以刨齒刀創成Helipoid齒輪之齒面數學模式與接觸分析

Abstract

目前工業上廣泛使用在交錯軸動力傳遞的齒輪主要有交錯軸螺旋齒輪﹙Crossed-axis helical gears﹚以及戟齒輪﹙Hypoid gears﹚。戟齒輪有高負載和高接觸率的優點，因此大量地使用在需要交錯軸動力傳遞的機械上。然而，由於戟齒輪之齒面複雜，製造時通常需要使用特殊的專用機以及有經驗之工程人員來操作，因此戟齒輪的生產成本較高。相較而言，交錯軸螺旋齒輪只要使用傳統的滾齒加工或是刨齒加工就能得到良好的成品，因此成本較戟齒輪低許多。但是由於交錯軸螺旋齒輪具有較小的負載能力以及接觸率，使得其可應用的範圍受到很大的限制。 Helipoid齒輪是一種新型的齒輪，由日本工業大學的長田重慶教授所提出，主要是設計來滿足成本與性能之間的平衡，也就是希望能以接近交錯軸螺旋齒輪的成本來達成更高的負載，創造出一個能結合戟齒輪與交錯軸螺旋齒輪兩者之優點的齒輪。又由於Helipoid齒輪能夠以滾齒或刨齒的方法來製造，因此製造成本也接近於交錯軸螺旋齒輪。 本論文應用齒輪創成原理，推導以刨齒創成之Helipoid齒輪的齒面數學模式，再依據所推導的Helipoid齒輪齒面方程式和齒面接觸分析技術，進行齒輪組之傳動誤差分析，並以齒面外形法分析其接觸齒印，最後再利用有限元素法應力分析軟體來分析其接觸應力。同時，本論文也根據分析之結果，探討創成Helipoid齒輪的刨齒刀齒數、螺旋角與接觸齒印、接觸應力以及傳動誤差之間的關係。

Crossed-axis helical gears and hypoid gears are two common types of crossed-axis power transmission devices. Hypoid gears offer a high load capability and a high contact ratio, and are used for rear-axle transmission in automobiles. However, hypoid gears should be manufactured by special machines with various machine-tool settings due to complex tooth surface geometries. A hypoid gear set can obtain good contact patterns and contact locations only with appropriate machine-tool settings. Accordingly, the manufacture of hypoid gear sets requires experienced and well-trained engineers. Therefore, the production and maintenance costs of a hypoid gear are relatively high. The manufacture of helical gears, however, requires only easily operated and conventional machines, and the production cost is lower. However, the load capability and the contact ratio are also lower. A new type of gear, named the helipoid gear, is proposed herein by Nagata, eminent professor of Nippon Instittude of Technology, in an attempt to achieve a better balance between gear performance and manufacturing cost than that of hypoid and crossed-axis helical gears. Helipoid gears are designed to exhibit the advantages of both hypoid and helical gears ─ higher load capability and contact ratio than those of a helical gear, and a lower manufacturing cost than that of a hypoid gear. A helipoid gear, like a helical gear, can be produced by two conventional gear manufacturing methods, hobbing and shaping methods. Thus, the manufacturing cost of the helipoid gear is similar to that of the helical gear and far less than that of the hypoid gear. In this paper, based on the theory of gearing the mathematical model for helipoid gears cut by a shaper cutter is developed. According to this mathematical model and the tooth contact analysis technique, transmission errors of the helipoid gear set are investigated. Furthermore, the gear set contact pattern is simulated by applying the contact surface topology method and the stress analysis software developed by applying the finite element method is also adopted for the tooth contact stress analysis. Besides, the relationships among the tooth numbers of the shaper cutter, helical angles and contact patterns and transmission errors are also investigated.