机械毕业设计英文外文翻译63超高速行星齿轮组合中内部齿轮的有限元分析

翻译部分

英文原文

Finite Element Analysis of internal Gear in High-Speed Planetary Gear Units Abstrac t: The stress and the elastic deflection of internal ring gear in high-speed spur planetary gear units are investigated. A rim thickness parameter is defined as the flexibility of internal ring gear and the gearcase. The finite element model of the whole internal ring gear is established by means of Pro/E and ANSYS. The loads on meshing teeth of internal ring gear are applied according to the contact ratio and the load-sharing coefficient. With the finite element analysis(FEA),the influences of flexibility and fitting status on the stress and elastic deflection of internal ring gear are predicted. The simulation reveals that the principal stress and deflection increase with the decrease of rim thickness of internal ring gear. Moreover, larger spring stiffness helps to reduce the stress and deflection of internal ring gear. Therefore, the flexibility of internal ring gear must be considered during the design of high-speed planetary gear transmissions.

Keywords: planetary gear transmissions; internal ring gear; finite element method High-speed planetary gear transmissions are widely used in aerospace and automotive engineering due to the advantages of large reduction ratio, high load capacity, compactness and stability. Great attention has been paid to the dynamic prediction of gear units for the purpose of vibration reduction and noise control in the past decades(1-8).as one of the key parts, internal gear must be designed carefully since its flexibility has a strong influence on the gear train’s performance. studies have shown that the flexibility of internal gear significantly affects the dynamic behaviors of planetary gear trains(9).in order to get stresses and deflections of ring gear, several finite element analysis models were proposed(10-14).however, most of the models dealt with only a segment of the internal ring gear with a thin rim. the gear segment was constrained with corresponding boundary conditions and appoint load was exerted on a single tooth along the line of action without considering the changeover between the single and double contact zone in a complete mesh cycle of a given tooth. A finite element/semi-analytical nonlinear contract model was presented to investigate the effect of internal gear flexibility on the quasi-static behavior of a

planetary gear set(15). By considering the deflections of all gears and support conditions of splines, the stresses and deflections were quantified as a function of rim thickness. Compared with the previous work, this model considered the whole transmission system. However, the method described in Ref. (15) requires a high level of expertise before it can even be successful.

The purpose of this paper is to investigate the effects of rim thickness and support conditions on the stress and the deflection of internal gear in a high-speed spur planetary gear transmission. Firstly, a finite element model for a complete internal gear fixed to gearcase with straight splines is created by means of Pro/E and ANSYS. Then, proper boundary conditions are applied to simulating the actual support conditions. Meanwhile the contact ratio and load sharing are considered to apply suitable loads on meshing teeth. Finally, with the commercial finite element code of APDL in ANSYS, the influences of rim thickness and support condition on internal ring gear stress and deflection are analyzed.

1 finite element model

1.1 example system

A three-planet planetary gear set (quenched and tempered steel 5140) defined in Tab. 1 is taken as an example to study the influence of rim thickness and support conditions.

As shown in Fig.1, three planets are equally spaced around the sun gear with 120·apart from each other. Here, all the gears in the gear unit are standard involute spur gears. The sun gear is chosen as the input member while the carrier, which is not indicated in Fig.1 for the sake of clarity, is chosen as the output member. The internal ring gear is set stationary by using 6 splines evenly spaced round the outer circle to constrain the rigid body motion of ring gear.

A dimensionless internal gear rim thickness parameter λis defined as the ratio of rim thickness to the tooth height as follows:

(1)

Where r0 ,r f ,r a are the outer , dedendum and addendum radius of internal gear, respectively.

A smaller λindicates a more flexible ring gear and vice versa . internal gears with different values of λ=1.0,1.5,2.0,2.5 are investigated in this paper. In all these cases, the widths of ring gear are 44mm, and the connecting splines are 34mm in length and 14 mm inwidth, while the heights of splines in each case are 5mm, 6mm,7mm and 8mm, respectively.

A finite element model for the internal gear with λ=1.5 is shown in Fig.2, which contains 69 813 elements and 112 527 nodes.

