Study on the Load Distribution and Dynamic Characteristics of a Thin-Walled Integrated Squirrel-Cage Supporting Roller Bearing
Study on the Load Distribution and Dynamic Characteristics of a Thin-Walled Integrated...
Mao, Yuze;Wang, Liqin;Zhang, Chuanwei
2016-12-14 00:00:00
applied sciences Article Study on the Load Distribution and Dynamic Characteristics of a Thin-Walled Integrated Squirrel-Cage Supporting Roller Bearing Yuze Mao, Liqin Wang * and Chuanwei Zhang School of Mechatronics Engineering, Harbin Institute of Technology, Harbin 150001, China; ma0yuze@163.com (Y.M.); zhchwei1984@163.com (C.Z.) * Correspondence: lqwanghit@163.com; Tel.: +86-451-8640-2012 Academic Editor: Zhong Tao Received: 7 October 2016; Accepted: 3 December 2016; Published: 14 December 2016 Abstract: Thin-walled integrated flexible support structures are the major trend in the development of current rolling bearing technology. A thin-walled, integrated, squirrel-cage flexible support roller bearing, quasi-dynamic iterative finite element analysis (FEA) model is established in this paper. The FEA model is used to calculate the structural deformation of the thin-wall rings and support structures; the dynamic characteristics of the bearing are analyzed using the noncircular bearing modified quasi-dynamic model. The influence of the integrated flexible support structure on the internal load distribution and the dynamic characteristics of the roller bearing are analyzed. The results indicate that with the support of a flexible squirrel-cage, the maximum contact load is decreased by 14.2%, the loading region is enlarged by 25%, the cage slide ratio is reduced by 24%, and the fatigue life is increased by more than 50%. In addition, as the ring wall thickness increased, the results increasingly approached those under a rigid assumption. Keywords: rolling bearing; thin wall; flexible support; structural deformation; load distribution; fatigue life 1. Introduction Advanced rolling bearing technology is one of the basic strategic industries and core technologies in high-speed systems and the aerospace field. The use of thin walls, lightweight structures, and component integration represent major trends in the development of advanced rolling bearing technologies [1]. As a main technological trend in the field of high-speed bearings, the integrated squirrel-cage flexible support roller bearing is a thin-wall bearing with a squirrel-cage structure connected to the outer ring and a hollow shaft assembled in the inner ring. In this type of structure, the hollow shaft, bearing ring, and squirrel-cage flexible support structures are all thin-walled, and, as a result, easily produce elastic structural deformations under load; this induces a change in the internal load distribution of the bearing, and leads to a significant influence on the characteristics of the bearing dynamics, which should not be neglected. Based on the Hertz theoretical calculation model, Jones [2] built a static model and calculated the load distribution of rolling bearing. Subsequently, many researchers studied the behaviors of rolling bearings [2–5], however, most of their methods are built using the rigid structure hypothesis, considering the rings as constant geometry parts, and ignoring the elastic deformations of the structures under load. Thus, these rigid structure assumption models cannot fully meet the demands of the continued development of rolling bearings, because the structural deformation of thin-walled rings and flexible supports is much more pronounced. Thus, it is essential to carry out dynamic performance analyses of the integrated squirrel-cage flexible support bearing in the study of current high-end bearing usage. Appl. Sci. 2016, 6, 415; doi:10.3390/app6120415 www.mdpi.com/journal/applsci Appl. Sci. 2016, 6, 415 2 of 11 Broschard [6] established a flexible bearing model for planetary gear, while Shen [7] built a flexible bearing model for harmonic gear. However, these two models are only for bearings in planetary gear transmission or harmonic gear transmission, which are quite different from the flexible support bearings in high-speed shaft systems. Lostado [8] analyzed the contact stress of the double-row tapered roller bearings using the finite element method; experimental analyses and an analytical model shows that the finite element method could be very accurate in bearing stress calculations. Yao [9] used the curved Timoshenko beam (CTB) theory to calculate the structural deformation of a thin-walled bearing ring, and analyzed the load distribution of thin-walled roller bearing; however, the model, assumed that the rings are supported at one or two azimuth positions only, and did not consider the specific form of the bearing support. Researchers, such as Ignacio [10], Kania [11], Olave [12], Shu Ju [13] and Liu [14], used the finite element method to analyze the structural deformation of thin-walled rings, but such a method cannot take into account the contact mechanics and the lubrication status between the rollers and the raceway. Moreover, a method in which linear springs are used to replace rolling elements cannot provide an accurate internal loading solution, because the load-deflection factor is non-linear, and the centrifugal effort in high-speed bearing is very difficult to simulate by a spring element, and it would be very complicated when the bearing has clearances because the number of loading rollers is unknown [15]; thus, it was difficult to precisely analyze the dynamic performance of the bearing. In this paper, an integrated squirrel-cage flexible support roller bearing iterative quasi-dynamic finite element analysis (FEA) model is established. This model is used to comprehensively consider the effects of contact mechanics, lubrication, and structural deformations. The model can accurately predict the dynamic performance of the integrated squirrel-cage flexible support roller bearing. The standard circular rolling bearing quasi-dynamic model is modified in order to obtain the modified noncircular rolling bearing quasi-dynamic model. The integrated squirrel-cage flexible support roller bearing finite element model is constructed using ANSYS software (ANSYS 12.0, ANSYS Corporation, Canonsburg, PA, USA), and is used to calculate the deformation of the bearing ring and squirrel-cage structure. The calculated deformations are inserted into the noncircular rolling bearing quasi-dynamic modified model in order to obtain load distribution and dynamic characteristic analysis results of the bearing. 