有限元分析 行星变速外文翻译

华北水利水电大学毕业设计附录三 外文原文International Journal of Engineering Research and Development ISSN: 2278-067X, Volume 1, Issue 5 (June 2012), PP.38-43 Influence of Friction on Contact Stress for Small PowerTransmission Planetary Gear BoxMr. J.R. Koisha, Mr. H.P. DoshiPost Graduate Student, Department Of Mechanical Engineering, L.D. College of Engineering, Ahmedabad, GujaratAssociated Professor, Department Of Mechanical Engineering, L.D. College of Engineering, Ahmedabad, Gujarat. AbstractIn planetary gear box, load is shared by planets, therefore to know the behaviour of contact is significant in it. In the present study parametric solid model of planetary gear box with three planets having involute profiled spur gears is modelled by Pro-Engineer Wildfire 5.0 software. Contact stress analysis of the three dimensional solid model of the planetary gear box requires huge computational resources, therefore plane stress frictionless contact analysis of the planetary gear box is carried out in ANSYS Workbench 12.0 and the results are verified by theoretical calculation of contact stress as per ISO 6336 standard. Then plane stress frictional contact analysis of the planetary gear box is done with the different values of coefficient of friction. The relationship of contact stress and coefficient of friction has showing a linear relationship. Further it is observed that at the low power transmission the slope of the graph is less. Keywordsplanetary gear box, contact analysis, FEA, frictional contact, plane stress analysis I. INTRODUCTIONDuring the recent past, significant progress in the field of contact analysis of gears has also been made, and finite element analysis (FEA) is gradually becoming established as an efficient tool in gear box design. Using the finite element analysis, which is a general and systematic computational procedure for approximately solving problems in physics and engineering, many contact problems, ranging from relatively simple ones to quite complicated ones, can be solved with high accuracy. The Finite Element Method can be considered the favourite method to treat contact problems, because of its proven success in treating a wide range of engineering problem in areas of solid mechanics, fluid flow, heat ransfer, and for electromagnetic field and coupled field problems. II. CONTACT ANALYSIS OF GEARSThe first works of development of Tooth Contact Analysis have been done by Litvin and Kai, and Baxter. Significant contributions to the development of Tooth Contact Analysis have been made by the engineers of the Gleason Works, Klingelnberg, and Oerlikon 1. O. Vogel et al, (2002) presented a constructive approach for the approximation free tooth contact analysis of hypoid bevel gears 2. A.R. Mijar and J.S. Arora (2004) developed an augmented Lagrangian optimization method and discussed for contact analysis problems that automatically update the user specified penalty values to obtain the final appropriate values. Further, to solve the frictional contact analysis problem accurately, a two phase formulation is proposed 3. Shuting Li (2005) presented three-dimensional, finite element methods to conduct surface contact stress and root bending stress