以盤狀圓弧刀具創成之曲線齒的數學模式推導與接觸分析研究

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(1)國立交通大學機械工程學系. 碩士論文. 以盤狀圓弧刀具創成之曲線齒的數學模式推導與接觸分析研究 Mathematical Model and Tooth Contact Analysis of Gears with Curvilinear-Teeth Generated by a Disk-Type Circular-Arc Cutter. 研究生：吳一正指導教授：徐瑞坤曾錦煥共同指導教授：蔡忠杓. 中華民國九十五年五月.

(2) 以盤狀圓弧刀具創成之曲線齒的數學模式推導與接觸分析研究. 研究生：吳一正. 指導教授：徐瑞坤、蔡忠杓. 國立交通大學機械工程學系. 摘要圓弧型曲線齒齒輪（circular-arc curvilinear-tooth gear）係指一種由圓弧型盤狀刀具所創成，具有曲線型齒線及圓弧型齒形特徵的齒輪。不同於以往傳統正齒輪與螺旋齒輪的線接觸型的傳動運轉方式，具曲線型齒線特徵的齒輪組是以點接觸的接觸型態來傳輸動力。點接觸的接觸型態不僅能使齒輪組避免齒緣的接觸及降低由軸向組裝誤差所引起的運動誤差和振動噪音，更可增強齒輪的接觸強度。除此之外，具有曲線型齒線之齒輪在運轉時亦不會產生軸向的推力。具圓弧型齒形的螺旋齒輪因為具有較一般傳統螺旋齒輪為高的負載能力及更佳的潤滑效果等特性，故常被使用於需傳輸高負載的情況，但是圓弧型螺旋齒輪對於中心距離的組裝誤差是十分敏感的。本研究係將漸開線型曲線齒齒輪的齒形改以圓弧型外形之齒刀來創成齒輪，亦即應用一把虛擬的圓弧型齒刀依循一曲線路徑而創成出該齒輪之齒面，並根據齒輪原理建立圓弧型曲線齒齒輪之齒面數學模式，同時也以轉位修正的方式來避免齒形過切的情況發生，亦利用齒輪接觸分析技術來探討具有裝配偏差時，齒輪組運動誤差、接觸比及接觸齒印。 i.

(3) Mathematical Model and Tooth Contact Analysis of Gears with Curvilinear-Teeth Generated by a Disk-Type Circular-Arc Cutter. Student: Yi-Zheng Wu. Advisor: Ray-Quan Hsu, Chung-Biau Tsay. Department of Mechanical Engineering National Chiao Tung University. Abstract The circular-arc curvilinear-tooth gear, which has curvilinear trace with circular-arc profile, is generated by a circular-arc disk-type cutter. Gears with curvilinear traces ensure that the gear pair is operated in a point contact condition, which is different from the line contact type of a helical gear pair or spur gear pair. The point contact condition can not only avoid the tooth edge contact, decreases the kinematical errors and vibro-acoustic due to axial misalignments, but also increases the bearing strength of the contact gears. Moreover, a gear pair with curvilinear-tooth traces operates without axial thrust force. The helical gears with circular-arc profile are used for transmitting with high load circumstance, because their loading capacity and lubricity are better than those of conventional helical gears. However, the circular-arc helical gear pair is quite sensitive to the central distance assembly errors. ii.

(4) This study adopts the rack cutter with circular-arc profile to modify the profile of involute-type curvilinear-tooth gear pair. The gear surface is generated by an imaginary circular-arc rack cutter with a curvilinear trace. According to theory of gearing, a mathematical model of the gears with circular-arc profile and curvilinear-tooth trace is developed. The undercutting of the gears can be avoided by a positive profile-shifted modification during the gear set generation process. The tooth contact analysis technique is utilized to the investigation on kinematical errors, contact ratios and contact patterns of the gear set under axial misalignments.. iii.

(5) Acknowledgement (誌謝). 我很幸運！能夠進入地球上最棒的實驗室－最佳化設計實驗室，在這邊渡過了碩士班的時光，能跟隨最慈祥的. 曾錦煥教授做學問。在實驗室學習與生活的這兩年中，我. 成長了許多，除了碩士班應有的訓練外，亦習得許多做人處事的大道理，聽起來像是老生常談，實驗室的伙伴們一定能對我所說的一言一句感同深受。最難忘的還是曾老師的咳嗽聲，輕撫著他視為孩子們的學生，與大家探討學術上的問題，大家將老師圍成一圈，聽著老師說著成功的道理及做事的方法。實驗室的孩子都是這樣成長茁壯，出了實驗室的大門，我們都能抬起胸膛，大聲的告訴大家：我和別人不一樣。這一切都要感謝. 曾錦煥老師。雖然已無法當著您的面，致上學生最誠摯的感. 謝之意，但學生已不知在心中默唸了數百次，希望您都能聽見。這篇論文的完成，除了最敬愛的曾老師外，也謝謝蔡忠杓教授在學生碩士班二年級這一年來的細心指導及對論文寫作不厭其煩的指正，蔡老師在研究學問及為人處世所展現的細心、執著和堅毅，都是學生十分佩服的。您的諄諄教誨，學生將永遠銘記於心。感謝在碩士班二年級指導我們的徐瑞坤教授，徐老師將我們當成自己的學生般，隨時不忘關心、照顧我們，並給予學生論文許多寶貴的建議，謝謝徐老師。感謝求學生涯中一直默默支持我的父母，因為有你們在背後的鼓勵，給我一個穩定無虞的良好學習環境，讓我能更專心地面對研究所生活的挑戰；如今，順利完成了碩士學位，這些成就都將歸功於我親愛的父母，謝謝你們。在應用最佳化實驗室的這短短兩年中，留下了許多難忘的回憶。謝謝林聰穎老師、羅接興老師及嘉宏、明達、炫慧、陽光．．．等學長姊們給我了許多學業及生活上的援助。謝謝美惠及筑妃同學，和優秀的你們一同學習讓我不敢怠慢。謝謝培峰、翊猷及岳. iv.

(6) 良學弟所給予的許多幫忙，讓我能有更多的時間完成這篇論文。也謝謝齒輪實驗室的學長、學弟，給了我許多論文上的建議及課業上的幫助。感謝實驗室的每位伙伴，我永遠不會忘記大家互相扶持，攜手走過的每一天。感謝你們的包容與協助，讓我能順利、充實、快樂的渡過這段時光。有你們真好！感謝論文的口試委員：清華大學宋震國教授及中正大學馮展華教授，百忙中不辭辛勞的撥冗參加學生的碩士論文口試，並對學生的論文提出了寶貴的建議及修正，讓學生的論文更加充實且完整，謝謝你們。這兩年的研究生活中，因為有你們的存在，讓我能一路走過從所未經歷過的挫折及難關。祝福這些長輩及朋友們能永遠平安、順利。. v.

(7) Table of Contents 摘要 ........................................................................................................................ i Abstract ................................................................................................................ii Acknowledgement (誌謝)................................................................................... iv List of Figures ..................................................................................................... ix List of Tables ..................................................................................................... xiv Nomenclature..................................................................................................... xv CHAPTER 1 Introduction.................................................................................. 1 1.1 Background .....................................................................................................................1 1.2 Literature Reviews .........................................................................................................3 1.3 Motivations......................................................................................................................4 1.4 Thesis Overviews ............................................................................................................5. CHAPTER 2 The Curvilinear-Tooth Gear and the Circular-Arc Tooth Gear ............................................................................................................................... 8 2.1 Introduction ....................................................................................................................8 2.2 The Curvilinear-Tooth Gear..........................................................................................9 2.3 The Circular-Arc Tooth Gear ......................................................................................10. vi.

(8) 2.4 Remarks.........................................................................................................................16. CHAPTER 3 Mathematical Model of Gears with Curvilinear-Teeth Generated by a Disk-Type Circular-Arc Cutter ................ 17 3.1 Introduction ..................................................................................................................17 3.2 Mathematical Model of Circular-Arc Gears with Curvilinear-Teeth......................17 3.2.1 Generation Method ...............................................................................................17 3.2.2 Equation of the Disk-Type Circular-Arc Cutter.................................................22 3.2.3 Computer Graphs..................................................................................................36 3.3 Remarks.........................................................................................................................37. CHAPTER 4 Tooth Undercutting Analysis .................................................... 42 4.1 Introduction ..................................................................................................................42 4.2 Conditions of Tooth Undercutting...............................................................................42 4.3 Numerical Examples for Tooth Undercutting of Circular-Arc Curvilinear-Tooth Gears..............................................................................................................................48 4.4 Remarks.........................................................................................................................60. CHAPTER 5 Tooth Contact Analysis.............................................................. 61 5.1 Introduction ..................................................................................................................61 5.2 Analysis on Kinematical Errors. .................................................................................62. vii.

(9) 5.3 Contact Ratio ................................................................................................................68 5.4 Numerical Examples for Kinematical Errors of Circular-Arc Curvilinear-Tooth Gears..............................................................................................................................69 5.5 Remarks.........................................................................................................................86. CHAPTER 6 Contact Pattern.......................................................................... 87 6.1 Introduction ..................................................................................................................87 6.2 Surface Topology Method ............................................................................................88 6.3 Numerical Simulation of Contact Ellipses of Circular-Arc Curvilinear-Tooth Gears ........................................................................................................................................94 6.4 Remarks.......................................................................................................................103. CHAPTER 7 Conclusions and Future Works .............................................. 104 7.1 Conclusions .................................................................................................................104 7.2 Future Works ..............................................................................................................106. References ........................................................................................................ 107. viii.

