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OPTIMUM DESIGN OF MULTISPEED GEARBOXES AND
MODELING OF TRANSMISSION COMPONENTS
Prof. dr Bo?idar 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 (Auto CAD, 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 designer’s 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) = 0.25πd13jI((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:
G（x）=Z·[(2·K·T1)/d13] ·(U1+1)/U1≤[SH]1/SH (2)
and, from the standpoint of the volume strength:
g``(x)=K·Y·(2T1)/(Ψ1·d12·m1) ≤[Φf]1/SF (3)
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:
h1(x)=u-u1·u2·u3·…·u(n)=0 (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
- tooth-root gear stress for II stage gear
Based on gear stress relations the value of gear module is determinated:
- for contact stress
- for contact stress
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.
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. 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 trochoid 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 de-terminated. Based on this analytic-kinematics model computer program is developed to define points of gear profiles [5].
Gears modeling 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 modeling, as ACAD, Mechanical Desktop, Pro Engineer and last years Solid Works, CATIA etc.But it can be seen that gear modeling (especially internal gears) with real profiles is more complicated compared with modeling of all other machine elements. Here will be presented the possibilities of cylindrical gears modeling 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 scatch made figure, or like simulation of production process. On Fig. 4 it is given a gear model made also by using scatch 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 modeling 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 modeled 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 modeled 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
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 multicriteria optimization problem is solved.
To resume the point of this modeling part of paper, here could be said that it presents only a brief of cylindrical gears modeling possibilities in CATIA software. Besides presentation of modeling in Part and Assembly design modules, at the end of this paper it is to add that CATIA is powerful and today may by completest design software in engineering with wide range of applications.
References：
[1] Rosi?B, 1993.: Parameter Investigation and Optimization of Planetary Gear Train Transmission, Ph.D Thesis, Mechanical Engineering Faculty, University of Belgrade
[2] Arora J.S, 1989.: Introduction to optimum design, McGraw–Hill Book Company, New York
[3] Rosi?B., Marinkovi?A.: Planetary gear transmission as a tribosystem: Efficiency calculation and simulation, ?TG Jahres Symposium, Wien, November 2003.
[4] Colbourne, J. R., 1987: The geometry of Involute gears, Springer-Verlag, New York
[5] Rosi?B., Rinkovec B., Marinkovi?A., Pavlovi?N.: The analytical-kinematics method for definition of internal cylindric gears,Yugoslav Conference “IRMES 2002”, Faculty of Mechanical Engineering Srpsko Sarajevo, Jahorina – BIH, September 2002, Proceedings, pp. 625-630.
[6] Rosi?B.: Planetary gear trains, Monography, Faculty of Mechanical Engineering, University of Belgrade, edited in year 2003.
[7] Rosi?B., Marinkovi?A., Vencl А, 2004.: Cylindrical Gears modeling using CATIA software,4th International Conference “RADMI 04“, Zlatibor, Serbia and Montenegro, August- September 2004., Proceedings on CD, pp. 73-77.
[8] Rosi?B., Marinkovi?A., Vencl А, 2004.: Modeling and Structural Optimization of Cylindrical Gears construction profiles,Yugoslav Conference “IRMES 04”, Faculty of Mechanical Engineering Kragujevac, Kragujevac, September 2004, Proceedings, pp. 173-178.
