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0 毕 业 设 计 外 文 资 料 翻 译学 院： 机械工程学院 专 业： 机械制造设计及其自动化 姓 名： 学 号： 外文出处： Design and investigation of gear drives with non-circular gears applied for speed variation and generation of functions附 件： 1.外文资料翻译译文；2.外文原文。 指导教师评语：翻译较为通顺、逻辑基本合理、无明显翻译错误、翻译质量较高。签名： 2015年 4月 22日 1 附件 1：外文资料翻译译文研究和设计应用于变速和函数生成的非圆齿轮传动Faydor L. Litvin a， Ignacio Gonzalez-Perez b， Alfonso Fuentes b， Kenichi Hayasaka ca 机械工程系，伊利诺伊大学在芝加哥，美国b机械工程系，卡塔赫纳理工大学，西班牙c 齿轮的研发部门，研发业务，雅马哈电机有限公司，日本摘要：论文介绍了非圆齿轮传动设计和函数生成的非圆齿轮：(a) 通过一个偏心渐开线齿轮的齿轮传动形成非圆齿轮;(b)通过函数生成非圆齿轮。并且用数值例子描述了开发理论。关键词：齿轮的设计 非圆齿轮 用函数生成偏心齿轮1 介绍随着非圆齿轮函数的发展(12，23-28)使生产非圆形齿轮和圆齿轮一样简单。因此非圆齿轮的传动和非圆齿轮的设计成为许多学科专家研究的主题也就不足为奇了1-37。以下是论文的主要内容的：(1) 偏心齿轮传动的设计和生成是通过一个偏心平面和螺旋渐开线齿轮或共轭圆齿轮形成的。偏心渐开线齿轮的主要特征是其旋转中心与几何中心不重合的。螺旋齿轮和直齿齿轮可以用于设计的主动轮传动 (见图 1)，接触轴承需要固定，以及通过小偏心设计，驱动器可以应用在减速器(作为齿轮传动敏感性降低失调)。(2) 函数生成非圆齿轮的应用 (第四节)以下是函数应用的步骤：(1)在其中获得不同信号下函数的导数，(2)瞬心迹是一条不闭合的曲线，(3)生成两对齿轮。在(3)中，这个设计需应用一个函数。是给定的已知函数， 一关于对非圆齿轮传动的函数 2 (3) 椭圆齿轮可以被应用如下的修改设计 (A.1.6 节)：(1)与非对称函数的齿轮进行传动；(2)叶(被修改瞬心迹的椭圆形)。(4) 非圆齿轮的齿面可以通过插齿机床或滚刀生成。有一个简单的方法可以避免非圆齿轮发生齿根圆相切现象。应用一个可以观察并可以检测配合的瞬心迹函数。下面图和图纸数值例子描述了开发理论。图1 偏心渐开线齿轮和共轭非圆齿轮:(a)螺旋驱动 (b)平面驱动；O 1，O 2旋转中心,O 几何中心。2 非圆齿轮和偏心渐开线齿轮形成的齿轮传动2.1 偏心齿轮传动瞬心迹2.1.1 前言下面讨论的齿轮传动(是偏心齿轮传动的缩写)是由一个偏心渐开线齿轮 1和共轭非圆齿轮 2组成。通过设计一个偏心齿轮的传动由直齿螺旋齿轮啮合生成非圆齿轮(图1)。偏心轮传动相比椭圆齿轮传动更具有竞争性，本文提出了滚刀（对于凸瞬心迹）和插齿机（如果瞬心迹是凹凸行）生成非圆齿轮的方法。 3 进行的研究主要涵盖了几何，设计和偏心轮传动。2.1.2 中心配合轨迹方程在图 2中，a 和 b显示的是瞬心迹 r1和 r2在初始和当前的位置，瞬心迹 r1的偏心圆半径在极坐标中表示的形式，如下(图 2b)其中参数 可以确定极轴 相对 的位置(图 2b)。因此我们将使用导数 表示瞬心迹(见附录部分 A.1.2)在 处，瞬心迹的旋转角；参数 可以确定极轴 相对 的位置 (见部分 A.1.2)。我们已知瞬心迹 r2 、r1 是由以下两个方程表示图 2 派生的瞬心迹：(a)初始位置；(b)当前位置。在这里，导函数为 4 其中要确定传递函数 需要进行其的数值积分。瞬心迹 r1是一个已知的封闭的曲线半径希尔(一圈)。瞬心迹 r2可能被观察是一个封闭的曲线形式方程：其中 n是齿的数量，运行的偏心齿轮 1是生成的非圆齿轮之一。瞬心迹 r1和 r2的推导，可以设计一个中心距不变的凸轮机构需要一个偏心圆和一个由函数生成的凸轮。偏心非圆齿轮传动的瞬心线是由凸轮确定的。2.1.3 中心轨迹 r2的曲率和应用 基本的推导。中心轨迹 r2的曲率的必要条件：(1)通过滚刀和插齿机选择生成齿轮的方法；(2)为避免根切齿。考虑到由极坐标曲线 的瞬心迹，我确定其曲率半径(20、27)：极曲线的凸性条件可以表示为或另一种不等式这里，角度 由位置矢量 和切线极曲线 t见27)确定的； 是极轴的曲率曲线。在本文中，考虑到瞬心线的曲率 r1是已知的，测定 2利用欧拉萨瓦公式里的方程（18） ，偏心齿轮传动的中心轨迹的 r1和 r2的曲率与曲率 1和 2的关系（图3）如下： 5 这里当 （图 3）方程式（11）变形保证瞬心迹 r2的是凸性的条件：图 3 推导的曲率半径 瞬心迹 r2。图 4说明了齿轮传动的函数 ：(a)凸瞬心迹 r2；(b)凸凹形 r2。曲率半径的 的位置是可视化的人通过应用一个四连杆的机构（图 3）的链接：（a）链接 1为 O1A1， （b）链接 2为 A1A2， （C）链接 3为 O2A2，和（D）链接 4为 O1O2，当 。 6 从已知的四杆机构运动学（F 点扩展链接 1和 3路口）就是瞬时中心对链接 2相对于 4旋转连接。向量 如果是垂直于它的大小被确定为 O1O2，它的大小被确定： 避免根切。图纸的图 5表示渐开线齿轮齿形角的齿条刀的变化瞬心迹是齿条刀的中线，齿轮的瞬心迹节圆半径 rp。为了避免渐开线齿轮发生根切需要在其中安装齿条刀27防止偏心非圆的齿轮发生根切是基于以下考虑：（1）主动齿轮 2有各种不同的齿面，但它们可能被抽象的表示为曲率半径（大约）自的环形齿面（图 6） 。（2）曲率中心 r2的半径 是点 A在齿面处的集合 。同样，齿面 表示为直齿圆柱齿轮的半径 ，等等（图 6） 。（3）当一个最小半径为 的圆柱直齿轮发生根切时，就用一个半径为 的齿轮替代（4）图 6a表示齿轮 2的剖视图其中心轨迹 r2是根据一个 n=3的齿轮设计的，这里的 n是偏心齿轮 1啮合的齿数，偏心齿轮 1是根据主动齿轮 2的中心轨迹发生改变的。（5）这个用圆齿轮代替的方法，可能会应用在所有中心轨迹为 r1和 r2的偏心齿轮上，为的是防止根切。因为这个原因，必须获得函数 ，这个函数 表示最小的曲率半径；A 是中心轨迹 r2上的点（图 6a） ，当 （见等式（6） ） 。（6）为了避免所有主动齿轮发生根切，要选择性的建立一个数学模型，该模型是函数 开平方与顶点 的成积。 （图 6b ） 7 图 4 函数 的插图和偏心齿轮驱动器的设计参数 8 图 5 对一个圆柱渐开线齿轮的根切的条件回避插图 9 图 6 插图：（a）非圆齿轮 2剖视图的表示（M 是瞬心线 R2和 N代圆弧齿轮瞬心线） ；（b）当 的根切线2.2 通过插齿和滚刀生成非圆齿轮2.2.1 介绍本节的目的是证明主动轮所产生的非圆齿轮和成型刀具（插齿，滚刀）的运动算法关系。