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yabo亚博问题
yabo亚博
Volume22,2021
一种R.T.一世cle Number 3.4.
页数) 11
迪伊 https://doi.org/10.1051/meca/2021030
在线发布 0.6.may 2021

© Y. Yu et al., Published by EDP Sciences 2021

Licence Creative Commons这是在Creative Commons归因许可证的条款下分发的开放式访问文章(HT.T.ps://creativecommons.org/licenses/by/4.0)提供任何介质中的不受限制使用,分发和再现,所以提供了正确的工作。

1在T.R.O.D.uction

谐波减速机广泛用于航空航天,医疗设备,机器人,CNC机床,包装设备,仪器等领域。作为主要的工作组件之一,柔性轴承(FB)的损坏是谐波驱动器的主要故障原因之一1-3.]。有必要研究柔性轴承的负载分布,高效率,高精度和谐波驱动的长寿命。

谐波减速器主要由三个组件组成:圆形花键(CS),Flexspine(FS)和波发生器(WG),其中WG由FB和波发生器凸轮(WG Cam)组成,如图所示F一世gure 1。当CS固定并且WG是驱动构件时,FS变为跟随器。WG使FS能够产生可控弹性变形,其力使得CS和FS的长轴的两端处的齿处于完全接合,而短轴两端的齿完全分离。当WG旋转时,FS依次啮合FS啮合,重复四种状态:接合,接合,接触和脱离,如图所示F一世gure 1。T.Hus, this staggered tooth transmission makes the harmonic reducer drive with great deceleration ratio.

这FB.一世s of great difference compared with the ordinary bearing. In structure, the thickness of FB is much smaller than that of ordinary bearing. After the FB is assembled on the WG cam, the inner and outer rings will have a certain deflection, and the shape depends on that of the WG cam. In loading form, when the harmonic reducer works, the flexible bearing will bear symmetrical external load at both ends of the major axis. Even there is no external load, the internal load distribution of FB is quite different from that of ordinary bearing due to the pre-deformation caused by WG cam.

这calculation and research of load distribution in rolling bearing is the analysis basis of kinematics, lubrication mechanics, contact fatigue life and efficiency of rolling bearing, which is necessary for improving the life and the working performance of rolling bearing. For the ordinary rolling bearing, the analysis and calculation of its load distribution has become the mature theory [4.5.]。然而,FB的变形,负载和故障模式与普通轴承的变形,负载和故障模式完全不同。因此,通过现有理论研究FB并不准确。为了深入研究FB的机械特性,主要步骤是获得其精确的内部载荷分布。

应用了几种理论方法,用于研究FB中的负载分布。梁和沉[6.] obtained the load distribution of the FB using the curved beam theory of elasticity and material mechanics, three-moment theory of continuous beam, and energy method theory. The load decomposition obtained by the model is in good agreement with the experimental results. But this method can only calculate the bearing load distribution of the FB mounted on a four-force action type cam. The deflection of the outer ring of the FB was calculated based on the circular radial deformation under a static equilibrium state and Shao and Wen [7.通过求解一组非线性变形配位方程,获得了滚动元件上的负荷。然而,在模型中讨论了奇数球的情况。建立单型薄壁环叠加模型以计算凸轮上配件的FB的负载分布[8.9.]。该方法可用于计算FB中的负载分布,随机数量的球,但在该模型中不考虑外部负载。建立了双重薄壁环超位置模型,以计算凸轮上配件的FB的负载分布[10.]。然而,这种方法只能用偶数球施加FB。因此,有必要提供一种新的理论方法,用于计算FB中的负载分布,随机数量的球,其在力平衡同时。

一种ccording to the general solution of the deformation of the thin-walled ring deduced by Liu and Chiu [11], Xiong et al. [12]开发了一种通用静态分析模型,分析了谐波减速机中不同FB的负载分布。不仅可以在该模型中考虑任意数量的球(偶数或奇数),而且可以在该模型中考虑任意径向对称的外部负载。

除了理论分析外,还采用了有限元方法(有限元)在FB的负荷分析中使用[13.14.]。CO.mpared with the theoretical methods, this kind of method requires a lot of computing time.

