以盤狀圓弧刀具創成之曲線齒的數學模式推導與接觸分析研究

Abstract

圓弧型曲線齒齒輪（circular-arc curvilinear-tooth gear）係指一種由圓弧型盤狀刀具所創成，具有曲線型齒線及圓弧型齒形特徵的齒輪。 不同於以往傳統正齒輪與螺旋齒輪的線接觸型的傳動運轉方式，具曲線型齒線特徵的齒輪組是以點接觸的接觸型態來傳輸動力。點接觸的接觸型態不僅能使齒輪組避免齒緣的接觸及降低由軸向組裝誤差所引起的運動誤差和振動噪音，更可增強齒輪的接觸強度。除此之外，具有曲線型齒線之齒輪在運轉時亦不會產生軸向的推力。 具圓弧型齒形的螺旋齒輪因為具有較一般傳統螺旋齒輪為高的負載能力及更佳的潤滑效果等特性，故常被使用於需傳輸高負載的情況，但是圓弧型螺旋齒輪對於中心距離的組裝誤差是十分敏感的。 本研究係將漸開線型曲線齒齒輪的齒形改以圓弧型外形之齒刀來創成齒輪，亦即應用一把虛擬的圓弧型齒刀依循一曲線路徑而創成出該齒輪之齒面，並根據齒輪原理建立圓弧型曲線齒齒輪之齒面數學模式，同時也以轉位修正的方式來避免齒形過切的情況發生，亦利用齒輪接觸分析技術來探討具有裝配偏差時，齒輪組運動誤差、接觸比及接觸齒印。

The circular-arc curvilinear-tooth gear, which has curvilinear trace with circular-arc profile, is generated by a circular-arc disk-type cutter. Gears with curvilinear traces ensure that the gear pair is operated in a point contact condition, which is different from the line contact type of a helical gear pair or spur gear pair. The point contact condition can not only avoid the tooth edge contact, decreases the kinematical errors and vibro-acoustic due to axial misalignments, but also increases the bearing strength of the contact gears. Moreover, a gear pair with curvilinear-tooth traces operates without axial thrust force. The helical gears with circular-arc profile are used for transmitting with high load circumstance, because their loading capacity and lubricity are better than those of conventional helical gears. However, the circular-arc helical gear pair is quite sensitive to the central distance assembly errors. This study adopts the rack cutter with circular-arc profile to modify the profile of involute-type curvilinear-tooth gear pair. The gear surface is generated by an imaginary circular-arc rack cutter with a curvilinear trace. According to theory of gearing, a mathematical model of the gears with circular-arc profile and curvilinear-tooth trace is developed. The undercutting of the gears can be avoided by a positive profile-shifted modification during the gear set generation process. The tooth contact analysis technique is utilized to the investigation on kinematical errors, contact ratios and contact patterns of the gear set under axial misalignments.