2. Model Design 2.1. Modified Noncircular Rolling Bearing Quasi-Dynamic Model Quasi-dynamic analysis is an accurate bearing analysis method, because it comprehensively considers the microscopic contact and the rheological characteristics of the lubricant. Quasi-dynamic analysis can accurately calculate the dynamic characteristics of a rigid rolling bearing, but is not suited for a flexible bearing, of which the raceways would be noncircular following structural deformation; thus, a modified quasi-dynamic model for noncircular rolling bearings is needed. Figure 1 shows a sketch of the force of roller bearing. Cutting the roller into slices, the k-th slice of the j-th roller experiences contact forces Q and Q , and drags forces T and T from the inner 1jk 2jk 1jk 2jk and outer raceways through oil films, normal force F and tangential force f from the cage, and cj cj oil–gas mixture frictional force F , as well as its own inertial force, F , and inertia moment M due gj zj x j to rolling. Appl. Sci. 2016, 6, 415 3 of 11 Appl. Sci. 2016, 6, 415 3 of 12 zj Q1j j-th zj Ck Fzj oj roller xj yj k-th T1jk P1j p2 Mx slice k ω rp'2p2 p'2 xj Inner xj ring P2j T2jk ro2oj ro2pj2 My Fgj Q2j Fcj z2 fcj yj o2 x2 y2 (a) (b) Figure 1. Sketch of the force of roller bearing: (a) Model of the roller–raceway interaction; (b) forces Figure 1. Sketch of the force of roller bearing: (a) Model of the roller–raceway interaction; (b) forces and moments acting on a roller. and moments acting on a roller. According to the Hertz theory, forces and between the roller slice and the raceways Q Q 1jk 2jk According to the Hertz theory, forces Q and Q between the roller slice and the raceways can 1jk 2jk be can obtained be obtain fred om from contact cont appr act oaches approaches and δ and of the δ slice of and the raceways, slice and racew respectively ays, respectively. . When the 1jk 2jk 1jk 2jk ring deforms, the raceway becomes a nonideal cylindrical surface. The deviations of the k-th slice When the ring deforms, the raceway becomes a nonideal cylindrical surface. The deviations of the of the j-th roller at the local surface from the ideal cylindrical surface can be expressed as and 1jkh k‐th slice of the j‐th roller at the local surface from the ideal cylindrical surface can be expressed as . These two deviations can be superimposed with approaches and , based on the small 2jkh 1jk 2jk δ and δ . These two deviations can be superimposed with approaches δ and δ , based 1jkh 2jkh 1jk 2jk deformation assumption. By further considering oil film thickness h and h at the contact area, 1jk 2jk 0 0 the actual contact approaches, and , between the roller slice and the noncircular raceway can 1jk 2jk h h 1jk 2jk on the small deformation assump0tion. By 0further considering oil film thickness and at be obtained. Thus, contact loads Q and Q of the rolling object and the noncircular raceway can be 1jk 2jk ' ' obtained, the contact as area, expressed the act inu Equation al contact (1). ap Cproaches, is the cor δ rection and of δ the roller , between radius, the considering roller slice the and Roller the 1jk 2jk Profile Modification and chamfer [16]. In addition, the roller balance equation can be established as Q Q 1jk 2jk Equation noncircular (2). ra The cewa lubricant y can be parameters obtained. in Thus, the model contact ar loa e calculated ds an using d the of Hertz the contact rolling theory object ,and the Dowson-Higginson the noncircular raceformula way can[ 17 be], obtained, and the five-parameter as expressed in rheological Equation (1 model ). [18 is ].the The correction five-parameter of the rheological model has been experimentally verified and contains the viscosity–pressure property and roller radius, considering the Roller Profile Modification and chamfer [16]. In addition, the roller the viscosity–temperature property of the oil. balance equation can be established as Equation (2). The lubricant parameters in the model are calculated using the Hertz contact theory, the Dowson‐Higginson formula [17], and the five‐parameter = + h + + C 1jk 1jk 1jkh k 1jk rheological model [18]. The five‐parameter rheological model has been experimentally verified and 1/0.9 0 0.8 1jk E contains the viscosity–pre 0 ssure property and the 1 viscosity–temperature property of the oil. Q = 1jk 3.81 2(1 ) = + h + + C 2jk 2jk 2jkh k 2jk δδhC δ 11 jk jk 1jk 1jkh k (1) 1/0.9 0.8 2jk E 0 1/0.9 ' 0.8 Q = 2 2jk δ 3.81 l π E 2(1 ) '1jk e 1 2 Q ( 1jk 2 l l l l +l e s e e s 3.81 2(1 υ) 0; (k 0.5) 1 2 n 2 C = 0.5 ' 0.5 2 2 2 l l l +l s e s e s δδhC δ (R ) (R x ) ; 0 < k < [ < k < l 22 jk jk 2jk 2jkh k 4 k 2 2 1/0.9 ' 0.8 δ l π E (1) ' 2jk n e Q 1 2jk (T T + P P ) + F + F + F = 0 å 1jk 2jk 1jk 2jk cj gj yj n 3.81 2(1 υ) 2 k=1 1 0 0 ll l l l es ( Q + Q ) e f +eF =s 0 cj zj 0; n 1j( kk 02.j5 k) k=1 22 n C n 1 2 ( l (T + T ) f ) + Mll = 0 l l å (2) 20.5 1jk 