在T.H一世s paper, a superposition model of three forces ring was proposed based on the theory of thin-walled ring and superposition principle. According to this mechanical model, the internal load distribution in the FB could be calculated. And then the radial deformation and the bending normal stress on the outer ring were also calculated and analyzed. Based on the calculation of this mechanical model, we investigated the influences of the number of balls, the angle of the ball position and the load torque on the ball load. And the influences of the numbers of balls and the load torque on the radial deformation and maximum bending normal stress of outer ring were subsequently studied.

T.Humbnail F一世g. 1

谐波减速器的工作原理示意图。

2三力环机械模型

2.1基本理论

一种R.在g whose cross-section size is much smaller than its radius is called a thin-walled ring. It is assumed that the cross-section of the ring is rectangular and the shape is unchanged along the circumference. And the load is uniformly distributed along the width of the ring. Under these conditions, stress and displacement are constant along the width, so it can be treated as a plane problem. For the simplification of calculation, the mechanical model has the following basic assumptions: ①It is assumed that the load distribution inside the FB is equal to that the outer ring bears multiple radial loads, as shown inF一世gure 2。②假设外圈的变形很小,因此可以使用材料力学和叠加原理的方法来研究外圈的负载分布。

在T.H一世s paper, three main formulas about the theory of thin-walled ring [15.16.] are given by:(1)(2)(3)

方程(1)一世s the elastic equation connecting bending momentm和径向位移W.那W.HereE.是材料的弹性模量和一世是环部分的惯性力矩。方程(2)一世s derived from no elongation hypothesis, and it indicates the relationship between radial displacementW.和circumferential displacementV.。方程(3)是正常角度的表达θO.f circular section.

这D.eformation of outer ring of flexible bearing can be analyzed by the theory of thin-walled ring because of its small thick-ness. In the following analysis and calculations, the shape of the ring is indicated by the neutral layer curve of the ring, and the load and deformation of points on the neutral layer are discussed.

T.Humbnail F一世g. 2

负载FB外环的简化。

2。2Distributed bending momentmab(φ)在环上

这structure diagram of the three-force ring model is shown inF一世gure 3。一世fF1 = F一世s supposed, then other two radial forcesF2F3.可以根据力系统的平衡关系知道。沿水平中线切割环,并拍摄环的上部进行分析,如图所示F一世gure 4

When the bending momentm一种mB.和R.adial force system act respectively on the half ring shown inF一世gure 4那T.He normal angles of section A and B could be calculated by the unit force method [17.]。这n the total normal angles of two sections are obtained by superposition. According to deformation compatibility condition, the values of these two normal angles are 0, so the bending moments are given by:(4)W.HereH1H2are given by:(5)

在T.He same way, when the bending momentm一种mB.和径向力系统在半环上行动,可以通过叠加来计算环的总分布弯矩。它由:(6)W.HereH3.H4.H5.are given by:(7)

T.Humbnail F一世g. 3

三个力作用下环的结构图。

T.Humbnail F一世g. 4

在T.ernal force of the symmetric section of the ring.

2。3.表达R.adial displacement

Substituting equation(6)进入等式(1)那D.一世fferential equations are given by:(8)

这general solution of two differential equations is expressed as:(9)(10)W.Here一种1一种2B.1B.2are integral constants.

由于结构的对称性,正常角度θ和圆周位移V.equal to 0 whenϕ = 0 andϕ = π。Substituting this symmetric condition into equation(3)一种1B.1are given by:(11)

然后根据平滑连接的条件,即何时ϕ = ϕ0.,方程式(9)(10)should get the same values and derivatives. The same result is given by:(12)

一世T.seems that the above conditions are not enough to get a solution, this is because of the lack of horizontal constraints for the structure inF一世gure 3。要完全确定四个集成常数,我们让B点的径向位移等于X0.F一世gure 3。根据方程式(9)(11)(12)一种2B.2表示X0.可以给出:(13)

表达X0.需要定义。比较双力环和三力环的机械模型,圆周位移ϕ = π/ 2在环上应为0,以确保点B相对于穿过圆的中心的垂直线的位移是恒定的。通过方程式(2),通过整合径向位移可以获得环的周向位移。通过整体计算,表达X0.一世s given by:(14)

最后,取代方程(11)(13)(14)进入等式s(9)(10),三力环的径向位移可以给出:(15)其中h(ϕ) is the coefficient of bending deformation, which is given by:(16)