(10) List of Figures Fig. 1-1 Classification of gears...................................................................................................1 Fig. 1-2 Various types of tooth traces for parallel axis gears .....................................................3 Fig. 2-1 Schema of gear profile and gear tooth trace of a spur gear ..........................................8 Fig. 2-2 Schema of curvilinear-tooth gear [3] ..........................................................................10 Fig. 2-3 Contact region of a helical gear pair ........................................................................... 11 Fig. 2-4 Contact region of a crowned teeth gear pair ...............................................................12 Fig. 2-5 Contact region of a curvilinear-tooth gear pair...........................................................13 Fig. 2-6 Schema of the normal section of rack cutters .............................................................14 Fig. 2-7 Imaginary of the separating rack cutters.....................................................................14 Fig. 3-1 The cutting process of a curvilinear-tooth gear with a disk-type tool [2]...................18 Fig. 3-2 The relationship between imaginary rack cutter and blank ........................................20 Fig. 3-3 Surfaces of the imaginary cutters ∑ g and ∑ p .........................................................21 Fig. 3-4 The axodes of the gears ..............................................................................................21 Fig. 3-5 Imaginary circular-arc rack cutters ∑ g and ∑ p .......................................................23. Fig. 3-6 Normal section of the circular-arc rack cutter ∑ g .....................................................24. Fig. 3-7 Formation schema of the imaginary rack cutter ∑ p and ∑ g ...................................26 ix.

(11) Fig. 3-8 Normal section of the circular-arc rack cutter ∑ p .....................................................27. Fig. 3-9 Kinematical relationship between the imaginary rack cutter Σ g and the gear..........29. Fig. 3-10 Kinematical relationship between the imaginary rack cutter Σ p and the pinion ....32 Fig. 3-11 Relationship between two tangent surfaces ..............................................................33 Fig. 3-12 Computer graph of the circular-arc gear with curvilinear-teeth ...............................38 Fig. 3-13 Computer graph of the circular-arc pinion with curvilinear-teeth ............................39 Fig. 3-14 Tooth surface profiles with different radii of circular-arc rack cutters .....................40 Fig. 3-15 Normal profiles at different cross sections of the gear .............................................41 Fig. 4-1 Schema of normal cross section of the circular-arc rack cutter ∑ g ...........................47 Fig. 4-2 Position of the undercutting line .................................................................................50 Fig. 4-3 Locations of undercutting points with different α ....................................................51 Fig. 4-4 Locations of undercutting points with different Rab ..................................................52 Fig. 4-5 Locations of undercutting points with different number of teeth ...............................53 Fig. 4-6 Pressure angles of rack cutters ∑ g and ∑ p ..............................................................54 Fig. 4-7 Flowchart for determination of the minimum teeth number. without tooth. undercutting ............................................................................................................56 Fig. 4-8 The minimum teeth number for non-undercutting......................................................57 x.

(12) Fig. 4-9 The schematic of shifted cutting .................................................................................58 Fig. 4-10 Flowchart for determination of the shifting coefficient x without tooth undercutting ....................................................................................................................................59 Fig. 4-11 The minimum shifting coefficient x versus number of gear teeth to avoid undercutting ............................................................................................................60 Fig. 5-1 Coordinate systems for simulation of a gear pair meshed with assembly errors........63 Fig. 5-2 Schematic of the relationship between two meshing gear tooth surfaces...................67 Fig. 5-3 Kinematical errors of the gear pair with different pressure angles. under ideal. assembly condition and assembly error ΔC = 0.1mm .............................................74 Fig. 5-4 Kinematical errors of the gear pair with different pressure angles. under axial. assembly misalignment Δγ h = 0.1 and Δγ v = 0.1 ................................................75. Fig. 5-5 Kinematical errors of the gear pair with different pressure angles under mixed assembly errors with ΔC = 0.1mm , Δγ h = 0.1 and Δγ v = 0.1 ............................76 Fig. 5-6 Kinematical errors of the gear pair with different R (i ) under ideal assembly condition and assembly error ΔC = 0.1mm ...............................................................................77 Fig. 5-7 Kinematical errors of the gear pair with different R (i ) under axial assembly misalignment Δγ h = 0.1 and Δγ v = 0.1 ................................................................78 Fig. 5-8 Kinematical errors of the gear pair with different pressure angles under mixed assembly errors with ΔC = 0.1mm , Δγ h = 0.1 and Δγ v = 0.1 ............................79. xi.

(13) Fig. 5-9 Kinematical errors of the gear pair with different ΔR under ideal assembly condition and assembly error ΔC = 0.1mm ...............................................................................80 Fig. 5-10 Kinematical errors of the gear pair with different ΔR under axial assembly misalignment Δγ h = 0.1 and Δγ v = 0.1 ..............................................................81 Fig. 5-11 Kinematical errors of the gear pair with ΔR under mixed assembly errors with ΔC = 0.1mm , Δγ h = 0.1 and Δγ v = 0.1 ...............................................................82. Fig. 5-12 Kinematical errors of the gear pair with different Rab under ideal assembly condition and assembly error ΔC = 0.1mm .............................................................................83 Fig. 5-13 Kinematical errors of the gear pair with different Rab under axial assembly misalignment Δγ h = 0.1 and Δγ v = 0.1 ..............................................................84 Fig. 5-14 Kinematical errors of the gear pair with ΔR under mixed assembly errors with ΔC = 0.1mm , Δγ h = 0.1 and Δγ v = 0.1 ...............................................................85. Fig. 6-1 Schematic relationship between the tooth surface and tangent plane.........................89 Fig. 6-2 Schematic relationship among the coordinate systems and tangent plane .................93 Fig. 6-3 Contact patterns of the gear tooth surface under ideal and error assembly conditions (Example 6-1) .............................................................................................................95 Fig. 6-4 Contact patterns of the gear tooth surface under horizontal axial misalignments (Example 6-1) ...........................................................................................................96 Fig. 6-5 Contact patterns of the gear tooth surface under. vertical axial misalignments. (Example 6-1) ...........................................................................................................97 xii.

(14) Fig. 6-6 Orientation and dimension of the contact ellipse........................................................98 Fig. 6-7 Contact patterns of the gear tooth surfaces under ideal assembly condition with different value of Rab (Example 6-2) ......................................................................99 Fig. 6-8 Effects of design parameters Rab versus the semi-major axis length a of contact ellipse ......................................................................................................................100 Fig. 6-9 Contact ellipses on the gear tooth surface (Example 6-3) ........................................102 Fig. 6-10 Relationship of design parameter R ( g ) to the length of the contact ellipse...........102. xiii.

(15) List of Tables Table 3-1 Some major design parameters for the gear set........................................................36 Table 4-1 Some major design parameters of the generated gear..............................................49 Table 4-2 Location of singular points on tooth surface Σ1 ......................................................49 Table 4-3 Some major design parameters of gear ....................................................................51 Table 4-4 Some major design parameters of the gear with different number of teeth .............53 Table 5-1 Contact Ratios under different design parameters....................................................70 Table 5-2 Some major design parameters of the circular-arc curvilinear-tooth gear pair ........71 Table 5-3 Bearing contacts and kinematical errors of the gear pair under ideal assembly condition .................................................................................................................72 Table 5-4 Bearing contacts and KE of the gear pair under center distance assembly error ΔC = 0.1 mm...........................................................................................................72. Table 5-5 Bearing contacts and kinematical errors of the gear pair under axial misalignments Δγ h = 0.1 .................................................................................................................73. Table 5-6 Bearing contacts and kinematical errors of the gear pair under axial misalignments Δγ v = 0.1 .................................................................................................................73. Table 6-1 Some major design parameters of the circular-arc curvilinear-tooth gear pair ........94 Table 6-2 Some major design parameters of the circular-arc curvilinear-tooth gear pair ......101. xiv.

(16) Nomenclature a. semi-major length of contact ellipse, as shown in Fig. 6-6. b. semi-minor length of contact ellipse, as shown in Fig. 6-6. A. initial point of circular-arc rack cutter Σ j ( j = p, g), as shown in Fig. 3-6. B. end point of circular-arc rack cutter Σ j ( j = p, g), as shown in Fig. 3-6. C( j). center of the circular-arc trace of rack cutter Σ j ( j = p, g), as shown in Fig. 3-7. C. ideal center distance of gear pair, C = r1 + r2. ΔC. center distance variation (in mm). C'. center distance of gear pair with variation ΔC , C ' = C + ΔC. di. perpendicular distance of tooth surface Σ j to common tangent plane T ( j = p, g) (in mm), as shown in Fig. 6-1. lc(i ). parameter of defining location of singular point on rack cutter Σ j ( j = p, g). Lij. 3 × 3 vector transformation matrix transforming from coordinate system S j to Si. mc. contact ratio, defined in Eq.(5.20). M ij. 4 × 4 homogeneous coordinate transformation matrix transforming from coordinate system S j to Si. xv.

