圓弧形螺旋齒輪之接觸分析與量測模擬

Abstract

在各型螺旋齒輪的動力傳遞系統中，以圓弧形螺旋齒輪的承載能力為最佳，相當於同等級之漸開線形螺旋齒輪的三到五倍[16]。但由於圓弧形螺旋齒輪對中心距裝配誤差有相當大的敏感性，刀具製作成本亦高，而且在以往的機械工業技術水平，由於精密度不高的情況下，其應用範圍受到相當大的限制。但現今由於製造與量測精密度之技術進步，已可克服圓弧形螺旋齒輪對中心距之高敏感度與刀具製作困難的缺點，逐漸地被重視且應用在高負載之傳動上。 本論文研究之主要動機係因1999年接受到工業技術研究院機械工業研究所執行經濟部科技專案研究的學界轉委託研究案：『圓弧形螺旋齒輪嚙合特性之研究』，利用成形磨齒機可輪磨出精密的圓弧形螺旋齒輪。因為圓弧形螺旋齒輪能承受較大的負荷，故所構成之齒輪組可望取代電梯變速齒輪箱中，用於減速之漸開線螺旋齒輪組。 在本論文中，吾人根據齒輪原理及微分幾何的觀念，推導出兩種圓弧形螺旋齒輪之齒面數學模式：其中一種為刀具之法向剖面為圓弧形之螺旋齒輪，另一種則為法向齒面為圓弧形之螺旋齒輪。其齒形可用軌跡法得之。基於所推導出之齒面數學模式，可進行齒面接觸分析以探討在不同裝配情況下的傳動誤差。並以齒面數學模式，依據嚙合軸定理，推導出輪磨圓弧形螺旋齒輪之磨輪數學模式。此外，利用齒面方程式也可推導出跨銷量測之公式。

The load capacity of circular-arc helical gears are three to five times that of involute helical gears[16]. During the past years, circular-arc helical gears are not widely used since they are quite sensitive to the central distance and assembly errors. Nowadays, the machine and technology for gear manufacturing become more accurate. Therefore, applications of the circular-arc helical gears become popular in high-loading power transmissions. Circular arc helical gears, which can be produced with high precision by the form grinding, are may to take replace of the conventional involute helical gears applied in the reduction mechanism of elevators due to their high load capacity. This thesis concenrated on the investigation of circular arc helical gears, which originate from the research project “Tooth Contact Analysis and measurement simulation of circular arc helical gears” granted by MIRL at ITRI. Firstly, the mathematical models of two types of circular arc helical gears have been developed based on the theory of gearing and differential geometry. One type of the circular arc helical gear is generated by a rack cutter with circular-arc normal section. The other type of the circular arc helical gear, possessing a circular arc normal section in the tooth profile, can be obtained by applying the locus method. Based on the developed mathematical models of the gears, tooth contact analysis has been performed to investigate the transmission errors of both types of circular arc helical gears under various assembly conditions. In addition, the axial profile of the grinding wheel, used to produce the circular arc helical gears, has also been derived. Furthermore, according to the developed derived gear geometry, the gear equations of overwire measurement have been investigated as well.