3柔性轴承的负载分布

3.1计算参数

When the FS is loaded, the loading diagram of outer ring of the FB withNB.alls (N≧ 5) is shown inF一世gure 5。FO.R.T.He simplification of calculation, we have the following assumptions: ①It is assumed that there is no friction between the CS and the FS. ②It is assumed that the rotational speed of the shaft has no influence on the radial force state of the FB. ③It is assumed that the centrifugal load of the balls on the outer ring is much smaller than the ball load itself, which can be ignored. The parameters related to external radial distributed load are given inT.able 1,并且柔性轴承的主要几何参数T.able 2。这V.alues ofΦ1Φ2Φ3.确定径向分布式负载的位置。在本文中,这三个变量的值是指相关论文[18.]。这main variables are the load torque and the number of balls, which can be changed in a certain range.

T.Humbnail F一世g. 5

Loading diagram of outer ring of the FB withNB.alls (N≧ 5).

T.able 1

R.elevant Parameters of radial distributed load.

T.able 2

main parameters of the FB.

3.2变形兼容性方程

在本文中,余弦凸轮用于分析FB的内部负荷。由这种凸轮引起的理想预变形:(17)

W.HenΦ2=Φ3.那T.He radial distributed load inF一世gure 5一世s given by [19.20.]:(18)W.Here问:T.最大限度一世s the maximal value of tangential distributed load;τ是剖面角;D.R.一世s the pitch diameter of the FS;B.W.一世s the working width of gear ring.

在径向分布式载荷下,外圈的径向变形由[15.]:(19)W.HereB.O.一世s the width of outer ring;R.一世s the neutral layer radius of outer ring;问:R.K.一世s the coefficient in theK.术语的系列。

F一世gure 5,钢球上的径向载荷表示F一世和each loading point is marked with a number in an anti-clockwise direction. And the angular position of the ith loading point is given by:(20)W.HereαR.是球的旋转角度,其决定了装载点的角度位置。

然后可以在每个负载点处建立相应的变形兼容性方程。ITH加载点的等式由以下方式提供:(21)W.HereW.cam,一世c引起的变形吗am assembly;W.问:一世一世s the deformation caused by the external radial load问:R.;W.一世是由内部径向载荷引起的变形F一世;P.D.一世s the radial clearance of the FB.

3.3三力系统的叠加算法

Unknown radial loadsF一世(i = 1,2,3,......,NN≧5)显示F一世gure 5可以分解成N未知的三队系统。每个力系统都显示在F一世gure 3,其中中间的径向力是X一世(i = 1,2,3,......,NN≧5)及其角度位置由等式确定(20)。通过方程可以获得每个装载点处不同力系统的径向变形(15)(16)。叠加所有力系统后,总径向变形N加载点可以用矩阵表示:(22)

W.Hereϕ0. = 2π/N。这above matrix can be simplified as:(23)在哪里 [W.] is the column matrix of radial deformation of outer ring; [一种]是变形系数矩阵;[X]是未知力的柱矩阵。

柱矩阵中加载点的变形[W.] can be calculated by equation(21),然后我们可以获得一组线性方程。通过解决方程,[X] is given by:(24)

这unknown radial loads inF一世gure 4can be calculated by superposing unknown forcesX一世at each loading point. The superposition process is written in the form of matrix, which is given by:(25)W.Here。这above matrix can be simplified as:(26)在哪里 [F] is the column matrix of radial loads; [C] is the transformation matrix.

一种ccording to the above linear algorithm, the internal radial load of the FB can be calculated. If there are negative results, it indicates that these steel balls are not bearing load. In order to eliminate the negative load, the force boundary conditionF一世 = 0 should be applied for the position where the ball is not under load. The above calculations can be programmed, and the diagram is shown inF一世gure 6

应当注意,三力系统的叠加算法不适合球的数量小于5.当有三个球时,由于装载时的径向变形,无法确定径向载荷方程式相同(15)(16)那一世T.一世s impossible to establish the linear equation. And when there are two or four balls, the three-force system is not balanced, but the radial load can be calculated by two-force ring model.

