HY A , WS A , LEI G B , et al. Study on mesh stiffness and sensitivity analysis of planetary gear system considering the deformation effect of carrier and bearing [J ] . Engineering Failure Analysis , 2022 , 135 : 106 - 146 .

HAN Y H , YANG L H , XU T F . Analysis of static stiffness fluctuation in radially loaded ball and roller bearings [J ] . Archive of Applied Mechanics , 2021 , 91 ( 4 ): 1 - 16 . DOI: 10.1007/s00419-020-01753-9 http://doi.org/10.1007/s00419-020-01753-9 This work proposes a method for determining the effective surface area of a rigid body induced by the wind. The proposed method assumes a variable surface effective area on which the wind acts. To determine the effective surface parameter, the dependence of normal vectors and the position change of individual vertices of the solid body were used. Cardan angles and its derivatives were used to determine the rigid body orientation in space. The free motion of the rigid body was analysed for two variants: influence of wind force and forced motion. During calculations, various cases were analysed differing in terms of wind speed and direction. A comparison of the developed method with the constant and projected surface area was presented. Various acceptable cases in which the surface area approximation is permissible were also shown. The proposed method can be used to calculate the wind load acting on a load of transport devices.

HEMOT X , SARTOR M , GUILLOT J . Calculation of the stiffness matrix of angular contact ball bearings by using the analytical approach [J ] . Journal of Mechanical Design , 2000 , 122 ( 1 ): 83 - 90 . DOI: 10.1115/1.533548 http://doi.org/10.1115/1.533548 https://asmedigitalcollection.asme.org/mechanicaldesign/article/122/1/83/443754/Calculation-of-the-Stiffness-Matrix-of-Angular https://asmedigitalcollection.asme.org/mechanicaldesign/article/122/1/83/443754/Calculation-of-the-Stiffness-Matrix-of-Angular The stiffness matrix of angular contact ball bearings is calculated by using the analytical approach in which the summation of ball-race loads is replaced by an integration. The matrix connected to the conventional model in two degrees of freedom is first presented. A practical application of this formulation is illustrated through the common problem of sizing a two bearings-shaft arrangement. Variations of displacements, axial forces, and bearing fatigue life related to preload are shown to be easily obtained. Then, the matrix connected to the model in five degrees of freedom is given. This may be coupled with finite elements which are generally used to model the shaft or the housing. [S1050-0472(00)01801-8]

ZHANG J H , FANG B , ZHU Y S , et al. A comparative study and stiffness analysis of angular contact ball bearings under different preload mechanisms [J ] . Mechanism and Machine Theory , 2017 , 115 : 1 - 17 . DOI: 10.1016/j.mechmachtheory.2017.03.012 http://doi.org/10.1016/j.mechmachtheory.2017.03.012 https://linkinghub.elsevier.com/retrieve/pii/S0094114X16306103 https://linkinghub.elsevier.com/retrieve/pii/S0094114X16306103

FANG B , ZHANG J H , YAN K , et al. A comprehensive study on the speed-varying stiffness of ball bearing under different load conditions [J ] . Mechanism and Machine Theory , 2019 , 136 : 1 - 13 . DOI: 10.1016/j.mechmachtheory.2019.02.012 http://doi.org/10.1016/j.mechmachtheory.2019.02.012 https://linkinghub.elsevier.com/retrieve/pii/S0094114X19302356 https://linkinghub.elsevier.com/retrieve/pii/S0094114X19302356

刘越 , 周广明 , 张祖智 , 等 . 多轴承支撑轴系的精确变形计算研究 [J ] . 兵工学报 , 2014 , 35 ( 3 ): 307 - 311 .

LIU Y , ZHOU G M , ZHANG Z Z , et al. Accurate calculation for deformation of multi-bearing shafting system [J ] . Acta Armamentarii , 2014 , 35 ( 3 ): 307 - 311 . (in Chinese)

罗祝三 , 吴林丰 . 多支点轴系中滚动轴承的拟静力学研究 [J ] . 南京航空航天大学学报 , 1995 , 27 ( 3 ): 291 - 298 .