这种关系是非线性方程组的代表，计算机是对非圆齿轮的生成的基础。该主动偏心齿轮通过观察制作的过程和相对于旋转中心的几何中心位置的装配，可以产生一个传统的渐开线齿轮。此外，为接触轴承定位的目的，有必要采用偏心齿轮式。这有通过一个插齿机床产生非圆齿轮的可能，和用偏心渐开线齿轮产生是一样的。通过观察用插齿机床生产非圆齿轮，只要在插齿机和被生产的非圆齿轮之间设定一个不变的的中心距。然而，这对于插齿机和偏心渐开线齿轮本身是不利的限制所以还没有被作者采用。2.2.2 由非偏心插齿机生产的非圆齿轮 插齿机所生产的非圆齿轮表面 的推导。我们要提醒的是，插齿机不是与主动偏心轮渐开线相同的，基于以下步骤进行推导：（1）被连接到插齿机和齿轮被确定的表面的两个坐标系统的 SS和 S2被认为是刚性的。 10 （2）一个渐开线齿面 是通过一个节圆半径 的插齿机进成的，这可能与有所不同。（3）坐标系统的 SS坐标系是旋转的而坐标系 S2是旋转和平移。运动系统的坐标系 SS和 S2是由运动学关系和中心轨迹决定的(见图 7b)。图 7。对用渐开线插齿机生成非圆齿轮的齿面 R2的推导。而坐标系 SS是围绕 旋转的， 是与极角 的相关的函数： 11 在式子中， 是一个关于偏心齿轮的极角 的函数（4）坐标系 SN可以被分解为大小为 ， 的坐标系 S2，而而坐标系 S2是围绕矢量 旋转的，这矢量表示是关于极角 的函数。函数 和 用极角 表达的函数表达式（5）通过考虑矩阵变化获得非圆齿轮面啮合方程然而，该方法是基于矩阵的推导（见下文） ，需要用计算机推导。 坐标系转换，将坐标系 SS转换成 S2. 公式（28）推导的过程如下：在这里 12 啮合矩阵方程 0的推导。矩阵变化式（30）用笛卡尔坐标系表达是下面给出的 3 X 3矩阵 R，是从 4 X 4的 M矩阵中获得的其中相关的速度 被给出如下以下是列出的是推导出导数 和 。看公式（28）的推导过程，很容易得到导数 和导数 。然后，对啮合方程可确定为或者为附件 2：外文原文（复印件） 13 of functionsFuentesUnitedEccentric gearsgenerationandasofresearch(2) Application of non-circular gears for generation of functions(Section 4) has been developed for the following cases: (i)wherein the derivative of the function is of a varied sign,(ii) the centrodes are unclosed curves, and (iii) two pair ofgears are applied for generation. In case (iii), the designrequires application of a functionalw/ff/; 1where w/ is the given function, f/ is the transmissionfunction of a pair of non-circular gears.2.1. Centrodes of eccentric gear drive2.1.1. Introductive commentsThe discussed below gear drive (called for the purpose of abbre-viation Eccentric drive) is formed by an eccentric involute pinion 1and conjugated non-circular gear 2 (Fig. 1). The non-circular gearsof the eccentric gear drive may be designed and generated withstraight and helical teeth.The eccentric drive is a competitive one to the one formed byelliptical gears 28. The approaches proposed in the paper allow* Corresponding author. Tel.: +34 968 326429; fax: +34 968 326449.Comput. Methods Appl. Mech. Engrg. 197 (2008) 37833802Contents lists availableAppl.E-mail address: ignacio.gonzalezupct.es (I. Gonzalez-Perez).The contents of the paper are the following ones:(1) Design and generation of an eccentric gear drive formed byan eccentric planar or helical involute gear and conjugatednon-circular gear has been developed. The main feature ofthe eccentric involute gear is that its center of its rotationdoes not coincide with the geometric center. The gear drivemay be designed with helical teeth (see Fig. 1) and straightteeth, the bearing contact may be localized, and, by designwith small eccentricity, the drive may be applied in reducers(as a gear drive with reduced sensitivity to misalignment).lar gear generated by a shaper or by a hob are proposed. Asimple approach for determination of avoidance of under-cutting of a non-circular gear is developed. A functional forobservation of identity of mating centrodes is proposed.The developed theory is illustrated with numerical examplesand with graphs and drawings.2. Gear