the theoretical base circle and the outer raceway is negative and its absolute value is larger than P0d .
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Fig. 4 203
Schematic diagram of a cylindrical roller bearing with a trilobe-raceway.
3.1. Force analysis of bearing parts
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3.1.1. Forces between a roller and raceways
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Tilt always happens when a cylindrical roller bearing with a trilobe-raceway operates under a radial force Fr , and the bending moment M and the forces and deformations between a roller and raceways are shown in Fig. 5. In Fig. 5, fO; X; Y; Zg is the inertial coordinate system of the bearing, which is a fixed coordinate system with the X axis coincident with the rotation axis of the bearing, and the YZ plane is parallel to the radial plane across the bearing center. Other coordinate systems are all established according to this inertial coordinate system; forj ; xrj ; yrj ; zrj g is the coordinate system of the jth roller center, whose original point orj is coincident with the mass center of the roller, the xrj axis is along with the roller axis, and the yrj zrj plane is parallel to the radial plane across the mass center of the roller. Every roller has its own local coordinate system which moves and rotates but doesn’t spin with the roller; a is the tilting angle of the inner ring; hj is the tilting angle of the jth roller at the azimuth angle uj . The classic slice method is applied to handle the contact issue between the crowned roller and raceways, where the roller is sliced to N sections along the axial direction, w ðw ¼ Ls =NÞ is the slice width, and Ls is the effective length of the roller. The elastic deformations between the kth slice of the jth roller and raceways at the azimuth angle uj are expressed as: 8 0 < di ¼ dr cos uj Pd Pdj d0 Cjk jk uj 2 2 ð3Þ : de ¼ Pdj þ d0 C jk jk uj 2
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Fig. 5
Forces and deformations between a roller and raceways.
where d0uj is the deformation of the roller center, and Cjk is the
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amount of crown drop of the kth slice of the jth roller. The normal contact forces between the kth slice of the jth roller and raceways are written as16 w iðeÞ iðeÞ1:11 qjk ¼ djk ð4Þ 0:11 A1:11 1 Ls
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where A1 is the elastic deformation coefficient, and A1 ¼ 1:36g0:9 ; g is the combined elasticity modulus of two contact bodies. The normal contact forces between the jth roller and raceways are expressed as:
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iðeÞ
Qj
¼
N X iðeÞ qjk
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ð5Þ 246
k¼1
The additional moments caused by normal contact forces between the jth roller and raceways are written as N X iðeÞ iðeÞ Ls wk ð6Þ MQj ¼ qjk 2 k¼1
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According to Eq. (5), the oil drag forces of the jth roller are written as
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iðeÞ
Tj
¼
N X
iðeÞ
Tjk ¼
k¼1
N X
iðeÞ
qjk lj
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ð7Þ
k¼1
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where lj is the drag coefficient of oil film (see Ref. 17 for detailed equation). The additional moments caused by additional moments are depicted as N X Ls iðeÞ iðeÞ wk ð8Þ MTj ¼ qjk lj 2 k¼1
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3.1.2. Forces between a roller and the cage
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The forces between a roller and the cage pocket are mainly considered as the hydrodynamic pressure between the roller and the beam of the cage pocket as well as elastic impact force due to contact deformation, which are also computed by the ‘‘slice method”. The relationship of displacement and deformation between the roller and the cage pocket is shown in Fig. 6. fopj; xpj:ypj; zpjg is the coordinate system of the cage pocket center, whose original point opj coincides with the geometric center of the cage pocket, the xpj axis is parallel to the
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rotation axis of the cage, and the ypj axis is along the radial direction when the zpj axis is along the circumferential direction of the pocket center. Every cage pocket has its own local coordinate system which moves and rotates with the cage. In Fig. 6, bj is the skewing angle of the jth roller, Dzcj is the displacement of the roller geometric center under the coordinate system of the cage pocket center. The deformation dcjk between the kth slice of the jth roller and the cage bar is expressed as follows: Ls w k tan bj Cjk dcjk ¼ Dcj þ ð9Þ 2
ing, the infinitely short bearing theory is applied in this paper to determine the acting forces Fcy, Fcz and moment Mcx between the cage and the guide surface of the ring (see Ref. 19 for detailed equation).
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3.1.4. Surface resistance and end-surface of the cage
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For a high-speed cylindrical roller bearing with a triloberaceway, cage rotation makes the external sides of the cage affected by the resistance from surrounding air/oil mist mixture shearing. See Ref. 19 for the calculation method of the retardation torque Mco of the cage surface and sides.
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The contact force qcjk between the kth slice of the jth roller and the cage bar is shown as ( ðdcjk 6 Cp Þ Kc dcjk c ð10Þ qjk ¼ 1:11 ðdcjk Cp Þ A1:11wL0:11 ðdcjk > Cp Þ
3.2. Dynamics differential equations of the bearing
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According to the above-mentioned analysis, when a cylindrical roller bearing with a trilobe-raceway is working, the roller is simultaneously applied by combined forces of inner, outer raceways and the cage, as shown in Fig. 7, where Fm j is the cen1 2 trifugal force of roller, and Fm ¼ m d x , in which xrbj is the j 2 r m rbj orbital angular velocity of the jth roller, and dm is the pitch diameter of the bearing. As seen in Fig. 7, dynamics differential equations of the jth roller are expressed as:
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1
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s
where Kc is a linear approximation constant obtained by experiments, and Kc ¼ 67=Cp ; Cp is the cage pocket clearance, and Cp ¼ 0:5ðDp Dw Þ; Dp is the cage pocket diameter; Dw is the roller diameter. The direction of force qcjk is estimated by dcjk . When dcjk P 0, the roller is pushed forward by the cage; otherwise, the roller is resisted by the cage. The contact force Qcj and additional moment Mcj between the jth roller and the cage bar are expressed as: Qcj ¼
N X qcjk
ð11Þ
k¼1
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N X Ls qcjk wk 2 k¼1
Mcj ¼
ð12Þ
The friction force between the jth roller and the cage bar is expressed as follows: N N X X Fcj ¼ Fcjk ¼ lcj qcjk k¼1
ð13Þ
k¼1
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where lcj is the friction coefficient between the roller and the cage pocket (see Ref. 18 for detailed equation).
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3.1.3. Forces between the cage and the guide surface
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When the cage is guided by the outer ring, the interaction between the guide surface of the ring and the cylindrical surface of the cage is mainly produced by the hydrodynamic effect of lubricating oil. Since the acting surface of the guide rib and the cylindrical surface of the cage are small and mutually slid-
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Fig. 6
Forces and deformations between a roller and the cage.
8 c mr y€j ¼ Qij cos uj Qej cos uj þ Qcj sin uj þ Fm > j cos uj Fj cos uj > > > > i e > Tj sin uj þ Tj sin uj > > > > > c > > mr z€j ¼ Qij sin uj þ Qej sin uj þ Qcj cos uj Fm j sin uj þ Fj sin uj > < i e Tj cos uj þ Tj cos uj > > > > Jxj x_ xj ¼ Tij D2w þ Tej D2w þ Fcj D2w > > > > > > > Jyj x_ yj ¼ MiQj sin uj þ MeQj sin uj Mcj cos uj MiTj cos uj MeTj cos uj > > > : Jzj x_ zj ¼ MiQj cos uj þ MeQj cos uj þ Mcj sin uj þ MiTj sin uj þ MeTj sin uj ð14Þ
where mr is the roller mass; y€j ; z€j are the displacement accelerations of the jth roller’s mass center in fO; X; Y; Zg; Jxj ; Jyj ; Jzj are the moments of inertia of the jth roller in fO; X; Y; Zg; x_ xj ; x_ yj ; x_ zj are the angular accelerations of the jth roller in fO; X; Y; Zg, respectively. When the bearing is working, the cage is simultaneously applied by the collision force of rollers, the guiding force of the outer ring, and the combined resistance of air/oil mist mixture to both cage ends and its surface, as shown in Fig. 8, where xrj is the spin angular velocity of the jth roller, and xm is the actual speed of the cage.
Fig. 7
Schematic diagram of roller forces.