(17) M. normal module. n. unit normal vector. ni( j ). unit normal vector of surface Σ j ( j = p, g, 1, 2) represented in coordinate system Si (i = c, f). N (cg ). normal vector of surface ΣC represented in coordinate system S g. OR(i ). center of the circular arc MN of rack cutter Σ j ( j = p, g), as shown in Fig. 3-6. ( r ,θT ) polar coordinate system on common tangent plane T, as shown in Fig. 6-1(a) ri. radius of operating pitch cylinder of the gear blank i ( i =1, 2) (in mm). R( j ). radius of circular-arc of the rack cutter Σ j ( j = p, g). R i( j ). position vector of surface j represent in coordinate system Si. Rab. radius of the disk-type circular-arc cutter. SG. tooth thickness measured along the pitch circle. Si ( X i , Yi , Z i. ). coordinate system i ( i = 1, 2, f, h, m, n, T, v,). Sc( j ) ( X c( j ) , Yc( j ) , Z c( j ) ). coordinate system i attached to rack cutter Σ j ( j = p, g). S r( j ) ( X r( j ) , Yr( j ) , Z r( j ) ). coordinate system r attached to rack cutter Σ j ( j = p, g). T (i ). teeth number of rack cuter Σ j ( j = p, g) xvi.

(18) V (12). relative velocity of body 1 and body 2. Vk(ij ). relative velocity of body i to body j represented in coordinate system Sk. Vi( j ). velocity of surface Σ j ( j = p, g, 1, 2) represented in coordinate system Si. Vr( i ). tangent velocity of contact point on body i ( i = g, 1). Vtr( i ). relative velocity of contact point on body i ( i = g, 1). x. coefficient of cutter shifting. W. face width (in mm). Σj. surface of rack cutter j ( j = p, g). Σi. surface of gear i ( i = 1, 2). α. normal pressure angle of rack cutter. α ptip. instance normal pressure angle on the tip of rack cutter Σ p , shown in Fig. 4-6. α gtip. instance normal pressure angle on the tip of rack cutter Σ g , shown in Fig. 4-6. ε. transformed angle of coordinate system (in degree). δ. transformed angle of coordinate system (in degree). δ bl. backlash of the gear. φ1E. rotation angle of final meshing position of the gear. xvii.

(19) φ1S. rotation angle of initial meshing position of the gear. φi. rotation angle of gear blank i ( i =1, 2) (in degree). φi'. rotation angle of gear gear i when meshing with each other ( i =1, 2) (in degree). Δφ2'. transmission error (in arc-sec.). γ ( j). trace parameter of circular-arc rack cutter j ( j = p, g). ( j) γ max. maximum value of γ ( j ) which corresponds to the final position of working part of disk-type rack cutter Σ j ( j = p, g). ( j) γ min. minimum value of γ ( j ) which corresponds to the initial position of working part of disk-type rack cutter Σ j ( j = p, g). Δγ h. horizontal axial misalignment (in degree), as shown in Fig. 5-1. Δγ v. vertical axial misalignment (in degree), as shown in Fig. 5-1. θ ( j). parameter of the circular-arc-lined cutting blade surface of Σ j ( j = p, g). ( j) θ max. maximum value of θ ( j ) which corresponds to the final position of working part of circular-arc normal section for rack cutter Σ j ( j = p, g). ( j) θ min. minimum value of θ ( j ) which corresponds to the initial position of working part of circular-arc normal section for rack cutter Σ j ( j = p, g). xviii.

(20) θuc( j ). value of the θ ( j ) where singular point is appeared on rack cutter Σ j ( j = p, g). ωt. rotational speed of the disk-type circular-arc cutter, as shown in Fig. 3-1. ω1. rotational speed of gear blank i ( i =1, 2). Δ1. equality equation for tooth undercutting, defined in Eq.(4.9) and Eq.(4.17). Δ2. equality equation for tooth undercutting, defined in Eq.(4.10) and Eq.(4.18). Δ3. equality equation for tooth undercutting, defined in Eq.(4.11) and Eq.(4.19). Δ4. equality equation for tooth undercutting, defined in Eq.(4.12) and Eq.(4.20). ΔR. difference between R ( g ) and R ( p ). FEA. finite element analysis. KE. kinematical errors. TCA tooth contact analysis. xix.

(21) CHAPTER 1 Introduction 1.1 Background Gears play a very important role in transmissions. One of the first documented uses of gears can be traced back to more than forty centuries ago when the Egyptians used gear transmissions in their camel-driven watering facilities. Today, gears are widely used in industry for power transmissions because of their high efficiency. Gear transmissions are used in a wide variety of products, such as printer mechanism, mechanical watches, power plants, vehicle transmissions, etc.. Gears. Intersected Axes. Parallel Axes. Spur Gear. Crossed Axes. Helical Gear. Single Helical Gear. Crossed Helical Gear. Herringbone Gear. Fig. 1-1 Classification of gears. 1. Hypoid Gear. Worm Gear.

(22) There are a lot of kinds of gears, such as spur gear, bevel gear, helical gear, worm gear and so on, which are used in many different ways. Gears are used to transmit the power between two shafts, and their classifications, are shown in Fig. 1-1, which are sorted by the axis relations. The classifications of the gears are parallel axes, intersecting axes, and crossed axes [1]. Cylindrical gears, such as spur gear, helical gear and herringbone gear, are widely used to transmit powers between parallel shafts and the shafts are rotated in opposite directions. The tooth traces of the spur gear are parallel to the rotational axis as shown in Fig. 1-2(a). The advantages of spur gears are easy to manufacture and inexpensive, but the spur gear pair tends to be noisy at high speed and is sensitive to the axial misalignments. Step gear has two or more spur gears fastened together and each gear is advanced relative to the adjacent one by a small amount of screw motion. Helical gear can be viewed as a stepped gear with an infinite number of steps. The tooth trace of the helical gear is shown in Fig. 1-2(b). The major advantage of helical gears is that they are engaged with a gradual contact between the teeth compared with spur gear, which make contact across the entire face at once during operation. This gradual contact results in less noisy and longer life cycle of the gear pair during operation. Whereas the contact force of helical gear pair is not perpendicular to its rotational axis, an axial thrust force is produced to push the gears apart. The herringbone gear made by two opposite directions helical gears are bolted together, has been developed to reduce or eliminate the axial thrust force. The trace of the herringbone gear is shown in Fig. 1-2(c). The advantages of herringbone gear include the advantages of helical gear and also have no axial thrust force during operation because of the balance of thrust force. A herringbone gear is bolted by two helical gears, thus the manufacture cost is higher.. 2.

(23) (a) Spur Gear. (b) Helical Gear. (c) Herringbone Gear. Fig. 1-2 Various types of tooth traces for parallel axis gears. 1.2 Literature Reviews Curvilinear-tooth gear has many excellent performances, such as higher bending strength, lower noise, better lubrication effect, no axial thrust force, etc. It will be introduced in the next chapter. The generation method is first proposed by Liu [2] with a face mill-cutter on a special machine. Tseng and Tsay [3] developed a mathematical model of cylindrical gears with curvilinear shaped teeth by using an imaginary rack cutter and investigated the tooth undercutting of curvilinear-tooth gears. However, in modern gear practice and manufacturing, the gears are usually generated by a hobbing or CNC cutting process. Tseng and Tsay [4] proposed a generating process of using a hob cutter which has a higher cutting efficiency than a rack cutter, the mathematical model for generation, the theoretical analysis on the tooth undercutting and secondary cutting of the gear teeth in the generation process were investigated with numerical examples. The kinematical errors (KE) induced by gear axial misalignments is an efficient factor to predict the contact behaviors of gear pair, such as the noise, vibration, etc. Tsay [5] applied 3.

(24) TCA (tooth contact analysis) techniques to simulate the meshing conditions for involute helical gears. The TCA technique was also used in cylindrical gears with curvilinear shaped teeth by Tseng and Tsay [6]. The circular-arc tooth gear is a kind of gears with a higher loading compatibility. Circular-arc helical gears have been proposed by Wildhaber [7] in 1926 and Nobikov [8] in 1956. These types of gears became very popular in 1960s because of their low contact stress. Litvin and Tsay [9] investigated the mathematical model and TCA for the circular-arc helical gears by considering the gears were generated by imaginary circular-arc rack cutters. Recently, Litvin et al. [10][11][12][13] proposed a concept of tooth surface topology that will reduce the KE and localize the bearing contact. The KE can be reduced by imposing a pre-designed parabolic-like KE on the gear tooth surfaces, which may absorb discontinuous KE caused by axial misalignments [10]. Meanwhile, the localization of bearing contact can be achieved by tooth modifications to obtain point contacts instead of line contacts, whether under an ideal or a misaligned meshing condition.. 1.3 Motivations Curvilinear-tooth gear pair can be used to transmit powers between two parallel shafts. It can sustain a higher bending and contacting strength, and can operate quietly and smoothly. Besides, no axial thrust force occurs for the curvilinear-tooth gear pair is during its operation. Tseng [3] also investigated the TCA simulation of the curvilinear-tooth gear. The gear designers and manufacturers always aspire to design and manufacture a gear with a high loading capacity and low sensitivity to assembly errors. In order to achieve this goal, researchers focus on the development of advanced materials and new methods of heat 4.