英文文獻(xiàn)中文翻譯
多速變速箱的優(yōu)化設(shè)計(jì)傳動(dòng)部件的建模
博日達(dá)爾·洛賽克教授，亞歷山大·馬林科維奇博士，亞歷山大溫瑟主席
摘要: 通過應(yīng)用優(yōu)化設(shè)計(jì)中的齒輪傳動(dòng)裝置設(shè)計(jì)領(lǐng)域，可以分別定義每個(gè)傳輸級(jí)的完整齒輪傳動(dòng)裝置的參數(shù)之間的最佳關(guān)系，并且。本文提出了一個(gè)標(biāo)準(zhǔn)程序，齒輪傳動(dòng)優(yōu)化和多標(biāo)準(zhǔn)優(yōu)化程序?yàn)槊總€(gè)傳輸階段。紙張的第二部分被集中在圓柱齒輪是常用的機(jī)械元件和齒輪傳動(dòng)裝置的主要部分的建模。這些模型使用的部分，并在CATIA V5R11軟件的裝配設(shè)計(jì)模塊進(jìn)行。對(duì)底紙的有限元分析和優(yōu)化模型的一些應(yīng)用程序也有所說明。
關(guān)鍵字：多級(jí)變速箱的優(yōu)化設(shè)計(jì)，計(jì)算機(jī)輔助設(shè)計(jì)，齒輪，建模，CATIA
1、簡(jiǎn)介
優(yōu)化與決策理論的概念是所有階段的設(shè)計(jì)中的一個(gè)重要的過程。將用一個(gè)多速變速箱來舉例說明優(yōu)化設(shè)計(jì)的理論和應(yīng)用方法。變速箱是重要組機(jī)器部件，它涉及了大量工程研究領(lǐng)域，同時(shí)它必須滿足非常嚴(yán)格的技術(shù)要求以達(dá)到其可靠性，效率，精密制造的齒輪，軸承等，此外，在該領(lǐng)域的最新研究技術(shù)和嚴(yán)謹(jǐn)?shù)臏y(cè)試制造已應(yīng)用于齒輪，軸承等的制造過程。
計(jì)算機(jī)技術(shù)的發(fā)展，與相應(yīng)的計(jì)算機(jī)程序（Auto CAD，Solid Works，CATIA，等），很快被發(fā)并用于研發(fā)高級(jí)的減速器設(shè)計(jì)系統(tǒng)。因此，變速箱的設(shè)計(jì)不再被看做是一個(gè)“日常工作”，而是設(shè)計(jì)者經(jīng)驗(yàn)和知識(shí)的結(jié)晶。
本文演示的應(yīng)用非線性多目標(biāo)優(yōu)化方法，以目的建立這樣一個(gè)強(qiáng)大的方法，當(dāng)一個(gè)模塊在變速箱設(shè)計(jì)專家系統(tǒng)。簡(jiǎn)介一些標(biāo)準(zhǔn)考慮到理想的性能，結(jié)合高質(zhì)量齒輪箱部件模型是對(duì)現(xiàn)實(shí)的一個(gè)重要步驟一個(gè)齒輪火車模型。
1． 變速箱分解
變速箱表示復(fù)雜的機(jī)械系統(tǒng)可以分解成相應(yīng)的與相應(yīng)的交互齒輪數(shù)。這
意味著對(duì)于多級(jí)變速箱的程序優(yōu)化也可以通過進(jìn)行相應(yīng)數(shù)量的階段。在第一個(gè)
優(yōu)化階段，特點(diǎn)是比較小的變量數(shù)，傳動(dòng)比的分配每箱的階段是在定義的條件
該齒輪體積最小集。在第二階段，多目標(biāo)優(yōu)化問題的求解通過引入更多的標(biāo)準(zhǔn)
表示的基本的變速箱性能。從而，必須滿足的限制以下幾個(gè)方面：負(fù)荷分布，應(yīng)力，運(yùn)動(dòng)學(xué)正確的共軛齒輪的行動(dòng)。