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As seen in Fig. 8, dynamics differential equations of the cage are expressed as: 8 Z1 X > > > ðFcj cos uj þ Qcj sin uj Þ þ Fcy cos wc þ Fcz sin wc > mc y€c ¼ > > > j¼1 > > > > Z1 X > > > > mc z€c ¼ ðFcj sin uj Qcj cos uj Þ Fcy sin wc þ Fcz cos wc > > > j¼1 > > > > Z1 < X Jcx x_ cx ¼ Qcj d2m Mcx Mco > > j¼1 > > > > Z1 > X > > > _ cy ¼ J ðMcj cos uj Þ cy x > > > j¼1 > > > > Z1 > X > > > ðMcj sin uj Þ > : Jcz x_ cz ¼ j¼1
ð15Þ
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where mc is the cage mass; y€c ; z€c are the accelerations of the cage’s mass center in fO; X; Y; Zg; Jcx, Jcy, and Jcz are the moments of inertia of the cage in fO; X; Y; Zg; x_ cx ; x_ cy ; x_ cz are the angular accelerations of the cage in fO; X; Y; Zg; forc ; yrc ; zrc g is the cage’s reference coordinate system; wc is the angle between foc ; yc ; zc g and forc ; yrc ; zrc g. Dynamics differential equations of the inner ring are expressed as: 8 Z1 X > > > € m ¼ ðQij cos uj þ Tij sin uj Þ þ Fr y i i > > > > j¼1 > > > > Z1 X > > > > mi z€i ¼ ðQij sin uj þ Tij cos uj Þ > > > j¼1 > > > > Z1 < X Jix x_ ix ¼ Tij D2w ð16Þ > > j¼1 > > > > Z1 > X > > J x_ ¼ > ðMiQj sin uj þ MiTj cos uj Þ yj yj > > > j¼1 > > > > Z1 > X > > > ðMiQj cos uj þ MiTj sin uj Þ > : Jzj x_ zj ¼ M j¼1
where mi is the mass of the inner ring; y€i ; z€i are the linear accelerations of the inner ring mass center in fO; X; Y; Zg; Jix, Jiy, and Jiz are the moments of inertia of the inner ring in
Fig. 9
Solution procedure of dynamics differential equations.
fO; X; Y; Zg; x_ ix ; x_ iy ; x_ iz are the angular accelerations of the inner ring in fO; X; Y; Zg; Z1 is the number of rollers.
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3.3. Solution procedure of dynamics differential equations
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The GSTIFF integer algorithm is based on backwarddifference formulae (BDF) and is a variable-step and multi-step integrator,20 which is applied to solve dynamics differential Eq. (16) of a cylindrical roller bearing with a trilobe-raceway in this study. The solution procedure of dynamics differential equations is shown in Fig. 9.
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(1) The solution duration, initial step, and convergence error of dynamics differential equations are set, firstly. In this study, the solution duration was set to 0.5 s, the initial step was set to 0.00001 s, and the convergence error was set to 1 103. (2) According to initial estimated values of a bearing component’s position and motion constraint, initial conditions of dynamics differential equations, namely, relative positions and motion vectors of various components are obtained by quasi-static analysis. (3) Given initial conditions of dynamics differential equations, forces and moments applied on bearing components are obtained according to the expressions in Section 3.1. (4) The dynamics differential equations are solved by GSTIFF, and then positions and motion vectors of bearing components are obtained. (5) Verify whether the error meets the convergence error set in Step (1). If yes, continue next solution after getting the outputs of motion parameters including displacements, velocities, and accelerated speeds of the inner ring, cage, and roller. If no, choose a smaller step value and repeat Step (4) until the solving error meets the convergence error.
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Fig. 8
Schematic diagram of cage forces.
4. Analysis on cage slip characteristics and basic rating life
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In this paper, a certain type of bearing is used to study the impacts of structural parameters and tolerance of the outer
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raceway, working conditions, and the outer ring installation method on cage slip characteristics and basic rating life. Main parameters of the bearing are shown in Table 1. The inner ring, outer ring, and roller are made of 8Cr4Mo4V, and the cage is made of 40CrNiMo, with a fixed outer ring and a cage guided by the outer ring. The cage slip ratio Sc and the basic rating life L10 of a cylindrical roller bearing with a trilobe-raceway are defined as follows: xm ð17Þ Sc ¼ 1 0 100% xm
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106 e9=8 8=9 L10 ¼ a1 aiso ½Li9=8 þ L10 60 Vi 10 " 9=8 #8=9 9=8 106 Qci Qce ¼ a1 aiso þ 60 Vi Qei Qee
ð18Þ
where x0m is the theoretical speed of the cage; a1 and aiso are correction coefficients, a1 = 1.0, aiso = 1.0; Vi is the inner ring speed, r/min; Li10 and Le10 are the basic rating lives of the inner and outer rings, respectively; Qci and Qce are the equivalent dynamic loads of the inner and outer rings, respectively; Qei and Qee are the dynamic load ratings of the inner and outer rings, respectively. 4.1. Impact of geometric parameters on cage slip characteristics and rating life As known from Ref. 1, the raceway contour of the outer ring after machining can be determined by the locations of high and low points, as shown in Fig. 10. Therefore, this paper takes the low-radius Re min and the D-value de as the geometric parameters to analyze their impacts on cage slip characteristics. 4.1.1. Impact of the low-radius on the cage slip ratio and rating life Fig. 11(a) shows the influence of the low-radius on the cage slip ratio when the rotating speed of the bearing is set to 45,000 r/ min, the radial force is 800 N, and the D-value is 0.098 mm. As seen in Fig. 11(a), with the increase of the low-radius, the cage slip ratio increases and the amplitude grows. Lh is the basic rating life. This is because when the low-radius increases, as shown in Fig. 12, the number of loaded rollers and the contact forces between rollers and the inner ring reduce, so the traction
Table 1
forces between rollers and the inner ring decrease, and the drop of the roller pushing force finally leads to a rise of the cage slip ratio. Meanwhile, with a small low-radius, the preloads applied on rollers from the outer raceway are quite large that the bearing rating life declines sharply, as shown in Fig. 11(b). Consequently, there must be a reasonable low-radius that can avoid early failure caused by cage slip and also meet the requirement of the bearing rating life.
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4.1.2. Impact of the D-Value on the cage slip ratio and rating life
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Fig. 13(a) shows the impact of the D-value on the cage slip ratio and rating life of a bearing when the rotating speed is 45,000 r/min, the radial force is 800 N, and the low-radius is 34.8625 mm. As shown in Fig. 13(a), the cage slip ratio rises linearly with the increase of the D-value. When the D-value increases, as shown in Fig. 14, the loaded zones of the bearing become larger, and there are more rollers pushing the cage forward because of the increased number of loaded rollers, thus the cage slip ratio reduces, and the bearing rating life also decreases, as shown in Fig. 13(b). Therefore, a smaller D-value is preferred on condition that the rating life of the bearing is guaranteed.
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4.2. Impact of working conditions on cage slip characteristics and rating life
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4.2.1. Impact of the bearing rotating speed on the cage slip ratio
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Assume that Fig. 15 shows the impact of the inner ring rotating speed on the cage slip ratio when the radial force of the bearing is 800 N, the D-value is 0.098 mm, and the lowradius is 34.8600, 34.8625, 34.8650, and 34.8675 mm, respectively. As shown in Fig. 15, the cage slip ratio rises with the acceleration of the bearing rotating speed, and the amplitude grows with the increase of the low-radius. The reason lies in the increased rollers centrifugal forces by the acceleration of the rotation speed, and diminished contact forces between rollers and the inner ring shown in Fig. 16 resulting in a rise of the cage slip ratio by inadequate traction forces on rollers by the inner ring. When the low-radius is small, the preloads on rollers from the outer raceway are very effective that the centrifugal forces of rollers are relatively tiny compared to the preloads. Therefore, bearings with a smaller low-radius tend
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Main parameters of a bearing.
Parameters
Value
Geometric
Outside diameter (mm) Bore diameter (mm) Inner raceway diameter (mm) Low-radius (mm) D-value (mm) Roller diameter (mm) Length of roller (mm) Roller number
73.4 50.027 55.75 69.725 0.098 7.0 7.0 18
Condition
Normal rotate speed (r/min) Rated radial force (N)
45,000 800
Fig. 10
Schematic diagram of trilobe-raceway contours.
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Fig. 11
Fig. 12 487 488
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Effects of the low-radius on the cage slip ratio and basic rating life.
