| 摘要: | 諧波齒輪因其高減速比、高精度與緊湊結構的特性，被廣泛應用於機器人、精密伺服系統與航太設備等高精度傳動領域。其剛輪與柔輪齒形的設計對傳動性能至關重要，因此本研究針對諧波齒輪的齒形進行優化與分析，以提升接觸特性、負載分佈與扭轉剛性。
本研究以雙貝茲曲線構建剛輪齒形，並透過包絡法推導出與其共軛的二維柔輪齒面數學模式，進而建立用於加工柔輪之二維假想齒條刀的數學模型。透過C++等程式開發自動化流程，使剛輪與柔輪能夠快速建立網格，並完成二維有限元素分析，包括嚙合齒數、扭轉剛性與應力分佈的計算。透過串聯ISIGHT軟體最佳化流程，針對二維齒形進行多目標優化，以最大化嚙合齒數、最小化扭轉角度（提升扭轉剛性）為目標，優化後的接觸齒數從16%提升至40%，扭轉角度由56.298×10⁻⁴ rad 降至 17.417×10⁻⁴ rad，有效改善齒形設計。
本研究進一步將最佳化後的二維齒形推導至三維空間，考量因錐化效應造成之齒頂干涉，提出一套前後端導程修整設計，並以變轉位係數參數式進行軸向修整，使接觸區域轉移至齒面中段，改善應力集中。三維有限元素分析結果顯示，修整後接觸齒數由 14% 提升至 26%，最大接觸應力由 800.8 MPa 降至 732.4 MPa，最大 von Mises 應力由 828.5 MPa 降至 779.4 MPa，並同步降低遲滯損失、傳動誤差與週期應力，扭轉剛性亦略有提升。
本研究建立了一套從二維齒形最佳化到三維齒形修整的完整流程，透過數值模擬驗證其可行性，並評估齒形設計對諧波齒輪各項指標的影響，包括接觸齒數、應力分佈、扭轉剛性、遲滯損失、傳動誤差及週期應力。
;Harmonic drives, known for their high reduction ratio, high precision, and compact structure, are widely used in high-precision transmission applications such as robotics, precision servo systems, and aerospace equipment. The tooth profile design of the circular spline (CS) and the flexspline (FS) plays a critical role in transmission performance. Therefore, this study focuses on the optimization and analysis of harmonic drive tooth profiles to enhance contact characteristics, load distribution, and torsional stiffness.
In this study, a CS tooth profile was constructed using a double Bézier curve. The conjugate two-dimensional FS tooth profile was derived through the envelope method, followed by the construction of a corresponding 2D imaginary rack cutter model for FS machining. An automated process was developed using C++ to efficiently generate mesh models for both CS and FS, enabling two-dimensional finite element analysis (FEA) to evaluate key performance indicators such as contact tooth count, torsional stiffness, and stress distribution. Through integration with ISIGHT software, a multi-objective optimization (MOO) was performed to maximize contact ratio and minimize torsional angle (i.e., enhance torsional stiffness). As a result, the contact ratio increased from 16% to 40%, and the torsional angle decreased from 56.298×10⁻⁴ rad to 17.417×10⁻⁴ rad, effectively improving the overall tooth design.
The optimized 2D tooth profile was further extended into a 3D model. To address tooth tip interference caused by the conical effect, a lead modification strategy was proposed, employing a variable profile shift coefficient to perform axial correction at both ends of the FS. This approach shifted the contact region toward the mid-width of the tooth surface and alleviated stress concentration. Results from 3D FEA showed that the contact ratio improved from 14% to 26%, the maximum contact stress decreased from 800.8 MPa to 732.4 MPa, and the peak von Mises stress dropped from 828.5 MPa to 779.4 MPa. Additionally, improvements were observed in terms of reduced hysteresis loss, transmission error, and cyclic stress, while torsional stiffness was slightly enhanced.
This study establishes a complete workflow from 2D tooth profile optimization to 3D profile correction. Numerical simulations validate the feasibility of the approach and comprehensively evaluate the influence of tooth profile design on critical harmonic drive performance metrics, including contact ratio, stress distribution, torsional stiffness, hysteresis loss, transmission error, and cyclic stress. |