2 2jk2 0.5 cj x j 2 n s es e s () R () Rx ;0k kl k=1 ke 4 22 1 0 0 å ( Q x + Q x ) + M = 0 k k yj n 1jk 2jk k=1 ( P x + P x ) + M = 0 zj 1jk k 2jk k k=1 Appl. Sci. 2016, 6, 415 4 of 11 Variable and can be obtained based on the geometric shape of the deformed raceways, 1jkh 2jkh thus, with the rollers’ loads on the inner ring and the applied load of the ring from the shaft, the ring balance equation can be obtained, as expressed in Equation (3). The cage balance equation can be obtained using the method of Cui [16]. Thus, the modified noncircular ring bearing quasi-dynamic model is established. N n 2jk F F sin F cos v sin f = 0 å å y lr lt j j=1 k=1 N n 2jk F F cos F sin v cos f = 0 å å z j lr lt j=1 k=1 Q [(k 0.5)v 0.5l ] 0 å e N n 2jk 2jk k=1 M v cos f = 0 å å y n j e (3) j=1 k=1 å Q 2jk k=1 Q [(k 0.5)v 0.5l ] 0 å e N n 2jk 2jk k=1 M v sin f = 0 å å z n j j=1 k=1 å Q 2jk k=1 N n M v (T + P ) = 0 x å å 2jk 2jk j=1 k=1 The proposed model can be used to accurately analyze the load distribution and dynamic characteristics of the deformed bearing by taking into consideration the influence of the noncircular raceway following structural deformation, and also the microscopic contact and the rheological lubricant properties in the bearing. By setting the values of and as zero, this model can also 1jkh 2jkh analyze load distribution without structural deformations, which can be the initial values of finite element analysis. 2.2. Finite Element Analysis of Integrated Squirrel-Cage Structural Deformation An integrated squirrel-cage flexible support roller bearing includes a roller bearing, a hollow shaft, and a squirrel-cage support structure, and all components are thin-walled, as shown in Figure 2a. The shaft is assembled in the inner ring. The outer ring and the squirrel-cage structure are one integrated component, which means the outer ring is a part of the squirrel-cage structure, and the outer raceway is manufactured on the internal surface of the cylinder of the squirrel-cage structure. A finite element model containing a thin-walled hollow shaft, an inner bearing ring, and an outer ring-squirrel-cage support structure was constructed in ANSYS. Boolean operation of the assembly was conducted in order to ensure satisfaction of the hexahedral meshing conditions, and three-dimensional object element meshing is achieved with SOLID45 elements. The mesh size has a great influence on the accuracy of the calculation [19]. The dimensions of contact areas are on the micrometer scale and the overall dimension of the bearing is in the millimeter scale. Thus, after meshing the whole assembly, the contact lines on the inner and outer raceway is re-mashed twice with smaller mesh sizes, so it is more accurate in imposing a load, as shown in Figure 2b. To accurately reflect the relationship between the inner ring and the shaft, contact pairs are placed between the inner surface of the inner ring and the outer surface of the shaft, and between the side of the inner ring and the shaft shoulder. The contact method is surface-to-surface, and the friction coefficient is 0.1 [13]. The outer ring and the squirrel-cage structure are one integrated component, thus there is no contact pair. Designating the inner surface and the side surface of the inner ring as the contacting surface and the corresponding shaft outer surface and shaft shoulder as the target surface, respectively, the mutual contacting portion is described by TARGE170 and CONTA174 surface–surface contact elements. Targe170 is used to represent various 3-D “target” surfaces for the associated contact elements, and CONTA174 is used to represent contact and sliding between 3-D “target” surfaces and a deformable surface defined by this element. Appl. Sci. 2016, 6, 415 5 of 12 SOLID45 elements. The mesh size has a great influence on the accuracy of the calculation [19]. The dimensions of contact areas are on the micrometer scale and the overall dimension of the bearing is in the millimeter scale. Thus, after meshing the whole assembly, the contact lines on the inner and outer raceway is re-mashed twice with smaller mesh sizes, so it is more accurate in imposing a load, as shown in Figure 2b. To accurately reflect the relationship between the inner ring and the shaft, contact pairs are placed between the inner surface of the inner ring and the outer surface of the shaft, and between the side of the inner ring and the shaft shoulder. The contact method is surface-to-surface, and the friction coefficient is 0.1 [13]. The outer ring and the squirrel-cage structure are one integrated component, thus there is no contact pair. Designating the inner surface and the side surface of the inner ring as the contacting surface and the corresponding shaft outer surface and shaft shoulder as the target surface, respectively, the mutual contacting portion is described by TARGE170 and CONTA174 surface–surface contact elements.Targe170 is used to represent various 3-D “target” surfaces for the Appl. Sci. 2016, 6, 415 5 of 11 associated contact elements, and CONTA174 is used to represent contact and sliding between 3-D target surfaces and a deformable surface defined by this element. The contact loads on the raceway from the rollers are calculated using the modified quasi- The contact loads on the raceway from the rollers are calculated using the modified quasi-dynamic dynamic model, and the calculated loads are added to the contact line in the FEA model, as shown model, and the calculated loads are added to the contact line in the FEA model, as shown in Figure 2c,d. in Figure 2c,d. The squirrel-cage base surface is restricted to zero degrees of freedom. The shaft’s two The squirrel-cage base surface is restricted