Effect of the low-radius on load distribution.
to diminish their sensitivity of cage slip to the rotating speed variation during operation. 4.2.2. Impact of the radial force on the cage slip ratio and rating life Fig. 17 shows the relationship between the radial force and the cage slip ratio when the rotating speed of a bearing is 45,000 r/ min, the D-value is 0.098 mm, and the low-radius is 34.8600, 34.8625, 34.8650, and 34.8675 mm, respectively. As shown in Fig. 17, the cage slip ratio remains unchanged with the increase of the radial force when the low-radius is
Fig. 13
34.8600 mm. When the low-radius is 34.8625 mm and 34.8650 mm, with the increase of the radial force, the cage slip ratio firstly remains unchanged, and then decreases after the radial force exceeds a threshold value. The cage slip ratio will decrease with the increase of the radial force when the lowradius is 34.8675 mm. The reason lies in the fact that the loaded zone of the bearing and load distribution are mainly determined by the preloads from the raceway when the lowradius is small, with unchanged raceway geometric parameters and preloads to the roller by the raceway. When the low-radius increases and the radial force exceeds a threshold value, the radial force has a greater impact on bearing load distribution than preloads by the raceway. As Fig. 18 shows, with the increases of the radial force, the contact forces between rollers and the inner ring as well as the number of loaded rollers, the cage slip ratio decreases as a result. Fig. 17 also shows that the larger the low-radius is, the more evident the influence of the radial force on the cage slip ratio is. In other words, bearings with a smaller low-radius tend to weaken their sensitivity of cage slip when they are working under a variable radial force.
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4.3. Impact of the outer ring installation angle on cage slip characteristics and rating life
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As shown in Fig. 1, when the outer ring is fixed and the direction of the radial force goes through the low point of a raceway contour, we define the installation angle at this time as 0 . Installation angle is defined as ue when the outer ring is rotated
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Effects of the D-value on the cage slip ratio and basic rating life.
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Cage slip characteristics of a cylindrical roller bearing
Effect of the D-value on load distribution.
Fig. 17
Effect of the radial force on the cage slip ratio.
Effect of the inner ring rotating speed on the cage slip
Fig. 18
Effect of the radial force on load distribution.
Fig. 14
Fig. 15 ratio.
Fig. 16
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Effect of the rotating speed on load distribution.
clockwise to an angle ue . When ue ¼ 60 , i.e., the installation angle is 60 , the radial force goes through the high point of a raceway contour, and the direction of the radial force changes over one period. Fig. 19(a) shows the impact of the installation angle on the cage slip ratio when the rotating speed of the bearing is 45,000 r/min, the radial force is 800 N, the D-value is 0.098 mm, and the low-radius is set to 34.8600, 34.8625, 34.8650, and 34.8675 mm, respectively. As shown in Fig. 19, when the installation angle increases, the cage slip ratio remains unchanged and the bearing rating life rises up a little.
This is because of the tiny dragging forces on the cage by rollers, with almost unchanged contact forces between rollers and the inner ring due to fixed raceway geometric parameters and radial force. As the installation angle changes from 0 to 60 , the loaded zone of the bearing as shown in Fig. 20 becomes larger, and the dynamic equivalent load decreases with the increased number of loaded rollers, leading to a slight rise of the bearing rating life. Hence, when installing a cylindrical roller bearing with a trilobe-raceway, if possible, make the direction of the radial force go through the high point of the raceway contour to prolong the bearing rating life.
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4.4. Design tolerance of the trilobe-raceway
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4.4.1. Determination of the tolerance of raceway radius parameters
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Fig. 21(a) and (b) shows the cage slip ratio and bearing rating life at different D-values when the rotating speed is 45,000 r/ min, the radial force is 800 N, and the low-radius is 34.8600, 34.8625, 34.8650, and 34.8675 mm, respectively. As shown in Fig. 21, when the low-radius is smaller than 34.8650 mm, the cage slip ratio and bearing rating life rise with the increase of the D-Value, just like the above-mentioned analysis. When the low-radius is 34.8675 mm, with the increase of the D-value, the cage slip ratio and bearing rating life increase at first and then remain unchanged afterwards. This is because when the low-radius is small, preloads provided by the outer raceway is very small; when the D-value exceeds a threshold value,
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Fig. 19
Fig. 20 560 561 562 563 564 565 566 567 568 569
Effects of the installation angle on the cage slip ratio and basic rating life.
Effect of the installation angle on load distribution.
the area of the loaded zone and load distribution are determined by the applied radial force which is set to be a constant value. Therefore, the cage slip ratio remains unchanged. Fig. 21(a) also shows that compared to the D-value, the low-radius has a greater impact on the cage slip ratio. Since the low-radius can affect not only the number of loaded rollers, but also the contact forces between rollers and the inner ring, once the low-radius is determined, the D-value has limited influence on the contact forces between rollers and the raceway within the loaded area; instead, it affects the cage slip
Fig. 21
ratio mostly by changing the number of loaded rollers through its impact on the loaded area. Therefore, to design a cylindrical roller bearing with a trilobe-raceway, it is recommended to determine firstly the range of the low-radius based on the demands of the cage slip ratio and bearing rating life, and then the range of the D-value according to the selected low-radius. As for the bearing concerned in this paper, the cage slip ratio under 10% is acceptable in actual practice. In order to meet the bearing rating life requirement of 300 h, the lowradius should be within the range of 34.8625 mm to 34.8650 mm. When the D-value is less than 0.095 mm, a bearing with a low-radius as 34.8625 mm can no longer meet the rating life requirement. When the D-value is larger than 0.105 mm, the cage slip ratio of a bearing with a low-radius of 34.865 mm will be over 10%. Therefore, the D-value should be within the range of 0.095–0.105 mm. To reduce the influences of the bearing rotating speed and radial force variation during operation, the low-point diameter of the outer raceway contour can be designed as 69:725þ0:005 mm, and the D-value 0 can be designed as 0:095þ0:01 mm. 0
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4.4.2. Impact of the base circle roundness on the cage slip ratio
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Fig. 22 shows the impact of the roundness of the base circle where the radius of the lowest point of the raceway contour locates on the cage slip ratio and bearing rating life, when the rotating speed is 45,000 r/min, the radial force is 800 N,
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Effects of the D-value on the cage slip ratio and basic rating life at different low-radius values.
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Fig. 22
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Effects of the roundness on the cage slip ratio and basic rating life at different low-radius values.
and the low-radius is 34.8625 mm and 34.8650 mm, respectively. As shown in Fig. 22, when the roundness increases, the cage slip ratio increases while the bearing rating life reduces. This is because when the roundness of the base circle increases, the radii of three low points of the raceway contour are no longer the same, so are those of three high points. Therefore, the radius of the lowest point decreases when the difference value between the highest and lowest points of the raceway contour increases. Contact forces between rollers and the inner ring increase in areas near the lowest points but decrease in other areas. The increasing amplitude of the total bearing load caused by the former is smaller than the decreasing amplitude of the total bearing load caused by the latter. As a result, the cage slip ratio tends to increase and the bearing rating life decreases gradually. As for the bearing concerned in this paper, when the roundness of the base circle at low points is larger than 0.006, a bearing with a low-radius of 69.725 mm and a D-value of 0.095 mm can no longer meet the rating life requirement, so the roundness of the base circle at low points shall be designed as 0.006.
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5. Comparison validation
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Experimental results in Ref. 21 are adopted to verify the validity of the dynamic model built in this paper. Main parameters of a test bearing are shown in Table 2. Fig. 23 shows the comparison of the cage slip ratio between experimental and simulation results when the applied radial force on the tested bearing is 800 N, and the rotating speed
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Table 2
Main parameters of a test bearing.
Geometric parameters
Value
Outside diameter (mm) Bore diameter (mm) Inner raceway diameter (mm) Low-radius (mm) D-value (mm) Roller diameter (mm) Length of roller (mm) Roller number
73.4 50 55.7 69.648 0.06 7.0 7.0 18
Fig. 23 results.
Comparison between experimental and simulation
is varying from 15,000 to 52,000 r/min. As shown in Fig. 23, the cage slip ratio rises with the acceleration of the bearing rotating speed, so simulation results are generally in line with experimental results, and the error is not more than 9%. When operating at lower speed, simulation results are smaller than experimental results, which may be caused by different lubricating oil used in the dynamic model and the experiment. In addition, when working at a higher speed, simulation results are larger than experimental results, because the dynamic model built in this paper does not include the impact of temperature on bearing geometric parameters. When the rotating speed is high, the radial internal clearance decreases because of the large heat productivity; hence, the cage slip ratio in the experiment is slightly smaller than that in the simulation.
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6. Conclusions
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(1) The cage slip ratio and bearing rating life of a cylindrical roller bearing with a trilobe-raceway reduce with the decrease of the low-radius and D-value, and the lowradius has a greater impact than the D-value. Therefore, on the premise of bearing rating life, a small low-radius and D-value are preferred, and the variation range of the low-radius should be determined before the determination of the D-value.
