外文翻譯計算機控制插齒刀加工橢圓齒輪齒形

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. al

5、l rights 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 know

6、n for providing 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

7、gearing 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

8、 of gear 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 pit

9、ch curve 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, wher

10、e this 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 sl

11、ack. moreover, 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 elliptica

12、l gears, 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 cu

13、tters, 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

14、analyzed 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

15、 gear average 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

16、addendum 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 ar

17、ound one 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. u

18、nder the 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

19、tooth profile 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

20、be the 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,

21、which rotate 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

22、elliptical 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 app

23、ear- ance 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

24、pointed 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 undercut

25、ting and 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

26、generated 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

27、a rack 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 pu

28、re roll 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, r

29、egions 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- dimensiona

30、l rack cutter 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 syste

31、m sc (xc , yc , 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 s

32、haper, b0 the 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

33、 vector of the 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-ax

34、is and translates 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 th

35、at the z-axis 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

36、 formed by the 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

37、-axis of the 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,

38、the locus (family) 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

39、also be obtained 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:andreferr

40、ing to fig. 3, 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 v

41、ector of the 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

42、the shaper cutter, 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 expresse

43、d as follows:中文翻譯計算機控制插齒刀加工橢圓齒輪齒形白炳文國立聯(lián)合大學機械工程系摘要 這個工作是模擬橢圓齒輪傳動,軸圍繞其幾何中心旋轉,用插齒刀加工。一個橢圓齒輪開發(fā)數(shù)學模型的建立是基于耗散結構產(chǎn)生的機理和齒輪嚙合齒輪理論。此外, 也基于發(fā)達的數(shù)學模型橢圓齒輪單位法向量和數(shù)值模擬方法對蝸桿傳動齒輪齒形進行了研究。幾何關系已發(fā)展和應用于防止橢圓齒輪發(fā)生尖齒。此外,本研究也開發(fā)了計算機仿真程序生成無根切和尖齒的橢圓齒輪。也對橢圓齒輪驅動的角速度變化做了研究。結果表明:該橢圓齒輪驅動有較大的變化量和角速度可以被用來作為抽油泵。2002卷。版權所有。關鍵詞:橢圓齒輪、根切、尖齒1簡介以一

44、個焦點為旋轉中心的橢圓齒輪傳動,在運動學上等效為交叉連接,并可用于制造不規(guī)則的旋轉。此外 ,它能提供許多是眾所周知的優(yōu)良的特性,如傳動精確,體積小、容易達到動態(tài)平衡。因此,橢圓齒輪傳動已經(jīng)成功地應用于各種類型的自動機械、包裝機械、急回機構,和印刷機1。這種類型的齒輪傳動,也可以用來發(fā)展高階橢圓曲線非圓齒輪。第二, 以一個焦點為旋轉中心的橢圓齒輪傳動,同時它的一個組件,可應用于儀器設計,如泵和流量儀表1。然而這些雙循環(huán)變異每一次波動都很劇烈,所以二階橢圓齒輪組不能作為穩(wěn)定吸油的油泵。橢圓齒輪的設計制造是困難的,因為其分度曲線是橢圓形的。一些研究2-6都聚焦在運動學分析和橢圓分度曲線的計算機輔助設

45、計上。kuczewski7用齒輪近似的橢圓齒輪。emura和荒川8用載波范圍很小的橢圓齒輪轉向機構進行探討。同樣,freudenstein、陳9發(fā)展可變傳動比鏈條驅動(例如橢圓齒輪傳動),都適用于自行車變速器和可變運動包括傳送帶驅動、磁帶驅動器、最少時間時區(qū)。而且,litvin10采用漸屈線概念的基礎上，形成齒廓曲線，并推導出了一個橢圓牙型漸屈線。常和tsay常使用插齒刀和應用反機制關系和方程的嚙合產(chǎn)生以其一個焦點為旋轉中心的橢圓齒輪的數(shù)學模型。同樣,常以及其他人12用一架刀具和相同的方法來生產(chǎn)條件的數(shù)學模型和同一類型的橢圓齒輪的根切條件。然而,當橢圓齒輪表面由插齒刀成型、銳利的牙齒可能會出現(xiàn)

46、，齒頂高也會減小。pedrero以及其他人13提出的一種近似方法補充和修飾齒頂高和嚙合系數(shù)，計算機模擬的結果也表明,齒輪嚙合系數(shù)取決于齒頂高。另外，liou以及其他人14 采用動態(tài)負荷齒輪動力學軟件danst分析了直齒輪低嚙合系數(shù)(嚙合系數(shù)低于2)和高接嚙合系數(shù)(等于或大于2)的情況。danst程序基于剛性平均確定的瞬時接觸的齒和嚙合系數(shù),通常被稱為嚙合剛度。最近, bair和tsay(15),提出了一種齒面接觸分析(tca)程序計算瞬時接觸牙齒和平均重合度的雙蝸輪驅動模型。danst tca的研究方法證明降低齒頂高會降低齒輪瞬時接觸齒數(shù)和平均嚙合系數(shù)。基于旋轉中心的位置來劃分、橢圓齒輪驅動可

47、分為兩類:橢圓齒輪的基于幾何中心旋轉(1型)、橢圓齒繞一個焦點旋轉。一個n -訂單驅動非圓齒輪軸的轉動,其中之一是它的組件,是一個橢圓齒輪傳動為執(zhí)行n轉革命的非圓齒輪驅動。一個二階橢圓齒輪(2)被定義為2例。在相同的偏心和主軸、尺寸的1型橢圓齒輪是低于2型。此外,如果在1型橢圓齒輪油泵,應用波的波動較小、開采石油比其他類型。橢圓齒輪的齒形通常是由滾刀冰或成型機與切刀或鐵架插齒刀。本研究對橢圓齒輪模擬生產(chǎn)上通過塑造成型機器切。眾所周知,如果一個齒輪是由一個成型機、剖面的插齒刀應該一樣的刀具的交配齒輪。因此,建立了相應的數(shù)學模型的創(chuàng)造者一樣刀具的齒輪,可從中生成機理與一架刀具。根據(jù)這個理論,建立了相應的數(shù)學模型,對齒輪的齒型材,橢圓齒輪旋轉關于他們的幾何中心(1型)的基礎上,提出了用插齒刀機理。由于這