to zero degrees of freedom. The shaft’s two end faces are end faces are restricted to zero degrees of freedom, and the inner ring is restricted on the shaft by restricted to zero degrees of freedom, and the inner ring is restricted on the shaft by contact pairs. contact pairs. There are no other constraints in this model, thus, it can deform under forces. Solving There are no other constraints in this model, thus, it can deform under forces. Solving the model, the the model, the flexure and torsion deformations of the squirrel-cage structure and the thin-walled flexure and torsion deformations of the squirrel-cage structure and the thin-walled ring, as well as the ring, as well as the resultant internal stress due to structural deformation, can be analyzed. In resultant internal stress due to structural deformation, can be analyzed. In addition, the values of 1jkh and in the modified quasi-dynamic model are obtained. additi 2on, the jkh values of δ and δ in the modified quasi-dynamic model are obtained. 1jkh 2jkh (a) (b) (c) (d) Figure 2. Preprocessor of the FEA (finite element analysis) model: (a) Flexible support structure with a Figure 2. Preprocessor of the FEA (finite element analysis) model: (a) Flexible support structure with a squirrel-cage; (b) refinement of the contact area; (c) loads on the outer ring; (d) loads on the contact line. squirrel-cage; (b) refinement of the contact area; (c) loads on the outer ring; (d) loads on the contact line. 2.3. Iterative Quasi-Dynamic FEA Model 2.3. Iterative Quasi-Dynamic FEA Model The initial values of internal bearing loads Q and Q are calculated using the modified 1jk 2jk quasi-dynamic model of noncircular bearing under the assumption of rigidity; the values of and 1jkh are set as zero. These loads are inserted into the FEA model in order to calculate the local offset 2jkh of each of the slice/raceway contact points ( and ). These offsets are substituted into the 1jkh 2jkh modified noncircular quasi-dynamic model via Equation (1) in order to recalculate raceway loads Q 1jk and Q , and they are then inserted into the finite element model once again. These steps are repeated 2jk until the load values satisfy the convergence precision. The quasi-dynamic iterative FEA model analysis flow is shown in Figure 3. When the results converge, the load distribution of the bearing, with consideration to the structural deformation of the integrated squirrel-cage flexible support, can be obtained. Furthermore, the dynamic characteristics of the bearing, and the stiffness of the entire structure, including squirrel-cage stiffness and the deformed bearing stiffness, can be obtained. The fatigue life can be calculated by the Harris improved Appl. Sci. 2016, 6, 415 6 of 12 2.3. Iterative Quasi‐Dynamic FEA Model Q Q 1jk 2jk The initial values of internal bearing loads and are calculated using the modified quasi‐dynamic model of noncircular bearing under the assumption of rigidity; the values of δ 1jkh and δ are set as zero. These loads are inserted into the FEA model in order to calculate the 2jkh local offset of each of the slice/raceway contact points (δ and δ ). These offsets are 1jkh 2jkh substituted into the modified noncircular quasi‐dynamic model via Equation (1) in order to ' ' Q Q 2jk 1jk recalculate raceway loads and , and they are then inserted into the finite element model once again. These steps are repeated until the load values satisfy the convergence precision. The quasi‐dynamic iterative FEA model analysis flow is shown in Figure 3. When the results converge, the load distribution of the bearing, with consideration to the structural deformation of the integrated squirrel‐cage flexible support, can be obtained. Furthermore, the dynamic Appl. charact Sci.e 2016 rist,ic 6,s415 of the bearing, and the stiffness of the entire structure, including squirrel‐ 6ca ofg 11e stiffness and the deformed bearing stiffness, can be obtained. The fatigue life can be calculated by the Harris improved Lundberg‐Palmgren (L‐P) theory. The Harris improved L‐P theory is based on Lundberg-Palmgren (L-P) theory. The Harris improved L-P theory is based on the L-P theory and the L‐P theory and contains the load distribution and lubrication of the bearing [2]. contains the load distribution and lubrication of the bearing [2]. Start Quasi-dynamic of rigid assumption Initial loads between rollers and rings FEA model analysis Deformation of squirrel and Distortion of rings Modified quasi-dynamic model of non-circular race Modified loads between rollers and i=i+1 rings (Qi) Error of Qi and Q(i-1) <0.5% Load distribution, sliding, fatigue life and stiffness End Figure 3. Flow diagram of the quasi-dynamic iterative FEA model. Figure 3. Flow diagram of the quasi‐dynamic iterative FEA model. 3. Results and Discussion 3. Results and Discussion An integrated squirrel-cage support roller bearing is used as an example to analyze the influence An integrated squirrel‐cage support roller bearing is used as an example to analyze the of structural deformation on the load distribution, dynamic behavior, and fatigue life of the bearing. influence of structural deformation on the load distribution, dynamic behavior, and fatigue life of The parameters of the bearing are given in Tables 1–3, and the numerical values of the analysis results the bearing. The parameters of the bearing are given in Tables 1–3, and the numerical values of the depend on the geometry of the bearing. analysis results depend on the geometry of the bearing. Table 1. Geometry parameters and working conditions of the bearing. Parameter Value/mm Parameter Value Inner ring 59.6 inner ring thickness (mm) 5 Outer ring 82.2 outer ring thickness (mm) 3.6 Roller