T.Humbnail F一世g. 6

FB的负载分布计算框图。

3.4柔性轴承分析模型的验证

为了将负载分布结果与雄根的结果进行比较,轴承LY-6025的主要参数采用上述模型[12]。这B.all load distribution curves with no external loads are shown inF一世gure 7。与广场代表球负载d一世stribution result calculated by the three-force ring superposition method, while the line with dots represents that by the static analysis model and the line with triangles represents that by the FEM simulation model in Xiong's paper [12]。一世T.can be seen from the figure that the result of three-force ring superposition method and is similar to that of the compared model. The maximal error between two numerical solutions is within 10%, which occurs around the major axis. And both of the number of no-load balls is 8. The reason for the differences may be the hertz contact deformation and the nonlinear part in the compared model. Therefore, the correctness and validity of the three-force ring superposition model are proved.

T.Humbnail F一世g. 7

这B.all load distribution comparison of the FB LY-6025.

3.5负载分布的特点

3.5.1球数量对最大球负荷的影响

为了获得负载说的特点T.R.一世B.ution, we took maximum ball load as study object, similar results are obtained for other ball loads. The relevant calculation parameters are listed inT.ables 12

一种ccording to the algorithm in 3.3, the relationship of maximum ball load and number of balls under load is derived, as shown inF一世gure 8。从图中可以看出,不同负载扭矩下的最大球负载随着球数的增加而始终减小。

T.Humbnail F一世g. 8

R.elationship of maximum ball load and number of balls under load.

3.。5.。2这在fluence of load torque on maximum ball load

WhenαR. = 0, the relationship of maximum ball load and load torque under different number of balls is derived, as shown inF一世gure 9。一世T.can be seen from figure that the maximum ball load increases linearly with the increase of the load torque. And the larger the number of balls, the smaller the slope of the line. This explains that when load torque is large, the maximum ball load of the FB with more balls is far less than that of the FB with less balls.

T.Humbnail F一世g. 9

R.elationship of maximum ball load and load torque under different number of balls.

3.。5.。3.这R.elationship of ball load and angular position

当球的数量为23时,可以通过改变球的旋转角度来获得不同负载扭矩下的连续载荷曲线,如图所示图10.。一世T.can be seen from figure that the amplitude of ball load is increased significantly by load torque. WhenT. = 0, the width of no-load area at the end of minor axis is 30.39°; WhenT. = 50 Nm, the width of no-load area increases to 44.1°. Apparently, the load torque narrows the load range of the flexible bearing, resulting in the reduction of the number of loaded balls. For the FB with 23 balls, the number of loaded balls decreases from 19 to 17 after the load torque increases from 0 to 50 Nm. Thus, high load torque on the FB should be avoided, which may cause the noise and the abnormal operation of the FB.

T.Humbnail F一世g. 10

载荷下球负荷和角度位置的关系。

4.这D.eformation of outer ring of the flexible bearing

4.1径向变形的表达

衍生出对环形结构变形的分析方法[21]。一种nd it assumes that the single load applied to the thin ring is balanced by a symmetric and tangential shear stress distribution, which is not suitable for this paper. In the former two chapters, the relationship between radial deformation and load of three-force ring was derived and the unknown loads were decomposed into several three-force systems. During the above process, the sizes and positions of these three-force systems were derived. Therefore, the total radial deformation of all points on the ring could be calculated by superimposing the radial deformation caused by each three-force system. For the FB withNB.alls, the radial loads on the outer ring can be decomposed intoNT.HR.ee-force systems, andX一世R.epresents the size of the ith force system. According to the equations(15)(16)(20)那T.He radial deformation caused by the ith force system is given by:(27)W.Hereβ一世一世s given by:(28)

因此,外圈的总径向变形N三力系统和径向分布式负载问:R.一世s given by:(29)

W.HereW.问:一世s determined by equation(19)

4.2外圈径向变形的特点

4.2.1球数对外环径向变形的影响

随着球数的增加,理论变形将更接近理想的预变形。为了研究球数对径向变形的影响,计算期望的差异和理论结果,由:(30)W.HereW. ⁡⁡ (ϕ)是理论径向变形,可以通过方程计算(29);W.camϕ)是理想的径向变形,由等式确定(17)

WhenαR. = 0 andT. = 0, the relationship curve of the maximum difference ΔW.最大限度和球的数量N一世s obtained, shown in图11.。一世T.can be seen from figure that the maximum difference is negative and it appears at the end of the major axis whenN ≤ 10. This will have an adverse effect on the engaging state of teeth here. And whenN> 10,最大差异是正的,它出现在短轴的末端。总体而言,随着球数的增加,最大差异减少。

