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附录多级变速箱的优化设计传动部件的建模 教授博士博伊达尔ROSI，DR亚历山大马林科维奇,vencl先生摘要: 利用优化设计的齿轮传动，能够确定最佳的齿轮传动参数之间的关系，以及各传输阶段区别。本文提出了齿轮传动的优化和多目标优化程序一一标准程序对于每个传输阶段。本文的第二部分是集中在圆柱齿轮，常用的机械零件，齿轮传动的主要零件。这些模型是用零件装配设计在CATIA软件模块v5r11。在有限元分析模型研究someapplications完成了优化。关键字：优化设计，常用的机械零件，计算机辅助设计，齿轮建模，CATIA1.简介从优化与决策理论的概念可以知道所有阶段的设计中的重要作用过程。优化设计的理论和应用方法将展示出一个多级变速箱的例子。变速箱是机器的一个重要组成成员，广泛应用于工程领域，它必须满足非常严格的技术要求，可靠性，效率，精密制造的齿轮，轴承等。精密的测试制造齿轮、轴承等该领域的最新成果技术已应用于制造过程。计算机技术的发展，与相应的计算机程序（AutoCAD，Solid Works，CATIA，等），有很多专家发现在高技术水平的地方减速器设计系统发展很快。因此，可以说，目前，变速箱的设计不再是一个“日常工作”，大多数情况下，基于设计者的经验和知识。本文演示应用非线性多目标优化方法，专家以当一个模块在变速箱设计系统为目的建立这样一个强大的方法。简介一些标准，考虑到的理想性能，结合高质量齿轮箱部件模型对一个火车齿轮模型实现一个重要步骤。2.变速箱分解 变速箱复杂的机械系统可以分解成相应的交互齿轮数。这意味着对于多级变速箱的程序优化也可以通过相应数量进行的阶段。在第一个优化阶段，特点是比较小的变量数，每箱传动比的分配的阶段是在定义的条件该齿轮体积最小。在第二阶段，多目标优化问题的求解通过引入更多的标准，表示变速箱的基本性能。因此，必须满足以下几个方面的限制：负荷分布，应力，运动学正确的共轭齿轮。多级变速箱的目标函数表示该齿轮组的体积，可以写在表格下面的关系 1 ：f(x) = 0.25d13jI(1+uI2)+jIId32/d12jI(1+uII2)+.) (1)注释：Ul，UII特定的传动比多级齿轮传动的阶段；D1，D3直径，主动齿轮；J=B/D1主动的齿轮直径宽度比驱动齿轮的运动学圈。对目标函数的声明，它也是必要的从功能限制的角度定义的第一阶段齿轮的表面强度，这可以写在下面的表格： G（x）=Z(2KT1)/d13 (U1+1)/U1SH1/SH (2)而且，从强度角度量：g(x)=KY(2T1)/(1d12m1) f1/SF(3)在完全类似的情况下，功能限制从传输阶段的表面的体积力确定了变速箱。从开始的对变速器传动比的技术要求，它也要确定在功能上的限制方程形式：h1(x)=u-u1u2u3u(n)=0 (4) 根据确定的目标函数和的限制，可以注意到这个问题属于非线性优化领域的不等式形式的限制。为解决这个问题的方案，计算机程序SUMT法，基于混合罚函数，已经被应用。图1显示结果的图形表示计算机程序SUMT。基于截面相应的功能的域对于多级最佳传动比变速箱是通过以下方式定义：图1：轮系总传动比和体积之间的关系完成此分析分解变速箱，这是增加了一个对的形式的限制不等式，基于应力的限制：-I级齿轮齿应力-II级齿轮齿应力基于齿轮应力关系价值的齿轮模块法：-对接触应力对弯曲应力图2显示的图形解释的关系（7）和（8）的齿数Z1功能。上两个在图2线路提出了齿轮模数值并对接触应力和较低的值的齿根应力的测定。线和在图2中可容空间。表明，接触应力与齿轮模数的关系（7）优先，是用于齿轮的尺寸测定。齿数Z1 图2：模数值与齿数图3.齿轮建模齿轮在今天是非常重要的机械零件，普遍使用在不同类型的变速箱和传动装置。特别是圆柱形的齿轮是最适用的，具有很高的传动效率和简单的生产。圆柱齿轮建模是非常重要的设计过程，为使齿轮箱实际模型，如齿轮和传动结构的分析优化。去年，这个过程可以很快速定性使用新的软件工具，如CATIA。这个软件是很复杂的，但一些主要模块部分的设计和装配设计中采用圆柱齿轮建模。主要问题是任何齿轮建模是定义一个真正的齿之后，将其导入到齿轮体的制备。圆柱形的齿轮建模包括几个阶段，取决于齿轮体及其生产方式：齿轮建模过程的第一阶段是定义渐开线齿轮齿廓。第二阶段，在切割或挤压齿轮箱体，部分是利用CATIA模块进行设计齿轮机构。第三阶段，只有在焊接方法齿轮本体，是将所有部件使用组件设计模块。所有这些阶段包括多种经营它将分别描述在后续的章节。每一章的主要内容为一般建模，使用一些具有特殊的操作CATIA软件在齿轮建模与实例不同的圆柱齿轮建模。内齿轮齿廓曲线有一节四个不同的线和外部环境分析，明确了齿轮的完整轮廓。所以有渐开线齿廓圆弧，圆弧形脚，齿顶圆弧摆线弧作为连接 4 。在分析运动学方法对轮廓定义是定义一个很大的限制设置和约束方程的参数轮廓圆弧角。经过矩阵变换接触线啮合矩阵参数方程齿轮的齿廓曲线可以确定。基于此解析运动学模型的计算机程序来定义齿轮廓 5 点。齿轮建模是非常有用的，使现实的齿轮传动的仿真，所以有很多其他的分析。不同的软件工具在今天使用的机械设计和机械零件的造型，如ACAD，机械工程师和最后的桌面，亲年工程制图，CATIA等，但可以看出齿轮建模（特别是内齿轮）与真实分布比较复杂的建模所有其他的机械元件。这里将介绍圆柱齿轮建模使用CATIA的可能性v5r11软件。取决于生产方式和齿轮体形式可以使用设计模块或部分CATIA软件组件设计。对于简单的圆柱齿轮设计（平）第一步是定义正确的草图，在渐开线齿形坐标（从第一阶段）应进口。后设计师可以将基于草图的特征（创建垫），把切齿轮模型是在图3所示。 图3：圆柱齿轮的简单模型向前一步是设计一个机齿轮体，可以通过旋转skatch使建模图，或是生产过程的模拟。在图4给出了一种齿轮模型也采用skatch很少基于草图的，打扮和转化特征。介绍了齿轮使用中常见的，他们有一个外部的渐开线齿廓。但在某些情况下，如行星齿轮传动设计，有必要做一个模型内部的异形齿轮。为此，设计师必须计算渐开线齿廓坐标的新表，由使用外部齿轮作为一种工具，使内部轮廓。正确的素描和其他特征，其他圆柱齿轮建模已被使用。图4：圆柱齿轮模型制造装配设计是另一个模块设计，这是在完成所有零件和标准已经在部分或形状建模元素的模块设计。此外，它是可能的插入新的在现有的组件和身体也做布尔操作之间的身体如果它是必要的。这些之间的布尔运算体组装体，相交体，等。使用装配设计的最佳样本圆柱齿轮的焊接数量的分离元素。这意味着这种类型的齿轮组成以许多元素为蓝本，零件设计的主要部件为渐开线齿形，外板焊接中央筒板和两侧两圈六加强筋（图5）。图5：焊接的圆柱齿轮齿轮建模是非常重要的因为许多应用程序可以做：完成组装后可以做运动学仿真，使用另一个CATIA模块单元。内外齿轮的模型可用于解决很多机械工程的问题，如结构分析，相应的齿轮之间接触压力和许多其他分析 8 。一个典型的这可能是以下结构为例采用有限元法进行分析，其中图6显示了齿轮模型77633这使得18965节点四面体。图7：在有限元网形齿轮模型应力值（图7）是在高负荷的齿轮重要的建设性的观点可以在设计和优化过程和程序也非常有用。图7：利用应力值加载齿轮模型计算及其结构分析4.结论 本文是一个简短的说明，更广泛的研究与建设强大的多目标优化方法引入变速箱设计专家系统。指出了分解的多级变速箱复杂机械系统的必要性。在此过程中，变速箱优化程序也通过了相应数量的阶段。在这第一次的优化阶段，实际的域应用齿轮箱的定义，然而，在第二阶段，多目标优化解决了问题。从这里介绍建模部分，可以说，它只提出一个简短的圆柱齿轮在CATIA的可能性模型软件。除了建模部分介绍和装配设计模块，在本文结束添加CATIA强大的今天可能的最完整的设计软件在工程设计中具有广泛的应用范围。V International Scientific Conference Heavy Machinery-HM05 Kraljevo, 28. June - 3. July 2005 I C.17 Faculty of Mechanical Engineering Belgrade, Kraljice Marije 16, 11120 Belgrade, Serbia, E-mail: amarinkovicmas.bg.ac.yu OPTIMUM DESIGN OF MULTISPEED GEARBOXES AND MODELING OF TRANSMISSION COMPONENTS Prof. dr Boidar Rosi, dr Aleksandar Marinkovic, mr Aleksandar Vencl Abstract: By applying the optimum design in the field of gear transmission design it is possible to define the optimal relations between the parameters of the complete gear transmission, and of each transmission stage separately. This paper presents a one criterion procedure for gear transmission optimization and multicriterion