quantity 30 length of squirrel-cage (mm) 10.25 Roller diameter 5 shift thickness (mm) 4 Roller length 5 radial load (N) 19,621.1 clearance 0.05 Speed (r/min) 38,000 Table 2. Material parameters of the bearing and squirrel-structure. Component Elastic Modulus Poisson Ratio () Density Thermal Expansion (E)/GPa ()/kg/m Coefficient () Shaft 179 0.281 8240 0.0138 Inner ring 203 0.28 7850 0.0112 Roller 209 0.3 7860 0.0119 Outer ring-squirrel Cage 203 0.28 7850 0.0112 Appl. Sci. 2016, 6, 415 7 of 12 Table 1. Geometry parameters and working conditions of the bearing. Parameter Value/mm Parameter Value Inner ring 59.6 inner ring thickness (mm) 5 Outer ring 82.2 outer ring thickness (mm) 3.6 Roller quantity 30 length of squirrel‐cage (mm) 10.25 Roller diameter 5 shift thickness (mm) 4 Roller length 5 radial load (N) 19,621.1 clearance 0.05 Speed (r/min) 38,000 Table 2. Material parameters of the bearing and squirrel‐structure. Component Elastic Modulus Poisson Ratio Density Thermal Expansion (E)/GPa (υ) (ρ)/kg/m Coefficient (α) Shaft 179 0.281 8240 0.0138 Inner ring 203 0.28 7850 0.0112 Roller 209 0.3 7860 0.0119 Appl. Sci. 2016, 6, 415 7 of 11 Outer ring‐squirrel Cage 203 0.28 7850 0.0112 Table 3. Temperature of bearing components. Table 3. Temperature of bearing components. Shaft Inner Ring Roller Outer Ring and Squirrel‐Cage 230 °C 220 °C 226 °C 218 °C Shaft Inner Ring Roller Outer Ring and Squirrel-Cage 230 C 220 C 226 C 218 C The temperature of the components can influence the rheological behavior of the oil and the internal clearance of the bearing; thus, it is necessary to consider the temperature of each of the The temperature of the components can influence the rheological behavior of the oil and the bearing components. Additionally, the FEA model is built with dimensions after heat expansion. internal clearance of the bearing; thus, it is necessary to consider the temperature of each of the bearing Consideration the influence of structural deformations, the abovementioned quasi‐dynamic components. Additionally, the FEA model is built with dimensions after heat expansion. iterative FEA model is used to calculate the load distribution of the bearing. The FEA model has 1.3 Consideration the influence of structural deformations, the abovementioned quasi-dynamic million elements, and the mesh size on the contact areas of raceway is 0.11 mm, while that of the iterative FEA model is used to calculate the load distribution of the bearing. The FEA model has bearing body is 0.8 mm. A computer with two CPU (Central Processing Unit) (Intel Xeon E5‐2650, 1.3 million elements, and the mesh size on the contact areas of raceway is 0.11 mm, while that of the octa‐core, 2.00 GHz, Intel Corporation, Santa Clara, CA, USA) and 64 GB Random‐Access Memory bearing body is 0.8 mm. A computer with two CPU (Central Processing Unit) (Intel Xeon E5-2650, (RAM) was used to simulate this model. 20 minutes were needed to preprocess the model, and, octa-core, 2.00 GHz, Intel Corporation, Santa Clara, CA, USA) and 64 GB Random-Access Memory during each iteration, 10 to 15 minutes were needed to solve. The quasi‐dynamic model is built (RAM) was used to simulate this model. 20 minutes were needed to preprocess the model, and, using MATLAB (MATLAB R2013b, MathWorks Corporation, Natick, MA, USA), and, during each during each iteration, 10 to 15 minutes were needed to solve. The quasi-dynamic model is built using iteration, two minutes were required to solve. After 73 iterations, the load distribution reached MATLAB (MATLAB R2013b, MathWorks Corporation, Natick, MA, USA), and, during each iteration, convergence. two minutes were required to solve. After 73 iterations, the load distribution reached convergence. ' ' 0 0 R R Figure 4 shows the deformation of the structure, and the equivalent radious R , R of the 1 raceways 2 Figure 4 shows the deformation of the structure, and the equivalent radious , of the 1 2 are calculated and depicted in Figure 5. The maximum radial deviation between the equivalent radius raceways are calculated and depicted in Figure 5. The maximum radial deviation between the and the original radius is 0.01428 mm. equivalent radius and the original radius is 0.01428 mm. (a) (b) Figure 4. Deformation of the rings and squirrel‐cage: (a) Deformation of squirrel‐cage and outer ring; Figure 4. Deformation of the rings and squirrel-cage: (a) Deformation of squirrel-cage and outer ring; (b) deformation of the shaft and inner ring. (b) deformation of the shaft and inner ring. Appl. Sci. 2016, 6, 415 8 of 12 (a) (b) R 0 Figure Figure 5. 5. Equ Equivalent ivalent radius radius of of raceways: raceways: ((aa)) Equ Equivalent ivalent radius radius R of of the the outer outer race; race; ( (b b) ) equivalent equivalent radius R of the inner race. radius of the inner race. The calculated results are compared to those based on the rigidity assumption, without consideration of the influence of the structural deformation. The load distribution of the bearing is shown in Figure 6. Figure 6. The contrast of load distribution between the rigid support and flexible support. It can be seen from the above results that, compared to the results calculated under the rigid support assumption, the maximum roller/raceway load decreased to 3035 N from 3537 N due to the influence of the deformation of the flexible squirrel‐cage support structure, representing a decrease of 14.2%. Moreover, the internal bearing load is distributed more evenly. The number of loaded rollers decreased to 9 from 11, so the loading range is enlarged to 120° (−60°~60°) from 96° (−48°~48°), representing an increase of 25%. Because bearing sliding is strongly dependent on the roller/raceway load, the