諧波齒輪減速機具有高傳動精度、高速比以及體積小的優點，半導體業、醫療設備以及機械手臂等領域對於諧波齒輪減速機的需求越來越多，目前市面上的諧波減速機為雙波型波產生器。由於三波型諧波齒輪有更好的傳動穩定性的優點，但是文獻鮮少討論三波型波產生器，因此本論文提出一個杯型之三波型諧波齒輪設計方法。
諧波齒輪主要有三個零件，波產生器、柔輪以及剛輪，理論上，剛輪齒形是根據柔輪齒形及剛輪柔輪間的運動關係求得，其中變形後的柔輪中性線是建立剛輪齒形非常重要的基礎。
本論文基於三波型波產生器建立一套剛輪齒面的數學模型。變形後的中性線位置是利用有限元素分析軟體求得，進一步得到其位置向量。杯型柔輪變形後會產生錐化齒面，導致與剛輪在開口端會有干涉現象，閉口端則是會有間隙產生。我們使用不同軸截面的輪廓建立三維的剛輪齒形，這些輪廓根據有限元素分析軟體得到的不同軸截面中性線求得。最後對剛輪進行齒面修形以此改善齒輪的嚙合性能。

Harmonic drive reducer (HD) has the advantages of high transmission accuracy, high-speed ratio, and small size. The demands for HD are rising in semiconductors, medical devices, and robotic arms. Currently, the wave generator is mostly a two-lobe type in the market. Three-lobe type HD has the advantage of better transmission stability, but the literature rarely discusses this wave generator. As a result, this paper proposes a design method for the cup-type three-lobe HD.
HD comprises three essential components: wave generator (WG), flex spline (FS), and circular spline (CS). Theoretically, the CS tooth profile is derived from the FS’s tooth profile and its movement relative to the CS. The neutral curve of FS after deformation is a fundamental basis for developing the CS tooth profile.
This paper proposes the mathematical model of the CS tooth surface based on the three-lobe wave generator. The neutral curve after deformation is determined from finite element analysis (FEM), and its position vector is then derived. The cup-type FS has a tapering tooth surface after deforming that causes interference near the open end and clearance near the closed end between the CS. Therefore, we use a group of different profiles along the gear axis to rebuild the three-dimensional tooth profile of CS. Those profiles are derived from the different neutral curves on transverse sections according to FEM. Finally, the flank modification of the CS tooth surface is performed to improve the contact performance of the gear pair.

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