修整型螺旋齒輪對之特性研究

Abstract

本論文提出一種修整型螺旋齒輪對，由一個漸開線形小齒輪與一個修整型圓弧形大齒輪所組成，此新式的修整型圓弧形螺旋齒輪可以目前產業界所使用之滾齒機或磨齒機與靠模板機構來實際加工成形。經分析結果顯示，在理想的嚙合狀況下，本研究所提出之修整型螺旋齒輪對的接觸齒印是落在齒面之中央區域而且具有拋物線型的傳動誤差，能有效地吸收因齒輪齒面誤差和組裝誤差所造成之傳動不連續，即使有水平軸或垂直軸裝配誤差存在時，此齒輪對也不會發生不連續的傳動誤差和齒緣接觸的情況。因此，本研究所提之修整型螺旋齒輪對確實具有產業上之應用性和優越性。本論文主要的研究主題包括以下六大項： 一、依據齒輪原理與創成機構以推導出此修整型圓弧形螺旋齒輪的齒面數學模式。 二、經由創成時刀具與齒胚的相對速度及其嚙合方程式，即可得到規則齒面存在的充分條件，亦即避免齒面發生過切的條件式。此條件式可供齒輪設計者選擇適當之齒輪設計參數與刀具設定，以避免齒面過切的產生。 三、進行修整型螺旋齒輪對之齒面接觸分析並配合齒面外形法，深入探討此修整型螺旋齒輪對的嚙合傳動誤差與接觸齒印。此外，本研究亦採用最佳化之數值方法來求得最佳之齒輪設計參數。 四、探討齒輪設計參數及各種裝配誤差對於齒輪對之傳動誤差、接觸齒印和接觸率的影響。 五、依據微分幾何與曲率分析理論，可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。 六、依據所推導出之修整型圓弧形螺旋齒輪的齒面方程式，發展建構三維的齒面網格分割程式，利用有限元素分析軟體ABAQUS，進行一對接觸齒的應力分析。

This thesis proposes a modified helical gear pair composed of an involute pinion and a modified circular-arc gear. This novel type of modified circular-arc helical gear can be manufactured on hobbing or grinding machines with a curved-template guide in practice. The analyses results reveal that the proposed modified helical gear set possesses a localized bearing contact and a parabolic type of transmission errors (TEs) under an ideal meshing condition. The parabolic TE can absorb the discontinuous TE resulting from the tooth profile errors and assembly errors. In addition, neither TE jump nor edge contact will occur in the case of axial misalignments. Therefore, the proposed modified helical gear set indeed possesses its applicability and superiority. The research subjects of this thesis include the following six major items: (1) The mathematical model of the modified circular-arc helical gear set is developed based on the theory of gearing and the generating mechanism. (2) The sufficient conditions for the existence of a regular tooth surface is derived by considering the relative velocity of the generating tool and the gear blank along with the equation of meshing. These conditions result in the equations for tooth undercutting and provide gear designers with useful information for choosing gear parameters and tool settings to avoid tooth undercutting. (3) The TE and contact patterns of the proposed modified helical gear set are investigated by tooth contact analysis (TCA) and the surface topology method. Additionally, the optimization technique is also adopted to determine the adequate design parameters for the gear set. (4) The influences of design parameters and assembly errors on the TE as well as contact patterns and contact ratio (CR) of the gear set are also studied. (5) Based on the differential geometry and curvature theory, the principal directions and curvatures of the mating gear tooth surfaces are investigated, and the orientations and dimensions of the contact ellipses are also studied. (6) An automatic mesh-generation computer program for the three-dimensional tooth model is developed based on the developed gear geometry. Meanwhile, finite element stress analysis of a pair of contact teeth is performed to investigate the stress distribution on the tooth surface.