LUO Z S , WU L F . Quasi-static analysis of multi-bearing shaft system [J ] . Journal of Nanjing University of Aeronautics Astronautics , 1995 , 27 ( 3 ): 291 - 298 . (in Chinese)

田程 , 周驰 , 桂良进 , 等 . 考虑轴承刚度耦合性和非线性的多支撑轴系有限元分析方法 [J ] . 机械工程学报 , 2015 , 51 ( 17 ): 90 - 95 . DOI: 10.3901/JME.2015.17.090 http://doi.org/10.3901/JME.2015.17.090 针对多支撑轴系分析的难点，提出一种考虑轴承刚度耦合性和非线性的有限元计算方法。建立一种轴承单元，考虑了轴承刚度的耦合性和非线性及轴承预紧、滚子锥角等细节因素，推导得到了轴承单元刚度矩阵的解析表达式。采用考虑了剪切变形的欧拉梁单元建立轴模型，将轴与轴承模型进行组合得到系统计算模型，并采用NR法进行求解。通过算例验证，所提出的方法能够准确地计算出给定外载荷下多支撑轴系中各轴承各方向的载荷和系统的变形，且具有较高的计算效率，适用于多支撑轴系问题的分析计算。

TIAN C , ZHOU C , GUI L J , et al. Method of finite element analysis for multi-support shaft system based on the coupling and nonlinearity of bearing stiffness [J ] . Journal of Mechanical Engineering , 2015 , 51 ( 17 ): 90 - 95 . (in Chinese) DOI: 10.3901/JME.2015.17.090 http://doi.org/10.3901/JME.2015.17.090 A non-linear finite element calculation method based on the coupling and nonlinearity of bearing stiffness is presented to solve the difficulties for the analysis of multi-support shaft system. A bearing element is built in the present method. The coupling and nonlinearity of bearing stiffness, the preload of bearing, the roller taper angle, and the other details are considered, and the analytical expression of bearing stiffness matrix is deduced. The shaft is modeled by Euler beam element with shear deformation, and system computational model is assembled by the model of shaft and bearings. The NR method is used to solve the model. An example shows that this method can calculate the load of each bearing in every direction and the shaft deformation for multi-support shaft system by given external load correctly. This method has high computational efficiency, and is suitable for the analysis of multi-support shaft system.

倪艳光 , 刘万强 , 邓四二 , 等 . 计及套圈变形的薄壁角接触球轴承性能分析 [J ] . 航空动力学报 , 2010 , 25 ( 6 ): 1432 - 1436 .

NI Y G , LIU W Q , DENG S E , et al. Performance analysis of thin-wall angular contact ball bearings considering the ferrule deformation [J ] . Journal of Aerospace Power , 2010 , 25 ( 6 ): 1432 - 1436 . (in Chinese)

毛宇泽 , 王黎钦 , 古乐 . 负游隙对高速高温薄壁圆柱滚子轴承动态性能的影响分析 [J ] . 航空动力学报 , 2016 , 31 ( 11 ): 2795 - 2800 .

MAO Y Z , WANG L Q , GU L . Effect of negative clearance on dynamic characteristics of high-speed high-temperature thin-walled cylindrical roller bearings [J ] . Journal of Aerospace Power , 2016 , 31 ( 11 ): 2795 - 2800 . (in Chinese)

SUN X , DENG S E , CHEN G D , et al. Calculation and analysis of groove elastic support's radial stiffness [J ] . International Journal of Engineering , 2017 , 30 ( 7 ): 1006 - 1073 .

李云峰 , 程亚兵 . 交叉圆柱滚子转盘轴承的滚子偏载分析及修形 [J ] . 机械工程学报 , 2015 , 51 ( 1 ): 161 - 166 .

