球形螺旋齒輪之特性研究

Abstract

球形齒輪(Spherical gear)是由日本三留謙一教授所提出的一種新型的齒輪。由其幾何外形來區分，球形齒輪可分為凸狀球形齒輪(Convex spherical gear)與凹狀球形齒輪(Concave spherical gear)。由球形齒輪所組成之齒輪組共有三種配對型式：凸狀球形齒輪配凸狀球形齒輪、凸狀球形齒輪配凹狀球形齒輪及凸狀球形齒輪配正(螺旋)齒輪。與一般常用之正齒輪組不同的是，球形齒輪組(Spherical gear set)具有可容許軸交角變動與軸裝配誤差且不發生齒形干涉之傳動特性。 基於球形齒輪在組裝上的優點，本論文提出一種結合球形齒輪及螺旋齒輪特性的球形螺旋齒輪(Spherical helical gear)。球形螺旋齒輪除了具有球形齒輪所有的幾何特色及傳動特性外，亦可透過球形螺旋齒輪之齒輪螺旋角(Helix angle)的設計，以交錯軸的組裝型式(Crossed axes mounting mode)進行傳動。因此，若能建立出球形螺旋齒輪的數學模式，則可利用此數學模式來進行球形螺旋齒輪之相關分析，以提供產業界更進一步了解球形螺旋齒輪之特性及應用上的限制。 由於滾齒加工方法具有切削效率高與成本低的加工特性，因此，本論文選用滾齒加工方法來模擬創成凸狀及凹狀球形螺旋齒輪，進一步分析並探討由滾齒加工所創成之球形螺旋齒輪所組成之齒輪組的接觸特性。首先，本論文建立一把ZN型蝸桿滾齒刀之齒面數學模式，接著依據滾齒加工之創成機構與齒輪原理推導出由此ZN型蝸桿滾齒刀所創成之凸狀及凹狀球形螺旋齒輪之齒面數學模式，並利用所建立之凸狀與凹狀球形螺旋齒輪之齒面數學模式進行後續的電腦模擬，包括球形齒輪之齒形過切分析(Tooth undercutting analysis)、齒形尖點分析(Tooth pointing analysis)、齒面接觸分析(Tooth contact analysis)及接觸橢圓(Contact ellipses)分析，最後再利用本論文所發展的球形螺旋齒輪有限元素網格產生軟體，自動產生一組於接觸狀態的球形螺旋齒輪組之有限元素接觸模型，接著再使用有限元素商用分析軟體ABAQUS/Standard進行球形螺旋齒輪組之應力分析(Stress analysis)。 齒形過切分析探討在何種齒輪設計參數及滾齒加工參數下，凸狀球形螺旋齒輪會發生齒形過切的現象及其齒形過切之發生位置，而齒形尖點分析則探討當凹狀球形齒輪發生齒形尖點時，其所對應的齒輪設計參數及滾齒加工參數，以提供適合的球形齒輪設計參數及加工參數。齒面接觸分析則探討三種配對型式的球形螺旋齒輪組，分別在平行軸及交錯軸組裝，且在具有裝配誤差及理想組裝狀況時之運動誤差、接觸點位置與接觸比。接觸橢圓分析則利用齒面外形法(Surface separation topology method)來探討三種配對型式的球形齒輪組在不同組裝條件下之接觸橢圓的位置、大小與平均長短軸比。此外，應力分析則模擬球形螺旋齒輪組在實際受負載情況時，其可能產生之齒面接觸應力及齒根彎曲應力。

The spherical gear is a new type of gears proposed by Mitome. Geometrically, the spherical gears have two types of gear teeth－convex tooth and concave tooth. The spherical gear sets have three types of mating combinations: convex tooth with concave tooth, convex tooth with convex tooth and convex tooth with spur gear tooth. Different from that of the conventional spur gear set, the spherical gear set is in point contact and allows variable transmission shaft angles and larger axial misalignments without gear interference during the gear drive meshing. Based on the advantages of the spherical gear, this study proposes a gear by considering the assembly and transmission characteristics of the spherical gear and helical gear, called spherical helical gear. The spherical helical gear has all geometry and transmission characteristics of the spherical gear, while the gear set can also be assembled in crossed axes mounting mode. Therefore, to develop a complete mathematical model of the spherical helical gears with convex and concave teeth can provide further investigation on the manufacturing conditions, transmission characteristics and application limits of the spherical helical gear for industry. In this study, hobbing method is considered for generation of spherical helical gears with convex and concave teeth due to its high cutting efficiency and low manufacturing cost. Based on the hobbing generation mechanism and theory of gearing, mathematical models of the spherical helical gears with convex and concave teeth can be developed. Firstly, the surface equation of a ZN-type worm-type hob cutter is derived, and then surface equations of the spherical helical gears with convex tooth and concave teeth cut by the hob cutter can be obtained. Sequentially, the tooth undercutting and tooth pointing condition equations for the convex and concave spherical helical gears are derived by utilizing the developed tooth surface equations of the gears, respectively. Therefore, the limit curves of the tooth non-undercutting of the convex spherical helical gear under different design parameters are investigated, while the Z cross-sections of tooth pointing beginning of the concave spherical helical gear are determined. Moreover, the tooth contact analysis (TCA) method is applied to determine the contact characteristics, such as kinematic errors, contact ratios and contact positions, of the spherical helical gear set with the three mating combinations (convex pinion mating with convex gear, convex pinion mating with concave gear and convex pinion mating with helical gear) and two assembly modes (parallel axes and crossing axes modes). Surface separation topology method is adopted to find the contact ellipses and bearing contacts of the spherical helical gear set, and the average ratio a/b of the major and minor axes of contact ellipses of the spherical helical gear set can also be obtained. Finally, an automatic mesh-generation program of the spherical helical gear sets is developed to investigate the stress analysis of the gear sets by utilizing the commercial FEA package, ABAQUS/Standard. Therefore, the contact and bending stress contours of the spherical helical gear sets under two axes mounting modes and three mating combinations can be determined.