drive formed by eccentric involute pinion andnon-circular gear1. IntroductionDevelopment of generation of non-circularmakes the manufacture of such gearsTherefore it is not surprising that designcircular gears became the subject of137.0045-7825/$ - see front matter C211 2008 Elsevier B.V. Alldoi:10.1016/j.cma.2008.03.001gears 12,2328easy as circular gears.gear drives with non-of many scientists(3) Application of modified elliptical gear (Section A.1.6) hasallowed design of: (i) a gear drive with an asymmetric trans-mission function, and (ii) lobes (which centrodes are modi-fied ellipses).(4) Algorithms for determination of tooth surfaces of non-circu-Non-circular gearsGeneration of functionsDesign and investigation of gear drives withapplied for speed variation and generationFaydor L. Litvina, Ignacio Gonzalez-Perezb,*, AlfonsoaDepartment of Mechanical and Industrial Engineering, University of Illinois at Chicago,bDepartment of Mechanical Engineering, Polytechnic University of Cartagena, Campus UniversitariocGear R (b) generation of functions by non-circular gears.with numerical examples.C211 2008 Elsevier B.V. All rights reserved.non-circular gearsat ScienceDirectMech. Engrg.drive,3784 F.L. Litvin et al./Comput. Methods Appl. Mech. Engrg. 197 (2008) 37833802to generate the non-circular gear by a hob (in case of a convex cen-trode) and by a shaper (if the centrode is a convexconcave one).The performed investigation covers the basic topics of geome-try, design, and generation of eccentric drives.2.1.2. Equations of mating centrodesFig. 2a and b shows centrodes r1and r2in initial and currentpositionsCentrode r1is an eccentric circle of radius rp1and is representedin polar form as (Fig. 2b)r1h1r2p1C0 e21sin2h112C0 e1cosh1; 2Fig. 1. Eccentric involute pinion and conjugated non-circular gear: (a) for helicalwhere parameter h1determines the position of r1h1 with respectto polar axis O1A1(Fig. 2b).Fig. 2. For derivation of centrodes: (a) initialHenceforth we will use representation of a centrode in terms ofderivative m12/1 (see Section A.1.2 of Appendix), where /i hi(i 1;2) is the angle of rotation of the centrodes; parameter hidetermines location of rihi with respect to polar axis OiA (see Sec-tion A.1.2). For r1, we haver1/1E1 m12/1: 3Centrode r2is represented by two following equationsr2/1Em12/11 m / ; 4(b) for a planar drive; O1and O2centers of rotation; O the geometric center.12 1/2/1Z/10d/1m12/1: 5position; (b) current position.alized by application of a four-bar linkage (Fig. 3) with links: (a)the gear is the pitch circle of radius rp. Undercutting of the involutegear is avoided by the installment of the rack-cutter wherein 27msin2acPrp: 17Avoidance of undercutting of non-circular gear of eccentric geardrive is based on following considerations:(i) Gear 2 of the drive has various tooth profiles, but they maybe represented (approximately) as tooth profiles of respec-tive circular gears with curvatures radii qA;qB; .;qK(Fig. 6).