change in the load distribution alters the bearing slide ratio. Compared to the rigid support, the bearing cage slide ratio under the flexible support decreased from 6.28% to 4.77%, representing a decrease of 24.05%. Figure 7 shows the slide ratio distribution of each roller. Appl. Sci. 2016, 6, 415 8 of 12 (a) (b) Figure 5. Equivalent radius of raceways: (a) Equivalent radius of the outer race; (b) equivalent Appl. Sci. 2016, 6, 415 8 of 11 radius of the inner race. The calculated results are compared to those based on the rigidity assumption, without The calculated results are compared to those based on the rigidity assumption, without consideration of the influence of the structural deformation. The load distribution of the bearing consideration of the influence of the structural deformation. The load distribution of the bearing is is shown in Figure 6. shown in Figure 6. Figure 6. The contrast of load distribution between the rigid support and flexible support. Figure 6. The contrast of load distribution between the rigid support and flexible support. It can be seen from the above results that, compared to the results calculated under the rigid It can be seen from the above results that, compared to the results calculated under the rigid support assumption, the maximum roller/raceway load decreased to 3035 N from 3537 N due to the support assumption, the maximum roller/raceway load decreased to 3035 N from 3537 N due to the influence of the deformation of the flexible squirrel‐cage support structure, representing a decrease influence of the deformation of the flexible squirrel-cage support structure, representing a decrease of of 14.2%. Moreover, the internal bearing load is distributed more evenly. The number of loaded 14.2%. Moreover, the internal bearing load is distributed more evenly. The number of loaded rollers rollers decreased to 9 from 11, so the loading range is enlarged to 120° (−60°~60°) from 96° decreased to 9 from 11, so the loading range is enlarged to 120 ( 60 ~60 ) from 96 ( 48 ~48 ), (−48°~48°), representing an increase of 25%. representing an increase of 25%. Because bearing sliding is strongly dependent on the roller/raceway load, the change in the load Because bearing sliding is strongly dependent on the roller/raceway load, the change in the load distribution alters the bearing slide ratio. Compared to the rigid support, the bearing cage slide ratio distribution alters the bearing slide ratio. Compared to the rigid support, the bearing cage slide ratio under the flexible support decreased from 6.28% to 4.77%, representing a decrease of 24.05%. Figure 7 under the flexible support decreased from 6.28% to 4.77%, representing a decrease of 24.05%. Figure 7 shows the slide ratio distribution of each roller. Appl. Sci. 2016, 6, 415 9 of 12 shows the slide ratio distribution of each roller. Figure 7. Slide ratio of rollers. Figure 7. Slide ratio of rollers. Figure 8 shows the changes observed for the bearing cage slide ratio due to variations in the Figure 8 shows the changes observed for the bearing cage slide ratio due to variations in the rotational speed. As the rotational speed increased, the cage sliding decreased by a greater rotational speed. As the rotational speed increased, the cage sliding decreased by a greater percentage. percentage. Figure 8. Influence of operation speed on cage sliding. Figure 9 shows the influence of the change in thickness of the outer ring on the load distribution of the bearing. As the outer ring increases in thickness, the load distribution more closely approached the calculation result obtained under the rigid assumption; conversely, as the outer ring became thinner, the influence of structural deformation increased significantly. Figure 9. Influence of ring thickness on load distribution. Figure 10 shows the cage slide ratio due to different ring thicknesses. It can be seen that under the rigid assumption, the ring wall thickness does not influence the cage sliding. After considering Appl. Sci. 2016, 6, 415 9 of 12 Appl. Sci. 2016, 6, 415 9 of 12 Figure 7. Slide ratio of rollers. Figure 7. Slide ratio of rollers. Figure 8 shows the changes observed for the bearing cage slide ratio due to variations in the Figure 8 shows the changes observed for the bearing cage slide ratio due to variations in the rotational speed. As the rotational speed increased, the cage sliding decreased by a greater rotational speed. As the rotational speed increased, the cage sliding decreased by a greater percentage. percentage. Appl. Sci. 2016, 6, 415 9 of 11 Figure 8. Influence of operation speed on cage sliding. Figure 8. Influence of operation speed on cage sliding. Figure 8. Influence of operation speed on cage sliding. Figure 9 shows the influence of the change in thickness of the outer ring on the load Figur Figeure 9 shows 9 shothe ws influence the influe ofnce the of change the change in thickness in thickness of the outer of th ring e outer on the rin load g on distribution the load of distribution of the bearing. As the outer ring increases in thickness, the load distribution more distribution of the bearing. As the outer ring increases in thickness, the load distribution more the bearing. As the outer ring increases in thickness, the load distribution more closely approached the closely approached the calculation result obtained under the rigid assumption; conversely, as the closely approached the calculation result obtained under the rigid assumption; conversely, as the calculation result obtained under the rigid assumption; conversely, as the outer ring became thinner, outer ring became thinner, the influence of structural deformation increased significantly. outer ring became