optimization procedure for each transmission stage. Second part of the paper is focused on modeling of cylindrical gears that are common used machine elements and main parts of gear transmissions. These models are made using part and assembly design module in CATIA V5R11 software. On the end of paper some applications of models in finite elements analysis and optimization are also described Keywords: optimum design, multistage gearbox, computer added design, gears modeling, CATIA 1. Introduction Concept from optimization and decision theory can play an important role in all stages the design process. The optimizing design theory applying and methodology will be illustrated on a multispeed gearbox example. Gearboxes present a very important group of machine members, which are utilized in a great number of engineering fields and which must satisfy very rigorous technical requirements regarding reliability, efficiency, precise manufacturing of gears, bearing, etc. In addition, the latest achievements in the fields of technology and testing of the preciseness of manufacturing gears, bearings, etc., have been applied to the manufacturing process. The development of the computer technology, together with the corresponding computer programs (AutoCAD, Solid Works, CATIA, etc.), have very quickly found their place in the development of the expert system for gearbox design at a high technical level. Thus, it can freely be said nowadays that the gearbox design is no longer a “routine job”, which in most cases based upon the designers experience and knowledge. This paper demonstrates the application of a nonlinear multicriteria optimization method, with the purpose to build such a powerful method as a module into the gearbox design expert system. The introduction of some criteria considering the desirable performances, combined with high quality gearbox component modeling represents a significant step towards the reality of a gear train model. 2. Gearbox decomposition Gearboxes represent complex mechanical systems that can be decomposed into the corresponding number of gears with corresponding interaction. This means that the procedure for multistage gearbox optimization can also be carried out through the corresponding number of stages. During the first optimization stage, characterized by comparatively small number variables, the distribution of transmission ratio per gearbox stages is defined from the conditions of the minimal volume of the gear sets. During the second stage, the multicriteria optimization problem is solved by introducing a greater number of criteria which represent the essential gearbox performances. Thereby, it is necessary to satisfy the restrictions from the following aspects: load distribution, stresses, kinematics and correct conjugate gear action. The target function for multistage gearbox representing the volume of the gear sets can be written in the form the following relation 1: f(x) = 0.25d13jI(1+uI2)+jIId32/d12jI(1+uII2)+.) (1) where: uI, uII the transmission ration for particular transmission stages of multistage gearbox; d1, d3 diameters of kinematics circles of the driver gears; j=b/d1 ratio of width of the gear and diameter of the driver gear kinematics circle. For the target function stated, it is also necessary to define the functional restrictions from the standpoint of the surface strength for the first stage of gearing, which can be written in the following form: 1 3 1 21 ( ) H II IH s K Tu g xZ duS + = (2) and, from the standpoint of the volume strength: 1 2 1 2 ( ) F I IIF T gxK Y dmS = (3) I C.18 In the exactly analogous way, the functional restrictions from the standpoint of the surface and volume strength for other transmission stages of gearboxes are determined. Commencing from the technical requirement concerning the transmission ratio of a gearbox, it is also necessary to determine the functional restriction in the form of the equation: ( ).0 IIIIN h xuuuu= (4) Basing upon the determined target function and the restrictions, it