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1、英文部分Computerized tooth profile generation of elliptical gears manufactured by shaper cuttersBiing-Wen BairDepartment of Mechanical Engineering, National Lien Ho Institute of Technology,AbstractThis work simulates an elliptical gear drive, the axis of rotation of which is coincident with its geometri
2、c center, manufactured by shaper cutters. The mathematical model of an elliptical gear is developed based on the theory of gearing and gear generation mechanisms. In addition, the tooth undercutting of the gear is also investigated based on the developed mathematical model of the elliptical gear, it
3、s unit normal vectors and a numerical method. A geometric relationship is developed and applied to prevent the occurrence of pointed teeth on elliptical gears. Further, this study also develops computer simulation programs to generate the tooth profile of elliptical gears without tooth undercutting
4、and pointed teeth. Comparison of the angular velocity variations of the elliptical gear drives is also made. The results show that the developed elliptical gear drive can be utilized as an oil pump with a larger pumping volume and less angular velocity variation. 2002 Elsevier Science B.V. All right
5、s reserved.Keywords: Elliptical gears; Undercutting; Pointed teeth1. IntroductionAn elliptical gear drive, the rotation center of which coincides with one of its foci, is kinematically equivalent to the crossed link, and can be used to produce irregular rotations. In addition, it is well known for p
6、roviding excellent characteristics such as accurate transmission, compact size, and ease of dynamic balance. Hence, elliptical gear drives have been applied successfully in various types of automatic machinery, quick-return mechanisms, packaging machines, and printing presses 1. This type of gearing
7、 can also be used to develop non-circular gears, which belong to high-order elliptical pitch curves. Second- order elliptical gear drives, the rotation center of which coincides with one of its foci, can find use in the design of instruments such as pumps and flow meters 1. However, this type of gea
8、r set has two speed changes for each revolution, these two-cycle variations inducing a wave fluctuation that is so severe that the second-order elliptical gear set cannot be used as oil pumps for steady oil pumping.The design and manufacture of an elliptical gear are difficult because the pitch curv
9、e of the gear is an ellipse.Some studies 26 have focused on kinematic analysis and computer-aided design of elliptical pitch curves. Kuczewski 7 used a spur gear to approximate the profile of an elliptical gear. Emura and Arakawa 8 used an elliptical gear to analyze a steering mechanism, where this
10、steering mechan- ism can turn a carrier with a small radius. Also, Freudenstein and Chen 9 developed variable-ratio chain drives (e.g. elliptical gear drives), which were applied to bicycles and variable motion transmissions involving band drives, tape drives, and time belts with a minimum slack. Mo
11、reover, Litvin 10 adopted the concept of evolute curves to form the tooth profile, and also derived the tooth evolute of an ellipse. Chang and Tsay 11 used a shaper cutter and applied the inverse mechanism relationship and the equation of meshing to produce the mathematical model of elliptical gears
12、, the rotation center of which turns around one of its foci. Also, Chang et al. 12 used a rack cutter and the same method to produce the mathematical model and undercut- ting conditions of the same type of elliptical gears. However, when the elliptical gear surfaces are generated by shaper cutters,