T.Humbnail F一世g. 11

空载下最大差分和球数的关系。

4.2.2负载扭矩对外圈径向变形的影响

WhenN = 23 andαR. = 0, the relationship curve of the difference of radial deformation ΔW.和angular positionϕ在下面T. = 0 andT. = 50 Nm is obtained, as shown in图12.。在T.He figure, the dash dotted line represents the theoretical deformation shape whenT. = 0 and the solid line represents the theoretical deformation shape whenT. = 50 Nm. And the expected deformation shape is indicated by the dotted line, which represents zero position. And the dots indicate the position of the balls. It can be seen from figure that the amplitude of ΔW.can be increased by load torque. WhenT. = 50 Nm, the maximum difference at the end of minor axis is 2.3 times of that under no-load. And the deformation shape is actually wavelike. The theoretical curve coincides with the expected curve at the loading points, and the shapes are similar around these points. But there is an obvious difference between the theoretical and expected deformation shape around the minor axis. And the shape is convex, which results in no-load balls. If the radial deformation here is too large, it may cause that the teeth around the minor axis of the FS mesh with the teeth of steel wheel, which should be avoided in harmonic drive. And it can be found that the number of no-load balls will increase under load torque, which is consistent with the result in 3.5.3.

T.Humbnail F一世g. 12

R.elationship of the difference of radial deformation and angular position under load.

5.这B.ending normal stress of outer ring of flexible bearing

5.1外圈上的弯曲正常应力分布

计算由球负荷引起的弯曲力矩的方法类似于4.1中的方法。根据方程式(6)(7)那T.He bending moment at any section of outer ringmFϕ)可以通过叠加由每个三力系统引起的弯曲力矩来计算,这是由以下给出的:(31)W.HereGmϕ) is given by:(32)

Substituting equation(19)进入等式(1),由径向分布式负载引起的弯曲瞬间由:(33)

因此,外圈上的总弯矩由:(34)

考虑到外圈的小壁厚,外圈的弯曲正常应力可以通过弯曲直射梁理论来计算[22], whose expression is given by:(35)W.HereR.是外环横截面的任何地方的半径;R.一世s the neutral layer radius of outer ring;一世一世s the inertial moment on cross section.

WhenN = 23 andαR. = 0, the bending normal stress of outer ring under no load can be calculated by equation(35),结果显示在图13.。这stress curves are smooth under no-load condition. For the bending normal stress outside the cross section of outer ring, the maximum tensile stress which appears at the end of major axis is 182.7 MPa, and the maximum compressive stress which appears at the end of minor axis is −171.6 MPa. And the distribution of the bending normal stress inside the cross section of outer ring is contrary.

加载扭矩时T. = 50 Nm, the curve of the bending normal stress distribution changes. It can be seen from figure that load torque will cause obvious extreme points on the stress curve. And the maximum bending normal stress at the end of the major axis increases apparently, but the maximum bending stress at the end of the minor axis barely change.

T.Humbnail F一世g. 13

弯曲正应力分布O.f outer ring of the FB.

5.2最大弯曲正常应力的特点

5.。2。1这在fluence of number of balls on the maximum bending normal stress

图14.展示横向和内侧外圈的最大弯曲正常应力与无负载下的球数之间的关系。横向应力表示外圈的横截面之外的最大弯曲正常应力,内侧应力表示在外环的横截面内。可以发现,横向和内侧应力具有相同的趋势,即长轴上的应力随着球的数量的增加而降低,但是短轴上的应力保持不变。例如,调查横向应力的曲线,我们可以发现,在球的数量从7到25增加后,最大拉伸应力从249.1到181.8MPa降低,这减少了27%。并且在短轴上的最大压缩应力保持在约170MPa。

T.Humbnail F一世g. 14

这在fluence of number of balls on the maximum bending normal stress.

5.2.2负载扭矩对最大弯曲正常应力的影响

这R.elationship between the maximum bending normal stress of lateral and medial outer ring and load torque is shown in图15.。可以发现,随着负载扭矩的增加,弯曲正常应力随负载扭矩的增加而增加,但是较小轴上的弯曲常规应力保持不变。例如,调查横向应力的曲线,在负载扭矩从0到100nm增加后,最大拉应力从182.7%增加到235.5MPa,这增加了28.9%。并且在短轴上的最大压缩应力保持在约-175MPa。

T.Humbnail F一世g. 15

这在fluence of load torque on the maximum bending normal stress.