can be noticed that this problem belongs to the field of nonlinear optimization with the restrictions in the form of inequalities. For the solution of this problem, the computer program SUMT, based on the mixed penalty functions, has been applied. Fig. 1 shows a graphic representation of the results of the computer program SUMT. Basing upon the section of the corresponding functions, the domains of the optimum transmission ratios for the multistage gearboxes are defined in the following way: Figure.1: The relation between the volume of gear train and overall gear ratio. To complete this analysis of decomposed gearbox, here are added a pair of restrictions in the form of inequalities, based on stress restrictions: - tooth gear stress for I stage gear F IF II F Smd T YKxg = 2 1 1 13 2 )( (5) - tooth-root gear stress for II stage gear F IIF IIII I F Smd uT YKxg = 2 3 1 34 2 )( (6) Based on gear stress relations the value of gear module is determinated: - for contact stress () 3 1 3 1 2 2 2 1 12 + zu SZuTK m IIHI HI (7) - for tooth-roth stress 3 1 2 1 1 2 IFI F z SYTK m (8) Fig. 2 shows graphical interpretation of relations (7) and (8) in function of tooth number Z1. Upper of two lines on the Fig. 2 presents values of gear module determinated on contact stress and lower one for values determinated on tooth-root stress. The lines and admissible space on Fig. 2. indicate that contact stress relation for gear module (7) is prior and is to be used for gear dimensions dermination. admissible space Tooth number Z1 Figure. 2: Diagram of module values up to tooth number 3. Gears modeling Gears are very important machine elements today and they are common used in different kinds of gearboxes and transmissions. Especially cylindrical gears are most applicable because of their very high efficiency and not complicated production. Modeling of cylindrical gears is very important process in machine design, as for making real model of gearbox, such for gear and transmission structure analysis and optimization. Last years this process can be done very fast and qualitative using new software tools such as CATIA. This software is very complex, but some main modules like Part design and Assembly design are in use for cylindrical gear modeling. The main problem in any gears modeling is to define a real gear tooth and after that to import it into gear body making. Cylindrical gears modeling consists of several phases, depends from gear body and kind of its production: The first phase of gear modeling is definition and making real involute gear teeth profile. The second phase, in case of cutting or pressed gear body, is to use Part design CATIA module to make gear body. The third phase, only in case of welding way made gear body, is to use Assembly design module to connect all its parts. All this phases consists of several operations and it will be described separately in followed chapters. Volume of material used for gears 300 250 200 150 100 50 0 0 5 10 1520 25 30 V/cm3 u I stage II stage III stage Overall gear ratio 2.00 4.00 0.00 15171921 23 25 27 Values of gear module m I C.19 Every chapter gives principal facts of general modeling, some special operations with advantages of using CATIA software in gears modeling and examples of different cylindrical gears that are modeled. In analysis of internal and external gear profiles there are four different lines in one pitch, which defines complete profile of gear. So there are the involute profile arc, profile foot circle arc, addendum circle arc and trohoide arc as a connection 4. In analytic-kinematics way for profile definition is to define a lot of restrictions and constrains for setting parameter equations each of this profile arcs and angles. After some matrix transformations matrix parameter equation for contact line of engaged gear tooth profiles can be determinated. Based on this analytic-kinematics model computer program is developed to define points of gear profiles 5. Gears modelling is very useful and important, as to