(25) treatment or on design of stronger tooth profiles and methods of gear manufacturing. There are many methods to design for a stronger tooth profile, such as adding circular root fillets [14], predicting the power loss for high contact ratio spur gears with nonstandard addendum proportions by using cutter elongation of tool shifting [15], or modifying the straight edge rack cutter into a circular-arc profile [16][17], etc. A gear pair with a higher loading can be designed by a higher contact ratio or a larger contact area of tooth surfaces. A higher contact ratio can reduce the stress by distributing load among neighboring teeth. It is good practice to maintain a gear contact ratio larger than 1.2. With a higher contact ratio, loadings on the gear teeth surface can be dispersed by more contact gear teeth. The contact area on tooth surface can efficiently disperse the contact stress during contact. It is said that the curvilinear-tooth gear has several advantages and characteristics. In order to obtain a curvilinear-tooth gear set which can endure a higher contacting strength, the profile of the involute curvilinear gear can be considered to be modified. The normal section of the rack cutter is designed as a circular-arc to produce the tooth profile. The tooth which generated by a circular-arc rack cutter is called a circular-arc tooth. Comparing the circular-arc tooth with involute tooth, the minor axis of the contact ellipse of the tooth generated by a circular-arc cutter is larger than that of an involute tooth. It means that there is a larger contact area for the modified curvilinear-tooth gear and this result in a smaller contact stress. In this thesis, two conjugate circular-arc rack cutters are used to generate the curvilinear gear pair respectively with circular-arc teeth to improve the bearing contact of the curvilinear gear.. 1.4 Thesis Overviews In this thesis, a gear with curvilinear teeth generated by a disk-type circular-arc cutter is 5.

(26) briefly proposed. In all, this thesis includes seven chapters, and the contents of each chapter are described as follows: 1. CHAPTER 1 is the introduction of the background, motivations, literature reviews and a brief overview of this thesis. 2. CHAPTER 2 introduces the curvilinear-tooth gear and the circular-arc tooth gear. Advantages and disadvantages of these two gears are briefly discussed. 3. The mathematical model of the circular-arc curvilinear-tooth gear set has been developed in CHAPTER 3. The circular-arc profile on its curvilinear-tooth gear is generated by a disk-type rack cutter with circular-arc normal section geometry. However, the generation method of the circular-arc curvilinear-tooth gear is simulated by a circular-arc profile rack cutter. Equations of the circular-arc curvilinear-tooth gear can be obtained by applying the theory of gearing and the equations of the cutter. Computer graph illustrates the shape of the gear pair, and it also proves the correctness of the developed mathematical model. In addition, the equations of the gear set are the bases for further investigations, such as tooth contact analysis, contact ratio and contact ellipses. 4. CHAPTER 4 includes the study of tooth undercutting of the proposed gear pair. The undercutting points are the singular points on the generated tooth surface. Consequently, it can be derived according the theory of gearing. Some tables and graphs are built up by applying the developed computer programs to give suggestions for the manufactures and designers to prevent the tooth undercutting. 5. The TCA technique is applied to find the KE of the circular-arc curvilinear-tooth gear in CHAPTER 5. Tooth contact simulation model includes the center distance 6.

(27) variation, horizontal axial misalignment and vertical axial misalignment of the gear set. The KE curve are tried to pre-design into a parabolic form in order to prevent the jump of the junction. Several numerical examples are presented to demonstrate the influences of gear design parameters and assembly errors on the KE and contact analysis of the mating gear pair. 6. Contact pattern is an important symbol of gear loading capability. The contact pattern can be simulated based on the developed mathematical model of the gear set. Some numerical examples express the relationships between the design parameters and contact patterns. In this thesis, the contact patterns are evaluated by applying the surface topology method. 7. CHAPTER 7 concludes this thesis by summarizing the major findings of the accomplished work, and also discusses some potential topics for future study.. 7.

(28) CHAPTER 2 The Curvilinear-Tooth Gear and the Circular-Arc Tooth Gear 2.1 Introduction The shape of a gear can be described into two ways: one mentioned the profile of the cross section of gear. Fig. 2-1(a) is the involute-shape gear profile of a spur gear shown in Fig. 2-1(b). As everyone knows that most applications of the gear profiles are in the involute shape.. Fig. 2-1 Schema of gear profile and gear tooth trace of a spur gear. The other description for a gear shape is the trace of its tooth, e.g. gears with trace parallel to the rotational axis as marked with shadow zones in Fig. 2-1(b) is called a spur gear. Different profiles and traces of gears have their own characteristics, the usage of gear is according to there characteristics. In this chapter, two kinds of gear types will be introduced.. 8.

(29) One is the curvilinear-tooth trace gear and another one is the circular-arc tooth profile gear.. 2.2 The Curvilinear-Tooth Gear Curvilinear-tooth gear is one kind of cylindrical gears as shown in Fig. 2-2, and has circular-arc tooth traces. Since the circular arc tooth trace is symmetrical, thus gear axial thrust force may be eliminated. Besides, the curvilinear-tooth gear has the characteristics as follows [2]: 1. Curvilinear-tooth gears endure a higher bending and contacting strength; thus, the size of transmission gear boxes may be reduced while transmitting the power or torque. 2. The teeth of curvilinear-tooth gear pair engaged simultaneously smoothly with low noise. 3. Lubrication oil is retained within the concave tooth surfaces, therefore there is always an oil film between the two engaged surfaces, resulting in good lubrication qualities. The spur, helical, and herringbone gear pairs with parallel axes are in line contact. However, the line contact of gear tooth surfaces can be realized only for an ideal contact of the gear drive. In reality, the gear assembly errors of axial misalignments and errors of lead angles result in the so-called edge contact, a specific instantaneous contact caused by curve-to-surface tangency as shown in Fig. 2-3. Therefore, the stress is concentrated at the contact edge. Gear manufacturers always cut gears with tooth crowning to make the line contact become a point contact, and to avoid the edge contact of gears as shown in Fig. 2-4. The point contact of gears will spread over a contact ellipse with loadings. When a gear set 9.

(30) has an axial misalignment without crowning, tooth edge contact will occur and this will induce a serious stress concentration. In this circumstance, the noise and vibration of the gear pair may occur during the gear meshing. The curvilinear-tooth gear also has a characteristic similar to a crowned tooth, as shown in Fig. 2-5. The contact type of the curvilinear-tooth gear is in a point contact situation no matter there is an axial misalignment or not.. Fig. 2-2 Schema of curvilinear-tooth gear [3]. 2.3 The Circular-Arc Tooth Gear The theory of circular-arc tooth gears is first proposed by Wildhaber [7] in U.S. patent, 1926. Later, Novikov [8], also proposed another patent for helical gears with circular-arc teeth. Therefore, the circular-arc tooth gear is also called the Wildhaber-Novikov (W-N) gear. The profiles of the Wildhaber’s and Novikov’s are quite similar to each other. It is noted that the circular-arc tooth gears are generated by two imaginary rack cutters, described in two 10.

(31) Line contact. Contact stress Rear end Fillet z. Top land Contact region. y. Front end x. (a) Helical gear pair without axial misalignment. Contact stress. Edge contact. Rear end Fillet z. Front end. y x. (b) Helical gear pair with axial misalignment. Fig. 2-3 Contact region of a helical gear pair. 11. Contact region Top land.

(32) Contact point. Contact stress Rear end Fillet. z y. Top land Contact region. x. Front end. (a) Helical gear with crowning teeth pair without axial misalignment. Contact point. Contact stress Rear end Fillet. z y. Top land Contact region. Front end. x. (b) Helical gear with crowning teeth pair with axial misalignment. Fig. 2-4 Contact region of a crowned teeth gear pair. 12.

(33) Contact point. Contact stress Rear end. Fillet. Top land. z y. Contact region. x. Front end (a) Curvilinear gear pair without axial misalignment. Contact stress. Contact point. Rear end. Fillet. Top land. z. Contact region. y. Front end. x. (b) Curvilinear gear pair with axial misalignment. Fig. 2-5 Contact region of a curvilinear-tooth gear pair. 13.

(34) circular-arc curves as shown in Fig. 2-6, where the R ( p ) and R ( g ) are the radii of the two cutters, where OR( p ) and OR( g ) are the center of these circular-arc. In order to generate a pair of conjugate gear surfaces, the rack cutters can be considers as two separate rack cutters which may be imagine as a mold and its corresponding cast express in Fig. 2-7. One of these rack cutters generates the pinion and the other generates gear.. (g). (p). R. R. (g). (p). OR. Rack cutter Σg. Rack cutter Σp. OR. Fig. 2-6 Schema of the normal section of rack cutters. Cutter Σ g. Rack cutter centrode. Cutter Σ p. Fig. 2-7 Imaginary of the separating rack cutters. 14.