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S. DENG et al. (2) The cage slip ratio rises with the acceleration of the inner ring rotating speed and drops with the increase of the radial force, and the changing amplitude increases along with the increase of the low-radius. That’s to say, a bearing with a smaller low-radius tends to diminish its sensitivity of cage slip when working under a variable condition. (3) When the installation angle of a cylindrical roller bearing with a trilobe-raceway increases within one period, the cage slip ratio remains unchanged while the bearing rating life rises up a little. Hence, the direction of the radial force should go through the high-point radius of the outer raceway contour when installing the bearing to prolong its rating life. (4) With the increase of the roundness of the base circle at three low points of the raceway contour, the cage slip ratio rises and the bearing rating life decreases. Thus, the roundness of the base circle should be reduced to the greatest extent under attainable machining precision.
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Acknowledgements
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This research was financially co-supported by the National Natural Science Foundation of China (U1404514), Henan Outstanding Person Foundation in China (144200510020), and the Collaborative Innovation Center of Major Machine Manufacturing in Liaoning, China.
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References
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1. Wang LQ, Chen GC. Pre-deformation machining technology of the noncircular raceway of aero roller bearings. Chin J Mech Eng 2005;41(9):223–7. 2. Harris TA. Analytical method to predict skidding in high speed roller bearings. ASLE Tran 1966;9:229–41. 3. Poplawski JV. Slip and cage forces in a high-speed roller bearing. J Lubr Tech 1972;94(2):143–50. 4. Cavallaro G, Nelias D. Analysis of high-speed inter-shaft cylindrical roller bearing with flexible rings. Tribol T 2005;61 (11):38–244.
5. Takafumi Y, Yasuyoshi T, Hisao M, Masaaki S. Analysis of cage slip in cylindrical roller bearings considering non-newtonian behavior and temperature rise of lubricating oil. Tribol T 2008;53(6):647–62. 6. Arthanari S, Marappan R. Experimental analysis of factors influencing the cage slip in cylindrical roller bearing. Int J Adv Manuf Technol 2011;53(5):635–44. 7. Takabi J, Khonsari MM. On the influence of traction coefficient on the cage angular velocity in roller bearings. Tribol T 2014;57 (5):793–805. 8. Li JB, Wu LF. Dynamic analysis of high-speed roller bearings. Chin J Aeronaut 1992;13(12):625–32. 9. Hu X, Luo GH, Gao DP. Quasi-static analysis of cylindrical roller inter-shaft bearing. Chin J Aeronaut 2006;21(6):1069–74. 10. Hu X, Luo GH, Gao DP. Performance analysis of aero-engine inter-shaft bearing. Chin J Aeronaut 2007;22(3):439–43. 11. Tu WB, Shao YM. An analytical model to investigate skidding in rolling element bearings during acceleration. J Mech Sci Technol 2012;26(8):2451–8. 12. Chen W, Li JN, Zhang LB. Skidding analysis of high speed rolling bearing considering whirling of bearing. Chin J Mech Eng 2013;49 (6):38–43. 13. Deng SE, Gu JF, Cui YC, Sun CY. Analysis on dynamic characteristics of cage in high-speed cylindrical roller bearing. Chin J Aeronaut 2014;29(1):492–8. 14. Fan DP, Yin ZQ. Grinding technology of high speed anti-skidding bearings with noncircular raceway. Bearing 1996;12:18–20. 15. Lopatukhin I, Khaskel L. Deformation of thin-walled rings under the action of n concentrated and distributed forces. Int J Adv Manuf Technol 2010;49:627–34. 16. Harris TA. The effect of misalignment on the fatigue life of cylindrical roller bearings having crowned rolling members. J Tribol 1969;91(2):294. 17. Wang YS, Yang BY, Wang LQ. Investigation into the traction coefficient in elasto-hydrodynamic lubrication. Tribotest 2004;11 (2):113–24. 18. Xiuhai Liu, Sier Deng. Dynamic stability analysis of cages in highspeed oil-lubricated angular contact ball bearings. Trans Tianjin Univ 2011;17(1):20–7. 19. Deng SE, Jia QY, Xue JX. Design principle of rolling bearings. 2nd ed. Beijing: Standards Press of China; 2014. p. 113–24. 20. Gear CW. Simultaneous numerical solution of differential-algebraic equations. IEEE Trans Circ Syst 1971;18(1):89–95. 21. Zuo JT. Study on a certain bearings of aircraft engine. Harbin: Harbin Institute of Technology; 2008.
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Chinese Society of Aeronautics and Astronautics & Beihang University
Chinese Journal of Aeronautics [email protected] www.sciencedirect.com
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Cage slip characteristics of a cylindrical roller bearing with a trilobe-raceway
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Sier DENG a,b, Yujia LU c,*, Wenhu ZHANG d, Xue SUN d, Zhenwei LU e
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a
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School of Mechanical Engineering, Henan University of Science and Technology, Luoyang 471003, China Collaborative Innovation Center of Major Machine Manufacturing in Liaoning, Dalian 116024, China c Patent Examination Cooperation Center of the Patent Office, SIPO, Henan, Zhengzhou 450002, China d School of Mechanical Engineering, Northwestern Polytechnical University, Xi’an 710071, China e Technology Center of LUOYANG LYC BEARING CO., LTD, Luoyang 471003, China
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Received 19 September 2016; revised 5 July 2017; accepted 5 July 2017
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b
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KEYWORDS
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Bearing rating life; Cage slip ratio; Dynamics; Raceway contour; Trilobe-raceway
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Abstract Based on dynamic analysis of rolling bearings, the nonlinear dynamic differential equations of a cylindrical roller bearing with a trilobe-raceway were established and solved by the GSTIFF (gear stiff) integer algorithm with a variable step. The influences of structural parameters and the tolerance of the trilobe-raceway, working conditions of the bearing, and the outer ring installation method on cage slip characteristics were investigated. The results show that: (i) The cage slip ratio and bearing rating life of a cylindrical roller bearing with a trilobe-raceway would reduce when the low-radius (radius of the outer raceway contour at the lowest point) and D-value (difference value between the high and low points of the outer raceway contour) decrease, and the former (low-radius) contributes more significantly. (ii) The cage slip ratio of a cylindrical roller bearing with a trilobe-raceway rises with the increase of the bearing speed, and decreases with the increase of the radial force; the variation range increases with the increase of the low-radius. (iii) When the installation angle of the outer ring increases in a period, the cage slip ratio remains unchanged while the bearing rating life rises up a little. Therefore, when installing a cylindrical roller bearing with a trilobe-raceway, the location of the maximum radius shall be under that of the radial force to improve the bearing rating life. (iv) With the increase of the roundness of the base circle where the radius of the lowest points of the trilobe-raceway contour locates, the cage slip ratio rises gradually and the bearing rating life decreases. Ó 2017 Production and hosting by Elsevier Ltd. on behalf of Chinese Society of Aeronautics and Astronautics. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/ licenses/by-nc-nd/4.0/).
* Corresponding author. E-mail address: [email protected] (Y. LU). Peer review under responsibility of Editorial Committee of CJA.
Production and hosting by Elsevier
1. Introduction
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For high-speed cylindrical roller bearings in the application of aircraft engines, cage slip always happens due to high-speed and light-load conditions, and serious cage slip will lead to early failure of such bearings, which has a profound effect
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http://dx.doi.org/10.1016/j.cja.2017.07.001 1000-9361 Ó 2017 Production and hosting by Elsevier Ltd. on behalf of Chinese Society of Aeronautics and Astronautics. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: DENG S et al. Cage slip characteristics of a cylindrical roller bearing with a trilobe-raceway, Chin J Aeronaut (2017), http://dx.doi. org/10.1016/j.cja.2017.07.001
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on the aircraft safety. One of the effective solutions to this light–load slip problem is to make the outer raceway of a cylindrical roller bearing to be a trilobe-raceway.1 There are three man-made preload locations circumferentially equally-spaced on the outer ring raceway of a cylindrical bearing to increase preload onto more rollers, decrease the rotational speed difference at different positions, reduce bearing skidding, and improve its service life. The skidding problem of bearings at high speed with insufficient load has always been a hot concern by scholars both at home and abroad. Harris2 developed a skid prediction model for a high-speed roller bearing by means of a quasi static analytical method, and investigated the effects of bearing load, rotating speed, and number of rolling elements on bearing skidding. Poplawski3 established a quasi dynamics analysis model for a high-speed roller bearing taking into consideration the guide surface between the cage and rings as well as the friction between the rolling element and the cage pocket, and analyzed the cage and roller slip as well as the film thickness and forces in the cage. Cavallaro and Nelias4 presented an analytical model for high-speed cylindrical roller bearings with flexible rings, and investigated the relationship between ring deformation and bearing load distribution as well as roller slipping speed and load distribution. Takafumi et al.5 proposed a theoretical model for cage slip of cylindrical roller bearings with the consideration of non-Newtonian fluid properties and temperature rise of lube oil, and analyzed the cage slip ratio under different speeds and loads. Arthanari and Marappan6 experimentally analyzed the effects of rotating speed, radial load, and the number of rollers on cage slip of a cylindrical roller bearing. Takabi and Khonsari7 built a dynamic model of a high-speed cylindrical roller bearing, and investigated the influence of different traction models on the sliding velocities and cage wear rate between rollers and races. Chinese scholars also did lots of research on bearing skidding. Li and Wu8 established a dynamic analytical model for highspeed cylindrical roller bearings, and dynamically simulated the roller skew, axial movement, roller and cage slip characteristics, etc. Hu et al.9,10 established a quasi static calculation model for inter-shaft cylindrical roller bearings, and analyzed the relationship between the roller slip ratio and radial load when bearing rings are under different working conditions. Tu and Shao11 considered the acceleration phase of roller bearings, and presented an analytical model to study bearings’ skidding characteristics under different forces and inner ring rotational accelerations. Chen et al.12 developed a bearing dynamic model under a whirling condition taking into consideration the bearing skidding of a high-speed rolling bearing under the whirling condition, and analyzed the effects of various outer loadings, whirling frequencies, and whirling radii on the bearing skidding. Deng et al.13 established a dynamic differential equation for a high-speed cylindrical roller bearing, and analyzed the effects of the clearance ratio of the cage, guiding type, etc. on the cage slip ratio and centroid trajectory. All the studies mentioned above have been focused on the performance analysis of cylindrical roller bearings under highspeed and light-load conditions, whereas studies on a cylindrical roller bearing with a trilobe-raceway have been focused on its raceway process technology,1,14,15 and little research has been done on the theoretical research of the dynamics of a cylindrical roller bearing with a trilobe-raceway.
