EDP科学标志
开放存取
金博宝问题
金博宝
体积22,2021
物品编号 34
页数) 11
迪伊 https://doi.org/10.1051/meca/2021030
Published online 06 May 2021

© 是的。Yu等人,EDP Sciences 2021出版

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简介

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一13]。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。

The harmonic reducer is mainly formed of three components: circular spline (CS), flexspline (FS) and wave generator (WG), where the WG consists of a FB and a wave generator cam (WG cam), shown in图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,如图所示图1. 因此,这种错齿传动使谐波减速器传动具有较大的减速比。

The FB is 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.

The 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。

根据刘和志力推导的薄壁环的变形的一般解[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]。Compared with the theoretical methods, this kind of method requires a lot of computing time.

本文根据薄壁环理论和叠加原理,建立了三力环的叠加模型。根据该力学模型,可以计算出结构内部的载荷分布。计算分析了外圈的径向变形和弯曲正应力。在该力学模型计算的基础上,研究了球数、球位角和负载力矩对球负载的影响。研究了钢球数和加载扭矩对外圈径向变形和最大弯曲正应力的影响。

T.Humbnail Fig. 1

Schematic diagram of the working principle of the harmonic reducer.

2 A mechanical model of three-force ring

2.1基本理论

A ring 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 in图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。

本文给出了薄壁环理论的三个主要公式[15,16]由以下公式给出:(1)(2)(3)

方程(1)一世s the elastic equation connecting bending momentM和径向位移W., whereE是材料的弹性模量和I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.和周向位移V.。方程(3)是正常角度的表达θ.圆形部分。

The deformation 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 Fig. 2

负载FB外环的简化。

2.2 Distributed bending momentMAB(φ)在环上

The structure diagram of the three-force ring model is shown in图3. 如果F1 = F假设还有另外两个径向力F2F3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,如图所示图4

当弯矩时刻MA,MB径向力系统分别作用在半环上,如所示图4用单位力法计算A、B截面的法向角[17]。然后通过叠加获得两部分的总正常角度。根据变形兼容性条件,这两种正常角度的值为0,因此弯曲的时刻由:(4)哪里H1H2are given by:(5)

在T.He same way, when the bending momentMA,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)哪里H3,H4H5are given by:(7)

T.Humbnail Fig. 3

三个力作用下环的结构图。

T.Humbnail Fig. 4

圆环对称截面的内力。

2.3 The expression of radial displacement

Substituting equation(6)进入等式(1),微分方程如下:(8)

The general solution of two differential equations is expressed as:(9)(10)哪里A1,A2,B1B2are integral constants.

由于结构的对称性,正常角度θ.和圆周位移V.等于0时φ. = 0 andφ. = π。将该对称条件替换为方程(3),A1B1are given by:(11)

然后根据connecti光滑的条件O.n, that is whenφ. = φ.0,方程式(9)(10)应该得到相同的值和导数。相同的结果如下:(12)

It seems that the above conditions are not enough to get a solution, this is because of the lack of horizontal constraints for the structure in图3。To completely determine the four integration constants, we let the radial displacement at point B equal toX0图3。根据方程式(9),(11)(12),A2B2表示X0can be given by:(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 on the ring should be 0 to ensure that the displacement of point B relative to the vertical line passing through the center of the circle is constant. And through equation(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), the radial displacement of three-force ring can be given by:(15)其中h(φ.) is the coefficient of bending deformation, which is given by:(16)

3 Load distribution in the flexible bearing

3.1计算参数

FS加载时,FB外圈加载图N球(N≧5)显示在图5。为了简化计算,我们具有以下假设:①假设CS和FS之间没有摩擦。②假设轴的旋转速度对FB的径向力状态没有影响。③假设外圈上球的离心载荷远小于球载自身,这可以被忽略。给出了与外部径向分布式负载相关的参数表1,并且柔性轴承的主要几何参数Table 2. 价值观Φ1,Φ2Φ3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]。The main variables are the load torque and the number of balls, which can be changed in a certain range.