calculations of a pair of spur gears with machining errors, assembly errors and tooth modifications 4. In the present study attempt is made to estimate effect of friction on contact stress in planetary gear box using non-linear FEA. III. PROBLEM FORMULATIONThe selected specification of the planetary gear box is as follow: TABLE IPower, P = 1500 W, Input Speed Ninpu= 500 r.p. m., Output Speed Noutput= 7.9365 r.p.m., Reduction Ratio, i =63:1, Two Stages, i1=7.5, i2=8.4, Gear Material = 16MnCr5 IV. CONTACT STRESS CALCULATION USING ISO 6336The Standard ISO 6336 Part2 (1996a) describes the following contact stress equation based on a different defined set of factors: Where, = it is the zone factor, which takes into account the flank curvatures at the pitch point and transforms tangential load at the reference cylinder to tangential load at the pitch cylinder. = it is the elasticity factor, which takes into account specific properties of the material. = it is the contact ratio factor, which takes into account the influence of the effective length of the lines of contact. = it is the application factor, which takes into account the load increment due to externally influenced variations of input or output torque. = it is the dynamic factor, which takes into account load increments due to internal dynamic effects. = it is the transverse load factor for contact stress, which takes into account uneven load distribution in the transverse direction resulting, for example, from pitch deviation. = it is the face load factor for contact stress, which takes into account uneven distribution of load over the face width, due to mesh misalignment caused by inaccuracies in manufacture, elastic deformations, etc. A. For the first stage:Where, = 1.5 (ISO 6336-2, clause-6 Graph) = 189.8 / (ISO 6336-2, clause-7 Table-1 for steel) Z = 0.7 (ISO 6336-2, clause-8 from fig-4) = 1 (ISO 6336-1, clause-5) = 1 (ISO 6336-2, clause-6, 6.1) = 1.5 (ISO 6336-1, Fig-16; Graph) = 1 (ISO 6336-6, Table C-2) = 309.05 N/B. For the second stage: Where, = 1.5 (ISO 6336-2, clause-6 Graph) = 189.8 / (ISO 6336-2, clause-7 Table-1 for steel) Z = 0.65 (ISO 6336-2, clause-8 from fig-4) = 1 (ISO 6336-1, clause-5) = 1 (ISO 6336-2, clause-6, 6.1) = 1.5 (ISO 6336-1, Fig-16; Graph) = 1 (ISO 6336-6, Table C-2) = 276.14 N/V. PARAMETRIC SOLID MODELLING OF THE PLANETARY GEAR BOXParametric means that the physical shape of the part or assembly is driven by the values assigned to the attributes (primarily dimensions) of its features. The parametric solid models of involute 20 full depth profiled gears are generated in Pro-Engineer Wildfire 5.0. and then assembled in the same. This model of the planetary gear box is transferred in ANSYS Workbench 12.0. VI. TEN STEPS OF FEA IN ANSYS WORKBENCH 12.01. In this present study, geometry is imported from Pro-Engineer Wildfire 5.0 modelling software. 