thinner, the influence of structural deformation increased significantly. the influence of structural deformation increased significantly. Figure 9. Influence of ring thickness on load distribution. Figure 9. Influence of ring thickness on load distribution. Figure 9. Influence of ring thickness on load distribution. Figure 10 shows the cage slide ratio due to different ring thicknesses. It can be seen that under Figure 10 shows the cage slide ratio due to different ring thicknesses. It can be seen that under the rigid assumption, the ring wall thickness does not influence the cage sliding. After considering Figure 10 shows the cage slide ratio due to different ring thicknesses. It can be seen that under the Appl. Sci. 2016, 6, 415 10 of 12 the rigid assumption, the ring wall thickness does not influence the cage sliding. After considering rigid assumption, the ring wall thickness does not influence the cage sliding. After considering the deformation of the structure, the cage slide ratio decreased by 8.3%~47.2%; in addition, as the ring the deformation of the structure, the cage slide ratio decreased by 8.3%~47.2%; in addition, as the wall thickness decreased, the cage slide ratio was reduced. ring wall thickness decreased, the cage slide ratio was reduced. Figure 10. Influence of ring thickness on cage sliding. Figure 10. Influence of ring thickness on cage sliding. Figure 11 shows the bearing fatigue life. It can be seen that under the rigid assumption, the ring Figure 11 shows the bearing fatigue life. It can be seen that under the rigid assumption, the ring thickness does not influence fatigue life. After considering the structural deformation, the fatigue life thickness does not influence fatigue life. After considering the structural deformation, the fatigue life increased by 33.8%~76.4%; in addition, as the ring thickness increased, the fatigue life was reduced, increasingly approaching that under the rigid assumption. Figure 11. Influence of ring thickness on fatigue life. Figure 12 shows the radial stiffness. When the influence of the deformation of the structure is considered, the stiffness of the bearing is decreased because the bearing stiffness depends on the contact mechanics of the rollers and the raceways, which change following structural deformation. The stiffness of the squirrel‐cage is only 8.3%~32.7% that of the bearing, reducing the stiffness of the entire structure by an order of magnitude compared to that under the rigid assumption. In addition, as the ring wall thickness increased, the stiffness approached that under the rigid assumption. Figure 12. Influence of ring thickness on stiffness. Appl. Sci. 2016, 6, 415 10 of 12 Appl. Sci. 2016, 6, 415 10 of 12 the deformation of the structure, the cage slide ratio decreased by 8.3%~47.2%; in addition, as the the deformation of the structure, the cage slide ratio decreased by 8.3%~47.2%; in addition, as the ring wall thickness decreased, the cage slide ratio was reduced. ring wall thickness decreased, the cage slide ratio was reduced. Figure 10. Influence of ring thickness on cage sliding. Figure 10. Influence of ring thickness on cage sliding. Appl. Sci. Fig2016 ure, 11 6, 415 shows the bearing fatigue life. It can be seen that under the rigid assumption, the 10 rin of 11g Figure 11 shows the bearing fatigue life. It can be seen that under the rigid assumption, the ring thickness does not influence fatigue life. After considering the structural deformation, the fatigue life thickness does not influence fatigue life. After considering the structural deformation, the fatigue life increased by 33.8%~76.4%; in addition, as the ring thickness increased, the fatigue life was reduced, increased by 33.8%~76.4%; in addition, as the ring thickness increased, the fatigue life was reduced, increased by 33.8%~76.4%; in addition, as the ring thickness increased, the fatigue life was reduced, increasingly approaching that under the rigid assumption. increasingly approaching that under the rigid assumption. increasingly approaching that under the rigid assumption. Figure 11. Influence of ring thickness on fatigue life. Figure 11. Influence of ring thickness on fatigue life. Figure 11. Influence of ring thickness on fatigue life. Figure 12 shows the radial stiffness. When the influence of the deformation of the structure is Figure 12 shows the radial stiffness. When the influence of the deformation of the structure is Figure 12 shows the radial stiffness. When the influence of the deformation of the structure is considered, the stiffness of the bearing is decreased because the bearing stiffness depends on the considered, the stiffness of the bearing is decreased because the bearing stiffness depends on the considered, the stiffness of the bearing is decreased because the bearing stiffness depends on the contact mechanics of the rollers and the raceways, which change following structural deformation. contact mechanics of the rollers and the raceways, which change following structural deformation. contact mechanics of the rollers and the raceways, which change following structural deformation. The stiffness of the squirrel‐cage is only 8.3%~32.7% that of the bearing, reducing the stiffness The stiffness of the squirrel‐cage is only 8.3%~32.7% that of the bearing, reducing the stiffness The stiffness of the squirrel-cage is only 8.3%~32.7% that of the bearing, reducing the stiffness of of the entire structure by an order of magnitude compared to that under the rigid assumption. In of the entire structure by an order of magnitude compared to that under the rigid assumption. In the entire structure by an order of magnitude compared to that under the rigid assumption. In addition, addition, as the ring wall thickness increased, the stiffness approached that under the rigid addition, as the ring wall thickness increased, the stiffness approached that under the rigid as the ring wall thickness increased, the stiffness approached that under the rigid assumption. assumption. assumption. Figure 12. Influence of ring thickness on stiffness. Figure 12. Influence of ring thickness on stiffness. Figure 12. Influence of ring thickness on stiffness. 