13、pointed teeth may appear and the tooth addendum is reduced. Pedrero et al. 13 proposed an approximation method for modifying the tooth addendum and contact ratio, and computer simulation results also show that the gear contact ratio depends on the tooth addendum. Additionally, Liou et al. 14 analyze
14、d spur gears with low contact ratios (a contact ratio of less than 2) and high contact ratios (equal or larger than 2) when subjected to dynamic loads by applying the NASA gear dynamics software DANST. The DANST program determined the instantaneous contact teeth and contact ratio based on the gear a
15、verage stiffness, commonly referred to as mesh stiffness. Recently, Bair and Tsay 15 proposed a tooth contact analysis (TCA) program to calculate the instantaneous contact teeth and average contact ratio of a dual-lead worm gear drive. The DANST and TCA methods confirm that reducing the gear addendu
16、m results in the decrease of the gear instantaneous contact teeth and the average contact ratio.Based on the position of the rotation center, elliptical gear drives fall into two types: the elliptical gear which rotates about its geometric center (type 1); the elliptical gear which rotates around on
17、e of its elliptical foci. An n- order driven non-circular gear, of which the rotation axis is one of its foci, is one in which the driving elliptical gear performs n revolutions for one revolution of the driven non- circular gear. A second-order elliptical gear (type 2) is defined as n = 2. Under th
18、e same eccentricity and major axis, the size of the type 1 elliptical gears is smaller than that of the type 2. Further, if the type 1 elliptical gear set is applied to oil pumps, the wave fluctuation of the pumping oil is smaller and smoother than that of the other type. The elliptical gear tooth p
19、rofile is usually produced by a hob- bing or shaping machine with a hob cutter or shaper cutter. This study simulates the manufacture of elliptical gears via a shaper cutter on a shaping machine. It is known that if a spur gear is produced by a shaper, the profile of the shaper cutter should be the
20、same as that of the mating spur gear. Therefore, the mathematical model of a shaper cutter is the same as that of a spur gear, which can be derived from the generation mechanism with a rack cutter. According to the theory of gearing, the mathematical models of elliptical gear tooth profiles, which r
21、otate about their geometric center (type 1), are developed based on the proposed generation mechanism with shaper cutters. Due to the complex characteristics of this type of elliptical gear, undercutting and pointed teeth may exist on its tooth profile. The tooth undercutting of this type of ellipti
22、cal gear is affected by its pressure angle, number of teeth, module, and major axis. The strength of the gear tooth root can be increased by applying a positive-shifted modification for the cutter during the gear generation. However, an over-positive-shifted modification may result in the appear- an
23、ce of pointed teeth. Pointed teeth are generated when the right- and left-side involute tooth profiles intersect on or below the addendum circle of the gear. Further, the pointed teeth are usually generated on the two major axis of an elliptical gear. If a profile index is defined to prevent pointed
24、 teeth generation on the two major axis of an elliptical gear, then no other pointed tooth will be generated for all elliptical gear profiles. Thus, the computer program developed here can calculate and provide proper design parameters for the designed elliptical gears to avoid tooth undercutting an