2. Choose proper type of analysis i.e., structural, thermal, etc. Frictional contact stress analysis is chosen. It is a type of static structural analysis. The behaviour of the stresses in planetary gear box having spur gears is plane stress. One can choose plane stress analysis when the normal stress and the shear stresses directed perpendicular to the plane are assumed to be zero. The main advantage of plane stress analysis is it requires very less computational resources for the same domain. In this problem Plane stress analysis can be selected with the facts: (i) the geometry of spur gear along the axis of rotation is uniform and there is absence of axial loading. (ii) the assembly error is not modelled in this study. (iii) the stresses generated in carrier are ignored because the here focus of the study is contact stress generated at teeth meshing. 3. Choose proper unit system which will be used to define various properties. 4. Define material properties which are necessary to solve the problem. Here the gear material is 16MnCr5 and the required material properties for this analysis are only Modulus of elasticity, Poisons Ratio and Density. The Modulus of elasticity of 16MnCr5 is 206 GPa. The Poisons Ratio of 16MnCr5 is 0.3. The Density of 16MnCr5 is 7870 kg/m3. 5. Generate new coordinate system if required. 6. Define contact type & formulation method of contact, if the problem has contact nonlinearity. The type of contact determines how the contacting bodies can move relative to one another. The types of connections available in ANSYS Workbench are Bonded, No separation, Frictionless, Rough and Frictional. Choosing the appropriate contact type depends on the type of problem one is trying to solve. If the stresses very nearer to a contact interface are important, use one of the nonlinear contact types. Here, in this analysis frictionless and frictional contact types are used. However, use of nonlinear contact types usually results in longer solution times and can have possible convergence problems due to the contact nonlinearity. Here, Augmented Lagrange formulation method is selected. It is best suitable method among available methods for the frictional contact type which is nonlinear type analysis. Fig.1 the contacts defined 7. Generate appropriate mesh using various meshing control parameters. Fig.2 Mesh Generation optionsMesh should be enough fine at the area of interest (location in the part where stresses are critical) and mesh should be enough coarse at the remaining areas. This combination of fine and coarse mesh gives accurate results with optimum use of computational resources. The figure 2 shows set of meshing option parameters used. The figure 3 shows generated mesh.Fig.3 Generated Auto Mesh8. Apply boundary conditions which really represent actual loading conditions. Boundary condition refers to the external load on the border of the structure. If one makes any mistake in simulation of actual boundary conditions, software will give wrong results. Therefore it is very critical to decide the boundary conditions in software. Here the four boundary conditions are applied on the geometry (shown in figure 4), (i) frictionless support at the sun gear centre. It means that sun gear is free to rotate on its axis. (ii) fixed support at the outer periphery of the annulus gear. It means that annulus gear is fixed. (iii) frictionless support at the planet gears centers. It means that planet gears are free to rotate on its axis but carrier arm is fixed, which generates resistance to the torque. (iv) moment is applied on the sun gear. It means that gearbox is transmitting power. Fig.4 boundary conditions9. Solve the FEA model using proper solver. 