小齒數螺旋齒輪之研究

Abstract

對於一般減速機構而言，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，因此，所需要的齒輪數目也會相對地增加，而整體變速齒輪箱的體積也會隨著齒輪數目的增加而增大，進而使得變速齒輪箱的構造複雜和組裝不易，同時造成損壞的機率也會相對地比尋常的齒輪組為多，除了成本增加之外，維修工作也將成為一項沉重的負擔。 為此，蝸桿蝸輪組可使用於交錯軸而小齒數螺旋齒輪則提供了平行軸齒輪組一個解決的方法，只需要二個齒數差異極大的齒輪相互組合，便可以達到高減速比的設計目的，同時也可以縮小整體變速齒輪箱的體積，而且構造簡單，組裝容易，並且大幅減少成本支出。 本論文是探討平行軸小齒數螺旋齒輪組，首先推導出小齒數螺旋齒輪大小齒輪之齒面數學模式，並依據所推得之齒面數學模式進行小齒輪外形之過切分析，再利用齒形轉位方法以及基本幾何外形修整方法來克服齒根過切之現象。另外，比較這兩種不同的方法所修整後的小齒輪外形，並將這兩種方法結合在一起，截長補短，以得到一個較佳的解決方法。 其次，利用修整後之小齒輪外形為成形刨齒刀，藉由座標轉換原理推導出與修整小齒輪相互嚙合之修整大齒輪。除此之外，並比照齒條刀創成小齒輪套用轉位係數的方法，於修整小齒輪刨削修整大齒輪的過程中套用轉位方法，便可設計出一組轉位係數和不為零之齒輪對。 最後，以齒輪接觸分析方法來計算齒形接觸率。由於整體接觸率包含齒形接觸率及導程方向接觸率，輔以螺旋齒輪之螺旋角的設定，適度地縮短齒寬，使整體之接觸率能符合一般螺旋齒輪之要求。

In order to obtain a high gear ratio gear reduction box, generally it needs more than two gears for multi-stage assembly. As a result, the cost, the total number of gears and the volume of the gearbox will be increased. Besides, the structure of the gearbox becomes complex and it is also difficult for assembly and maintenance. The worm gear set for crossed-axes gearing while the small teeth number of helical gear set for parallel-axes gearing provide a very good solution to this question. To achieve the purpose of high gear ratio, two gears with a huge difference between teeth number can be considered. Thus the total volume of the gearbox is small, the structure of transmission mechanism is simple and the assembly is easier. Besides, the cost is not as expensive as that of the multi-stage gearbox. This thesis investigates the small teeth number of helical gear set for parallel axes. Mathematical models of the gear’s and pinion’s profiles with small teeth number of helical gear set are developed. The investigation on the profile undercutting of the pinion is studied according to the developed gear’s and pinion’s mathematical models. Based on the methods of tooth profile shifting and basic geometry modification, the problem of tooth undercutting is solved. By comparing the results of these two methods, a combination of tooth modification is proposed to obtain the advantage of these two methods and receive a better result. Second, according to the theory of coordinate system transformation, the profile of the modified pinion is chosen as the shaper to generate the modified gear. According to shifting coefficients for the pinion generation by rack cutters, the concept by using the profile of the modified pinion is considered a shaper to produce the modified gear with shifting coefficients and design a gear set with the sum of the pinion and gear shifting-coefficients is not zero. Finally, calculation of the contact ratio of the gear set is based on the tooth contact analysis method. The total contact ratio is determined as the sum of profile and longitudinal contact ratios. The longitudinal length of the gear set can be calculated to obtain a satisfactory total contact ratio for a helical gear set.