(35) Since the normal section of each rack cutter has circular-arc edges, thus the tool needs a higher precision requirement. However, the technology is brilliant enough to produce high precision circular-arc cutter now. It is known that a circular-arc gear has the following two major advantages: 1. Reducing the contacting stress. Theoretically, the gear surfaces are contact when two gears are meshed with each other. However, the contact area spread over a contact ellipse due to elastic deformation under loading. After operating for a period of time, the two meshing surfaces fit each other into a suitable shape for meshing. It is said that the curvatures on the contact point of these two gears tend to equal gradually. The contact surfaces enlarge quickly with the operation loading. Hence, the loading limit is three to five times higher than an involute gear pair [18]. 2. Better conditions of lubrication. The sliding velocity of every point on the gear surface are equal in a slight value, it means the wear problem of the circular-arc tooth gear is not serious. Rotation speed of the circular-arc tooth gear is high on the vertical direction of the gear surface, so the oil- membrane can be formed easily. Thus the lubrication of the circular-arc gear pair is ten times better than an involute gear pair [19]. With the better lubrication, the life time of this type of gears becomes longer and abrasion becomes less serious. The circular-arc gear can provide power transmission with a high loading, and it is widely used in the transmission system of hoists, elevators, mechanisms for mining industry, and many transmission systems with a high load/weight ratio. Although the circular-arc gear tooth surfaces are in point contact at every instant, the contact area is also larger than that of involute gear pair under load with elastic deformation.. 15.

(36) Nevertheless, there are also three major disadvantages of the circular-arc gear as below: 1. The profile of the rack cutter is in a circular-arc shape, and it is manufactured by several processes requiring high precision, and the heat treatment process may somewhat cause distortion of the cutter shape. Therefore, the menufacturing cost of this cutter is higher. 2. There are no international standards for the circular-arc gears, e.g.. the radius of circular-arc edge. Hence, the circular-arc gear cannot be exchanged facilely. It means the comparability with the circular-arc gear is not good. 3. The kinematical error occurs if there has error of center distance. However, the variation of center distance takes no effects for involute gear pair.. 2.4 Remarks Gears are used in power transmissions. Therefore, good gears always include two essential factors: high loading capacity and low sensitivity to assembly errors. The goal can be achieved by designing the profile and tooth-trace for the gear pair. Two types of gears are introduced in this chapter, one is the curvilinear-tooth gear which has a curvilinear tooth-trace, and the other one is the circular-arc tooth gear which is generated by two conjugate rack cutters. Both of these two gears benefit to developing the performance of the gear pair.. 16.

(37) CHAPTER 3 Mathematical Model of Gears with Curvilinear-Teeth Generated by a Disk-Type Circular-Arc Cutter 3.1 Introduction In this chapter, the mathematical model of curvilinear-teeth generated by the disk-type circular-arc cutter will first be developed. Based on the theory of gear [10][20], the surface equation of circular-arc gears with curvilinear-teeth is derived as the envelope of the locus of circular-arc cutter. Hence, an imaginary rack cutter surface with circular-arc normal section will be first established to simulate the generation of the gear tooth.. 3.2 Mathematical Model of Circular-Arc Gears with Curvilinear-Teeth 3.2.1 Generation Method The curvilinear-tooth gear is generated by the cutting machine as shown in Fig. 3-1. Gear teeth are produced by a rotating disk-type cutter. The spindle of the disk-type cutter with radius Rab rotates on the axis B − B with an angular velocity ωt and translating velocity. ω1r1 to the right, where r1 is the pitch radius of the gear blank and ω1 is the angular velocity of it. The cutting process of a curvilinear-tooth gear was developed by Liu [2] as steps below: 1. The gear blank rotates with an angular velocity ω1 in clockwise. At this moment, the disk-type cutter rotates with an angular velocity ωt in counterclockwise and translates with a velocity r1ω1 to right. A curvilinear-tooth space may be generated on the gear blank by this process. 17.

(38) W ωt z y Rab. B. Disk-type cutter r 1ω 1. B r1. Work piece ω1. Circular-arc shape cutter x y. Fig. 3-1 The cutting process of a curvilinear-tooth gear with a disk-type tool [2]. 18.

(39) 2. After generating the tooth space of the gear, the gear blank stop rotating and then spin to the next working position with considering adjustment for the backlash δ bl . 3. In order to cut the other tooth space, the disk-type cutter rotates with an angular velocity ωt in clockwise and translates with a velocity r1ω1 to left, the gear blank rotates in the counterclockwise with angular velocity ω1 in the mean time (contrary to the Fig. 3-1), then the other tooth space is generated by this process. In order to prevent the interfering, the radius of the disk-type cutter used to generate the different side of the tooth surface should be modified. In this thesis, the difference between the different side of the gear tooth is defined as S G . 4. Generating cycle is repeated and this sequence is continued until all the spaces and teeth are formed. Obviously, the structure of cutting machine to produce the curvilinear-tooth gear is complex, and the assembling of the cutter is so inconvenient. Adjusting and grinding is also arduous for the curvilinear-tooth gear because of its special trace. Therefore, the manufacturing cost of curvilinear-tooth gears is higher than that of spur gears or helical gears. In this thesis, the cutting machine is adopted and the straight line cutter shape is replaced by the circular-arc cutter shape. Gears made by this process are called the curvilinear-tooth gears generated by a disk-type circular-arc cutter. Although the gear is generated by a disk-type cutter, however it can be considered that the gear is generated by an imaginary rack cutter as shown in Fig. 3-2. The surfaces of the two cutters are labeled as ∑ g and ∑ p , and we may imagine that the two surfaces are rigidly connected to each other and are in tangency along the curve a − a as shown in Fig. 3-3. The normal section of each rack cutter is a circular arc. Fig. 3-4 shows the relationships among the. 19.

(40) pitch plan of rack cutter and the axode of the two gears. The locus of an instantaneous axis of rotation represented in a coordinate system that is attached to a movable body is known as the body axode.. r. ω. 1. 1. rω 1. 1. Fig. 3-2 The relationship between imaginary rack cutter and blank. 20.

(41) Σ. g. a Π. a Σ. p. Fig. 3-3 Surfaces of the imaginary cutters ∑ g and ∑ p. Pinion axode. ω2 r2. I. O2. V. Pitch plane of the rack cutter. I r1 ω1. O1 Gear axode. Fig. 3-4 The axodes of the gears. 21.

(42) The two gears are rotating with an angular velocity ω1 and ω2 in opposite directions about their respective rotation axes. The radii of the two axodes are r1 =. V. ω1. , r2 =. V. ω2. . The. tangent line I − I of the two axodes is called the instantaneous axis of rotation. The tangent plane to the axodes is also the pitch plane of the rack cutters. The gear is generated by the cutter ∑ g and the pinion is generated by the cutter ∑ p . Namely, the mathematical model of the gear is generated by the surface ∑ g , and another one is generated by ∑ p . Finally, the mathematical model of the generated gear tooth surface is the combination of the meshing equation and the locus equation of imaginary rack cutter surface. Points on gear surface can be calculated by solving the developed gear mathematical model by using numerical methods.. 3.2.2 Equation of the Disk-Type Circular-Arc Cutter The teeth surfaces of a pair of conjugate circular-arc with curvilinear-teeth gear can be generated by two imaginary circular-arc rack cutters with curvilinear trace. These two rack cutters of ∑ g and ∑ p are shown in Fig. 3-5(a). The normal sections of the cutters are also shown in Fig. 3-5(b). Parameters A and B determine the initial and end points of the circular-arc curve, respectively, as shown in Fig. 3-6. OR( g ) is the center of the circular-arc MN with a radius of R ( g ) ; S G is the tooth thickness measured along the pitch line of the. rack cutter; θ ( g ) is the design parameter of the rack cutter which determines the point on the circular arc MN . The normal section of the circular-arc rack cutter is rigidly attached to coordinate system S r( g ) ( X r( g ) , Yr( g ) , Z r( g ) ) with its origin Or( g ) , as shown in Fig. 3-6. The circular-arc curve MN can be represented in coordinate system S r( g ) ( X r( g ) , Yr( g ) , Z r( g ) ) as follows: xr( g ) = − R ( g ) ( sin α − sin θ ( g ) ) , 22.

(43) ⎧ SG ⎫ yr( g ) = ± ⎨ R ( g ) ( cos α − cos θ ( g ) ) − ⎬, 2 ⎭ ⎩. (3.1). zr( g ) = 0 ,. and (g) ⎡ R ( g ) sin α − B ⎤ sin α + A ⎤ (g) −1 ⎡ R , = θ sin max ⎥ ⎢ ⎥, (g) R R( g ) ⎣ ⎦ ⎣ ⎦. (g) (g) (g) θ min ≤ θ ( g ) ≤ θ max , θ min = sin −1 ⎢. (g). Xr. (g). Yr ΣgR. ΣgL. Σg. (p). Xr. ΣpL. ΣpR (p). Yr. Σp. (a). (b). Fig. 3-5 Imaginary circular-arc rack cutters ∑ g and ∑ p. 23. (3.2).

