少齒數螺旋齒輪之齒形設計

Abstract

對於平行軸減速機構而言，要達到高減速比的要求，往往需要用數個齒輪做多段的排列組合而成，實務上常基於成本、齒輪數目和齒輪箱體積方面的考量，而無法使用多階式減速之齒輪組設計，因此，必須考慮使用一階式減速齒輪組以達到高減速比的目的。然而具高減速比的一階式減速齒輪對，其中之一個齒輪的齒數必須非常少，也因為齒數少而使得齒輪之齒形會有過切現象產生。本研究係根據齒輪原理及基本微分幾何的觀念，探討具少齒數之平行軸擺線螺旋齒輪與漸開線螺旋齒輪之齒形設計。 本研究使用具轉子外形之擺線形齒輪做為少齒數擺線形螺旋齒輪之線形。此擺線形螺旋齒輪之主動齒輪其齒冠為圓弧形，齒根則是由一個同樣具圓弧形齒冠之被動齒輪所創成而得。根據齒輪原理及齒輪創成機構推導少齒數擺線形螺旋齒輪之數學模式，對於齒輪之齒形的過切問題亦加以探討。此外，根據齒形幾何的分析，探討齒冠與齒根間齒形之連續性。同時研究過切指標 對於齒形過切與連續性之影響。齒形特性分析的結果，能幫助齒輪設計者設計一個既不產生齒形過切且齒輪之齒形也連續的齒輪。本論文的研究結果不僅可用於少齒數擺線形螺旋齒輪，亦可做為魯氏泵及由兩個相同齒輪帶動的螺旋壓縮機之轉子使用。 本研究亦推導修整型少齒數漸開線螺旋齒輪之數學模式，也依據所推導的齒輪齒形數學模式探討其齒形之過切問題，再以齒形轉位的方法減緩少齒數齒輪的齒根過切現象。其次根據基本幾何觀念，以圓弧線形來修整少齒數漸開線螺旋齒輪的齒根過切，因而可得到不具齒形過切的少齒數漸開線螺旋齒輪。本研究亦同時使用齒形轉位切削與齒根修整的雙重方法來改善齒根之過切，經與只使用齒形轉位切削方法所獲得之齒形比較後顯示，本論文所提出之同時考量齒形轉位切削與齒根圓弧修整的雙重方法，將可以得到較好的結果。利用齒頂經修整後之小齒輪外形做為成型刨齒刀之外形，並考慮齒條刀創成小齒輪的轉位切削，來創成與修整小齒輪相互嚙合之修整大齒輪。分析之結果發現，此嚙合齒輪對之中心距之改變量僅與成型刨齒刀創成大齒輪之轉位係數有關。 最後，本論文發展了一套電腦模擬程式，並分別繪出所設計的少齒數擺線與漸開線螺旋齒輪之電腦模擬齒形。本研究之成果不僅可以做為少齒數擺線螺旋齒輪與漸開線螺旋齒輪之齒形設計之參考，亦可應用於具高減速比的一階式減速擺線與漸開線螺旋齒輪的齒形設計，不但可以減少齒輪箱之體積、簡化傳動機構之構造並且可以使得齒輪組裝更為容易，所需成本也較低。

In order to obtain a gear reduction box with high gear ratio, usually it needs more than two gears for multi-stage assembly. Based on the considerations of the cost, the total number of gears and the volume of a gearbox, the use of multi-stage gears to obtain a reduction gear set with high gear ratio is not allowable. Instead, an one-stage reduction gear set should be considered and one of the gears must be extremely small. However, tooth undercutting on the generated gear may occur when the gear has a small number of teeth. This thesis investigates the tooth-profile design on both the helical cycloidal and involute gear sets with small number of teeth for parallel axes, based on the gear theory and the concept of differential geometry. In this research, a cycloidal gear with rotor profiles is used for the tooth profile of helical cycloidal gears with small number of teeth. The addendum profile of the driving gear of cycloidal helicoids is a circular arc, and the dedendum profile of the driving gear is generated by the addendum of the driven gear tooth-profile that is also a circular arc. A complete mathematical model of the cycloidal helicoid is developed, and tooth undercutting of the generated gear is investigated. Moreover, the tooth profile continuity between the addendum and dedendum of the gear is also investigated according to the geometric analysis. The influences of the undercutting index on tooth undercutting and profile continuity are studied. The characteristic analysis results are most helpful to the designers for avoiding the tooth undercutting and keeping the continuity of tooth profile. The results of this research can be applied not only to cycloidal helicoids with a small number of teeth but also to screw compressors and Root’s Blowers that rotors are driven by two equal gears mounted on the shafts of the rotors. The mathematical model of the modified helical involute gear with small number of teeth is developed based on the tooth-profile shifting method and basic geometry modification. Tooth-profile undercutting of this type of gears with small number of teeth is examined by using the developed mathematical model and the tooth-profile shifting method. Furthermore, an alternative method for lessening the tooth-profile undercutting is also presented by considering a modification of the basic fillet geometry using a modified rack cutter. A third method, combining the aforementioned two methods for the design of helical gears with small number of teeth, is also proposed to yield a gear set without tooth undercutting. The mating gear with tooth-profile shifting is generated using the pinion as a shaper. The tip fillet and root fillet are modified and a clearance between the pinion and the mating gear is also included in the design. Analysis result indicates that the change of gear set center distance depends only on the tooth-profile shifting of the mating gear. Results of this research can be used in designing spur and helical gear sets with small number of teeth. Finally, computer simulation programs are developed and computer graphs are also displayed for both the helical involute and cycloidal gear sets with small number of teeth. The results of this research can be used not only to the tooth design of cycloidal and involute gears with small number of teeth, but also to the design of a gearbox with one-stage high gear ratio gear pairs. Consequently, the total volume of the gearbox can be reduced, the structure of the gear transmission mechanism can also be simplified, the gear assembly made easier and the cost is lower.