Fig.2 Finite element model of internal ring gear

1.2 loads and boundary conditions

The internal gear is fixed to gearcase through splines and meshes with planet gears. Assuming that the load is evenly distributed to each planet and all frictions are negligible, the

meshing force between each planet and the ring is as follows:

Where T c is the overall output torque; i sc is the overall reduction ratio; r s is the radius of sun gear; n p denotes the number of planets; is the pressure angle.

In addition, by considering the contact ratio and load sharing factors, we can finally determine the mesh positions and the proportions of the load carried by each tooth of the ring. The load state of the ring is shown in Fig.3.

Here, the phase angle between each planet is 120。and F i(1,….,6) is the normal meshing force acting on the teeth of internal gear. For clarity purpose, only the teeth in mesh are plotted in Fig.3. after obtaining the meshing forces acting on internal gear, we can apply them to the finite element model. To be specific, the meshing forces are evenly distributed to the corresponding nodes along the line of engagement.

As support conditions can be very complicated if considering the contact problems, special substitute must be made to model the actual contacts at the splines. In this paper, the splines are coupled with the ring by the overlapped nodes and six springs equally spaced between the outer surface of the ring and the housing surface are applied to simulating the support conditions. The support condition between the ring and the housing is indicated through the stiffness of these springs. The process can be detailed as follows. A single node needs to be defined for each spline-to-housing connection. This is achieved suing COMBINE 14 elements at each spline position, which connect the splines to the points at the housing surface with an infinite stiffness. All degree of freedoms (DOFs ) of these predefined nodes

are constrained. At the other end of each spring element is a common node connected with spline whose DOFs except in radial direction are all constrained. In addition, the nodes on the loaded surface of each spline are constrained in circumferential DOF . And the axial DOF of the ring is constrained.

The support condition simulated with springs is shown as Fig.4

2 FEA results

By applying proper loads and boundary conditions, a finite element analysis can be conducted to figure out the effects of rim thickness and support conditions on internal gear stress and deflection. As to the example system, the stress and deflection are predicted at 24 discrete angular positions with an increment of 5。,which span a 120.. rotation of the carrier . this ensures that any tooth of internal gear goes through a complete meshing cycle because the number of planets is 3.

2.1 effect of rim thickness on internal gear stress and deflection

In Fig.5, the maximum principal stress (Mises stress) of the ring at each discrete position is plotted against the carrier rotation angle for four different ring rim thickness ( =1.0,1.5,2.0,2.5). here, the spring stiffness is 33N/mm.

From Fig.5, we can see that with the decrease of λ, the maximum stress in the ring increases . hence, the rim thickness of the ring cannot be too small for the sake of gear durability. And further investigations reveal that the critical point at which the maximum stress occurs moves from the fillet region to the root of tooth when λdecrease.

Fig.6 shows the deflection shapes of rings with different rim thickness. The ring deflections for λ=1.0 and λ=2.0 are demonstrated in Fig.7 with the same deflection magnification factor of 50.

Obviously, when λincreases , the deflection of ring decreases. The amount of radial deflection of the ring in both outward and inward direction is plotted as a function of λin Fig.7. here, the positive amounts denote the outward deflections while the negative ones denote the inward deflection. When λ=1.0, the maximum out-ward and inward radial deflections are predicted to be 0.139 and 0.122 mm, respectively. If the ring si permitted to deflect so much, those manufacturing errors associated with the internal gear such as the

roundness error and run-out error can be tolerated as long as their magnitudes are less the amount of deflection .

2.2 effect of spring stiffness on internal gear stress and deflection

The maximum principal stress of the ring with varied spring stiffness k is shown in Fig.8. here, the unit of stiffness is N/mm. obviously, the maximum principal stress of the ring with λ=1.0 is much more sensitive to the support stiffness than that of the ring with λ=2.5. and for a ring with a given λ, the maximum principal stress increases with the decrease of spring stiffness.