T.Humbnail Fig. 5

FB外环加载图N球(N≧ 5).

表1

Relevant Parameters of radial distributed load.

Table 2

FB的主要参数。

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,径向分布荷载图5一世s given by [19,20]:(18)哪里问:T.最大限度一世s the maximal value of tangential distributed load;τ.一世s profile angle;D.R.一世s the pitch diameter of the FS;B.W.是齿圈的工作宽度。

Under the radial distributed load, the radial deformation of the outer ring is given by [15]:(19)哪里BO.一世s the width of outer ring;R是外圈的中性层半径;问:rk公司是系数K.术语的系列。

图5,钢球上的径向载荷表示F一世每个加载点都标有逆时针方向的数字。第i个加载点的角位置由下式给出:(20)哪里αR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)哪里W.凸轮,一世是凸轮组件引起的预变形;W.问:,一世是由外部径向载荷引起的变形问:R.;W.一世一世s the deformation caused by the internal radial loadF一世;PD.是FB的径向间隙。

3.3三力系统的叠加算法

Unknown radial loadsF一世(i = 1,2,3,......,N,N≧5)显示图5可以分解成N未知的三队系统。每个力系统都显示在图3,其中中间的径向力是X一世(i = 1,2,3,......,N,N≧ 5) and its angular position is determined by equation(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)

哪里φ.0 = 2π/N。上述矩阵可以简化为:(23)在哪里 [W.]是外圈径向变形的列矩阵[A] is the deformation coefficient matrix; [X] is the column matrix of unknown forces.

柱矩阵中加载点的变形[W.]可以通过方程计算(21),然后我们可以获得一组线性方程。通过解决方程,[X]由下式给出:(24)

The unknown radial loads in图4可以通过叠加未知力来计算X一世at each loading point. The superposition process is written in the form of matrix, which is given by:(25)哪里。上述矩阵可以简化为:(26)在哪里 [F] is the column matrix of radial loads; [C]是变换矩阵。

根据上述线性算法,可计算出FB的内部径向载荷。如果结果为负,则表明这些钢球没有承受载荷。为了消除负荷载,采用了力边界条件F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 in图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),无法建立线性方程是不可能的。当有两到四个球时,三力系统不平衡,但是径向载荷可以通过双力环模型计算。

T.Humbnail Fig. 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]。The ball load distribution curves with no external loads are shown in图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]。It 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 Fig. 7

FB-LY-6025的球载荷分布比较。

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 inTables 12

根据3.3中的算法,导出了最大球载荷与载荷下球数的关系,如所示图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 Fig. 8

Relationship of maximum ball load and number of balls under load.

3.5.2 The influence of load torque on maximum ball load

WhenαR = 0,导出了不同球数下最大球载荷与载荷转矩的关系,如所示图9。It 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 Fig. 9

在不同数量的球下最大球负荷和负载扭矩的关系。

3.5.3球负荷和角度位置的关系

When the number of balls is 23, the continuous load curves under different load torque can be obtained by changing the rotation angle of balls, as shown in图10.。It 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 Fig. 10

载荷下球负荷和角度位置的关系。

4 The deformation 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]。它假设施加到薄环的单个负载通过对称和切向剪切应力分布平衡,这是不适合本文的。在前两章中,推导出径向变形和三力环的负载之间的关系,并且未知负载分解成几种三力系统。在上述过程中,导出了这三个力系统的尺寸和位置。因此,可以通过叠加由每个三力系统引起的径向变形来计算环上所有点的总径向变形。对于fb与N外环上的径向载荷可以分解为N三力系统,以及X一世表示第i个力系的大小。根据方程式(15),(16)(20), the radial deformation caused by the ith force system is given by:(27)哪里β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)

哪里W.问:由等式确定(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)哪里W. ⁡⁡ (φ.)是理论径向变形,可以通过方程计算(29);W.cam(φ.) is the ideal radial deformation, which is determined by equation(17)