S. DENG et al. Therefore, this paper analyzes the dynamics of a cylindrical roller bearing with a trilobe-raceway based on the dynamic analysis of the roller bearing, establishes nonlinear dynamics differential equations for a cylindrical roller bearing with a trilobe-raceway, and then adopts the GSTIFF (gear stiff) integer algorithm with a variable step to solve these equations. The focus of this paper is to study the influences of structural parameters of the raceway and parameters of working conditions as well as the relationship between the outer ring installation method and cage slip. The present paper provides some theoretical basis for the structure design of a cylindrical roller bearing with a trilobe-raceway.
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2. Calculation model of the trilobe-raceway
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An outer ring with a trilobe-raceway can be obtained by a predeformation machining method,1 and the principle of the method is shown in Fig. 1. As shown in Fig. 1(a), a uniformly-distributed load F is applied to the 1/3 symmetrical outer surface of the roughcast of the bearing ring along the circumferential direction, which leads to an elastic pre-deformation shown in Fig. 1(b). Keep the applied load constant while grinding the raceway to its theoretical base circle dimension as shown in Fig. 1(c), and the required trilobe-raceway contour can be obtained as a result of the deformation resilience after releasing the load, as shown in Fig. 1(d). The radius of the trilobe-raceway at different azimuth angles after machining can be expressed as:
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Re ðuÞ ¼ Regm Reg ðuÞ þ Reo
ð1Þ
where Reo is the original radius before machining; Reg ðuÞ is the radius of the raceway at different azimuth angles u after the elastic deformation of the outer ring is generated because of the applied force; Regm is the grinding radius, i.e., the theoretical base circle radius.
Fig. 1
Sketch map of the pre-deformation machining principle.
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2.1. Finite element analytical model of the outer ring with the trilobe-raceway Take a certain type of bearing as an example to study the relationship between the trilobe-raceway’s structural parameters and the application of loads. The finite element analytical model of the outer ring with the trilobe-raceway is established by adopting 20-node hexahedron units. The model established in Ref. 1 was simplified based on the geometrical relationship analyzed above, and the deformation caused by the grinding force in unloaded areas of the outer ring as well as the elastic deformation of the mold can be neglected. Based on the loads distribution needed for forming a trilobe-raceway, when under ideal processing conditions, gradient loads are applied to the 1/3 symmetrical outer surface of the roughcast of the bearing ring along the circumferential direction, and the included angles between loaded areas are 60°; one of the loaded areas is rotated at a slight degree uf to simulate non-ideal processing conditions, and by this way, the included angles between loaded areas become 60°, 60° + uf, and 60° uf. The finite element analytical model of the outer ring with the triloberaceway is shown in Fig. 2. Fig. 3 shows the impact of the loads rotating degree uf on the trilobe-raceway’s structural parameters when the original radius before machining is 34.86 mm and the applied loads are 2.0 MPa. As shown in Fig. 3, with an increase of the loads rotating degree, the high point of the trilobe-raceway counter increases and the low point decreases, i.e., the roundness of the base circle at low and high points rises, and the D-value between the highest and lowest points is also increased. This is because when a loaded area is rotated, the magnitudes of loads also take suitable changes to achieve the balance of applied loads; in the area where loads are increased, the deformation of the outer ring under applied loads increases, and when the grinding radius remains unchanged, the radius at the low point of the raceway rises while the radius at the high point reduces after molding; likewise, in the area where loads are decreased, the radius of the low point reduces and the radius of the high point rises after molding.
Fig. 2 Finite element analytical model of the outer ring with the trilobe-raceway.
Fig. 3 loads.
Wave shape of the raceway imposed with asymmetrical
2.2. Mathematical model of the outer ring with the triloberaceway
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As shown in Fig. 3, with the changes of the azimuth angle, the raceway radius is in accordance with a sinusoidal curve. In order to facilitate the expression of relationship between the trilobe-raceway radius and the azimuth angle, for an ideal trilobe-raceway, the obtained data can be fitted into a sinusoidal function in which the azimuth angle u is regarded as the independent variable and the radius of the raceway Re is regarded as the dependent variable as follows:
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Re ðuÞ ¼ A þ B cosðCu þ DÞ
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ð2Þ
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Meanwhile, as for a non-ideal trilobe-raceway, the obtained data can be fitted into six sinusoidal functions in the same form as Eq. (2), where the azimuth angle u is regarded as the independent variable, the radius of the raceway Re is regarded as the dependent variable, and A, B, C, and D are real constants.
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3. Dynamic analytical model of a cylindrical roller bearing with a Trilobe-raceway
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The schematic diagram of a cylindrical roller bearing with a trilobe-raceway is shown in Fig. 4. The outer ring is fixed, and the inner ring rotates at a constant speed x and bears a radial force Fr. The cage is guided by the outer ring, and the bearing face is lubricated by oil jets. Assume that the surfaces of bearing components are absolutely smooth with an ideal geometric shape, and the center of mass is coincident with the center of form. In Fig. 4, Oi and O0i are the inner ring centers of the bearing before and after loading, respectively; P0d is the radial clearance of the theoretical base circle of the outer ring; uj is the azimuth angle of the jth roller; Pdj is the radial clearance between the theoretical base circle and the outer raceway at the azimuth angle uj , and Pdj ¼ Re ðuj Þ Regm ; Re can be obtained from Eq. (2); dr is the radial displacement of the inner ring. As can be seen from Fig. 4, contact deformation between the raceway and part of rollers occurs before applying the radial force, where the radial clearance between the theoretical base circle and the outer raceway is negative and its absolute value is larger than P0d .
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Fig. 4 203
Schematic diagram of a cylindrical roller bearing with a trilobe-raceway.