(ii) Radius qA2of curvature of centrode r2for point A corre-sponds to profiles of the tooth notified as RA. Similarly, pro-files of tooth RBare represented as ones of spur gears ofradius qB2, and so on (Fig. 6).(iii) Undercutting occurs for a tooth with the smallest radius q2of the substituting circular gear; it is the substituting gearof radius qA2.(iv) Fig. 6a shows representation of tooth profiles of gear 2 withcentrode r2designed for a gear drive with n 3. Here, n isthe number of revolutions of eccentric gear 1 that is per-formed for one revolution of driven gear 2 with centrode r2.(v) The idea of application of substituting circular gears may beMech.Here, the derivative function ism12/1E C0 r1/1r1/1c11 C0 e21sin2/112C0 e1cos/16and c1Erp1;e1e1rp1.Function /2/1 is the transmission function and its determina-tion requires numerical integration.Centrode r1is already a closed form curve (a circle of radius rp1).Centrode r2might be a closed form curve by observation ofequation/2/12pnZ2p0d/1m12/1; 7wherein n is the number of revolutions that eccentric gear 1 per-forms for one revolution of the non-circular gear.Derivation of centrodes r1and r2enables to design as well acam mechanism with a constant center distance between the fol-lower as an eccentric circle and a cam for generation of functions.The cam is determined as the centrode of the non-circular gear ofthe eccentric drive.2.1.3. Curvature of centrode r2and applications. Basic derivations. Knowledge of curvature of centrode r2isnecessary for: (i) choosing the method of generation of the gears bya hob or by a shaper, and (ii) for avoidance of undercutting of toothprofiles.Considering that a centrode is represented by a polar curve rh,its curvature radius my be determined as 20,27:qhr2drdhC0C12hi32r2 2drdhC0C12C0 rd2rdh2: 8The condition of convexity of the polar curve may be represented asj 1q r2 2drdhC18C192C0 rd2rdh2P0 9or by an alternative inequalitysinl r2 C0 sin2lC0sin2ld2rdh2“#P0: 10Here, angle l arctanrdr=dhC16C17is formed by position vector rh andtangent t to the polar curve (see 27); j is the curvature of the polarcurve.In this paper, taken into account that the curvature of centroder1is known, determination of j2is obtained by using EulerSavaryequation 18, that relates curvatures j1and j2of centrodesr1and r2of the eccentric gear drive (Fig. 3) as follows:1q11q21r1/11r2/1C20C21sinl1: 11Here1q11rp1 j1;1q2 j2; 12where q2jIA2j (Fig. 3),r2/1E C0 r1/1; Fig:3: 13Eq. (11) yieldsj21r1/11E C0 r1/1C20C21sinl1C0 j1: 14Convexity of centrode r2is guarantied ifF.L. Litvin et al./Comput. Methods Appl.1r1/11E C0 r1/1C20C21sinl1C0 j1P0: 15link 1 as O1A1, (b) link 2 as A1A2, (c) link 3 as O2A2, and (d) link 4as O1O2, where jA1A2jq1 q2.It is known from kinematics of four-bar linkage that point F (ofintersection of extended links 1 and 3) is the instantaneous centerof rotation of link 2 with respect to link 4. Vector IF is perpendic-ular to O1O2and its magnitude is determined asjIFjr1/1C0q1sinl1q1r1/1jq2sinl1C0 r2/1jq2r2/1: 16. Avoidance of undercutting. Drawings of Fig. 5 show genera-tion of an involute spur gear with a rack-cutter of profile angle ac.The centrode of the rack-cutter is its middle line, the centrode ofFig. 