25、d pointed teeth.2. Mathematical model of the elliptical gear surfacesShaper cutters are used to generate elliptical gears, and the profiles of shaper cutters are the same as those of spur gears. Hence, the mathematical model of the shaper cutter is the same as that of the spur gear, which is generat
26、ed from rack cutters. A complete elliptical gear tooth profile consists of three surface regions, i.e. the working region, the fillet and the bottom land. Therefore, the profile parameters of a shaper cutter can be represented by the parameters of a rack cutter. Fig. 1 shows three regions of a rack
27、cutter 2p including the working region, the fillet and the top land, used for shaper cutter and elliptical gear generations. When the shaper cutter creates the elliptical gear in a cutting mechanism, its center rotates along the Zc-axis and translates along the Xc and Yc-axes, performing a pure roll
28、 without sliding on the pitch ellipse, and the gear blank is rotated about its geometric center 01 as in Fig. 2.2.1. Working region of shaper cutter profileFig. 1 presents the design of the normal section of rack cutter 2p, where regions 3 and 4 are the left- and right-side working regions, regions
29、2and 5 are the left- and right-side fillets, and regions 1 and 6 are the left- and right-side top lands. Meanwhile, parameter p = M0 M1 is a design parameter, expressing the distance measured from the initial point M0 to an arbitrary point M1 in the working region. The three- dimensional rack cutter
30、 profile can be obtained by translating its normal section, presented in Fig. 1, along the Zr-axis with a displacement parameter Up. Therefore, by applying the theory of gearing, the mathematical model of the working region of the shaper cutter can be represented in the coordinate system Sc (Xc , Yc
31、 , Zc ) by the following equation (Litvin, 1989): Fig. 1. Normal section of a rack cutter 2p for generating the driving shaper cutter. Fig. 2. Kinematic relationship between the shaper cutter and the generated gear.where A0 is the design parameter used to determine the addendum of the shaper, B0 the
32、 tooth width of the shaper, rs the pitch radius of the shaper, cc the generated angle of the shaper and i/in the pressure angle as shown in Fig. 1. In Eq. (1), the upper sign indicates the right-side shaper surface while the lower sign represents the left-side shaper surface. The normal vector of th
33、e working region of the shaper cutter surface can be obtained as follows:2.2. Locus of the shaper cutterFig. 2displays the kinematic relationship between the shaper cutter and the generated elliptical gear. During the elliptical gear generation, the shaper cutter rotates about the Zc-axis and transl
34、ates along a curve that keeps the shaper centrode and elliptical pitch of the generated gear in tangency at their instantaneous pitch point, I. The coordinate systems displayed in Fig. 2are the Cartesian coordinate system with right-handed three mutual perpendicular axes. It is noted that the Z-axis
35、 is not shown for simplicity. Coordinate system Sc (Xc , Yc , Zc ) is attached to the shaper cutter, and coordinate system S1 (X1 , Y1 , Z1 ) is attached to the generated elliptical gear of which the rotation center is coincident with the gear geometric center. Parameter y1 is the angle formed by th
36、e X1-axis and the tangent line which corresponds to the tangency of the shaper centrode and elliptical pitch at their instantaneous pitch point, I. The rotation angle of the elliptical gear is 7t/2 y1 , and angle c1 is a function of y1. Parameter cs, measured from the line0c I to the Yc-axis of the
37、shaper cutter, represents the rota- tional angle of the shaper. Let R1 denotes the position vector of the generated elliptical gear profile and Rc represents theposition vector of the shaper cutter surface. By applying thefollowing homogeneous coordinate transformation matrix equation, the locus (fa