3.1. Force analysis of bearing parts
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3.1.1. Forces between a roller and raceways
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Tilt always happens when a cylindrical roller bearing with a trilobe-raceway operates under a radial force Fr , and the bending moment M and the forces and deformations between a roller and raceways are shown in Fig. 5. In Fig. 5, fO; X; Y; Zg is the inertial coordinate system of the bearing, which is a fixed coordinate system with the X axis coincident with the rotation axis of the bearing, and the YZ plane is parallel to the radial plane across the bearing center. Other coordinate systems are all established according to this inertial coordinate system; forj ; xrj ; yrj ; zrj g is the coordinate system of the jth roller center, whose original point orj is coincident with the mass center of the roller, the xrj axis is along with the roller axis, and the yrj zrj plane is parallel to the radial plane across the mass center of the roller. Every roller has its own local coordinate system which moves and rotates but doesn’t spin with the roller; a is the tilting angle of the inner ring; hj is the tilting angle of the jth roller at the azimuth angle uj . The classic slice method is applied to handle the contact issue between the crowned roller and raceways, where the roller is sliced to N sections along the axial direction, w ðw ¼ Ls =NÞ is the slice width, and Ls is the effective length of the roller. The elastic deformations between the kth slice of the jth roller and raceways at the azimuth angle uj are expressed as: 8 0 < di ¼ dr cos uj Pd Pdj d0 Cjk jk uj 2 2 ð3Þ : de ¼ Pdj þ d0 C jk jk uj 2
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231
Fig. 5
Forces and deformations between a roller and raceways.
where d0uj is the deformation of the roller center, and Cjk is the
232
amount of crown drop of the kth slice of the jth roller. The normal contact forces between the kth slice of the jth roller and raceways are written as16 w iðeÞ iðeÞ1:11 qjk ¼ djk ð4Þ 0:11 A1:11 1 Ls
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where A1 is the elastic deformation coefficient, and A1 ¼ 1:36g0:9 ; g is the combined elasticity modulus of two contact bodies. The normal contact forces between the jth roller and raceways are expressed as:
239
iðeÞ
Qj
¼
N X iðeÞ qjk
234 235 236 238
240 241 242 243 244
ð5Þ 246
k¼1
The additional moments caused by normal contact forces between the jth roller and raceways are written as N X iðeÞ iðeÞ Ls wk ð6Þ MQj ¼ qjk 2 k¼1
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According to Eq. (5), the oil drag forces of the jth roller are written as
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iðeÞ
Tj
¼
N X
iðeÞ
Tjk ¼
k¼1
N X
iðeÞ
qjk lj
248 249
251
253 254
ð7Þ
k¼1
256
where lj is the drag coefficient of oil film (see Ref. 17 for detailed equation). The additional moments caused by additional moments are depicted as N X Ls iðeÞ iðeÞ wk ð8Þ MTj ¼ qjk lj 2 k¼1
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3.1.2. Forces between a roller and the cage
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The forces between a roller and the cage pocket are mainly considered as the hydrodynamic pressure between the roller and the beam of the cage pocket as well as elastic impact force due to contact deformation, which are also computed by the ‘‘slice method”. The relationship of displacement and deformation between the roller and the cage pocket is shown in Fig. 6. fopj; xpj:ypj; zpjg is the coordinate system of the cage pocket center, whose original point opj coincides with the geometric center of the cage pocket, the xpj axis is parallel to the
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rotation axis of the cage, and the ypj axis is along the radial direction when the zpj axis is along the circumferential direction of the pocket center. Every cage pocket has its own local coordinate system which moves and rotates with the cage. In Fig. 6, bj is the skewing angle of the jth roller, Dzcj is the displacement of the roller geometric center under the coordinate system of the cage pocket center. The deformation dcjk between the kth slice of the jth roller and the cage bar is expressed as follows: Ls w k tan bj Cjk dcjk ¼ Dcj þ ð9Þ 2
ing, the infinitely short bearing theory is applied in this paper to determine the acting forces Fcy, Fcz and moment Mcx between the cage and the guide surface of the ring (see Ref. 19 for detailed equation).
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3.1.4. Surface resistance and end-surface of the cage
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For a high-speed cylindrical roller bearing with a triloberaceway, cage rotation makes the external sides of the cage affected by the resistance from surrounding air/oil mist mixture shearing. See Ref. 19 for the calculation method of the retardation torque Mco of the cage surface and sides.
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The contact force qcjk between the kth slice of the jth roller and the cage bar is shown as ( ðdcjk 6 Cp Þ Kc dcjk c ð10Þ qjk ¼ 1:11 ðdcjk Cp Þ A1:11wL0:11 ðdcjk > Cp Þ
3.2. Dynamics differential equations of the bearing
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According to the above-mentioned analysis, when a cylindrical roller bearing with a trilobe-raceway is working, the roller is simultaneously applied by combined forces of inner, outer raceways and the cage, as shown in Fig. 7, where Fm j is the cen1 2 trifugal force of roller, and Fm ¼ m d x , in which xrbj is the j 2 r m rbj orbital angular velocity of the jth roller, and dm is the pitch diameter of the bearing. As seen in Fig. 7, dynamics differential equations of the jth roller are expressed as:
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1
305 306 307
309
s
where Kc is a linear approximation constant obtained by experiments, and Kc ¼ 67=Cp ; Cp is the cage pocket clearance, and Cp ¼ 0:5ðDp Dw Þ; Dp is the cage pocket diameter; Dw is the roller diameter. The direction of force qcjk is estimated by dcjk . When dcjk P 0, the roller is pushed forward by the cage; otherwise, the roller is resisted by the cage. The contact force Qcj and additional moment Mcj between the jth roller and the cage bar are expressed as: Qcj ¼
N X qcjk
ð11Þ
k¼1
302
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5
N X Ls qcjk wk 2 k¼1
Mcj ¼
ð12Þ
The friction force between the jth roller and the cage bar is expressed as follows: N N X X Fcj ¼ Fcjk ¼ lcj qcjk k¼1
ð13Þ
k¼1
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where lcj is the friction coefficient between the roller and the cage pocket (see Ref. 18 for detailed equation).
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3.1.3. Forces between the cage and the guide surface
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When the cage is guided by the outer ring, the interaction between the guide surface of the ring and the cylindrical surface of the cage is mainly produced by the hydrodynamic effect of lubricating oil. Since the acting surface of the guide rib and the cylindrical surface of the cage are small and mutually slid-
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Fig. 6
Forces and deformations between a roller and the cage.
8 c mr y€j ¼ Qij cos uj Qej cos uj þ Qcj sin uj þ Fm > j cos uj Fj cos uj > > > > i e > Tj sin uj þ Tj sin uj > > > > > c > > mr z€j ¼ Qij sin uj þ Qej sin uj þ Qcj cos uj Fm j sin uj þ Fj sin uj > < i e Tj cos uj þ Tj cos uj > > > > Jxj x_ xj ¼ Tij D2w þ Tej D2w þ Fcj D2w > > > > > > > Jyj x_ yj ¼ MiQj sin uj þ MeQj sin uj Mcj cos uj MiTj cos uj MeTj cos uj > > > : Jzj x_ zj ¼ MiQj cos uj þ MeQj cos uj þ Mcj sin uj þ MiTj sin uj þ MeTj sin uj ð14Þ
where mr is the roller mass; y€j ; z€j are the displacement accelerations of the jth roller’s mass center in fO; X; Y; Zg; Jxj ; Jyj ; Jzj are the moments of inertia of the jth roller in fO; X; Y; Zg; x_ xj ; x_ yj ; x_ zj are the angular accelerations of the jth roller in fO; X; Y; Zg, respectively. When the bearing is working, the cage is simultaneously applied by the collision force of rollers, the guiding force of the outer ring, and the combined resistance of air/oil mist mixture to both cage ends and its surface, as shown in Fig. 8, where xrj is the spin angular velocity of the jth roller, and xm is the actual speed of the cage.
Fig. 7
Schematic diagram of roller forces.
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As seen in Fig. 8, dynamics differential equations of the cage are expressed as: 8 Z1 X > > > ðFcj cos uj þ Qcj sin uj Þ þ Fcy cos wc þ Fcz sin wc > mc y€c ¼ > > > j¼1 > > > > Z1 X > > > > mc z€c ¼ ðFcj sin uj Qcj cos uj Þ Fcy sin wc þ Fcz cos wc > > > j¼1 > > > > Z1 < X Jcx x_ cx ¼ Qcj d2m Mcx Mco > > j¼1 > > > > Z1 > X > > > _ cy ¼ J ðMcj cos uj Þ cy x > > > j¼1 > > > > Z1 > X > > > ðMcj sin uj Þ > : Jcz x_ cz ¼ j¼1
ð15Þ
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where mc is the cage mass; y€c ; z€c are the accelerations of the cage’s mass center in fO; X; Y; Zg; Jcx, Jcy, and Jcz are the moments of inertia of the cage in fO; X; Y; Zg; x_ cx ; x_ cy ; x_ cz are the angular accelerations of the cage in fO; X; Y; Zg; forc ; yrc ; zrc g is the cage’s reference coordinate system; wc is the angle between foc ; yc ; zc g and forc ; yrc ; zrc g. Dynamics differential equations of the inner ring are expressed as: 8 Z1 X > > > € m ¼ ðQij cos uj þ Tij sin uj Þ þ Fr y i i > > > > j¼1 > > > > Z1 X > > > > mi z€i ¼ ðQij sin uj þ Tij cos uj Þ > > > j¼1 > > > > Z1 < X Jix x_ ix ¼ Tij D2w ð16Þ > > j¼1 > > > > Z1 > X > > J x_ ¼ > ðMiQj sin uj þ MiTj cos uj Þ yj yj > > > j¼1 > > > > Z1 > X > > > ðMiQj cos uj þ MiTj sin uj Þ > : Jzj x_ zj ¼ M j¼1
where mi is the mass of the inner ring; y€i ; z€i are the linear accelerations of the inner ring mass center in fO; X; Y; Zg; Jix, Jiy, and Jiz are the moments of inertia of the inner ring in
Fig. 9
Solution procedure of dynamics differential equations.
fO; X; Y; Zg; x_ ix ; x_ iy ; x_ iz are the angular accelerations of the inner ring in fO; X; Y; Zg; Z1 is the number of rollers.