Fig.9 demonstrates the influence of spring stiffness on the maximum radial deflection of the ring. Similarly the maximum radial deflections of the ring withλ=1.0 is much more sensitive to the support stiffness than that of the ring with λ=2.5. and for a ring with a given λ, the maximum deflection increase with the decrease of spring stiffness.

3 conclusions

In this paper, a finite element analysis model is employed to investigate the effect of flexibility of internal ring gear on stresses and deflections. Based on the results presented above, some conclusions are as follows.

(1)The rim thickness of ring is influential to its stresses. With the decrease of rim

thickness, the maximum principal stress of internal ring gear increases and the

critical point at which the maximum stress occurs moves from fillet region to

the root of tooth.

(2)The rim thickness also influences internal ring gear deflections. A ring with a

thin rim produces larger deflections than a ring with thick rim. When the

deflection is large enough, some manufacturing errors associated with internal

ring gear such as roundness error and run-out error can be tolerated.

(3)The spring stiffness both affects the stress and deflection of internal ring gear.

An internal gear ring with larger spring stiffness tends to produce smaller stress

and deflection.

An alternative way of using gears to transmit torque is to make one or more gears, i.e., planetary gears, rotate outside of one gear, i.e. sun gear. Most planetary reduction gears, at conventional size, are used as well-known compact mechanical power transmission systems [1]. The schematic of the planetary gear system employed is shown in Figure。Since SUMMiT V designs are laid out using AutoCAD 2000, the Figure 1 is generated automatically from the lay out masks (Appendix [1]). One unit of the planetary gear system is composed of six gears: one sun gear, a, three planetary gears, b, one fixed ring gear, c, one rotating ring gear, d, and one output gear. The number of teeth for each gear is different from one another except among the planetary gears. An input gear is the sun gear, a, driven by the arm connected to the micro-engine. The rotating ring gear, d, is served as an output gear. For example, if the arm drives the sun gear in the clockwise direction, the planetary gears, b, will rotate counter-clockwise at their own axis and at the same time, those will rotate about the sun gear in clockwise direction resulting in planetary motion. Due to the relative motion between the planetary gears, b, and the fixed ring gear, c, the rotating ring gear, d, will rotate counterclockwise direction. This is so called a 3K mechanical paradox planetary gear [1].

中文翻译

超高速行星齿轮组合中内部齿轮的有限元分析

摘要：超高速行星齿轮组合中内部齿轮的应力和弹性变形的调查。环的厚度参数的定义是内部齿圈和齿轮箱的弹性。整个内部齿圈的有限元模型是用Pro/E and ANSYS的方式确定的。内部齿圈轮齿的载荷取决于啮合系数和载荷分布系数。依靠有限元分析（有限元分析），可以预测内部齿圈的应力和弹性变形对其灵活性和装配情况的影响。模拟表明，主应力和挠度随着内齿圈齿厚的减少而增加的。此外，较大的弹簧刚度有助于减少内部齿圈的应力和挠度。因此，在设计的高速行星齿轮传动时，内部齿圈的弹性必须加以考虑。

关键词：行星齿轮传动;内部齿圈;有限元方法

由于大减速比，高承载能力，高压实度和高稳定性的优势,超高速行星齿轮传动被广泛应用于航空航天和汽车工程。动态预测齿轮单位为目的的减振及噪音管制在过去数十年已经被给予高度重视。（ 1-8 ）作为其中的关键部件，内部的齿轮设计必须小心，因为它的灵活性，对齿轮传动系的性能，具有很强的影响。研究表明，内部齿轮的弹性对行星齿轮系的动态行为有显著的影响（ 9 ）。为得到齿圈应力和挠度，提出了几个有限元分析模型（ 10月14日）。不过，大部分模型只能处理薄环内齿圈的一段。齿轮部分受相应的边界情况约束，在没有考虑到一个假设轮齿完整的啮合循环中的单、双接触带完全不同时，额定载荷等于线运动中单个轮齿受到的载荷。有限元/半解析非线性接触模式被提交去调查准静态行为的行星齿轮组中的内部齿轮的灵活性的影响。考虑到所有齿轮的挠度和齿条的支撑情况，其应力和挠度是关于环厚度的一个函数。与过去的工作相比，这种模式被考虑成整个传动系统。不过，标准的描述方法（ 15 ），需要一个高层次的专业知识，才可以更成功。