4 illustrates function q2/1 for gear drives with: (a) a convexcentrode r2; (b) a convexconcave r2.The location and orientation of curvature radius q2/1 is visu-Fig. 3. For derivation of curvature radius q2/1 of centrode r2.Engrg. 197 (2008) 37833802 3785applied for avoidance of undercutting for all eccentric geardrives with centrodes r1and r2. For this purpose, it is neces-sary to obtain functions qA2e;n (Fig. 6b) that represents theMech.3786 F.L. Litvin et al./Comput. Methods Appl.minimal curvature radius of qA2; A is the point of centrode r2(Fig. 6a) where q2 q2;min; e erp(see Eq. (6).(vi) Avoidance of undercutting for all eccentric gear drives isobtained by choosing such a module with which functionsqA2mwill be out of the square with the height 1=sin2ac(Fig. 6b).Fig. 4. Illustration of function q2/1 for eccentric gear drives with designFig. 5. Illustration of avoidance of undercutting of a spurEngrg. 197 (2008) 378338022.2. Generation of the non-circular gear by shaper and hob2.2.1. IntroductionThe purpose of this section is to derive the algorithms that re-late the motions of the generating tool (shaper, hob) and non-cir-cular of the drive being generated. Such relations are representedparameters rp1 22:35 mm; n 4, and (a) e 0:2, (b) e 0:7.involute pinion as the condition: m 0.In the case wherein the derivative y0x is of a varied sign in theinterval of derivation, the following process of generation is ap-plied (Fig. 16):(a) Instead of function yx assigned for generation, is generatedfunctiony1xyxbx; x16 x 6 x2; 81y2xis subtractedgear mechanis(b) circula/I 2/cC0 /II; /cC17 /2; /IIC17 /4; 84Fig. 16. Illustration of functions: y1xsinx bx;y2xx, yxasinx.F.L. Litvin et al./Comput. Methods Appl. Mech. Engrg. 197 (2008) 37833802 3793Fig. 14. Illustration of generation of function yx1x,16 x 6 3;/1;max/2;max 5p.Fig. 15. Schematic illustration of generation of function wag2g1aC138 by twopairs of non-circular gears that generate respectively: b g1a, d g2b.gears I and II (Fig. 17) 27/I /II 2/c: 83Angle of rotation /cis equivalent to /2, and angle of rotation /IIisequivalent to /4. Eq. (83) yields thatFig. 17.gears I3 and 4.between the angles of rotation of the carrier c andtion (82).(iii) Rotation of gears 2 and 4 are provided: (a) to the carrierc of the satellite s of the gear differential, and (b) to gearII of the differential, respectively.(iv) The differential provides the following relationsapproach illustrated by Fig. 17:(i) Functionbx 82from y1x by application of a gear differential. Twoms formed: (a) by non-circular gears 1 and 2, andr gears 3 and 4, are applied.