4. Conclusions (1) In this paper, an integrated squirrel-cage flexible support roller bearing quasi-dynamic iterative finite element analysis (FEA) model is established. The influence of deformed raceways is added to the bearing quasi-dynamic model, from which a noncircular raceway roller bearing quasi-dynamic-modified model is obtained. The modified model is coupled with a finite element model, which can calculate the elastic deformation of the squirrel-cage and the rings. (2) Analyses of an integrated squirrel-cage flexible support roller bearing indicate that the squirrel-cage support structure can effectively reduce the maximum contact load between the rollers and raceways, and can load more rollers, thereby distributing the load more evenly and prolonging fatigue life. (3) The proposed model is used to analyze the influence of ring thickness on the dynamic performance of the bearing. The results indicate that, the thicker the ring wall is, the closer the load distribution is to the calculation results obtained under the rigid assumption; in contrast, as the ring wall becomes thinner, the influence of the structural deformation becomes increasingly significant. (4) The integrated squirrel support structure significantly reduces stiffness; thus, when used in a high-speed shaft system, the loading performance of the bearing and the impact on the overall stiffness of the integrated squirrel-cage support structure must also be comprehensively considered. Appl. Sci. 2016, 6, 415 11 of 11 Acknowledgments: This work was supported in part by the National Natural Science Foundation of China (51375108), and the National Key Basic Research Program (2013CB632305). Author Contributions: All authors made significant contributions to this article. Yuze Mao wrote the source code and revised the article; Liqin Wang was mainly responsible for checking the data and performing the revision of the paper; Chuanwei Zhang was responsible for developing the FEA model. Conflicts of Interest: The authors declare no conflicts of interest. References 1. Zhao, L.; Sun, Y. The fault diagnosis of aero-engine main shaft bearings. Aircr. Des. 2010, 4, 46–50. 2. Harris, T.A. Rolling Bearing Anaysis; John Wiley & Sons. Inc.: New York, NY, USA, 2007. 3. Ebert, F.-J. An Overview of Performance Characteristics, Experiences and Trends of Aerospace Engine Bearings Technologies. Chin. J. Aeronaut. 2007, 20, 378–384. [CrossRef] 4. Hakan, A.; Aydin, B.; Hasan, S. An investigation of tribological behaviors of dynamically loaded non-grooved and micro-grooved journal bearings. Tribol. Int. 2013, 58, 12–19. 5. Oswald, F.B.; Zaretsky, E.V.; Poplawski, J.V. Interference-Fit Life Factors for Ball Bearings. Tribol. Trans. 2010, 54, 1–20. [CrossRef] 6. Broschard, J. Analysis of an improved planetary gear transmission bearing. ASME Trans. J. Basic Eng. 1964, 86, 457–461. 7. Shen, Y.; Zhang, T. Vibration Analysis of Flexible Rolling Bearing. Mech. Sci. Technol. Aerosp. Eng. 1995, 5, 1–6. 8. Yao, T.; Chi, Y.; Huang, Y. Research on flexibility of bearing rings for multibody contact dynamics of rolling bearings. Procedia Eng. 2012, 31, 586–594. [CrossRef] 9. Lostado, R.; Martinez, R.F. Determination of the contact stresses in double-row tapered roller bearings using the finite element method, experimental analysis and analytical models. J. Mech. Sci. Technol. 2015, 29, 4645–4656. [CrossRef] 10. Ignacio Amasorrain, J.; Sagartzazu, X.; Damian, J. Load distribution in a four contact-point slewing bearing. Mech. Mach. Theory 2003, 38, 479–496. [CrossRef] 11. Kania, L. Modelling of rollers in calculation of slewing bearing with the use of finite elements. Mech. Mach. Theory 2006, 41, 1359–1376. [CrossRef] 12. Olave, M.; Sagartzazu, X.; Damian, J.; Serna, A. Design of four contact-point slewing bearing with a new load distribution procedure to account for structural stiffness. J. Mech. Des. 2010, 132. [CrossRef] 13. Shu, J.; Wang, L.; Mao, Y.; Ding, Y. Iterative FEA Method for Load Distribution of Flexible Supporting Thin-section Ball Bearing. J. Harbin Inst. Technol. 2015, 22, 9–14. 14. Ni, Y.; Liu, W.; Deng, S.; Jiao, Y.; Liang, B. Performance analysis of thin wall angular contact ball bearings considering the ferrule deformation. J. Aerosp. Power 2010, 25, 1432–1436. 15. Lostado, R.; Martínez-De-Pisón, F.J.; Pernía, A.; Alba, F.; Blanco, J. Combining regression trees and the finite element method to define stress models of highly non-linear mechanical systems. J. Strain Anal. Eng. Des. 2009, 44, 491–502. [CrossRef] 16. Cui, L.; Wang, L.; Zheng, D. Analysis on dynamic characteristics of aero-engine high-speed roller bearings. Acta Aeronaut. Astronaut. Sin. 2008, 29, 492–498. 17. Harris, T.A. An Analytical Investigation of Cylindrical Roller Bearings Having Annular Rollers. Tribol. Trans. 1967, 10, 235–242. [CrossRef] 18. Wang, Y.; Yang, B. Investigation into the Traction Coefficient in Elastohydro-dynamic Lubrication. Tribotest 2004, 11, 113–124. [CrossRef] 19. Lostado, R.; García, R.E. Optimization of operating conditions for a double-row tapered roller bearing. Int. J. Mech. Mater. Des. 2016, 56, 1–21. [CrossRef] © 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).
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