WhenαR = 0 andT = 0, the relationship curve of the maximum difference ΔW.最大限度和T.He number of ballsN如中所示图11.。从图中可以看出,最大差异是负的,它出现在主轴的末端时N ≤ 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 Fig. 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.. 在图中,虚线表示当T = 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 = 50nm,短轴末端最大差值为空载时的2.3倍。变形形状实际上是波浪形的。理论曲线与加载点处的预期曲线一致,且这些点周围的形状相似。但短轴附近的理论变形形状与预期变形形状存在明显差异。而且形状是凸的,这就产生了空载球。如果此处径向变形过大,可能导致FS短轴周围的轮齿与钢轮轮齿啮合,在谐波传动中应避免。结果表明,在负载转矩作用下,空载球的数量会增加,这与3.5.3的结果是一致的。

T.Humbnail Fig. 12

Relationship of the difference of radial deformation and angular position under load.

5 The bending 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),外圈任意截面的弯矩MF(φ.) can be calculated by superimposing the bending moment caused by each three-force system, which is given by:(31)哪里GM(φ.) 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)哪里R.是外环横截面的任何地方的半径;R是外圈的中性层半径;I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.。The 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 Fig. 13

FB外圈弯曲正应力分布。

5.2最大弯曲正常应力的特点

5.2.1钢球数量对最大弯曲正应力的影响

The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 Fig. 14

The influence of number of balls on the maximum bending normal stress.

5.2.2 The influence of load torque on the maximum bending normal stress

The relationship 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 Fig. 15

The influence of load torque on the maximum bending normal stress.

6 Conclusions

本文提出了FB三力环的载荷分析模型,主要结论如下:

  • 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。

  • According 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.

The results show that the maximum ball load always increases with the increase of number of balls and it increases linearly with the increase of the load torque. For the FB with 23 balls, the width of no-load area at the end of minor axis increases from 30.39° to 44.1° after the load torque increases from 0 to 50 Nm. Consequently, the load torque narrows the load range of the FB, resulting in the reduction of the number of loaded balls. Thus, the load torque should be limited to avoid the noise and abnormal operation of the FB.

  • The theoretical 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。

  • The alternating bending normal stress on outer ring of the FB was studied. On the outer side, the maximum tensile stress appears on major axis and the maximum compressive stress appears on minor axis. On the inner side, the maximum tensile stress appears on minor axis and the maximum compressive stress appears on major axis. The radial distributed load produced by load torque will cause obvious extreme points on the stress curve. In addition, the bending normal stress on major axis will decreases with the increase of the number of balls and increases with the increase of the load torque. But the stress on minor axis is hardly affected by these two factors.

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:外圈上的径向载荷

W.AB:三力环的径向变形

ΔW.最大限度: Maximum difference of radial deformation

H(φ.): Coefficient of bending deformation

Acknowledgments

This 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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Cite 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)

所有表格

表1

Relevant Parameters of radial distributed load.

Table 2

FB的主要参数。

All Figures

T.Humbnail Fig. 1

Schematic diagram of the working principle of the harmonic reducer.

在文中
T.Humbnail Fig. 2

负载FB外环的简化。

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T.Humbnail Fig. 3

三个力作用下环的结构图。

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T.Humbnail Fig. 4

圆环对称截面的内力。

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T.Humbnail Fig. 5

FB外环加载图N球(N≧ 5).

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T.Humbnail Fig. 6

FB的负载分布计算框图。

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T.Humbnail Fig. 7

FB-LY-6025的球载荷分布比较。

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T.Humbnail Fig. 8

Relationship of maximum ball load and number of balls under load.

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T.Humbnail Fig. 9

在不同数量的球下最大球负荷和负载扭矩的关系。

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T.Humbnail Fig. 10

载荷下球负荷和角度位置的关系。

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T.Humbnail Fig. 11

空载下最大差分和球数的关系。

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T.Humbnail Fig. 12

Relationship of the difference of radial deformation and angular position under load.

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T.Humbnail Fig. 13

FB外圈弯曲正应力分布。

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T.Humbnail Fig. 14

The influence of number of balls on the maximum bending normal stress.

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T.Humbnail Fig. 15

The influence of load torque on the maximum bending normal stress.

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