曲線齒圓柱型齒輪之齒面數學模式與接觸分析

Abstract

摘要 一般正齒輪或平行軸螺旋齒輪之齒面若未經適當修整，則其接觸型態屬於線接觸，因此，齒輪組對於裝配誤差所造成的運動誤差較為敏感；齒輪組若存在軸偏差，則齒輪組將會發生齒緣接觸，因而造成齒面應力集中及較大的振動與噪音。曲線齒圓柱型齒輪之切削製造方法由Liu於1988年提出，並發現曲線齒具有高強度，低噪音，潤滑良好及無軸向推力等優點。本文利用具有圓弧齒線之假想齒條刀，推導出曲線齒大小齒輪之齒面數學模式，並以電腦繪圖程式繪出曲線齒之齒形。曲線齒具有左右齒面不相同之特性，而且在齒面寬中央之齒厚較大，而兩端之齒厚較小。 本論文依據所推導之曲線齒圓柱型齒輪齒面數學模式配合微分幾何原理及數值方法，探討齒線齒齒面過切的問題，可精確地分析出齒面上是否發生奇異點或奇異線及其發生之位置，亦即可探討曲線齒之齒面過切特徵。本論文亦探討曲線齒之轉位問題，研究以轉位切削齒輪之方式，以避免曲線齒之齒面發生過切。 本文亦應用齒輪接觸分析技術，探討曲線齒圓柱型齒輪組在具有裝配偏差時，其運動誤差及曲線齒之接觸比與接觸齒印。本文所分析之曲線齒圓柱型齒輪組之接觸型態為點接觸，因此對於裝配偏差所造成的運動誤差較不敏感；曲線齒之接觸點都位於齒面寬中央截面附近，因而可避免發生齒緣接觸之現象；對於接觸型態為點接觸之曲線齒，其接觸比之大小與一般正齒輪相近。曲線齒圓柱型齒輪組之接觸齒印可利用較為簡易的齒面外形法求得，利用此方法發現曲線齒之接觸點可能不是接觸橢圓之中心點，而且影響曲線齒接觸橢圓大小之最重要設計參數為切削曲線齒之刀盤半徑。

ABSREACT The contact type of helical gears with parallel axes or spur gears is line contact, and the kinematic errors are sensitive to the gear axial misalignment. The gear set exists axial misalignments, the tooth edge contact will be occurred and this results in stress concentration, vibration and noise. The manufacture of cylindrical gears with curvilinear shaped teeth has been proposed by Liu in 1988. The advantages of curvilinear toothed gears are higher bearing and contacting strength, lower noise, better lubrication condition and no axial thrust forces. In this paper, the mathematical model of curvilinear toothed gears has developed based on the generation mechanism with an imaginary rack-cutter of circular-arc tooth profile. The tooth surface of curvilinear toothed gears can be displayed by using computer graphics and the developed computer simulation program. The right-side and left-side tooth surfaces of curvilinear toothed gears are different and the tooth thickness of central section is larger than those of other sections. According to the developed gear tooth mathematical model, location of the singular point or singular line on tooth surfaces can be obtained by applying the concept of differential geometry and numerical method, i.e. the undercutting of curvilinear toothed gears can be analyzed. In this study, the profile-shifted generation has been also investigated. The undercutting of curvilinear toothed gears can be avoided by using a positive profile-shifted modification during the gear set generation process. The tooth contact analysis technology has been utilized to the investigation on kinematic errors, contact ratios and contact patterns of the gear set under ideal and error assembly conditions. Since the contact type of the curvilinear toothed gear proposed in this paper is point contact, the gear set is not as sensitive to the gear axial misalignments, and the kinematic errors are quite small. The tooth edge contact can be avoided because the contact paths on gear tooth surfaces are located near the central section of face width. The contact ratio of point contact type curvilinear toothed gears is similar to that of spur gears. The contact pattern of curvilinear toothed gears can be obtained by applying the contact surface topology method. It is found that the contact point of the mating tooth surfaces may not be the center of contact ellipses, and the most important design parameter which influences the dimensions of gear contact ellipses is the nominal radius of face mill cutter.