多級(jí)變速箱的目標(biāo)函數(shù)表示該齒輪組的體積可以寫在表格下面的關(guān)系[ 1 ]：
f(x) = 0.25πd13jI((1+uI2)+jIId32/d12jI(1+uII2)+...) (1)
注釋：
Ul，UII–特定的傳動(dòng)比多級(jí)齒輪傳動(dòng)的階段；D1，D3–直徑的司機(jī)，運(yùn)動(dòng)學(xué)界
齒輪；J?=?B?/?D1的齒輪直徑寬度–比驅(qū)動(dòng)齒輪的運(yùn)動(dòng)學(xué)圈。
對(duì)目標(biāo)函數(shù)的聲明，它也是必要的從的角度定義的功能限制第一階段為齒輪的表面強(qiáng)度，這可以寫在下面的表格：
G（x）=Z·[(2·K·T1)/d13] ·(U1+1)/U1≤[SH]1/SH (2)
而且，從強(qiáng)度角度量：
g``(x)=K·Y·(2T1)/(Ψ1·d12·m1) ≤[Φf]1/SF (3)
在完全類似的方式，功能限制從表面的立場(chǎng)其他的傳輸階段的體積力確定了變速箱。
從開始的技術(shù)要求對(duì)變速器傳動(dòng)比，它也要確定在功能上的限制該方程形式：
h1(x)=u-u1·u2·u3·…·u(n)=0 (4)
根據(jù)確定的目標(biāo)函數(shù)和的限制，可以注意到這個(gè)問題屬于非線性優(yōu)化領(lǐng)域的
不等式的形式的限制。為解決方案這個(gè)問題，計(jì)算機(jī)程序SUMT法，基于混合罰函數(shù)，已經(jīng)被應(yīng)用。圖1顯示的結(jié)果的圖形表示計(jì)算機(jī)程序SUMT。基于截面相應(yīng)的功能，的域?qū)τ诙嗉?jí)最佳傳動(dòng)比變速箱是通過以下方式定義：
圖。1：輪系總傳動(dòng)比和體積之間的關(guān)系
完成此分析分解變速箱，這是增加了一個(gè)對(duì)的形式的限制不等式，基于應(yīng)力的限制：-I級(jí)齒輪齒應(yīng)力
-II級(jí)齒輪齒應(yīng)力
基于齒輪應(yīng)力關(guān)系價(jià)值的齒輪模塊法：--對(duì)接觸應(yīng)力
-齒羅斯應(yīng)力
圖2顯示的圖形解釋的關(guān)系（7）和（8）的齒數(shù)Z1功能。上兩個(gè)在圖2線路提出了齒輪模數(shù)值并對(duì)接觸應(yīng)力和較低的值對(duì)齒根應(yīng)力的測(cè)定。線和在圖2中可容空間。表明，接觸應(yīng)力齒輪模塊的關(guān)系（7）優(yōu)先，是用于齒輪的尺寸測(cè)定。
齒數(shù)Z1
圖。2：模塊值達(dá)齒數(shù)圖
3。齒輪建模
在今天它是非常重要的機(jī)械零件，齒輪他們是普遍使用的不同類型的變速箱和傳動(dòng)裝置。特別是圓柱形的齒輪是最適用的具有很高的效率和不復(fù)雜的生產(chǎn)。建模圓柱齒輪在機(jī)非常重要的過程設(shè)計(jì)，為使齒輪箱實(shí)際模型，如齒輪和傳動(dòng)結(jié)構(gòu)的分析優(yōu)化。去年的這個(gè)過程可以很快速定性使用新的軟件工具，如CATIA。這個(gè)軟件是很復(fù)雜的，但一些主要模塊部分的設(shè)計(jì)和裝配設(shè)計(jì)中采用圓柱齒輪建模。主要問題任何齒輪建模是定義一個(gè)真正的齒和之后，將其導(dǎo)入到齒輪體的制備。圓柱形的齒輪建模包括幾個(gè)階段，取決于齒輪體及其生產(chǎn)方式：
齒輪建模過程的第一階段是定義和真正的漸開線齒輪齒廓。