38、mily) of the shaper cutter represented in coordinate system S1 can be obtained as follows: Substituting Eq. (1) into Eq. (3) provides the locus of shaper surfaces represented in coordinate system S1 as follows: Fig. 3. Tangent line of the ellipse.and the corresponding normal vector can also be obtai
39、ned by: WhereIn Eq. (6), parameter 21 is the eccentricity of the ellipse, a1 the major semi-axis and b1 the minor semi-axis. Expressing the pitch curve of the elliptical gear, r1(c1), using the Cartesian coordinate system, the x1 and y1 components along the coordinate axes are:AndReferring to Fig. 3
40、, the unit tangent vector to the pitch curve at point I is positive in the fourth quadrant. The tangent vector of the pitch curve can be obtained by differentiating Eqs. (7) and (8) with respect to parameter c1 and then normalizing the results. This process results in the unit tangent vector of the
41、pitch curve as follows:As Fig. 3 indicates, the unit tangent vector t1 of the pitch curve can also be represented in terms of angle y1 by the following equation:According to Eqs. (9) and (10), angle y1 can be expressed in terms of c1 as follows：And As shown in Fig. 2, the arc length of the shaper cu
42、tter, measured from the starting point N to the instantaneous pitch point I along the circular pitch, is equal to the arc length measured from the starting point M to the instantaneous pitch point I along the pitch ellipse. According to integral operation, the arc length can be expressed as follows:
43、中文翻译计算机控制插齿刀加工椭圆齿轮齿形白炳文国立联合大学机械工程系摘要 这个工作是模拟椭圆齿轮传动,轴围绕其几何中心旋转,用插齿刀加工。一个椭圆齿轮开发数学模型的建立是基于耗散结构产生的机理和齿轮啮合齿轮理论。此外, 也基于发达的数学模型椭圆齿轮单位法向量和数值模拟方法对蜗杆传动齿轮齿形进行了研究。几何关系已发展和应用于防止椭圆齿轮发生尖齿。此外,本研究也开发了计算机仿真程序生成无根切和尖齿的椭圆齿轮。也对椭圆齿轮驱动的角速度变化做了研究。结果表明:该椭圆齿轮驱动有较大的变化量和角速度可以被用来作为抽油泵。2002卷。版权所有。关键词:椭圆齿轮、根切、尖齿1简介以一个焦点为旋转中心的椭圆齿轮
44、传动,在运动学上等效为交叉连接,并可用于制造不规则的旋转。此外 ,它能提供许多是众所周知的优良的特性,如传动精确,体积小、容易达到动态平衡。因此,椭圆齿轮传动已经成功地应用于各种类型的自动机械、包装机械、急回机构,和印刷机1。这种类型的齿轮传动,也可以用来发展高阶椭圆曲线非圆齿轮。第二, 以一个焦点为旋转中心的椭圆齿轮传动,同时它的一个组件,可应用于仪器设计,如泵和流量仪表1。然而这些双循环变异每一次波动都很剧烈,所以二阶椭圆齿轮组不能作为稳定吸油的油泵。椭圆齿轮的设计制造是困难的,因为其分度曲线是椭圆形的。一些研究2-6都聚焦在运动学分析和椭圆分度曲线的计算机辅助设计上。Kuczewski7
45、用齿轮近似的椭圆齿轮。Emura和荒川8用载波范围很小的椭圆齿轮转向机构进行探讨。同样,Freudenstein、陈9发展可变传动比链条驱动(例如椭圆齿轮传动),都适用于自行车变速器和可变运动包括传送带驱动、磁带驱动器、最少时间时区。而且,Litvin10采用渐屈线概念的基础上，形成齿廓曲线，并推导出了一个椭圆牙型渐屈线。常和Tsay常使用插齿刀和应用反机制关系和方程的啮合产生以其一个焦点为旋转中心的椭圆齿轮的数学模型。同样,常以及其他人12用一架刀具和相同的方法来生产条件的数学模型和同一类型的椭圆齿轮的根切条件。然而,当椭圆齿轮表面由插齿刀成型、锐利的牙齿可能会出现，齿顶高也会减小。Pedr
46、ero以及其他人13提出的一种近似方法补充和修饰齿顶高和啮合系数，计算机模拟的结果也表明,齿轮啮合系数取决于齿顶高。另外，Liou以及其他人14 采用动态负荷齿轮动力学软件DANST分析了直齿轮低啮合系数(啮合系数低于2)和高接啮合系数(等于或大于2)的情况。DANST程序基于刚性平均确定的瞬时接触的齿和啮合系数,通常被称为啮合刚度。最近, Bair和Tsay(15),提出了一种齿面接触分析(TCA)程序计算瞬时接触牙齿和平均重合度的双蜗轮驱动模型。DANST TCA的研究方法证明降低齿顶高会降低齿轮瞬时接触齿数和平均啮合系数。基于旋转中心的位置来划分、椭圆齿轮驱动可分为两类:椭圆齿轮的基于几
47、何中心旋转(1型)、椭圆齿绕一个焦点旋转。一个n -订单驱动非圆齿轮轴的转动,其中之一是它的组件,是一个椭圆齿轮传动为执行n转革命的非圆齿轮驱动。一个二阶椭圆齿轮(2)被定义为2例。在相同的偏心和主轴、尺寸的1型椭圆齿轮是低于2型。此外,如果在1型椭圆齿轮油泵,应用波的波动较小、开采石油比其他类型。椭圆齿轮的齿形通常是由滚刀冰或成型机与切刀或铁架插齿刀。本研究对椭圆齿轮模拟生产上通过塑造成型机器切。众所周知,如果一个齿轮是由一个成型机、剖面的插齿刀应该一样的刀具的交配齿轮。因此,建立了相应的数学模型的创造者一样刀具的齿轮,可从中生成机理与一架刀具。根据这个理论,建立了相应的数学模型,对齿轮的齿
48、型材,椭圆齿轮旋转关于他们的几何中心(1型)的基础上,提出了用插齿刀机理。由于这种类型的复杂特征,抢椭圆齿轮和锐利的牙齿可能存在于它的齿形。牙消解这种类型的椭圆齿轮受其压力角、牙齿、模块、主轴。轮齿根系的强度增加可以利用正补的刀在齿轮的产生。然而,一个过大的正补可能会导致出现特性的锐利的牙齿。尖牙时会产生正确的而左侧渐开线齿或以下型材交叉齿顶圆的齿轮。进一步,锐利的牙齿通常产生于两主轴的椭圆齿轮。如果一个侧面指标的定义是防止锐利的牙齿上一代的两个主要轴线椭圆齿轮,就没有其他尖牙都将产生椭圆齿轮型材。因此,计算机程序,可以在这里计算提供合理的设计参数设计,避免了椭圆齿轮的根切和尖齿。2椭圆齿轮表面的数学模型插齿刀用来生成椭圆齿轮、插齿刀轮廓类似于直齿轮。因此,该数学模型对插齿刀一样,这是直齿圆柱齿轮齿条刀具产生。一个完整的椭圆齿轮齿形由三面区域,如下。工作区域,鱼片和底部降落。因此,剖面参数成型机刀可以由一架切的参数。图1显示的三个地区的一架刀具,包括工作区域2p顶部圆角,土地,用于成型机刀具、椭圆齿轮的后代。当插齿刀中创造了椭圆齿轮刀具切削机理,其中心旋转Zc-轴和翻译的Yc-轴Xc,纯净,而在球场上滚动滑,和齿轮的椭圆空白旋转关于它的几何中心在图1,2。2.1 工作区域的创造者。刀具图1提出设计的正常的部