本文件的目的是调查环的厚度和支撑条件对一个高速的行星齿轮传动的内部齿轮应力和挠度影响。首先，一个完整的内部齿轮用直齿条固定齿轮箱上的有限元模型是依靠Pro / E和ANSYS的方式创造的。其次，适当的边界条件适用于模拟实际的支撑条件。同时，啮合系数和载荷分布系数被认为同样适用于相啮合轮齿的载荷。最后，借助于ANSYS中的商业有限元APDL编码，可以分析环厚度和支撑情况对内齿圈应力和挠度的影响。

1有限元模型

1.1系统举例

表1所示的3个行星轮的行星齿轮组（调制钢5140）用来举例研究环厚度和支撑情况的影响。

表1为系统参数

项目太阳轮行星轮内齿圈

齿数23 22 67

模数/mm 3 3 3 压力角/(·) 20 20 20

杨氏模量/GPa —205 —泊松比—0.3 —

密度/(t·m3) —7.8 —

如图1，3个行星轮两两间距120度围绕太阳轮等空间布置。这里的所有齿轮都是标准的渐开线齿轮。太阳轮作为输入件的同时，为表达清晰，图1没有表示出作为输出件的支撑件。内齿圈外圆均匀布置的可约束齿圈刚性运动的6齿花键使其得以固定。

图1为系统图例

平面内齿圈环厚系数λ被定义为环厚与轮齿高度的比，如下

(1)

r 0 ,r

f

,r

a

分别为内齿圈的分度圆、齿根高和齿顶高。

λ值越小说明齿圈越灵活，λ值越大说明齿圈越不灵活。本文章研究λ取

不同值时λ=1.0,1.5,2.0,2.5的内齿圈.在所有这些情况下，内齿圈的宽为44毫米，连接花键长34毫米、宽14毫米、高度分别为5、6、7、8毫米。

图2所示的λ=1.5的内齿圈有限元模型包含了69813个元件和112527个节点。

图2。内齿圈有限元模型

1.2载荷和边界条件

内齿圈通过键与箱体连接，与行星轮啮合。假设每个行星轮上的载荷是均匀分布的，而且所有的摩擦力可以忽略，那么每个行星轮和太阳轮间的啮合力如下：

T C 是输出的扭矩和，i

sc

是全部减速比，r

s

是内齿圈半径，n

p

是行星轮个数，

是压力角。

此外，考虑到啮合系数以及载荷分布因数，最后确定啮合的位置和齿圈每个齿的承载比例。图3所示为环的受载状态。

图3为内齿圈的受载状态

这里，每两个行星轮的相位角是120度。F

i

(1,….,6)是作用在内齿圈轮齿上正常啮合力。为明确的目的，只有轮齿画在图3中。得到内齿圈轮齿啮合力后，我们可以将其加到有限元模型中。具体说，啮合力是均匀分布在沿啮合线上相应的节点上的。

因为支撑情况特别复杂，如果考虑到接触问题，必须建立键实际接触的特殊代替模型。这里，用交替的节点和6个空间均匀布置的键与内齿圈联合在一起，在内齿圈外表面和机架表面间模拟支撑状况。内齿圈与机架间的支撑状况说明了这些凸起的刚度。其过程可详列如下。单一的节点需要详细说明每个内齿圈与机架的连接。在每个键的位置用14个联合元件完成，在无限刚度的机架表面，联合元件与键是点连接。这些定义节点的自由度是受约束的。在每个弹性元件的令一端是径向自由度全部受约束键连接的常见节点。此外，每个键的受载荷表面上的节点是受圆周自由度的约束的。环的轴向自由度也是受约束的。

模拟环的支撑情况如图4所示。

图4为键支撑情况示意图

2有限元分析的结果

运用适当的载荷和边界条件，有限元分析可以计算出内齿圈厚度和支撑情况对应力和挠度的影响。以系统为例，可以预测在旋转支座120度范围内，以5度增量的24个离散角度位置的应力和挠度。行星轮数目为3，这将保证内齿圈的每个轮齿都能有一个完整的啮合周期。