(ii) Gears 1 and 3 are mounted on the same shaft and angles/1;/3are proportional to the variable x. The performeddesign of centrodes of gears 1 and 2 provides that angleof rotation of gear 2 is proportional to function (81); sim-ilarly, angle of rotation of gear 4 is proportional to func-and this allows to observe the requirement ofy01xP0:(b) Function yx assigned for generation will be obtained by theStructure of gear mechanism formed by a bevel gear differential (leveland II, satellite s, and carrier c), non-circular gears 1 and 2, and circular gearswhere /2is proportional to y1x represented by function (81) and/4is proportional to y2x represented by function (82). We assignfor the design that/4 2b/3; /3C17 /1y2x2x: 85(v) Eqs. (84) and (85) yield that angle /Iof rotation of gear Iof the differential will be obtained as proportional tofunction yx assigned for design/I 2/2C0 /4 2yxbxC02bx 2yx: 86Variation of function yx by the magnitude and sign will cause var-iation of angle /Iof rotation of gear I of the differential. This meansthat gear I will be rotated with varied angular velocity in twodirections.4.2. Problem 1: generation of function yxasinx;0 6 x 6 2pThe derivative y0x is varying its sign and therefore we applythe scheme of generation represented by Fig. 17.Non-circular gears 1 and 2 have to be designed for generation offunctiony1xsinx bx; 0 6 x 6 2p: 87Design of non-circular gears 1 and 2 (Fig. 17) has to cover determi-nation of their centrodes by application of the following procedure.(i) The angles of rotation of gears 1 and 2 are represented by theequations/1 k1x C0 x1; /2 k2y1xC0y1x1C138; x1 0: 88Here, k1and k2are scale coefficients determined ask1/1;maxx2C0 x1; k2/2;maxy1x2C0y1x1; x2 2p; x1 0: 89Taking that gears 1 and 2 will perform in the process of generationturns on /1;max /2;max 2p, we obtain thatk1 1; k21b: 90(ii) The coefficient b may be determined by observation of thefollowing conditions: (a) y0x 0, and (b) centrodes of gears1 and 2 have to be the convex ones. Condition (a) is observedby b 1. Detailed derivations for observation of condition(b) is represented in 25. The final result is thatblimP1:707: 91Observation of condition blim 1:707 means that centrode 1 willhave a point with curvature j1 0.(iii) Eqs. (88) with coefficients (90) yield the following transmis-sion function/2/1/11bsin/1; 0 6 /16 2p: 92The derivative function ism12/1d/1d/211 1bcos/1: 93(iv) Equations (A.5) and (A.6), (A.7) yield the following equationsfor centrodes r1and r2of non-circular gears 1 and 2(Fig. 17): For r1, we have3794 F.L. Litvin et al./Comput. Methods Appl. Mech. Engrg. 197 (2008) 37833802Fig. 18.b 1:707.Illustration of: (a) centrodes of non-circular gears for generation of function y1x with coefficient b 1:707; (b) transmission function /2/1 wherein coefficientr1/1E1 m12/1 E1 1bcos/12 1bcos/1: 94For r2, we haver2/1E12 1bcos/1; /2/1/11bsin/1: 95Center distance E is just a scale coefficient. Centrodes r1and r2determined with coefficients b 1:707 and b 1:400, are repre-sented in Figs. 18 and 19. Centrode 1 in Fig. 19 is a convexconcaveone.