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3.3. Solution procedure of dynamics differential equations
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The GSTIFF integer algorithm is based on backwarddifference formulae (BDF) and is a variable-step and multi-step integrator,20 which is applied to solve dynamics differential Eq. (16) of a cylindrical roller bearing with a trilobe-raceway in this study. The solution procedure of dynamics differential equations is shown in Fig. 9.
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(1) The solution duration, initial step, and convergence error of dynamics differential equations are set, firstly. In this study, the solution duration was set to 0.5 s, the initial step was set to 0.00001 s, and the convergence error was set to 1 103. (2) According to initial estimated values of a bearing component’s position and motion constraint, initial conditions of dynamics differential equations, namely, relative positions and motion vectors of various components are obtained by quasi-static analysis. (3) Given initial conditions of dynamics differential equations, forces and moments applied on bearing components are obtained according to the expressions in Section 3.1. (4) The dynamics differential equations are solved by GSTIFF, and then positions and motion vectors of bearing components are obtained. (5) Verify whether the error meets the convergence error set in Step (1). If yes, continue next solution after getting the outputs of motion parameters including displacements, velocities, and accelerated speeds of the inner ring, cage, and roller. If no, choose a smaller step value and repeat Step (4) until the solving error meets the convergence error.
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375 376 377 378 379
381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404
Fig. 8
Schematic diagram of cage forces.
4. Analysis on cage slip characteristics and basic rating life
405
In this paper, a certain type of bearing is used to study the impacts of structural parameters and tolerance of the outer
406
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raceway, working conditions, and the outer ring installation method on cage slip characteristics and basic rating life. Main parameters of the bearing are shown in Table 1. The inner ring, outer ring, and roller are made of 8Cr4Mo4V, and the cage is made of 40CrNiMo, with a fixed outer ring and a cage guided by the outer ring. The cage slip ratio Sc and the basic rating life L10 of a cylindrical roller bearing with a trilobe-raceway are defined as follows: xm ð17Þ Sc ¼ 1 0 100% xm
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106 e9=8 8=9 L10 ¼ a1 aiso ½Li9=8 þ L10 60 Vi 10 " 9=8 #8=9 9=8 106 Qci Qce ¼ a1 aiso þ 60 Vi Qei Qee
ð18Þ
where x0m is the theoretical speed of the cage; a1 and aiso are correction coefficients, a1 = 1.0, aiso = 1.0; Vi is the inner ring speed, r/min; Li10 and Le10 are the basic rating lives of the inner and outer rings, respectively; Qci and Qce are the equivalent dynamic loads of the inner and outer rings, respectively; Qei and Qee are the dynamic load ratings of the inner and outer rings, respectively. 4.1. Impact of geometric parameters on cage slip characteristics and rating life As known from Ref. 1, the raceway contour of the outer ring after machining can be determined by the locations of high and low points, as shown in Fig. 10. Therefore, this paper takes the low-radius Re min and the D-value de as the geometric parameters to analyze their impacts on cage slip characteristics. 4.1.1. Impact of the low-radius on the cage slip ratio and rating life Fig. 11(a) shows the influence of the low-radius on the cage slip ratio when the rotating speed of the bearing is set to 45,000 r/ min, the radial force is 800 N, and the D-value is 0.098 mm. As seen in Fig. 11(a), with the increase of the low-radius, the cage slip ratio increases and the amplitude grows. Lh is the basic rating life. This is because when the low-radius increases, as shown in Fig. 12, the number of loaded rollers and the contact forces between rollers and the inner ring reduce, so the traction
Table 1
forces between rollers and the inner ring decrease, and the drop of the roller pushing force finally leads to a rise of the cage slip ratio. Meanwhile, with a small low-radius, the preloads applied on rollers from the outer raceway are quite large that the bearing rating life declines sharply, as shown in Fig. 11(b). Consequently, there must be a reasonable low-radius that can avoid early failure caused by cage slip and also meet the requirement of the bearing rating life.
447
4.1.2. Impact of the D-Value on the cage slip ratio and rating life
455
Fig. 13(a) shows the impact of the D-value on the cage slip ratio and rating life of a bearing when the rotating speed is 45,000 r/min, the radial force is 800 N, and the low-radius is 34.8625 mm. As shown in Fig. 13(a), the cage slip ratio rises linearly with the increase of the D-value. When the D-value increases, as shown in Fig. 14, the loaded zones of the bearing become larger, and there are more rollers pushing the cage forward because of the increased number of loaded rollers, thus the cage slip ratio reduces, and the bearing rating life also decreases, as shown in Fig. 13(b). Therefore, a smaller D-value is preferred on condition that the rating life of the bearing is guaranteed.
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4.2. Impact of working conditions on cage slip characteristics and rating life
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4.2.1. Impact of the bearing rotating speed on the cage slip ratio
470
Assume that Fig. 15 shows the impact of the inner ring rotating speed on the cage slip ratio when the radial force of the bearing is 800 N, the D-value is 0.098 mm, and the lowradius is 34.8600, 34.8625, 34.8650, and 34.8675 mm, respectively. As shown in Fig. 15, the cage slip ratio rises with the acceleration of the bearing rotating speed, and the amplitude grows with the increase of the low-radius. The reason lies in the increased rollers centrifugal forces by the acceleration of the rotation speed, and diminished contact forces between rollers and the inner ring shown in Fig. 16 resulting in a rise of the cage slip ratio by inadequate traction forces on rollers by the inner ring. When the low-radius is small, the preloads on rollers from the outer raceway are very effective that the centrifugal forces of rollers are relatively tiny compared to the preloads. Therefore, bearings with a smaller low-radius tend
471
Main parameters of a bearing.
Parameters
Value
Geometric
Outside diameter (mm) Bore diameter (mm) Inner raceway diameter (mm) Low-radius (mm) D-value (mm) Roller diameter (mm) Length of roller (mm) Roller number
73.4 50.027 55.75 69.725 0.098 7.0 7.0 18
Condition
Normal rotate speed (r/min) Rated radial force (N)
45,000 800
Fig. 10
Schematic diagram of trilobe-raceway contours.
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Fig. 11
Fig. 12 487 488
489 490 491 492 493 494 495 496
Effects of the low-radius on the cage slip ratio and basic rating life.
Effect of the low-radius on load distribution.
to diminish their sensitivity of cage slip to the rotating speed variation during operation. 4.2.2. Impact of the radial force on the cage slip ratio and rating life Fig. 17 shows the relationship between the radial force and the cage slip ratio when the rotating speed of a bearing is 45,000 r/ min, the D-value is 0.098 mm, and the low-radius is 34.8600, 34.8625, 34.8650, and 34.8675 mm, respectively. As shown in Fig. 17, the cage slip ratio remains unchanged with the increase of the radial force when the low-radius is
Fig. 13
34.8600 mm. When the low-radius is 34.8625 mm and 34.8650 mm, with the increase of the radial force, the cage slip ratio firstly remains unchanged, and then decreases after the radial force exceeds a threshold value. The cage slip ratio will decrease with the increase of the radial force when the lowradius is 34.8675 mm. The reason lies in the fact that the loaded zone of the bearing and load distribution are mainly determined by the preloads from the raceway when the lowradius is small, with unchanged raceway geometric parameters and preloads to the roller by the raceway. When the low-radius increases and the radial force exceeds a threshold value, the radial force has a greater impact on bearing load distribution than preloads by the raceway. As Fig. 18 shows, with the increases of the radial force, the contact forces between rollers and the inner ring as well as the number of loaded rollers, the cage slip ratio decreases as a result. Fig. 17 also shows that the larger the low-radius is, the more evident the influence of the radial force on the cage slip ratio is. In other words, bearings with a smaller low-radius tend to weaken their sensitivity of cage slip when they are working under a variable radial force.
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4.3. Impact of the outer ring installation angle on cage slip characteristics and rating life
517
As shown in Fig. 1, when the outer ring is fixed and the direction of the radial force goes through the low point of a raceway contour, we define the installation angle at this time as 0 . Installation angle is defined as ue when the outer ring is rotated
519
Effects of the D-value on the cage slip ratio and basic rating life.
