| 摘要: | 本文針對高減速比戟齒輪對，建立一套可在五軸CNC上實施之面銑式創成到接觸分析的完整數學模型，並建立齒面接觸數學模型，由齒胚幾何參數設計出發，推導產形輪、刀具與機台運動關係，完成泛用機到五軸CNC之參數轉換與單變數創成軌跡方程式，以生成連續、可加工之三維齒面，其後以曲面擬合與裝配座標轉換，構築齒面接觸數學模型，並結合Abaqus有限元素分析於無負載條件下驗證接觸線、接觸齒印與傳動誤差。方法上，本文參照文獻所揭露之面銑雙面雙切削設定，於Mathematica中建立參數化刀盤與工件轉換矩陣結合，求得齒胚範圍內之嚙合包絡解即齒面輪廓，並將面銑泛用機九項參數轉換為立式五軸CNC六項自由度軌跡，使創成軌跡成為創成角的單變數多項式，利於加工路徑展開與幾何求解。接觸分析方面，藉由齒面輪廓點建立B-spline擬合曲面，並建立裝配座標系統求解接觸性能指標接觸線、接觸齒印與傳動誤差。方法驗證方面，以HyperMesh切製齒輪網格模型在Abaqus有限元素分析軟體中使用近似剛體接觸，配合Python擷取接觸點、接觸齒印與傳動誤差曲線數據，重現至數學模型座標系，比對接觸性能。數值驗證採三組高齒數比案例展示齒胚模型穩健性與泛用性，本文建立之接觸分析方法也與有限元素結果相符，顯示本文所建齒面與接觸模型具良好幾何傳動性能與工程適用性。綜上，研究完成胚參數設計至齒面接觸分析之完整計算鏈，突破專用機台侷限，為高減速比戟齒輪之自主設計與CNC製程提供一致化、可驗證的理論基礎。;This study develops a complete mathematical framework for face-milling generation and contact analysis of high-reduction Hypoid gears implementable on five-axis CNC machines. Starting from gear blank parameter design, deriving the kinematic relations among the generating gear, cutter, and machine, establishing a parameter mapping from a generic milling setup to a five-axis CNC, and formulating a single-variable generating trajectory to produce continuous, manufacturable 3D tooth surfaces. The resulting surfaces are then fitted and transformed into the assembly coordinate system to build a tooth-surface contact model, which is validated under no-load conditions in Abaqus finite element analysis by comparing contact lines, contact footprints, and transmission error. Methodologically, referencing dual-flank, dual-cut face-milling configurations in the literature, implementing parameterized cutter and workpiece transformation matrices in Mathematica to solve the enveloping equations within the gear-blank bounds and obtain the tooth-surface geometry. The nine parameters of the generic face-milling machine are converted into a six-degree-of-freedom trajectory for a vertical five-axis CNC, making the generating path a single-variable polynomial of the generating angle, which facilitates toolpath expansion and geometric solution. For contact analysis, B-spline surfaces are fitted from the tooth-surface points, and the assembly coordinate system is used to compute contact performance contact lines, contact footprints, and transmission error. For verification, HyperMesh is used to create layered gear meshes; near-rigid contact is modeled in Abaqus to extract contact points, contact footprints, and TE curves, which are then mapped back to the mathematical model coordinates via Python for direct comparison. Three high gear-ratio case studies demonstrate the robustness and generality of the gear-blank and contact models; the proposed contact predictions agree with finite-element results, confirming good geometric transmission performance and engineering applicability. In sum, the work establishes an end-to-end computational chain—from blank parameter design to tooth-surface contact analysis—that removes dependence on specialized machines and provides a consistent, verifiable theoretical basis for autonomous design and CNC manufacturing of high-reduction Hypoid gears. |
