ZN型單包絡與雙包絡蝸桿蝸輪組之研究

Abstract

蝸桿蝸輪組屬於空間交錯軸(Cross Axes)間之動力傳動元件，可以提供一般傳統平行軸之正齒輪及螺旋齒輪無法達成之單段(Single Stage)大減速比效果。因此，蝸桿蝸輪組常用於交錯軸傳動且需要大減速比的動力傳動的場合，因其具有傳動轉速比高、工作平穩、噪音低、結構緊湊等優點，故被廣泛地使用於各種減速傳動系統之環境。 在本論文中探討兩種類型的蝸桿蝸輪組，分別為非90°交錯角ZN型單包絡(Single-Enveloping)蝸桿蝸輪組與ZN型雙包絡(Double-Enveloping)蝸桿蝸輪組，而非90°交錯角單包絡蝸桿和雙包絡蝸桿皆可由具有導程角之直邊刀具所切削而成，而非90°交錯角單包絡及雙包絡之蝸輪則是分別利用前述之非90°交錯角單包絡蝸桿型滾刀與雙包絡蝸桿型滾刀所創成。依據蝸桿蝸輪之創成機構與齒輪原理，即可分別推導出非90°交錯角ZN型單包絡蝸桿蝸輪組與ZN型雙包絡蝸桿蝸輪組之兩組齒面數學模式，再利用此兩組數學模式及電腦輔助繪圖軟體(AutoCAD)，即可分別建構並繪製出兩組蝸桿蝸輪之三維模型。本論文亦依據所推導之非90°交錯角單包絡蝸桿蝸輪齒面數學模式，利用齒形過切分析方法，分析蝸輪齒面之過切線分佈情形。另外，亦根據雙包絡蝸桿蝸輪齒面數學模式，探討在不同設計參數下，ZN型雙包絡蝸輪之齒面工作範圍並探討與ZA型雙包絡蝸輪之齒面的差異。 一對嚙合之齒輪其齒輪彎曲應力、接觸應力以及接觸比是設計時很重要的考量因數，具有較高接觸率的齒輪組有助於減少齒輪組的應力與運動誤差。在本研究中亦使用齒面接觸分析(Tooth Contact Analysis)法來探討非90°交錯角之ZN型單包絡蝸桿蝸輪組以及雙包絡蝸桿蝸輪組之齒面接觸，並分別計算兩組蝸桿蝸輪組之運動誤差(Kinematic Error)。另外，蝸桿蝸輪組之平均接觸比(Average Contact Ratio)及瞬時接觸齒數(Instantaneous Contact Teeth)也是齒輪強度與動態負載設計時必須要考量的一個重要因素，因此，也可利用齒輪接觸分析電腦模擬，同時計算出蝸桿蝸輪組之瞬時接觸齒數與平均接觸比。本研究利用所發展之電腦模擬程式，也探討了非90°交錯角單包絡蝸桿蝸輪組及雙包絡蝸桿蝸輪組在具有裝配誤差時的運動誤差、瞬時接觸齒數、平均接觸比及接觸比(Contact Ratio)。本論文亦舉幾個例題，用以論證齒輪組在各種裝配條件下，齒輪參數對於齒輪組之瞬時接觸齒數、平均接觸比、接觸比以及運動誤差的影響。 ZK型蝸桿是以創成法製造，一般是在銑床或磨床上用盤狀環面刀具加工出來，用這種方法加工的蝸桿齒面為刀具面的包絡面(Envelope)齒形。由於ZK型蝸桿可利用磨輪研磨的方式來提高其精度，因而可製造出較高精度之蝸桿，並廣為工業界所使用。因此，本研究也同時選用ZK型蝸桿與ZN型蝸輪嚙合，並探討此類組合之蝸桿蝸輪組在具有裝配誤差時齒輪參數對於運動誤差、瞬時接觸齒數、平均接觸比及接觸比的影響。

The worm gear set is composed of a worm and worm wheel, and is one of the most important devices for transmitting torques between spatial crossed axes. The worm gear set provides high gear ratios in a single stage that spur and helical gear sets do not offer. Due to its high transmission ratio, working steady, low noise and compact structure, the worm gear set is widely used in gear-reduction mechanisms. This dissertation discusses non-ninety-degree crossing angles of the ZN-type worm gear set and the ZN-type hourglass worm gear set, respectively. This study generated both worms by straight-edged blade cutters possessing a lead angle, and generated both worm wheels by a worm-type hob cutter and worm-type hourglass hob cutter, with a non-ninety-degree crossing angle, respectively. This work derived mathematical models of both single-enveloping and double-enveloping worm gear sets based on their generation mechanisms and the theory of gearing. According to the developed mathematical models of both worm gear sets and utilizing CAD software, the current study plotted the three-dimensional model of both worm gear sets. This investigation used the tooth undercutting analysis method to observe the distribution of undercutting lines on the ZN-type worm wheel surfaces with a non-ninety-degree crossing angle. The current work also investigated and compared the ZN-type hourglass worm wheel tooth geometry as well as its tooth working-surface with those of the ZA-type hourglass worm wheel. Contact ratios (CRs), gear bending and contact stresses are important gear design factors. A higher contact ratio reduces the contact stress and kinematic error (KE) of the gear set during its meshing. This dissertation calculated kinematic errors of both the ZN-type worm gear set and the ZN-type hourglass worm gear set, with a non-ninety-degree crossing angle, by applying the tooth contact analysis (TCA) method. The TCA method can also be used to calculate the average contact ratios (ACR) and instantaneous contact teeth (ICT) of the worm gear set when the gear set in a machine is subjected to large impact forces and dynamic loads. The developed computer simulation programs are applied to investigate the KE, ICT, ACR and CR of the ZN-type worm gear set and the ZN-type hourglass worm gear set with assembly errors. Illustrative examples are also presented to demonstrate the effects of gear parameters on ICT, CRs, ACR, and KEs under various assembly conditions. The ZK-type worm is generated by a cone-shaped disk-type grinding wheel with two straight-edged revolving surfaces in a milling or grinding machine, and the worm surface is the envelope to the family of cone-shaped disk-type grinding wheel surfaces. The grinding wheel process increases the precision of the ZK-type worm surface, and is widely used in industry. Hence, the proposed worm gear set, composed of the ZK-type worm and the ZN-type worm wheel, is also investigated. Interpretable examples also describe the influences of gear parameters on ICT, CRs, ACR, and KEs under various assembly conditions.