10. Analyse the results using various displays. VII. RESULTSThe study was undertaken to investigate the effect of coefficient of friction on contact stress. First analysis is done without friction means with frictionless contact and that result is compared with ISO, which validates the results of ANSYS Workbench. Here in the table II the results obtained by ISO standard and ANSYS Workbench are compared. TABLE III: COMPARISON FRICTIONLESS CONTACT STRESS The stress calculated by ANSYS Workbench match with the stress calculated by ISO standard. The variation is up to 3%. This shows that ANSYS Workbench gives accurate result and one can use the same procedure to calculate frictional contact stress by ANSYS Workbench. The stress pattern at different place is shown in figure 5. Fig.5 stress patternFrictional contact analysis is done with different values of coefficient of friction ranging from 0.05 to 0.15 and the results are shown in table III. TABLE IIIII : RESULTS OF FRICTIONAL CONTACT ANALYSISThe trend lines of the above results are plotted in excel sheet which are shown in figure 6 and 7 for both stages of gearbox. Fig.6 effect of coefficient of friction on contact stress of stage 1Fig.7 effect of coefficient of friction on contact stress of stage 2VIII. CONCLUSION value on chart is called correlation coefficient or coefficient of determination and it shows that the reliability of the trend and accuracy of the forecast which is greater than 0.95. Therefore trend line shows that frictional contact stress follows linear trend with coefficient of friction and with the help of it we can find value of frictional contact stress for different values of coefficient of friction for the particular stage of gearbox. The slope of the graph is less for low power transmission planetary gear box. REFERENCES1. Faydor L. Litvin , Galina I. Sheveleva , Daniele Vecchiato , Ignacio Gonzalez-Perez , Alfonso Fuentes , Modified approach for tooth contact analysis of gear drives and automatic determination of guess values, Gear Research Center, Department of Mechanical and Industrial Engineering, University of Illinois at Chicago, Department of Theoretical Mechanics, Moscow State University of Technology 3A, Vadkovsky per., Moscow, 101472, Russia, Polytechnic University of Cartagena, Department of Mechanical Engineering, Campus Universitario Muralla del Mar, C/Doctor Fleming, Cartagena 30202, Spain, 2004. 2. O. Vogel , A. Griewank , G. Bar , Direct gear tooth contact analysis for hypoid bevel gears, Institute of Scientific Computing, Technical University of Dresden, 01062 Dresden, Germany, Institute of Geometry, Technical University Dresden, 01062 Dresden, Germany, 2002. 3. A.R. Mijar and J.S. Arora, An augmented Lagrangian optimization method for contact analysis problems, 1: formulation and algorithm, Optimal Design Laboratory, College of Engineering, The University of Iowa, Iowa City, IA 52242, USA, 2004. 