2．1齿环厚度对内齿圈应力和挠度的影响

在图5中，每一个离散的位置的最大主应力（ Mises应力）在机架的旋转角度的四种不同的环厚度(λ=1.0,1.5,2.0,2.5)的绘图。在这里，弹簧刚度是33n/mm。

图5为不同值时环的最大应力

从图5我们可以看出，随着λ的减少，环的最大应力在增加。因此，为了保证齿轮的耐久性，齿轮环的厚度不能太小。进一步调查显示，当λ减少时最大应力的产生点是从圆角处到齿根。

图6所示不同环厚时，环的挠度形状。在λ=1.0 和λ=2.0时，当用同样的放大偏转因数50，图7所示环的挠度。

图6为不同λ值时环的偏斜形状

很明显，当λ增加时，环的挠度就减少。图7所示的函数是，在环外和里方向上的变位度绘制的。这里，正数表示外面的挠度，而负数则表示里面的挠度。当λ=1时，外面和里面的最大挠度估算分别得0.139和0.122毫米。如果环允许偏离了这么多，只要总的挠度小于额定的，那些制造误差与内部齿轮如圆度误差和跳动误差可以容忍的。

图7为最大径向偏转关于的λ函数

2．2内齿圈的应力和挠度对弹簧刚度的影响

图8所示为具有不同弹簧刚度k的环的最大主应力。在这里，刚度的单位是N/mm。很明显，λ=1.0时环的最大主应力比λ=2.5时的最大主应力有更明显的支撑硬度。对于环有一个特定的λ值时，最大主应力随着弹簧刚度的减少而增加。

图8为在不同弹簧刚度下内齿圈的最大应力

图9所示为，环最大的径向偏转对弹簧刚度的影响。同样地，λ=1.0时环的最大径向偏转比λ=2.5时的最大径向偏转有更明显的支撑硬度。对于环有一个特定的λ值时，最大挠度随着弹簧刚度的减少而增加。

图9为在不同弹簧刚度下的最大径向偏转

3结论

本文的有限元分析是研究内齿圈灵活度对其应力和挠度的影响。基于上述的一些结果，得到一些结论如下：

（1）环的厚度对其应力有影响。随着环厚度的减少，内齿圈的最大主应力增加，危险点发生在最大应力产生的地方，即在圆角与齿根之间移动。（2）环的厚度也影响了内部齿圈的挠度。具有薄环的环比具有厚环的环能产生更大的挠度。当挠度足够大，一些内齿圈的制造误差，如圆度误

差和跳动误差是允许的。

（3）弹簧刚度影响内齿圈应力和挠度。具有较大的弹簧刚度的内齿圈，往往产生较小的应力和挠度。

使用齿轮传输转矩的其它可行的方法是将一个或者多个的齿轮,也就是, 行星齿轮,在另一个齿轮的外面旋转,也就是太阳轮。按照传统的尺寸设计的行星齿轮减速器是使整体结构紧凑的常用的传输系统。图1是上述的行星齿

轮的示意图。自从用AutoCAD设计SUMMiT V以来,图（1）可以通过软件自动产生(附[1])。一个完整的行星齿轮系统是由六个齿轮组成的: 一个太阳齿轮a，三个行星齿轮b，一个固定的内齿圈c,一个旋转的内齿圈d,和一个输出齿轮e。除了行星齿轮之外,每个齿轮的齿数都不相同。太阳齿轮a是输入齿轮，由与微引擎连接的机械手驱动。内齿圈d,被视为输出齿轮。举例来说,如果机械手驱动太阳轮按照顺时针方向方向旋转, 那么行星轮b, 将绕着它们自己的轴按照逆时针方向宣战，同时也将绕着太阳轮按照顺时针方向的方向旋转，这样就形成了行星运动。由于多个行星齿轮b和固定内齿圈c 之间的运动相似，所以旋转的内齿圈d将按照逆时针方向旋转。这也被叫做3K行星齿轮。