Open Access
yabo亚博问题 yabo亚博 Volume22,2021 3.4. 11 https://doi.org/10.1051/meca/2021030 0.6.may 2021

## 1在T.R.O.D.uction

 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

 F一世g. 2负载FB外环的简化。

### 2。2Distributed bending momentmab（φ）在环上

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）

 F一世g. 3三个力作用下环的结构图。
 F一世g. 4在T.ernal force of the symmetric section of the ring.

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

## 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.

 F一世g. 5Loading diagram of outer ring of the FB withNB.alls (N≧ 5).
T.able 1

T.able 2

main parameters of the FB.

### 3.2变形兼容性方程

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.

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。

### 3.3三力系统的叠加算法

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。

 F一世g. 6FB的负载分布计算框图。

### 3.4柔性轴承分析模型的验证

 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

 F一世g. 8R.elationship of maximum ball load and number of balls under 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.

 F一世g. 9R.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

 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

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

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。

 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.

 F一世g. 12R.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

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）

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.

 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

 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

 F一世g. 15这在fluence of load torque on the maximum bending normal stress.

## 6.CO.nclusions

• 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.

• 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.

• 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

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

T.able 2

main parameters of the FB.

## 所有数字

 F一世g. 1谐波减速器的工作原理示意图。 在文中
 F一世g. 2负载FB外环的简化。 在文中
 F一世g. 3三个力作用下环的结构图。 在文中
 F一世g. 4在T.ernal force of the symmetric section of the ring. 在文中
 F一世g. 5Loading diagram of outer ring of the FB withNB.alls (N≧ 5). 在文中
 F一世g. 6FB的负载分布计算框图。 在文中
 F一世g. 7这B.all load distribution comparison of the FB LY-6025. 在文中
 F一世g. 8R.elationship of maximum ball load and number of balls under load. 在文中
 F一世g. 9R.elationship of maximum ball load and load torque under different number of balls. 在文中
 F一世g. 10载荷下球负荷和角度位置的关系。 在文中
 F一世g. 11空载下最大差分和球数的关系。 在文中
 F一世g. 12R.elationship of the difference of radial deformation and angular position under load. 在文中
 F一世g. 13弯曲正应力分布O.f outer ring of the FB. 在文中
 F一世g. 14这在fluence of number of balls on the maximum bending normal stress. 在文中
 F一世g. 15这在fluence of load torque on the maximum bending normal stress. 在文中

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