LI Y F , CHENG Y B . Unbalance loading analysis and profile modification of roller of crossed cylindrical roller slewing bearings [J ] . Journal of Mechanical Engineering , 2015 , 51 ( 1 ): 161 - 166 . (in Chinese) DOI: 10.3901/JME.2015.01.161 http://doi.org/10.3901/JME.2015.01.161 For the problem of unbalance loading effect and edge loading effect of the contact stress distributions between the rollers and the raceway of crossed cylindrical roller slewing bearing under combined loads, the effective method for improving the roller and raceway contact state is put forward. This method is realized through the profile modification design of rollers under the premise of reasonable bearing clearance. The whole mechanical system of the bearing is modeled and solved, and the effect of the bearing clearance on the unbalance loading of the roller is researched by using the finite length line contact theory. Under the reasonable bearing clearance, the contact stress distributions between the rollers with different crowning values and the raceway are researched. The results show that the contact stress distribution of the roller appears asymmetric along the contact line after the roller suffers unbalanced load, the stress on the heavy end increases while on the light end decreases, and this trend tends to be more significant with the increase of the bearing clearance value. The unbalance loading effect can be improved and the edge loading effect can be eliminated by proper profile modification of the roller. The larger the crowning value of the roller, the more significant effect gets, but there exists the trend of the load concentrating to the central section of the roller. On the basis of considering these influence, the best profile modification parameters of the roller is determined for the crossed cylindrical roller slewing bearing under combined loads.

LACROIX S , NÉLIAS D , LEBLANC A . Experimental study of four-point contact ball bearing with deformable rings [J ] . Tribology Transactions , 2015 , 58 ( 6 ): 963 - 970 . DOI: 10.1080/10402004.2015.1017894 http://doi.org/10.1080/10402004.2015.1017894 http://www.tandfonline.com/doi/full/10.1080/10402004.2015.1017894 http://www.tandfonline.com/doi/full/10.1080/10402004.2015.1017894

王亚珍 , 赵坤 , 姜祎 , 等 . 预应力对柔性薄壁轴承疲劳寿命的影响 [J ] . 润滑与密封 , 2017 , 42 ( 3 ): 54 - 58 ,123. DOI: 10.3969/j.issn.0254-0150.2017.03.011 http://doi.org/10.3969/j.issn.0254-0150.2017.03.011 谐波减速器的柔性轴承装配到波发生器椭圆凸轮上时存在较大的预应力和预变形。为研究预应力的影响，利用ANSYS Workbench建立柔性轴承和波发生器凸轮多体接触模型，在实际工作载荷和预应力及预变形作用下，对柔性轴承内外套圈进行分析，得到其变形、应力以及载荷分布规律。分析结果表明，内外圈沟道处的等效应力主要取决于其受到的周向应力；内外圈的最大等效应力和最大径向应力发生在长轴处的球与内外沟道接触的区域。依据L-P理论计算出柔性轴承的疲劳寿命，结果表明，考虑预应力情况下柔性轴承的疲劳寿命降低了30%，研究结果对柔性薄壁轴承的设计及分析提供理论参考和实践指导。

WANG Y Z , ZHAO K , JIANG Y , et al. Influence of prestress on fatigue life of flexible thin wall bearings [J ] . Lubrication Engineering , 2017 , 42 ( 3 ): 54 - 58 ,123. (in Chinese) DOI: 10.3969/j.issn.0254-0150.2017.03.011 http://doi.org/10.3969/j.issn.0254-0150.2017.03.011 There are large prestress and pre-deformation when the flexible bearing of the harmonic reducer is fitted to the elliptical cam of the wave generator.To study the effect of prestress and pre-deformation,the multi-body contact model were built by ANSYS Workbench,considering the pre-deformation and pre-stress of inner and outer rings of the flexible pre-assembled bearing.The deformation,stress and load distribution of the flexible inner and outer bearing rings were ob-tained under practical working condition.The analysis result shows that the equivalent stress of the inner and outer ring is mainly determined by the circumferential stress.The maximum equivalent stress and the maximum radial stress of the inner and outer ring occur in the contact zone between the ball and the inner and outer rolling track at the major axis.According to L-P theory,the fatigue life of flexible bearing was calculated.It is shown that the fatigue life of flexible bearing is re-duced by 30% in the case of prestress.The research results provide the theoretical reference and guidance for the design and prediction the fatigue life of flexible bearing.

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刘宏 , 王三民 , 刘海霞 . 弹性支承下弧齿锥齿轮系统参数平面中的域界结构研究 [J ] . 振动与冲击 , 2010 , 29 ( 12 ): 173 - 176 .

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