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Cage slip characteristics of a cylindrical roller bearing
Effect of the D-value on load distribution.
Fig. 17
Effect of the radial force on the cage slip ratio.
Effect of the inner ring rotating speed on the cage slip
Fig. 18
Effect of the radial force on load distribution.
Fig. 14
Fig. 15 ratio.
Fig. 16
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9
Effect of the rotating speed on load distribution.
clockwise to an angle ue . When ue ¼ 60 , i.e., the installation angle is 60 , the radial force goes through the high point of a raceway contour, and the direction of the radial force changes over one period. Fig. 19(a) shows the impact of the installation angle on the cage slip ratio when the rotating speed of the bearing is 45,000 r/min, the radial force is 800 N, the D-value is 0.098 mm, and the low-radius is set to 34.8600, 34.8625, 34.8650, and 34.8675 mm, respectively. As shown in Fig. 19, when the installation angle increases, the cage slip ratio remains unchanged and the bearing rating life rises up a little.
This is because of the tiny dragging forces on the cage by rollers, with almost unchanged contact forces between rollers and the inner ring due to fixed raceway geometric parameters and radial force. As the installation angle changes from 0 to 60 , the loaded zone of the bearing as shown in Fig. 20 becomes larger, and the dynamic equivalent load decreases with the increased number of loaded rollers, leading to a slight rise of the bearing rating life. Hence, when installing a cylindrical roller bearing with a trilobe-raceway, if possible, make the direction of the radial force go through the high point of the raceway contour to prolong the bearing rating life.
534
4.4. Design tolerance of the trilobe-raceway
545
4.4.1. Determination of the tolerance of raceway radius parameters
546
Fig. 21(a) and (b) shows the cage slip ratio and bearing rating life at different D-values when the rotating speed is 45,000 r/ min, the radial force is 800 N, and the low-radius is 34.8600, 34.8625, 34.8650, and 34.8675 mm, respectively. As shown in Fig. 21, when the low-radius is smaller than 34.8650 mm, the cage slip ratio and bearing rating life rise with the increase of the D-Value, just like the above-mentioned analysis. When the low-radius is 34.8675 mm, with the increase of the D-value, the cage slip ratio and bearing rating life increase at first and then remain unchanged afterwards. This is because when the low-radius is small, preloads provided by the outer raceway is very small; when the D-value exceeds a threshold value,
548
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Fig. 19
Fig. 20 560 561 562 563 564 565 566 567 568 569
Effects of the installation angle on the cage slip ratio and basic rating life.
Effect of the installation angle on load distribution.
the area of the loaded zone and load distribution are determined by the applied radial force which is set to be a constant value. Therefore, the cage slip ratio remains unchanged. Fig. 21(a) also shows that compared to the D-value, the low-radius has a greater impact on the cage slip ratio. Since the low-radius can affect not only the number of loaded rollers, but also the contact forces between rollers and the inner ring, once the low-radius is determined, the D-value has limited influence on the contact forces between rollers and the raceway within the loaded area; instead, it affects the cage slip
Fig. 21
ratio mostly by changing the number of loaded rollers through its impact on the loaded area. Therefore, to design a cylindrical roller bearing with a trilobe-raceway, it is recommended to determine firstly the range of the low-radius based on the demands of the cage slip ratio and bearing rating life, and then the range of the D-value according to the selected low-radius. As for the bearing concerned in this paper, the cage slip ratio under 10% is acceptable in actual practice. In order to meet the bearing rating life requirement of 300 h, the lowradius should be within the range of 34.8625 mm to 34.8650 mm. When the D-value is less than 0.095 mm, a bearing with a low-radius as 34.8625 mm can no longer meet the rating life requirement. When the D-value is larger than 0.105 mm, the cage slip ratio of a bearing with a low-radius of 34.865 mm will be over 10%. Therefore, the D-value should be within the range of 0.095–0.105 mm. To reduce the influences of the bearing rotating speed and radial force variation during operation, the low-point diameter of the outer raceway contour can be designed as 69:725þ0:005 mm, and the D-value 0 can be designed as 0:095þ0:01 mm. 0
570
4.4.2. Impact of the base circle roundness on the cage slip ratio
590
Fig. 22 shows the impact of the roundness of the base circle where the radius of the lowest point of the raceway contour locates on the cage slip ratio and bearing rating life, when the rotating speed is 45,000 r/min, the radial force is 800 N,
591
Effects of the D-value on the cage slip ratio and basic rating life at different low-radius values.
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Fig. 22
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Effects of the roundness on the cage slip ratio and basic rating life at different low-radius values.
and the low-radius is 34.8625 mm and 34.8650 mm, respectively. As shown in Fig. 22, when the roundness increases, the cage slip ratio increases while the bearing rating life reduces. This is because when the roundness of the base circle increases, the radii of three low points of the raceway contour are no longer the same, so are those of three high points. Therefore, the radius of the lowest point decreases when the difference value between the highest and lowest points of the raceway contour increases. Contact forces between rollers and the inner ring increase in areas near the lowest points but decrease in other areas. The increasing amplitude of the total bearing load caused by the former is smaller than the decreasing amplitude of the total bearing load caused by the latter. As a result, the cage slip ratio tends to increase and the bearing rating life decreases gradually. As for the bearing concerned in this paper, when the roundness of the base circle at low points is larger than 0.006, a bearing with a low-radius of 69.725 mm and a D-value of 0.095 mm can no longer meet the rating life requirement, so the roundness of the base circle at low points shall be designed as 0.006.
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5. Comparison validation
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Experimental results in Ref. 21 are adopted to verify the validity of the dynamic model built in this paper. Main parameters of a test bearing are shown in Table 2. Fig. 23 shows the comparison of the cage slip ratio between experimental and simulation results when the applied radial force on the tested bearing is 800 N, and the rotating speed
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Table 2
Main parameters of a test bearing.
Geometric parameters
Value
Outside diameter (mm) Bore diameter (mm) Inner raceway diameter (mm) Low-radius (mm) D-value (mm) Roller diameter (mm) Length of roller (mm) Roller number
73.4 50 55.7 69.648 0.06 7.0 7.0 18
Fig. 23 results.
Comparison between experimental and simulation
is varying from 15,000 to 52,000 r/min. As shown in Fig. 23, the cage slip ratio rises with the acceleration of the bearing rotating speed, so simulation results are generally in line with experimental results, and the error is not more than 9%. When operating at lower speed, simulation results are smaller than experimental results, which may be caused by different lubricating oil used in the dynamic model and the experiment. In addition, when working at a higher speed, simulation results are larger than experimental results, because the dynamic model built in this paper does not include the impact of temperature on bearing geometric parameters. When the rotating speed is high, the radial internal clearance decreases because of the large heat productivity; hence, the cage slip ratio in the experiment is slightly smaller than that in the simulation.
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6. Conclusions
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(1) The cage slip ratio and bearing rating life of a cylindrical roller bearing with a trilobe-raceway reduce with the decrease of the low-radius and D-value, and the lowradius has a greater impact than the D-value. Therefore, on the premise of bearing rating life, a small low-radius and D-value are preferred, and the variation range of the low-radius should be determined before the determination of the D-value.
Please cite this article in press as: DENG S et al. Cage slip characteristics of a cylindrical roller bearing with a trilobe-raceway, Chin J Aeronaut (2017), http://dx.doi. org/10.1016/j.cja.2017.07.001
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S. DENG et al. (2) The cage slip ratio rises with the acceleration of the inner ring rotating speed and drops with the increase of the radial force, and the changing amplitude increases along with the increase of the low-radius. That’s to say, a bearing with a smaller low-radius tends to diminish its sensitivity of cage slip when working under a variable condition. (3) When the installation angle of a cylindrical roller bearing with a trilobe-raceway increases within one period, the cage slip ratio remains unchanged while the bearing rating life rises up a little. Hence, the direction of the radial force should go through the high-point radius of the outer raceway contour when installing the bearing to prolong its rating life. (4) With the increase of the roundness of the base circle at three low points of the raceway contour, the cage slip ratio rises and the bearing rating life decreases. Thus, the roundness of the base circle should be reduced to the greatest extent under attainable machining precision.
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Acknowledgements
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This research was financially co-supported by the National Natural Science Foundation of China (U1404514), Henan Outstanding Person Foundation in China (144200510020), and the Collaborative Innovation Center of Major Machine Manufacturing in Liaoning, China.
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References
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No. of Pages 12
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Please cite this article in press as: DENG S et al. Cage slip characteristics of a cylindrical roller bearing with a trilobe-raceway, Chin J Aeronaut (2017), http://dx.doi. org/10.1016/j.cja.2017.07.001