4. Shuting Li, Finite element analyses for contact strength and bending strength of a pair of spur gears with machining errors, assembly errors and tooth modifications, Nabtesco Co. Ltd. Oak-hills No. 202, Heki-cho 7028-2, TSU-shi, Mie-ken 514-1138, Japan, 2005. 附录四 中文翻译国际工程技术研究和开发ISSN：2278-067X，第1卷，第5期（2012年6月）:38-43摩擦接触应力对小功率行星齿轮变速箱的影响摘要：在行星齿轮箱中，负载由行星齿轮共同承担，因此，知道它的接触应力是很重要的。在本研究中参数化实体模型与三个渐开线异形直齿圆柱齿轮行星齿轮箱是以ENGINEER Wildfire野火版5.0软件为蓝本。三维实体模型行星齿轮箱的接触应力的分析需要庞大的计算资源，因此以行星齿轮箱中摩擦接触应力进行ANSYS Workbench的分析和理论计算结果进行了验证（按ISO 6336标准的接触应力）。这样一来，平面应力的行星齿轮箱的摩擦接触分析完成。当摩擦系数的不同时，接触应力和摩擦系数成线性关系。此外，观察到，在低功率时传输线图的斜率小。关键词：行星齿轮箱 接触分析 有限元分析 摩擦接触 平面应力分析一 简介 近些年以来，在齿轮接触分析领域里也取得了显著进展，有限元分析（FEA）逐渐成为齿轮箱设计的一个有效的工具。采用有限元分析，可以解决物理科学与工程技术中一般的和系统计算过程中的问题，范围从相对简单的到相当复杂的许多接触问题，可以做到高准确度。由于其在固体力学，流体流动，传热领域、转换磁场和耦合场等领域中解决问题的成功经验，因此有限元分析是最流行的解决接触问题的方法。二 齿轮接触应力分析 利特温凯和巴克斯特首先完成齿面接触应力的分析。格里森的工程师克林根贝格和欧瑞康已对齿面接触应力分析的发展做出了重大贡献。 O. Vogel等（2002）提出了一个建设性的方法近似准双曲面锥齿轮的齿面接触分析2。 A.R. Mijar和J.S.阿罗拉（2004）开发了一个增强拉格朗日最优化方法和接触应力分析问题，讨论了自动更新的用户指定惩罚值，以得到最终的适当的值。此外，为了解决准确的摩擦接触分析问题，一个两相制剂提出3。书亭李（2005）提出了三维有限元方法进行表面接触应力和根弯曲应力计算一对直齿圆柱齿轮的加工误差，装配误差和牙齿的修改4。在本研究中行星齿轮箱的摩擦接触应力的影响尝试采用非线性有限元分析。三 提出问题所选择的行星齿轮箱规格如下：表I行星齿轮箱的规格功率P =1500 W，输入转速N=500转每分，输出转速N=7.9365 转每分，减速比i=63:1，分两级= 7.5，= 8.4，齿轮材料为16MnCr5四 按ISO 6336计算接触应力 ISO 6336标准第2部分（1996年）描述了接触应力方程，根据不同的一组因素定义： 其中，是区域因素，其中考虑到间距点的侧翼曲率和参考筒转换成变桨圆筒的切向载荷。是弹性系数，这里需要考虑到特定属性的材料的弹性系数。是接触比例因子，其中考虑到接触线的有效长度的影响。是应用系数，考虑到由于受外部影响而变化的输入或输出转矩所导致的负载的增加。是动态因素，这需要考虑由于内部的动态效果导致的荷载增量。是横向载荷接触应力因素，这需要考虑横向方向负荷分布不均匀产生的影响，例如，从间距偏差。是表面负荷接触应力因素，这需要考虑由于网格错位所造成的不精确制造，弹性变形等所导致的表面宽度上的负载分布不均匀。A. 第一阶段： 其中，= 1.5（ISO6336-2，第6图）=189.8/（ISO6336-2，第7对钢材的表1）= 0.7（ISO6336-2，第8从图4）= 1（ISO6336-1，第5）= 1（ISO6336-2，第6，6.1）= 1.5（ISO6336-1，图16，图）= 1（ISO6336-6，表C-2） = 309.05 N/B. 对于第二个阶段： 其中，= 1.5（ISO6336-2，第6图）=189.8/（ISO6336-2，第7对钢材的表1）= 0.65（ISO6336-2，第8从图4）= 1（ISO6336-1，第5）= 1（ISO6336-2，第6，6.1）= 1.5（ISO6336-1，图16，图）= 1（ISO6336-6，表C-2） = 276.14N/五 行星齿轮箱的参数化实体建模参数化是指零件或装配体的物理形状是由指定值的属性来确定（主要尺寸）的特点。渐开线20的全深度异形齿轮的参数化实体模型的生成在ENGINEER Wildfire野火版5.0，然后用同样的方法装配。此行星齿轮箱模型转换到ANSYS Workbench12.0中。六 十步有限元分析ANSYS WORKBENCH12.01在目前的研究中，几何建模是从ENGINEER Wildfire野火版5.0引进的。2选择正确的类型来分析诸如结构，热等等，摩擦接触应力分析是适当的选择。这是一个静态结构分析的类型。具有直齿圆柱齿轮的行星齿轮箱中的应力是平面压力。一个可以选择的平面应力分析，当正常的应力和剪切应力，垂直于该平面时假定为零。平面应力分析的主要优点是它需要同一个域中的计算资源非常少。在这个问题上的平面应力分析，可以选择的事例如下：（）沿旋转轴线的直齿圆柱齿轮的几何形状是均匀的，有没有轴向加载。 （ii）该组件的错误不是模仿了这项研究。 （）载波中产生的应力被忽略因为这里研究的重点是齿轮啮合产生的接触应力。3选择适当的用于定义各种性质的单元系统。4定义材料的特性是必要的，以解决该问题。在这里，齿轮材料为16MnCr5，在此分析中所需的材料属性仅有弹性模量，泊松比和密度。16MnCr5的弹性模量为206 GPA，泊松比是0.3，密度是7870 kg/m3。5生成新的坐标系是必要的。6如果这个问题是非线性问题，我们需要定义接触类型和接触的配置方法。接触的类型决定接触体如何相对于彼此移动。在ANSYS Workbench中的连接类型有键，未分离，无摩擦，粗糙和摩擦。选择适当的接触方式取决于一个人试图解决的问题的类型。如果一个非常接近接触界面的应力是重要的不可忽略的，那么把它当作非线性接触类型来对待。在此分析中用到了无摩擦和有摩擦接触类型。然而，使用非线性接触类型通常会导致较长的解决方案时间，并且有可能涉及到非线性接触而导致的收敛性问题。在这里将用到拉格朗日增益。这是分析非线性摩擦接触类型最合适的方法。 图1 相关的定义7使用不同的网格控制参数生成合适的网格。 图2 网格生成选项在我们感兴趣的区域内的网格必须是极其精细的（指的是位于危险界面上的应力），而剩余区域的网格应该是比较粗的。这种粗细组合的网格，充分利用计算资源进而给出了准确的结果。如图2所示为一组选项参数。图3所示为生成的网格。图3 生成的自动网8应用真正代表实际负载条件的边界条件。边界条件指的是边界上结构的外部负载。如果一个人在模拟实际边界条件的过程中犯了错，软件会给错误的结果。因此，决定边界条件是非常关键的。在这里，4个边界条件应用到几何模型（图4中所示）中，（i）太阳轮中心上的摩擦，这意味着，太阳轮是绕它的轴线自由旋转。 （ii）固定内齿圈，这意味着，内齿圈固定。 （iii）行星轮中心的摩擦，这意味着，行星齿轮是自由的，绕它的轴旋转，但行星架是固定的，从而产生转矩。 （）扭矩被施加在太阳轮上，它意味着变速箱的输出功率。图4 边界条件9使用适当的解决方法解决有限元分析模型。10使用不同的显示器分析结果。七 结果 这项研究旨在调查摩擦系数对接触应力的影响。首先分析没有摩擦对接触应力的影响，并与ISO的结果进行了比较，验证的结果在ANSYS Workbench中。在表II中将ISO标准与ANSYS Workbench所获得的结果进行了比较。表三：