分階式雙漸開線形與圓弧形螺旋齒輪對之接觸分析

Abstract

現今齒輪發展的新趨勢，是以設計點接觸之嚙合方式為主而非線接觸之嚙合方式，以減少齒輪嚙合時對於裝配誤差的敏感性。利用兩不相匹配之曲面作為大、小齒輪之齒面，則可使其嚙合方式以點接觸形式取代線接觸以避免齒緣接觸。故本論文所研究之齒輪對，即使用兩把不同齒面幾何形狀的假想齒條刀來創成大、小齒輪，而不同於一般以同一把齒條刀創成的漸開線螺旋齒輪對。 由於被創成之齒輪外形具有階梯狀，所以其齒根處之齒厚較一般之漸開線齒輪為大，因此可以提高輪齒的彎曲強度，且這樣的設計可以使齒輪對的齒面獲得兩個嚙合接觸區域，而不是傳統漸開線螺旋齒輪對的一個接觸區域，可藉此分散齒面之接觸應力。 因為裝配誤差而引起的傳動誤差，是齒輪對在嚙合過程中引起噪音與振動的主要原因，為了減少噪音與振動的現象，本研究所採用之方法是在齒形上預先設計了二次式的傳動誤差函數，藉以吸收齒輪對因裝配誤差所引起的不連續傳動誤差。 另外，本研究也利用電腦數值分析來模擬齒輪對的傳動與接觸的過程，藉此證明前述之齒形修整設計方法，可製造出在裝配誤差之下能具有二次傳動誤差曲線特性之齒輪對。在製造大齒輪方面，被創成齒輪與刀具之間的非線性新創成關係，可利用電腦數值控制(CNC)機器來製造。對於小齒輪的製造方面，則仍屬於傳統的線性創成關係，所以利用傳統的製造機器即可製造完成。 由於齒輪在負載之下傳動與齒面彈性變形的關係，因此，接觸點將延伸成為一橢圓形之接觸區域，在本研究中，將利用齒輪齒面外形法及曲率分析法，針對接觸齒印之長、短軸與方向作深入的研究。其中曲率分析法必須先自齒條刀著手，根據齒輪的創成過程而推導，求得大、小齒輪接觸齒面上之主軸方向向量與主軸曲率，進而求得接觸橢圓。

The new trend of the gear design is to provide the gear teeth with surfaces being in contact at every instant at a point but not at a line to reduce the gear sensitivity to gear axial misalignments. By applying the mismatch of pinion-gear tooth surfaces, the tooth contact becomes a point contact instead of a line contact. The bearing contact is located at the middle region of the gear tooth surfaces and thus edge contact can be avoided. Unlike the generation of conventional involute helical gears, two imaginary rack cutters, instead of one, are applied for the generation of the proposed helical gear pair in this thesis. Since the profile of the proposed gear is in a ladder-shaped, the tooth thickness of the dedendum is larger than that of the conventional involute helical gears. This enables the gear set to obtain a two-zone of meshing instead of only one-zone of meshing existing in the conventional involute helical gears, and thus the gear has higher gear strength as well. The transmission error caused by gear axial misalignments is the main source of gear noise and vibration. Reduction of gear noise and vibration can be achieved by applying a predesigned function of transmission errors of a parabolic type. Such a predesigned parabolic function can absorb a discrete linear function of transmission errors caused by gear axial misalignments. Computerized simulations on gear meshing and contact of the designed gears demonstrate that the proposed design method for gear modifications produce a pair of gears with a parabolic transmission error function when axial misalignments are present. For the manufacture of the gear, the new relation between the rotational motions of the gear and cutting tool is nonlinear. This can be accomplished by the application of a computer numerical controlled (CNC) machine. For the pinion, a conventional manufacturing machine can be used for manufacturing since the relation between the rotational motion of the pinion and the cutting tool is linear. The contact of gear surfaces is spread over an elliptical area under the load due to the elasticity of the gear tooth surfaces. In this study, the dimensions and orientation of the instantaneous contact ellipse will be determined by applying the method of surface topology and method of gear curvature analysis. The latter one requires the determination of principal curvatures and directions of the contacting pinion-gear tooth surfaces, which can be obtain from the generation process of gears by rack cutters.