具漸遠作用之雙倍齒高蝸桿蝸輪組的特性研究

Abstract

摘要 大部份的齒輪是用於傳遞馬力或扭力，少部份的齒輪，如定位齒輪(Indexing gear)，是用於精密的定位或傳遞運動。定位齒輪使用的場合，有時不像濕式潤滑(Wet lubrication)那麼充份潤滑的場合，而會是會暴露於大氣或是只能使用油脂(Grease)的潤滑。Buckingham [1]解釋了齒輪嚙合的過程中，漸遠作用(Recess action，RA)比漸近作用(Approach action)有更低的摩擦係數，齒輪被設計成具有比一般齒輪有更多的漸遠作用則被稱為RA齒輪。一些特殊的精密定位裝置係採用全漸遠作用之蝸桿蝸輪對，比起傳統的蝸桿蝸輪對，此種齒輪對的齒形修正成雙倍齒高、低壓力角與全部是漸遠作用。這樣的齒形修正會造成蝸輪的增大齒數與多齒接觸，而多齒接觸的好處不僅可以降低齒面接觸力，更可以減少齒輪組之運動誤差(Kinematic error)或平均所有齒形總合的誤差，因此，此類型之齒輪組可以應用於潤滑條件較差及需要精密傳動的環境場合。 本論文首先提出了由ZN型蝸桿滾刀創成具有漸遠作用的雙倍齒高蝸輪數學模式。依創成機構與齒輪原理，推導出半漸遠作用、全漸遠作用與標準比例齒高蝸輪齒面數學方程式，依照所推導的蝸輪齒面數學模式，繪出半漸遠作用、全漸遠作用與標準比例齒高蝸輪蝸桿對嚙合的比較圖，以及在不同的軸向斷面其齒形的差異性，而在節圓與喉圓上的齒厚也加以計算與比較。 眾所皆知，齒形過切會降低齒輪的強度與減少接觸比，因此具有雙倍齒高之半漸遠作用、全漸遠作用與標準比例齒高蝸輪的齒形過切是值得探討的。特別是此半漸遠作用、全漸遠作用與標準比例齒高蝸輪的齒形已經修正成小壓力角、雙倍齒高與不同程度或全部是漸遠作用，可能更容易造成齒形過切。本研究中，可以求出具有雙倍齒高之半漸遠作用、全漸遠作用與標準比例齒高蝸輪齒形過切限制曲線(Limit curve)，亦可將齒形過切線(Tooth undercutting line)繪在蝸輪齒形上加以比較。同時，不同程度或全部是漸遠作用的蝸輪齒形，其最小整數不過切的齒數亦加以探討。 最後本研究再依據所推導的漸遠作用蝸桿蝸輪對做齒面接觸分析，求得其運動誤差、接觸比、平均接觸比與瞬時接觸齒數。在本研究中，亦探討使用增徑滾刀創成具有雙倍齒高之半漸遠作用、全漸遠作用與標準比例齒高蝸輪對之齒面接觸分析，求出運動誤差、接觸比、平均接觸比、瞬時接觸齒數與接觸路徑。除此之外，也應用齒面外形法(Surface Topology Method)求得接觸橢圓，並且比較增徑滾刀節徑半徑對接觸橢圓面積的影響。 本研究分析的結果顯示，具全漸遠作用之蝸輪在喉圓的橫向弦齒厚是最大，相反地，在節圓時是最小。具有雙倍齒高之半漸遠作用、全漸遠作用與標準比例齒高蝸輪的齒形，其左齒面及右齒面的右端容易產生較嚴重的齒形過切現象，而且其平均接觸比都比傳統定義的接觸比稍大。不論是理想組裝狀況或是有裝配誤差，具有全漸遠作用之蝸桿蝸輪組的接觸比、平均接觸比都是最大的，而且與半漸遠作用之蝸桿蝸輪組相同，都有最大的瞬時接觸齒數。具有全漸遠作用蝸桿蝸輪組的中心距裝配誤差若從正值改成負值，其運動誤差蝸會變成平滑的拋物線形狀，而且接觸橢圓會完整的在齒面的右測。具有全漸遠作用之蝸桿蝸輪組，其接觸橢圓的長軸尺寸是隨著增徑滾刀的節圓半徑的增加而變大，但是其短軸尺寸幾乎不會有太大的變化。 傳統的圓柱型蝸桿蝸輪組嚙合的接觸齒數是3至4齒，而本研究中具有雙倍齒高之半漸遠作用、全漸遠作用與標準比例齒高的蝸桿蝸輪組，在其嚙合過程中大都有7至8齒的接觸齒數，所以作用在齒面上的正向力會降低。其中具全漸遠作用之蝸桿蝸輪組不但是多齒接觸，可以增加接觸比、降低運動誤差與平均所有齒形總合的誤差，而且其接觸皆是漸遠作用之低摩擦係數，故作用在齒面上的正向力與低摩擦係數乘積，即是作用於齒面上的摩擦力會明顯的降低，於是在大氣中或是只有在少許的潤滑情況下操作是可行的。

