圓弧齒線圓柱型齒輪之特性研究

Abstract

曲線齒輪(具有圓弧齒線之圓柱型齒輪)通常是由專用機配合盤狀銑刀來加工製作，其切削效率較低，本研究提出另一種製造方式，即利用六軸CNC滾齒機配合一般滾刀來切製曲線齒輪。由於滾齒切削過程為連續分度，其切削效率將大幅提高。使用六軸CNC滾齒機來創成曲線齒輪是一新的製造方式，在創成的過程中，滾刀座之角速度必須是可以控制的，而使得滾刀座之旋轉角速度可與滾刀軸向進給速度互相配合來滾製曲線齒輪。然而，本研究所提以滾刀來創成曲線齒輪之新方法，必須克服齒面過切及齒面二次切削等問題。 本文依據六軸CNC滾齒機之創成機構及滾刀之幾何外形來建立曲線齒輪之齒面數學模式，並探討製造過程中的齒面過切現象及齒面二次切削等問題。其中，曲線齒輪之齒面過切問題係以Litvin教授所提之方法來分析，而本文也建立分析齒面二次切削之程式，討論齒面發生二次切削時滾刀外徑與圓弧齒線半徑之間的關係。由於本研究所探討之曲線齒輪其齒面為雙參數之包絡曲面，對於齒面之曲率分析，目前文獻中並無相關之計算式，故本文亦推導曲線齒輪之齒面曲率分析計算式及其電腦模擬程式。齒輪接觸分析技術將應用於討探曲線齒輪對之齒面嚙合特性。對於曲線齒輪對在具有水平軸、垂直軸及中心距組裝偏差時，其齒面接觸位置、運動誤差及接觸齒印之變化情形均加以研究。最後，依據所推導出之曲線齒輪數學模式，發展建構具有三對齒之有限單元應力分析模型程式，並利用有限單元分析軟體ABAQUS，進行三對齒之齒面接觸應力及齒根彎曲應力分析。 分析結果顯示曲線齒輪之齒面寬兩端容易產生較嚴重的齒面過切。在相同的齒線圓弧半徑下，使用較大的壓力角或較小的滾刀外徑可以避免齒面發生二次切削。本研究之曲線齒輪組為點接觸，故齒輪組對於組裝偏差並不敏感。應力分析之結果則顯示當接觸點接近節點時，齒面將有最大的接觸應力及齒根彎曲應力，而使用較大的齒線圓弧半徑將可降低齒面接觸應力及齒根彎曲應力。

In general, the tooth surfaces of the curvilinear-tooth gear (cylindrical gears with circular arc tooth trace) are generated by the special machine with gear milling cutters. However, the efficiency of this type of gear generation is lower than that of hobbing. The curvilinear-tooth gear cut by a hob cutter is proposed in this study. Since the generation of gear teeth is a continuous-indexing process, the cutting efficiency of the curvilinear-tooth gear generated by hobbing will be increased substantially. Therefore, the proposed hobbing method is applicable to mass production for the curvilinear-tooth gear. The proposed method by applying a 6-axis CNC hobbing machine to generate the curvilinear-tooth gear is a novel method. The angular velocity of the hob’s swivel must be controllable and correlative with the linear velocity of the axial feeding motion in the process of gear generation. However, the problems of tooth undercutting and secondary cutting of tooth surfaces by a hob cutter in the proposed novel generating process need to be solved. In this study, the geometry of a hob cutter was constructed at first, and then the mathematical model of the curvilinear-tooth gear cut by a hob cutter based on proposed the cutting mechanism for a 6-axis CNC hobbing machine was developed. Sequentially, the theoretical analysis on the tooth undercutting and secondary cutting of the gear teeth in the generating process were also investigated. The method proposed by Litvin was used to analyze the tooth undercutting of the curvilinear-tooth gear. The computer algorithm for investigation on the relationship between the outside diameter of the hob cutter and nominal radius of circular arc tooth trace without secondary cutting was developed. Because the tooth surfaces of the curvilinear-tooth gear are the envelope as two-parameter family of surfaces, literature search for the curvature analysis of the proposed curvilinear-tooth gear is not available. An algorithm for computerized determination of principal curvatures and directions of the curvilinear-tooth gear, as the envelope to two-parameter family of surfaces, was also proposed. Tooth contact analysis was applied to find the contact characteristic of the curvilinear-tooth gear pair such as bearing contacts, kinematic errors, contact ellipses of the curvilinear-tooth gear having center distance assembly errors, horizontal axial- misalignments, and vertical axial misalignments. Finally, The input file for the commercial software, ABAQUS/Standard, was generated automatically by the integrated computer programs. A pair of three-meshing-teeth finite element model was formed to investigate the contact stresses and bending stresses of the curvilinear-tooth gears. The analysis results show that the serious tooth undercutting is at the both-end sections of face width of the curvilinear-tooth gear, and increasing the normal pressure angle or decreasing the outside diameter of the hob cutter can avoid secondary tooth cutting under the same nominal radius of circular arc tooth traces. The KE of the gear pair is not sensitive to axial misalignments because the contact type of the proposed curvilinear-tooth gear pair is in point contacts. The maximum contact stress and bending stress occur when the contact point is near the pitch point, and increasing the nominal radius of circular arc tooth trace reduces the contact stresses and bending stresses.