第二階段，在切割或擠壓齒輪箱體，部分是利用CATIA模塊進(jìn)行設(shè)計(jì)齒輪機(jī)構(gòu)。
第三階段，只有在焊接方法齒輪本體，是使用組件設(shè)計(jì)模塊將所有部件。
所有這些階段包括多種經(jīng)營(yíng)它將分別描述在后續(xù)的章節(jié)。每一章的主要事實(shí)的一般建模，具有使用一些特殊的操作CATIA軟件在齒輪建模與實(shí)例不同的圓柱齒輪，建模。
內(nèi)齒輪齒廓曲線有一節(jié)四個(gè)不同的線和外部環(huán)境分析，明確了齒輪的完整輪廓。所以有漸開線齒廓圓弧，圓弧形腳，齒頂圓弧擺線弧作為連接[ 4 ]。在分析運(yùn)動(dòng)學(xué)方法對(duì)輪廓定義是定義一個(gè)很大的限制設(shè)置和約束方程的參數(shù)輪廓圓弧角。經(jīng)過矩陣變換接觸線嚙合矩陣參數(shù)方程齒輪的齒廓曲線可以確定?；诖私馕鲞\(yùn)動(dòng)學(xué)模型的計(jì)算機(jī)程序來定義齒輪廓[ 5 ]點(diǎn)。
齒輪建模是非常有用和重要，使真正的齒輪傳動(dòng)的仿真，所以很多其他的分析。不同的軟件工具在今天使用的機(jī)械設(shè)計(jì)和機(jī)械零件的造型，如ACAD，機(jī)械工程師和最后的桌面，親年工程制圖，CATIA等，但可以看出齒輪建模（特別是內(nèi)部齒輪）與真實(shí)分布比較復(fù)雜的建模所有其他的機(jī)械元件。這里將介紹圓柱齒輪建模使用CATIA的可能性v5r11軟件。取決于生產(chǎn)方式和形式齒輪體可以使用設(shè)計(jì)模塊或部分CATIA軟件組件設(shè)計(jì)。
對(duì)于簡(jiǎn)單的圓柱齒輪設(shè)計(jì)（平）第一步是定義正確的草圖，在漸開線齒形
坐標(biāo)（從第一階段）應(yīng)進(jìn)口。后設(shè)計(jì)師可以將基于草圖的特征（創(chuàng)建墊），把切齒輪模型是在圖3所示。
圖3：圓柱齒輪的簡(jiǎn)單模型
向前一步是設(shè)計(jì)一個(gè)機(jī)齒輪體，可以通過旋轉(zhuǎn)命令建模圖，或是生產(chǎn)過程的模擬。在圖4給出了一種齒輪模型采用很少基于草圖的，打扮和轉(zhuǎn)化特征。介紹了齒輪使用中常見的，他們有一個(gè)外部的漸開線齒廓。但在某些情況下，如行星齒輪傳動(dòng)設(shè)計(jì)，有必要做一個(gè)模型內(nèi)部的異形齒輪。為此，設(shè)計(jì)師必須漸開線齒廓坐標(biāo)計(jì)算的新表，由使用外部齒輪作為一種工具，使內(nèi)部輪廓。正確的素描和其他特征，其他后圓柱齒輪建模已被使用。
圖4：圓柱齒輪模型制造
裝配設(shè)計(jì)是另一個(gè)模塊設(shè)計(jì)這是在目標(biāo)完成所有零件和標(biāo)準(zhǔn)已經(jīng)在部分或形狀建模元素模塊設(shè)計(jì)。此外，它是可能的插入新的在現(xiàn)有的組件和身體也做布爾操作之間的身體如果它是必要的。這些之間的布爾運(yùn)算體組裝體，相交體，等。
使用裝配設(shè)計(jì)的最佳樣本圓柱齒輪的焊接數(shù)量的分離元素。這意味著這種類型的齒輪組成許多元素為藍(lán)本，在零件設(shè)計(jì)。的主要部件為漸開線齒形，外板焊接中央筒板和兩圈六加強(qiáng)筋的兩側(cè)（圖5）。
圖5：焊接的圓柱齒輪
齒輪建模是非常重要的因?yàn)樵S多應(yīng)用程序可以做：