(44) Xr. (g). M A. Yr. (g). Or. (g). B N α. (g). R. G. S. (g). θ. (g). α. θmax. OR (g). (g). θmin. Fig. 3-6 Normal section of the circular-arc rack cutter ∑ g. (g) (g) to θ max , where θ ( g ) is a design parameter of the circular-arc rack cutter, ranging from θ min. and α denotes the normal pressure angle defined in Fig. 3-6, and R ( g ) represents the radius of the circular-arc MN . The symbol “ ± ” represents the different side of the cutter ∑ g , where “ − “ indicates the right side circular-arc rack cutter ∑ gR and “ + “ indicates ∑ gL . To form a circular-arc rack cutter with curvilinear-trace, the normal section of the circular-arc (Fig. 3-6) should attach to coordinate system S r( g ) ( X r( g ) , Yr( g ) , Z r( g ) ) , as shown in Fig. 3-7. It is noted that the circular-arc ab is the cutting path of the disk-type rack cutter. The cutting path of the cutter consequently causes a crowning effect on the generated tooth flank. Coordinate system Sc( g ) ( X c( g ) , Yc( g ) , Z c( g ) ) is rigidly attached to the middle of transverse section of the imaginary rack cutter. Coordinate system S r( g ) ( X r( g ) , Yr( g ) , Z r( g ) ) is attached to. 24.

(45) coordinate system Sc( g ) ( X c( g ) , Yc( g ) , Z c( g ) ) with a variable angle γ ( g ) . The center of the curvilinear-trace is located at point C ( g ) with a radius of Rab and W represents the width of the gear pair. The. imaginary. rack. cutter. surface. ∑g. represented. in. coordinate. system. Sc( g ) ( X c( g ) , Yc( g ) , Z c( g ) ) can be obtained by applying the following homogeneous coordinate. transformation matrix equation as follows:. R (cg ) = M cr R (r g ) , 0 ⎡1 ⎢0 cos γ ( g ) where M cr = ⎢ ⎢0 − sin γ ( g ) ⎢ 0 ⎣0. (3.3) 0 ⎤ (g) ⎥ Rab (1 − cos γ ) ⎥ . Rab sin γ ( g ) ⎥ ⎥ 1 ⎦. 0 sin γ cos γ ( g ) 0 (g). (3.4). Matrix M cr is a homogeneous coordinate transformation matrix transforming from coordinate system S r( g ) ( X r( g ) , Yr( g ) , Z r( g ) ) to Sc( g ) ( X c( g ) , Yc( g ) , Z c( g ) ) . Using the Eq.(3.1),. Eq.(3.3), and Eq.(3.4) the mathematical model of the disk-type circular-arc cutter represented in coordinate system Sc( g ) ( X c( g ) , Yc( g ) , Z c( g ) ) can be obtained as following:. R c( g ). ⎡ xc( g ) ⎤ ⎡ ⎤ xr( g ) ⎢ (g) ⎥ ⎢ (g) (g) (g) (g) (g) ⎥ y y cos γ + zr sin γ + Rab (1 − cos γ ) ⎥ , = ⎢ c( g ) ⎥ = ⎢ r ( g ) ⎢ zc ⎥ ⎢ yr sin γ ( g ) + zr( g ) cos γ ( g ) + Rab sin γ ( g ) ⎥ ⎢ ⎥ ⎢ ⎥ 1 ⎣ 1 ⎦ ⎣ ⎦. (3.5). and ⎡ W ⎤ (g) −1 ⎡ W ⎤ ⎥ , γ max = sin ⎢ ⎥. ⎣ 2 Rab ⎦ ⎣ 2 Rab ⎦. (g) (g) (g) γ min ≤ γ ( g ) ≤ γ max , γ min = − sin −1 ⎢. 25. (3.6).

(46) Substituting Eq.(3.1) into Eq.(3.5) yields the imaginary rack cutter surface represented in coordinate system S r( g ) ( X r( g ) , Yr( g ) , Z r( g ) ) is as follows:. R (cg ). ⎡ ⎤ − R ( g ) ( sin α − sin θ ( g ) ) ⎢ ⎥ G ⎥ ⎡ xc( g ) ⎤ ⎢ ⎛ ⎞ S (g) (g) + R ( g ) ( cos α − cos θ ( g ) ) ⎟ ⎥ ⎢ ( g ) ⎥ ⎢ R ab (1 − cos γ ) ± cos γ ⎜ − y ⎝ 2 ⎠⎥ . = ⎢ c( g ) ⎥ = ⎢ ⎥ ⎢ zc ⎥ ⎢ ⎛ ⎞ ⎛ SG (g) (g) (g) ⎞ ⎥ ⎢ ⎥ ⎢ − R ( cos α − cos θ ) ⎟ ⎟ sin γ ⎜ Rab ± ⎜ ⎣ 1 ⎦ ⎢ ⎥ 2 ⎝ ⎠⎠ ⎝ ⎢ ⎥ 1 ⎢⎣ ⎥⎦. (3.7). The normal cross section of rack cutter Σ p is shown in Fig. 3-8. Similarly, the mathematical model of the rack cutter surface Σ p can be established by following the above-mentioned steps:. (i). Xc. i = g, p. X(i)r Z(i)r Z(i)c a O(i) r. (i). Oc. W 2. γ(i) Rab. Yc(i) C(i). W Yr(i). b Fig. 3-7 Formation schema of the imaginary rack cutter ∑ p and ∑ g. 26.

(47) SG Xr. (p). M B. Yr. (p). Or. (p). A. N. (p). R. (p). θ. α. α. OR (p). (p). θmax. (p). θmin. Fig. 3-8 Normal section of the circular-arc rack cutter ∑ p. R (c p ). ⎡ ⎤ − R ( p ) ( sin α − sin θ ( p ) ) ⎢ ⎥ G ⎥ ⎡ xc( p ) ⎤ ⎢ ⎛ ⎞ S ( p) ( p) + R ( p ) ( cos θ ( p ) − cos α ) ⎟ ⎥ ⎢ ( p ) ⎥ ⎢ Rab (1 − cos γ ) ± cos γ ⎜ − y ⎝ 2 ⎠⎥ . = ⎢ c( p ) ⎥ = ⎢ ⎥ ⎢ zc ⎥ ⎢ ⎛ ⎛ SG ⎞⎞ ⎥ ⎢ ⎥ ⎢ − R ( p ) ( cos θ ( p ) − cos α ) ⎟ ⎟ sin γ ( p ) ⎜ Rab ± ⎜ ⎣ 1 ⎦ ⎢ ⎥ ⎝ 2 ⎠⎠ ⎝ ⎢ ⎥ 1 ⎥⎦ ⎣⎢. (3.8). and the unit normal vector of the generating surface Rc( g ) is obtained by:. N c( g ) =. ∂R c( g ) ∂R c( g ) × . ∂θ ( g ) ∂γ ( g ). (3.9). and. 27.

(48) n c( g ) =. N c( g ) . N (cg ). (3.10). Eqs.(3.7), (3.9) and (3.10) result in the unit normal vector of the generating surface represented in the coordinate system Sc( g ) ( X c( g ) , Yc( g ) , Z c( g ) ) as follow:. n c( g ). ⎡n (Xcg ) ⎤ ⎡ ⎤ sin θ ( g ) ⎢ (g) ⎥ ⎢ (g) (g) ⎥ = ⎢nYc ⎥ = ⎢ ±(− cos γ cos θ ) ⎥ , ⎢n (Zcg ) ⎥ ⎢ ± sin γ ( g ) cos θ ( g ) ⎥ ⎣ ⎦ ⎣ ⎦. (3.11). where the upper sign of symbol “ ± ” indicates the left-side of the rack cutter surfaces. Similarly, the unit normal vector of the rack cutter surface ∑ p can be also obtained by following the similar process:. n (c p ). ⎡n (Xcp ) ⎤ ⎡ ⎤ sin θ ( p ) ⎢ ( p) ⎥ ⎢ ( p) ( p) ⎥ = ⎢nYc ⎥ = ⎢ ±(− cos γ cos θ ) ⎥ . ⎢n (Zcp ) ⎥ ⎢ ± sin γ ( p ) cos θ ( p ) ⎥ ⎣ ⎦ ⎣ ⎦. (3.12). The relative velocity of the gear with respect to the rack cutter ∑ g can be obtained by considering the gear generation mechanism as shown in Fig. 3-9. The rack cutter translates to the left tangent to the axode of the gear. Therefore, the velocity of the point on the rack cutter represented in coordinate system S f ( X f , Y f , Z f. V. (1) f. ⎡ 0 ⎤ = ⎢⎢ −ω1r1 ⎥⎥ , ⎢⎣ 0 ⎥⎦. ). is:. (3.13). where r1 is the pitch radius of the gear blank, and ω1 is the rotation speed of the gear blank during its generation. 28.

(49) (g). Xc. Xf. (g). Zc. I1. V. Z f, Z 1. r1f1 (g). Oc. r2 I1. X1 f1. r1 Y (g) 1. Of ,O. (C) c. Pitch plane of the rack cutter. Y1. ω1 Yf Gear axode. Fig. 3-9 Kinematical relationship between the imaginary rack cutter Σ g and the gear. 29.