ABSTRACT Majority of gears are used to transfer power or torque. Few of gears, such as indexing gears, are applied to transmit precise control of angular motion, and transferring of power or torque becomes their secondary consideration. Some of indexing gears usually operate in the open air, or only use grease lubrication. It is well-known that recess action gears (abbreviated to RA gears) have less wear with lower friction and less noise [1]. Some specific worm gear drives for precise indexing are actually modified into a full RA worm gear drive with double-depth teeth and low pressure angle. Compared with the conventional type of worm gear drive, these modifications result in a large number of teeth in contact and high recess action inducing lower friction. Benefit of multiple-tooth contact not only reduces kinematic errors but also averages the sum of all tooth transmission errors. In this study, the surface equations of recess action (RA) worm gears with double-depth teeth, generated by a ZN worm-type hob cutter, are proposed. Based on the generation mechanism and theory of gearing, a mathematical model for a series of worm gears, semi RA, full RA and standard proportional tooth types with double-depth teeth, is developed as the function of design parameters of the ZN worm-type hob cutter. According to the derived tooth surface equations, computer graphs of a series of RA worm gears with double-depth teeth are plotted. Tooth surface variations of the generated RA worm gears due to the varying pitch line, pressure angle and tooth height of the hob cutter are also investigated. In this study, tooth undercutting of the RA worm gear is investigated by applying the theory of gearing, tooth surface equations and the developed computer simulation programs. Limit curves of tooth undercutting can thus be plotted, and the minimum integer numbers of teeth for tooth non-undercutting of the full RA, semi RA and standard proportional tooth worm gears are studied. Besides, tooth undercutting lines of the full RA, semi RA and standard proportional tooth worm gears with double-depth teeth are also investigated. Moreover, based on the developed mathematical model of a series of RA worm gear drives (i.e. semi RA, full RA and standard proportional tooth types) with double-depth teeth, the tooth contact analysis (TCA) technique is utilized to investigate the kinematic error (KE), contact ratio (CR), average contact ratio (ACR), instantaneous contact teeth (ICT) under different assembly conditions. Several numerical examples are presented to demonstrate the influence of the assembly errors and design parameters of the RA worm gear drive on the KE, CR, ACR, ICT and contact patterns. The worm gear of an RA worm gear drive is generally generated by a standard ZN worm-type hob cutter. It results in line contact and easily induces edge contact of the gear drive meshing. In this study, the RA worm gear is generated by an oversize ZN worm-type hob cutter. Then, the worm gear drive becomes in point contact under ideal and even error assembly conditions. The bearing contact and contact ellipse are studied for semi RA, full RA and standard proportional tooth worm gear drives with double-depth tooth by applying the surface topology method under assembly errors. Comparisons of contact ellipse sizes of the full RA worm gear drive with double-depth teeth, that its meshed RA worm gears are generated by different oversize hob cutters, are investigated. Analysis results of this research show that the transverse chordal tooth thickness at pitch circle is the largest for the standard proportional tooth worm gear. Reversely, the transverse chordal tooth thickness of the standard proportional tooth worm gear at its throat circle becomes the smallest. Tooth undercutting is easier to occur at their right-side regions of the left-side and right-side tooth surfaces for the RA worm gears with double-depth teeth. ACRs are slightly larger than CRs, which are calculated by conventional definition. With negative center distance assembly error for full RA worm gear drive with double-depth teeth, contact ellipses are all completely located at the right-side of the full RA worm gear face width and KEs become the smooth parabolic shape. Full RA worm gear drives have maximum ICTs under the same ideal assembly or error assembly condition, and CRs and ACRs are the same as those of the semi RA worm gear drive. For full RA worm gear drive, the major-axis length of contact ellipse can be adjusted by changing the pitch radius of an oversize ZN worm-type hob cutter, but the minor-axis length of contact ellipse has not too much affected by the pitch radius variations of an oversize ZN worm-type hob cutter. A conventional worm gear drive usually has approximately 3 to 4 teeth in contact. In this study, the RA worm gears with double-depth teeth have multiple teeth contacts of almost 7 to 8 teeth for semi RA, full RA and standard proportional tooth worm gear drives. Advantages of multiple teeth contacts are the increase of CRs and decrease of KEs, and averaging the sum of all tooth transmission errors. It is noted that multiple teeth contacts of a mating gear pair not only reduces KEs but also averages the sum of all tooth transmission errors, besides, it also results in a lower normal bearing force on each contact tooth surfaces of worm and worm gear. And, the full RA worm gear drive with double-depth teeth has all recess action that induces a lower coefficient of friction during the meshing of worm gear drives. Therefore, the friction force acting on the tooth surfaces of full RA worm gear drive becomes lower. It enables the full RA worm gear drive to operate in a dry film lubrication or the open air.