漸開線錐形齒輪對之特性研究

Abstract

本論文依據齒輪原理與創成機構推導出傳統型漸開線錐形齒輪以及凹面漸開線錐形齒輪之數學模式，並利用所推導之數學模式進行齒面接觸分析、嚙合幾何分析、曲率分析、接觸橢圓模擬及有限元素應力分析以驗證此一齒輪對在平行軸、相交軸以及交錯軸嚙合狀況下之之接觸特性。模擬之結果反映了漸開線錐形齒輪所特有的接觸特性。凹面漸開線錐形齒輪不但可藉由增大接觸橢圓之面積來改善傳統漸開線錐形齒輪負荷過低的缺點，同時仍能保留傳統漸開線錐形齒輪對裝配誤差不敏感的特性。因此，本論文所提出之凹面漸開線錐形齒輪具有產業上之應用性和優越性，並可適用於高負載及高精密情形下之傳動。本論文主要的研究主題包括以下四大項： 一、 依據齒輪原理與推拔滾削之機構以推導出傳統漸開線錐形齒輪的齒面數學模式，並經由創成時刀具與齒胚的相對速度及其嚙合方程式，探討齒面過切的條件。 二、 進行傳統漸開線錐形齒輪對在平行軸、相交軸以及交錯軸嚙合狀況下之齒面接觸分析及嚙合幾何分析，並依據微分幾何與曲率分析理論，求得兩嚙合齒面之主軸曲率與主軸方向，進而探討齒輪接觸橢圓的大小及方向。 三、 凹面漸開線錐形齒之數學模式分別依據三留 謙一教授所提出之輪磨法以及作者所提出之新型輪磨法推導出來，並利用所推導出之數學模式進行齒輪接觸模擬，以探討凹面漸開線錐形齒輪之接觸特性。模擬之結果顯示作者所提出之新型輪磨法可有效改善利用三留 謙一教授之方法所創成之螺旋凹面漸開線錐形齒輪對具有傳動誤差之缺點。和傳統漸開線錐形齒輪相比較，凹面漸開線錐形齒輪不但具有較大的接觸橢圓，同時在非平行軸嚙合狀況下對裝配誤差亦不敏感。 四、 依據所推導出之凹面漸開線錐形齒輪對之數學模式，發展建構三維的齒面網格自動分割程式，並利用有限元素分析軟體，進行一對接觸齒的應力分析。

In this thesis, the mathematical models of both conventional beveloid gears and concave beveloid gears are derived based on the theory of gearing and the generating mechanism. Investigations including tooth contact analysis, meshing geometry analysis, curvature analysis, contact ellipse simulations and finite element stress analysis are performed to examine the contact characteristics of the beveloid gear pairs with parallel, intersected and crossed axes. The simulation results reflect the special contact nature of beveloid gears . Compared with conventional beveloid gear pairs, concave beveloid gear pairs not only solve the problems associated with low-load capacity by enlarging the contact ellipses, but also retain the special property of insensitivity to assembly errors under non-parallel axes meshing. The concave beveloid gear proposed in this study indeed possess its applicability and superiority, thus fits the requirements of high load and high precision motion transmission between non-parallel axes. The research subjects of this thesis include the following four major items: (1) The mathematical model of the conventional beveloid gear is developed based on the theory of gearing and the taper hobbing mechanism. Meanwhile, the conditions of tooth undercutting is investigated by considering the relative velocity of the generating tool and the gear blank along with their equation of meshing. (2) Tooth contact analysis and meshing geometry analysis are performed to examine the meshing and bearing contact of the conventional beveloid gear pairs with intersected, crossed and parallel axes. Based on the differential geometry and curvature theory, the principal directions and curvatures of the mating tooth surfaces are investigated, and the orientations and dimensions of the contact ellipses are also studied. (3) Two mathematical models of concave beveloid gears are derived according to Mitome’s grinding method and the novel grinding method proposed by the author, respectively. Based on the developed mathematical models, the contact simulations are performed and the characteristics of concave beveloid gear pairs are investigated. Simulation results indicate that the proposed novel grinding method ameliorates the drawback of Mitome’s grinding method by eliminating the transmission error of the helical concave beveloid gear pairs. In contrast to conventional beveloid gear pairs, the gears ground by the proposed novel grinding method not only have larger contact ellipses, but also mesh conjugately with non-parallel axes, although assembly errors exist. (4) An automatic mesh-generation computer program for the three-dimensional tooth model is developed based on the mathematical model of the concave beveloid gear. Meanwhile, finite element stress analysis of a pair of contact teeth is performed to investigate the stress distribution on the tooth surface.