完成組裝后可以做運(yùn)動(dòng)學(xué)仿真，使用另一個(gè)CATIA模塊單元。
內(nèi)外齒輪的模型可用于解決很多問題在機(jī)械工程，如結(jié)構(gòu)分析，接觸壓力相應(yīng)的齒輪之間還熱和許多其他分析[ 8 ]。一個(gè)典型的這可能是以下結(jié)構(gòu)為例采用有限元法進(jìn)行分析，其中圖6顯示了齒輪模型77633這使得18965節(jié)點(diǎn)四面體。
圖7：在有限元網(wǎng)形齒輪模型
應(yīng)力值（圖7）是重要的在高負(fù)荷的齒輪是建設(shè)性的觀點(diǎn)可以在設(shè)計(jì)和優(yōu)化也非常有用過程和程序。
圖7：應(yīng)力值加載齒輪模型計(jì)算及其利用結(jié)構(gòu)分析
4.結(jié)論
本文是一個(gè)簡(jiǎn)短的說明一個(gè)更廣泛的研究與建設(shè)的目的強(qiáng)大的多目標(biāo)優(yōu)化方法引入變速箱設(shè)計(jì)專家系統(tǒng)。指出了分解的多級(jí)變速箱的必要性復(fù)雜機(jī)械系統(tǒng)。在路上，變速箱優(yōu)化程序也通過了相應(yīng)數(shù)量的階段。在這第一次的優(yōu)化階段，實(shí)際的域應(yīng)用齒輪箱的定義，然而，在第二階段，多目標(biāo)優(yōu)化問題解決了。
從這里恢復(fù)本建模部分點(diǎn)，可以說，它只提出一個(gè)簡(jiǎn)短的圓柱齒輪在CATIA的可能性模型軟件。除了建模部分介紹和裝配設(shè)計(jì)模塊，在本文結(jié)束添加CATIA強(qiáng)大的今天可能的最完整的設(shè)計(jì)軟件在工程設(shè)計(jì)中具有廣泛的應(yīng)用范圍。
參考文獻(xiàn)：
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[ 2 ] Arora J.，1989：優(yōu)化設(shè)計(jì)介紹，麥格勞–希爾圖書公司，紐約
[ 3 ]?ROSI?B，marinkovi?答：行星齒輪傳輸作為一個(gè)摩擦系統(tǒng)：效率計(jì)算和仿真，?TG詞匯研討會(huì)，維也納，十一月2003。
[ 4 ]科爾伯恩，J. R.，1987：漸開線的幾何齒輪，施普林格出版社，紐約
[ 5 ] ROSI?B，B rinkovec，marinkovi?A.，巴甫洛夫.：定義分析運(yùn)動(dòng)學(xué)方法內(nèi)部圓柱齒輪，南斯拉夫會(huì)議“irmes?2002”，機(jī)械工程學(xué)院 薩拉熱窩，亞霍里納–波黑，2002九月，訴訟，第625-630。
[ 6 ] ROSI?B：行星輪系，專論，機(jī)械工程學(xué)院，大學(xué)貝爾格萊德，在2003年的編輯。
[ 7 ] ROSI?B，marinkovi?A，venclА，2004。：圓柱齒輪采用CATIA軟件建模，第四國(guó)際會(huì)議”radmi 04”，茲拉蒂博爾，塞爾維亞和黑山，八月—九月，2004。錄光盤，pp. 73-77.。
[ 8 ]?ROSI?B，marinkovi?A，vencl?。航?，2004與圓柱齒輪的結(jié)構(gòu)優(yōu)化建筑型材，南斯拉夫會(huì)議“irmes?04”，機(jī)械工程學(xué)院克拉古耶瓦茨，克拉古耶瓦茨，九月2004，
訴訟，第173-178。
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