6.CO.nclusions

一种load analysis model of three-force ring for the FB was proposed in this paper and the main conclusions were as follows:

  • 基于薄壁环理论,建立了FB三力环的机械模型,从而导出了外圈的径向变形与负荷之间的关系。然后通过叠加三力环获得FB中的负载分布。该三力环的叠加算法适用于球的数量超过4.并且该线性算法的计算速度远高于FEM。

  • 一种ccording to the superposition algorithm of three-force ring, the influence of number of balls, load torque and angular position on the maximum ball load was studied.

结果表明,随着球数的增加,最大球负荷总是随着负载扭矩的增加而增加。对于具有23个球的FB,在负载扭矩从0到50nm增加后,短轴末端的无负载区域的宽度从30.39°增加到44.1°。因此,负载扭矩变窄了FB的负载范围,导致装载球的数量的减少。因此,负载扭矩应限于避免FB的噪声和异常操作。

  • 这T.Heoretical deformation of outer ring of the FB is closer to the expected deformation with the increase of the number of balls. The load torque will increase the difference of the expectation and the theoretical radial deformation.

变形形状实际上是波状的。理论曲线与装载点处的预期曲线一致,它们在这些点周围的形状类似。但是在短轴周围的理论和预期变形形状之间存在明显的差异。并且形状是凸的,这导致空载球。应该注意的是,在谐波驱动中应避免此处的大径向变形。

  • 研究了FB的外圈上的交替弯曲正常应力。在外侧,最大拉伸应力出现在主轴上,最大压缩应力出现在短轴上。在内侧,最大拉伸应力出现在短轴上,并且最大压缩应力出现在长轴上。由负载扭矩产生的径向分布式负载将对应力曲线产生明显的极端点。另外,在长轴上的弯曲正常应力随着球的数量的增加而降低并且随着负载扭矩的增加而增加。但短轴的压力几乎不会受到这两个因素的影响。

命名法

N:球的数量

R.:外圈中性半径

m:弯矩

T.:负载扭矩

X一世:三力系统中的中径向力

F一世: The ith radial load on outer ring

W.ab:三力环的径向变形

ΔW.最大限度:径向变形的最大差异

Hϕ): Coefficient of bending deformation

一种cknowledgments

T.H一世s project is supported by cooperative foundation of BEIJING CTKM HARMONIC DRIVE CO., LTD and Shanghai university (Grant No: D.71-0109-18-076).

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C一世T.e this article as: Y. Yu, E. Zhu, X. Chen, Y. Wang, Load analysis and deformation research of the flexible bearing based on a three-force ring superposition method, Mechanics & Industry22,34(2021)

一种ll Tables

T.able 1

R.elevant Parameters of radial distributed load.

T.able 2

main parameters of the FB.

所有数字

T.Humbnail F一世g. 1

谐波减速器的工作原理示意图。

在文中
T.Humbnail F一世g. 2

负载FB外环的简化。

在文中
T.Humbnail F一世g. 3

三个力作用下环的结构图。

在文中
T.Humbnail F一世g. 4

在T.ernal force of the symmetric section of the ring.

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T.Humbnail F一世g. 5

Loading diagram of outer ring of the FB withNB.alls (N≧ 5).

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T.Humbnail F一世g. 6

FB的负载分布计算框图。

在文中
T.Humbnail F一世g. 7

这B.all load distribution comparison of the FB LY-6025.

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T.Humbnail F一世g. 8

R.elationship of maximum ball load and number of balls under load.

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T.Humbnail F一世g. 9

R.elationship of maximum ball load and load torque under different number of balls.

在文中
T.Humbnail F一世g. 10

载荷下球负荷和角度位置的关系。

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T.Humbnail F一世g. 11

空载下最大差分和球数的关系。

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T.Humbnail F一世g. 12

R.elationship of the difference of radial deformation and angular position under load.

在文中
T.Humbnail F一世g. 13

弯曲正应力分布O.f outer ring of the FB.

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T.Humbnail F一世g. 14

这在fluence of number of balls on the maximum bending normal stress.

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T.Humbnail F一世g. 15

这在fluence of load torque on the maximum bending normal stress.

在文中

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