make real gear transmission simulation, so for lot of other analysis. Different software tools are in use today for machine design and machine elements modelling, as ACAD, Mechanical Desktop, Pro Engineer and last years Solid Works, CATIA etc. But it can be seen that gear modelling (especially internal gears) with real profiles is more complicated compared with modelling of all other machine elements. Here will be presented the possibilities of cylindrical gears modelling using CATIA V5R11 software. Depends of production way and form of gear body it is possible to use Part design module or Assembly design module of CATIA software. For designing simplest cylindrical gear (flat) first step is to define correct sketch, where involute profile tooth coordinates (from first phase) should be imported. After that designer can apply Sketch based features (Create pad), to get cuted gear model as is shown at Fig. 3. Figure 3: Simplest model of cylindrical gear One step forward is designing a press made gear body, that could be modeled by rotating skatch made figure, or like simulation of production process. On Fig. 4 it is given a gear model made also by using skatch and few Sketch-Based, Dress-Up and Transformation Features. Presented gears are common in use and they have an external involute profile. But in some cases, like planetary gear train designing, it is necessary to make a model of internal profiled gear. For this purpose designer has to calculate a new table with involute profile coordinates, by using external gear as a tool for making internal profile. After that properly sketch and other features as for other cylindrical gears modelling has to be used. Figure 4: Press made model of cylindrical gear Assembly design is another module in CATIA which is in use in aim to complete all parts and standard elements that are already modelled in Part or Shape design modules. Besides that it is possible to insert new bodies in existing assembly and also to do Boolean Operations between bodies if it is necessary. These Boolean operations between bodies are Assemble Bodies, Intersect Bodies, Add Bodies, Remove Bodies, Trim Bodies, Remove Lumps, etc. The best sample of using Assembly design is cylindrical gear made by welding number of separated elements. It means that this type of gear consists of many elements that are modelled in Part design. The main part is outer plate with involute profiles that are welded with central cylinder with two circle plates and six stiffeners at both sides (Fig. 5). Figure 5: Cylindrical gear made by welding I C.20 A gear modeling is very significant because of many applications that could be done: After completing assembly it is possible to do kinematics simulations, using another CATIA module DMU. Internal and external gears models can be used for solving a lot of problems in mechanical engineering, such as structural analysis, contact pressure between corresponding gears and also thermal and many other analyses 8. A typical example for this could be following structural analysis made using finite element method, where Fig. 6 shows gear model made of 77633 tetrahedrons which makes 18965 nodes. Figure 7: Gear model in form of finite element net Stress values (Fig. 7) represent critical constructive points where gear is high loaded which could be also very useful in design and optimization process and procedure. Figure 7: Stress values of loaded gear model calculated in structural analysis 4. Conclusion The paper represents a brief illustration of a wider study undertaken with the aim of building the powerful multicriteria optimization methods into the expert system for gearbox design. It points out the necessity of decomposition multistage gearboxes as complex mechanical systems. In the way, the gearboxes optimization procedure is also carried out through the corresponding number of stages. In this first optimization stage, the domains of the practical application of gearboxes are defined, whereas, during the second stage, the multicrit