(v) We assign for the design that/4 2/3; /3 /1: 96(iv) Function yxasinx will be obtained from gear I of the dif-ferential as/I 2/cC0 /II 2/2C0 /4 2 /11bsin/1C18C19C0 2/12bsin/1;97wherein2b a.Non-circular gears with the developed centrodes may be gener-ated by the enveloping method by using a hob for centrodes shownin Figs. 18 and 19.4.3. Problem 2: Generation of function yx1x,x16 x 6 x24.3.1. IntroductionThe specific features of centrodes applied for generation are: (i)centrodes r1and r2are represented as unclosed curves, (ii) theyare identical, (iii) the gears may perform rotation on angles / 2p (Fig. 14), while performing simultaneously axial translation.4.3.2. Centrodes and transmission functionThe angles of rotation /1and /2of centrodes 1 and 2 are pro-portional to variable x and function yx, respectively. Thus wehave/1 k1x C0 x1; 98/2 k2y1C0 yk21x1C01xC18C19; 99where k1and k2are scale coefficients determined ask1/1;maxx2C0 x1; k2x1C0 x2/2;maxx2C0 x1: 100The derivative function ism12a3 a4/12a2a3; 101wherea2 k2; a3 k1x21; a4 x1: 102F.L. Litvin et al./Comput. Methods Appl. Mech. Engrg. 197 (2008) 37833802 3795Fig. 19. Illustration of: (a) centrodes of non-circular gears for generation of function y1b 1:400.x with coefficient b 1:400; (b) transmission function /2/1 wherein coefficientMech.The centrodes are represented as followsr1/1E11 m12/1 Ea2a3a2a3a3 a4/12; 103r2/1E C0 r1/1Ea3 a4/12a2a3a3 a4/12; 104/2/1F/1a2/1a3 a4/1: 105The centrodes are identical ones by observation the conditions/1;max /2;max, and satisfaction of the functional (see SectionA.1.7) that is represented for discussed example of design asa2/1;maxC0 F/1a3 a4/1;maxC0 F/1 /1;maxC0 /1: 106The conditions above are satisfied and the identical centrodes arerepresented by Fig. 14.4.4. Generation of function by gear drive by application of two pairs ofnon-circular gears4.4.1. IntroductionGeneration of given function wa;a C17 /1, by a gear drive withtwo pairs of gears (1, 2) and (3, 4) (instead of one pair) has the fol-lowing advantages:(i) a larger variation of derivativeowoamay be provided.(ii) lesser pressure angle of each of the two pairs of non-circulargears may be obtained.Fig. 15 shows schematic of the gear drive formed by the cen-trodes of the gears of the drive. Each of the centrodes 1, 2, 3, 4 per-forms rotation about point Oi(i 1;2;3;4). The relative motion ofeach of centrode i with respect to the mating centrode of a pair ofcentrodes (1, 2) and (3, 4) is pure rolling.Depending on the type of function wa to be generated, the cen-trodes might be closed curves or unclosed ones.We may consider initial and final positions of the centrodes thatcorrespond to the beginning of motion (where /i 0, i 1;2;3;4),and the end of motion (where /i /i;max).In the case of centrodes as closed curves, we have: /1;max/2;max /3;max /4;max 2p. In the case of unclosed curves, wehave: /1;max /2;max /3;max /4;max2p if in the process of motion the gears perform correlatedaxial motions, in additio