(50) The velocity of the point on the rack cutter ∑ g can be expressed in the coordinate system S f ( X f , Y f , Z f. V f( g ). ). as follow:. ⎡( yc( g ) − r1φ1 ) ω1 ⎤ ⎢ ⎥ = ω1 × Rc( g ) + Oc( g )O f × ω1 = ⎢ − ( xc( g ) + r1 ) ω1 ⎥ . ⎢ ⎥ 0 ⎢ ⎥ ⎣ ⎦. (3.14). The relative velocity of the gear with respect to the left and right side of the cutter ∑ g can be attained by subtracting Eq.(3.13) and Eq.(3.14):. V f( p1) = V f( p ) − V f(1). ⎡( r1φ1 − yc( p ) ) ω1 ⎤ ⎢ ⎥ xc( p )ω1 =⎢ ⎥, ⎢ ⎥ 0 ⎢⎣ ⎥⎦. (3.15). where. xc( g ) = − R ( g ) ( sin α − sin θ ( g ) ) ,. (3.16). ⎛ SG ⎞ + R ( g ) ( cos α − cos θ ( g ) ) ⎟ . yc( g ) = Rab (1 − cos γ ( g ) ) ± cos γ ( g ) ⎜ − ⎝ 2 ⎠. Similarly, the relative velocity of the pinion with respect to the left and right sides of the rack cutter ∑ p can also be attained by applying the same steps according to the kinematical relationships shown in Fig. 3-10:. V f( p 2) = V f( p ) − V f(2). ⎡( r2φ2 − yc( p ) ) ω2 ⎤ ⎢ ⎥ xc( p )ω2 =⎢ ⎥, ⎢ ⎥ 0 ⎢⎣ ⎥⎦. (3.17). where, xc( p ) = − R ( p ) ( sin α − sin θ ( p ) ) ,. (3.18) 30.

(51) ⎛ SG ⎞ + R ( p ) ( cos θ ( p ) − cos α ) ⎟ . yc( p ) = Rab (1 − cos γ ( p ) ) ± cos γ ( p ) ⎜ − ⎝ 2 ⎠. Fig. 3-11 illustrates the geometrical relationship between the contact surfaces and tangent point. The contact points on gear surfaces ∑1 , ∑ 2 are M 1 and M 2 , respectively. T is the tangent plane of these two surfaces. Relative velocity V (12) is defined in physical terms as the velocity of point M 1 of ∑1 as seen by an observer at point M 2 of ∑ 2 . Theoretically, in the generation process, gear and cutter are in pure rolling or sliding on the contact surfaces. It means that these two contact surfaces never embed into each other. Thus, the relative velocity of the gear with respect to the cutter along their common normal direction is equal to zero. Then, it can be said that the relative velocity V (12) lies on the common tangent surface and perpendicular to the unit normal vector n at the common tangent point. Therefore, the following equation must be observed:. n ⋅ V (12) = 0 .. (3.19). Eq.(3.19) is called the “meshing equation”. Based on the meshing equation, parameters. φ1 and φ2 can be expressed in the following implicit form:. and. f (θ ( g ) , γ ( g ) , φ1 ) = 0 f (θ ( p ) , γ ( p ) , φ2 ) = 0. ,. (3.20). where φ1 and φ2 are the rotation angle of the gear and the pinion, respectively. The meshing equation of the rack cutter ∑ g with the gear can be obtained by solving the equation: n (fg ) ⋅ V f( g1) = 0 ,. (3.21). 31.

(52) Xc. (p). Xf Zc. (p). ω2 f2 X2 V. I2. r2 O 2 (p). I2. r1. Y2. Z2. Pinion axode. Pitch plane of the rack cutter. Yc. (p). Zf. r2f2 Of. Yf. Fig. 3-10 Kinematical relationship between the imaginary rack cutter Σ p and the pinion. 32.

(53) n. Σ1 M1 M2. T V. (12). Σ2. Fig. 3-11 Relationship between two tangent surfaces. substituting Eqs.(3.11) and (3.15) into Eq.(3.21) yields: f (γ. (g). ,θ. (g). R( g ) , φ1 ) = − (∓2 Rab − S G + 2 R( g ) cos α − 2 R ( g ) cos θ ( g ) ) 4. {2 cos γ. (g). ⎡⎣ r1 (−1 + ω1 ) cos θ ( g ) + R ( g )ω1 sin(α − θ ( g ) ) ⎤⎦. }. + sin θ ( g ) ⎡⎣ ∓2 Rabω1 ± 2r1φ1 + (±2 Rab + S G )ω1 cos γ ( g ) ⎤⎦ = 0 .. (3.22). Similarly, the equation of meshing about the rack cutter ∑ p and the gear can be obtained by the same step: f ( γ ( p ) , θ ( p ) , φ2 ) = −. R( p ) (∓2 Rab − S G + 2 R ( p ) cos α − 2 R ( p ) cos θ ( p ) ) 4. {2 cos γ. ( p). ⎡⎣ r2 (−1 + ω2 ) cos θ ( p ) + R ( p )ω2 sin(α − θ ( p ) ) ⎤⎦. }. + sin θ ( p ) ⎡⎣ ∓2 Rabω2 ± 2r2φ2 + (±2 Rab + S G )ω2 cos γ ( p ) ⎤⎦ = 0 .. (3.23). According to the gear theory, the gear mathematical model can be obtained by representing the locus equation of the rack cutter on the coordinate system of the generated gear 33.

(54) and the meshing equation. The locus equation of the rack cutter represented in the gear coordinate system S1 ( X 1 , Y1 , Z1 ) can be attained by applying the following homogeneous coordinate. transformation. from. the. coordinate. system. Sc( g ) ( X c( g ) , Yc( g ) , Z c( g ) ). to. S1 ( X 1 , Y1 , Z1 ) is: R1 = M1c R (cg ) ,. (3.24). and the homogeneous coordinate transformation matrix: ⎡cos φ1 ⎢ sin φ1 M1c = ⎢ ⎢ 0 ⎢ ⎣ 0. − sin φ1 0 r1 ( cos φ1 + φ1 sin φ1 ) ⎤ ⎥ cos φ1 0 r1 ( sin φ1 − φ1 cos φ1 ) ⎥ . ⎥ 0 1 0 ⎥ 0 0 1 ⎦. (3.25). Using the equations Eqs.(3.18), (3.24) and (3.25), the locus of the rack cutter represented in the gear coordinate system S1 ( X 1 , Y1 , Z1 ) can be obtained as:. R1 = M1c R c( g ). ⎡cos φ1 − sin φ1 ⎢ sin φ1 cos φ1 =⎢ ⎢ 0 0 ⎢ 0 ⎣ 0. 0 r1 ( cos φ1 + φ1 sin φ1 ) ⎤ ⎡ xc( g ) ⎤ ⎥ ⎥⎢ 0 r1 ( sin φ1 − φ1 cos φ1 ) ⎥ ⎢ yc( g ) ⎥ , ⎥ ⎢ zc( g ) ⎥ 1 0 ⎥ ⎥⎢ 0 1 ⎦⎣ 1 ⎦. thus (g) ⎡ x1 ⎤ ⎡ F2 cos φ1 + ⎣⎡( − F3 + F1 cos γ ) sin φ1 ⎦⎤ ⎤ ⎥ ⎢ ⎥ ⎢ (g) ⎢ y F F F − + cos γ cos φ sin φ 1 ) 1 2 1 ⎥⎥ , 1 R1 = ⎢ ⎥ = ⎢ ( 3 ⎢ z1 ⎥ ⎥ F1 sin γ ( g ) ⎢ ⎥ ⎢ ⎢ ⎥ ⎣1⎦ ⎣ 1 ⎦. where. 34. (3.26).

(55) ⎡ SG ⎤ ± R ( g ) ( cos θ ( g ) − cos α ) + Rab ⎥ , F1 = ⎢ ± ⎣ 2 ⎦ (g) (g) (g) F2 = ( r1 − R sin α + R sin θ ) ,. and F3 = ( Rab − r1φ1 ) . The mathematical model of the gear with curvilinear-teeth generated by a disk-type circular-arc rack cutter can be attained by solving the locus equation Eq.(3.26) with the meshing equation Eq.(3.22). Similarly, the locus of the rack cutter surface ∑ p represented in the pinion coordinate system S 2 ( X 2 , Y2 , Z 2 ) can be obtained as follows:. R 2 = M 2 c R (c p ). ⎡G2 cos φ2 + ⎡( G3 − G1 cos γ ( p ) ) sin φ2 ⎤ ⎤ ⎣ ⎦⎥ ⎢ ( p ) ⎢ ⎥ = ⎢ ( G3 − G1 cos γ ) cos φ2 − G2 sin φ2 ⎥ , ⎢ ⎥ G1 sin γ ( p ) ⎢ ⎥ 1 ⎣ ⎦. (3.27). where ⎡ SG ⎤ ± R ( p ) ( cos θ ( p ) − cos α ) + Rab ⎥ , G1 = ⎢ ± ⎣ 2 ⎦ ( p) ( p) ( p) G2 = ( − r2 − R sin α + R sin θ ) , G3 = ( Rab − r2φ2 ) .. and the homogeneous coordinate transformation matrix is:. M 2c. ⎡ cos φ2 ⎢ − sin φ2 =⎢ ⎢ 0 ⎢ ⎣ 0. sin φ2 cos φ2 0 0. 0 − r2 ( cos φ2 + φ2 sin φ2 ) ⎤ ⎥ 0 r2 ( sin φ2 − φ2 cos φ2 ) ⎥ . ⎥ 1 0 ⎥ 0 1 ⎦. 35. (3.28).

(56) The mathematical model of the gear with curvilinear-teeth generated by a disk-type circular-arc cutter can be attained by solving the locus equation Eq.(3.27) with the equation of meshing Eq.(3.23).. 3.2.3 Computer Graphs. The equations of the circular-arc gear and pinion with curvilinear-teeth generated by a disk-type circular-arc cutter are expressed in Eqs.(3.26) and (3.27) with meshing equations Eqs.(3.22) and (3.23), respectively. Table 3-1 lists the major design parameters of a circular-arc gear pair with curvilinear-teeth generated by disk-type circular-arc cutters. Base on the developed gear mathematical models, three-dimensional tooth profiles of the gear and pinion are plotted by commercial software CATIA® as displayed in Fig. 3-12 and Fig. 3-13, respectively.. Table 3-1 Some major design parameters for the gear set Design parameters. Number of Teeth ( T (i ) ) Normal Module (M). Pinion. Gear. 18. 36 3mm/tooth 20∘. Normal Pressure Angle ( α ) Radius of the Disk-Type Cutter. 30mm. ( Rab ) (Fig. 3-7) Radius of Rack Cutter Normal. 40mm. Section ( R (i ) ) (Fig. 3-6 & Fig. 3-8) Face Width ( W ) (Fig. 3-7). 40mm 30mm. 36.

(57) In addition, Fig. 3-14 illustrates the tooth surface deviation of various radii of the circular-arc rack cutters. Fig. 3-14(a) depicts the tooth surface deviation of the normal profile at the middle section of the tooth flank ( Z = 0 mm) with different radii of convex rack cutter Σ p , and Fig. 3-14(b) depicts the tooth surfaces of concave rack cutter Σ g .. 3.3 Remarks. The normal section of the rack cutter is designed as a circular-arc for the generation of circular-arc gears with curvilinear-teeth gear. A new topology for tooth surfaces has been achieved by applying the imaginary circular-arc rack cutter with a curvilinear-trace generation mechanism. The mathematical model of the circular-arc gear with curvilinear-teeth generated by a disk-type circular-arc cutter has been derived based on the theory of gearing. Furthermore, a computer program applicable to the generation of the tooth profile has been developed on the basis of the derived equations. Further characteristics of the circular-arc curvilinear-tooth gear, such as TCA and contact patterns, can also be performed with the aid of the developed mathematical models. Based on the study of this chapter, the following geometric characteristics of the circular-arc curvilinear-tooth gear can be drawn: 1. The involute gear is a special case of the circular-arc gear when the radius of the rack cutter profile R ( g ) tends to infinity. 2. The spur gear and helical gear is a special case of the curvilinear-tooth gear when the radius of the disk-type rack cutter (i.e. the Rab in Fig. 3-1 and Fig. 3-7) tends to infinity, the curvilinear-tooth gear becomes a spur gear.. 37.

(58) Y1. X1. Z1. 50. 0. (mm). Fig. 3-12 Computer graph of the circular-arc gear with curvilinear-teeth. 38.

(59) Y2 X2. Z2. 50. 0. (mm). Fig. 3-13 Computer graph of the circular-arc pinion with curvilinear-teeth. 39.

(60) 2.5922mm 2.6111mm 2.6280mm 2.6580mm. Σp Tip. 15:1 (p). R =40mm (p) R =80mm (p) R =160mm (p) R =5000mm. Root. (a) Rack cutter Sp for gear generation. 2.7395mm 2.7736mm 2.7892mm 2.8037mm. 0mm. 15:1. 5mm. Tip. (g). R =5000mm (g) R =160mm (g) R =80mm. Σg. (g). R =40mm. Root. (b) Rack cutter Sg for gear generation. Fig. 3-14 Tooth surface profiles with different radii of circular-arc rack cutters. 40.

(61) 3. The curvilinear-tooth gear has no axial thrust force during operation similar to the herringbone gear because of its symmetrical geometric characteristic. 4. The transverse gear chordal thickness measured at the middle section is larger than those of other sections as shown in Fig. 3-15.. (mm). 8. (Tip) 6. Pitch circle. 4. (1). Z = 0 mm (1) Z = 7.5 mm (1) Z = 15 mm. 2. 0. 4. 2. (Root). 6. 8 (mm). Fig. 3-15 Normal profiles at different cross sections of the gear. 41.

(62) CHAPTER 4 Tooth Undercutting Analysis 4.1 Introduction. Tooth undercutting is an important issue for gear manufacture and design. When tooth undercutting occurs, the gear thickness near the tooth fillet will be decreased. The load capacity and the length action line are consequently reduced. Undercutting also induced noise and vibration. Besides, the contact ratio also decreases with undercutting. Actually, tooth undercutting is the singular points appeared on the tooth surface. Thus, the problem of tooth non-undercutting is the existence of singular points on the generated gear tooth surface. However, gear with a bit of tooth undercutting sometimes can be a benefit for solving the gear interference problem and storing of lubricating oil. In this chapter, the tooth undercutting conditions among the cutting parameters and gear design parameters on the generated tooth profiles are investigated based on the theory of gearing proposed by Litvin [10][20].. 4.2 Conditions of Tooth Undercutting. The concept for checking the undercutting of a gear tooth surface is to check the appearance of singular points on the generated gear tooth surfaces. Singularities of the generated surface ∑1 occurs when the relative velocity of the contact point over the generated surface equals to zero. The gear surface of the circular-arc curvilinear-tooth gear is generated by the imaginary rack cutter ∑ g . The position vectors of ∑1 and ∑ g at the instantaneous contact points should equal to each other if they are expressed in the same 42.

(63) coordinate system in the process of generation. Therefore, if the rack cutter surface ∑ g and the generated gear surface ∑1 are expressed in the fixing coordinate system S f ( X f , Y f , Z f ) shown in Fig. 3-9, the relationship will be found: R (fg ) = R (1) f .. (4.1). By differentiating Eq.(4.1) with respect to time, it results in: Vtr( g ) + Vr( g ) = Vtr(1) + Vr(1) .. (4.2). After transposition, the Vr(1) can be expressed as:. Vr(1) = Vtr( g ) + Vr( g ) − Vtr(1) = Vr( g ) + Vtr( g1) ,. (4.3). where Vtr( g ) and Vtr(1) are the transfer velocities move of the cutter and blank at the contact point, respectively, and Vr( g ) and Vr(1) represent the relative velocity of the cutter surface and tooth surface at the contact point, respectively. A singular point is occurred when the derivatives of Vr(1) becomes to zero. Therefore, the necessary conditions of tooth undercutting which allows the determination of the limit on the rack cutter surface ∑ g can be expressed by:. Vr( g ) + Vtr( g1) = 0 ,. (4.4). and the differential meshing equation:. d f ( γ ( g ) , θ ( g ) , φ1 ) = 0 . dt. (4.5) 43.

(64) Tooth undercutting curve can be found by solving Eqs.(4.4) and (4.5). This undercutting curve is expressed on the rack cutter surface. To avoid tooth undercutting of the generated gear tooth surfaces, the generating rack cutter surface must be limited with the constraints. Now, Eqs.(4.4) and (4.5) yield that: ∂R (cg ) dθ ( g ) ∂R (cg ) d γ ( g ) + (g) = − VcF 1 , (g) dt dt ∂θ ∂γ. ∂f ∂θ ( g ). (4.6). ∂f d γ ( g ) ∂f dφ1 dθ ( g ) + (g) =− . dt dt ∂γ ∂φ1 dt. (4.7). In Eqs.(4.6) and (4.7), R (cg ) represents the equation of rack cutter surface ∑ g expressed in coordinate system Sc( g ) ( X c( g ) , Yc( g ) , Z c( g ) ) , and VcF 1 means the relative velocity between two surfaces ∑ g and ∑1 at the contact point. Eqs.(4.6) and (4.7) can be also expressed by the matrix form as follow: ⎡ ∂x ( g ) ⎢ c( g ) ⎢ ∂θ ⎢ ∂y ( g ) ⎢ c( g ) ⎢ ∂θ ⎢ ∂z ( g ) ⎢ c( g ) ⎢ ∂θ ⎢ ∂f ⎢ (g) ⎣ ∂θ. ∂xc( g ) ⎤ ∂γ ( g ) ⎥⎥ ⎡ −Vxc( g1) ⎤ (g) ⎥ (g) ∂yc ⎡ dθ ⎤ ⎢ ⎥ −Vyc( g1) ⎥ (g) ⎥ ⎢ ⎢ ⎥ ∂γ ⎥ dt ⎢ ( g ) ⎥ = ⎢ −Vzc( g1) ⎥ . (g) ⎥ ⎥ ∂zc ⎢ d γ ⎥ ⎢ ∂f dφ1 ⎥ ⎢ ⎥ (g) ⎥ ⎢ ∂γ ⎥ ⎣ dt ⎦ ⎢ − ⎣ ∂φ1 dt ⎥⎦ ∂f ⎥ ⎥ ∂γ ( g ) ⎦. (4.8). Based on the theory of Linear Algebra, the sufficient conditions for Eq.(4.8) existing, the only solution is the rank of the augmented matrix of Eq.(4.8) is two. It means the following four equations must be conformed in the meantime:. 44.