本文提出螺旋齒輪對自動網格產生方法並分析螺旋齒輪對的動態響應。將採用近於連體幾何有限元素模式來探討螺旋齒輪對動態特性、製造誤差與齒輪修整之關係。漸開線螺旋齒輪系統具有高轉速運轉平順、低振動噪音、高精密傳動、高傳遞功率-體積比與加工易達高精密等優良特性，但其動態特性會受到加工與組裝誤差、嚙合背隙、嚙合齒對數改變等因素的影響而劣化。在設計時上述之劣化因素可運用各種齒形與隆齒修整手段予以改善之。 本研究不以傳統等效彈簧質量的離散模式，將採用連體幾何模式來探討螺旋齒輪對動態特性。首先將應用齒條形刀具之輪廓方程式館，以齊次座標轉換與齒輪嚙合方程式，推導出可應用於多種齒輪形式之齒形方程式，包括漸開線、齒底圓角等輪廓方程式，然後不經CAD軟體，撰寫C程式，直接以參數化的快速產生廣泛形式齒輪網格元素，解決以連體模式之齒輪系統分析時，高品質網格模式建立不易的問題。然後以LS-DYNA求解齒輪對動態，計算出動態嚙合齒根應力、接觸力與應力、以及傳動誤差。並以正齒輪動態齒根應力與文獻的實驗結果做比較，間接驗證數值結果的正確性。最後則探討裝配誤差、隆齒修整以及齒頂導角對於齒輪對動態特性之影響，建立齒輪動態、誤差、修整齒頂導角與過切間之關係，希望作為設計螺旋齒輪系統之參考。

This thesis presents an approach to automatically generating high quality meshing elements of gears and also analyzes dynamic responses of helical gear pairs. Using the finite element method, dynamic characteristics under the consideration of manufacturing errors and gear modifications are investigated. As commonly understanding, helical gearings have excellent features of high precision, high speed, high power to weight ratio, and low vibration and noise. Nevertheless, numerous factors, such as machining and assembly errors, backlash, and change of number of meshing tooth pairs, may deteriorate the dynamic performance. Fortunately, through suitable design technique, the negative effect can be improved such as profile or crowing modifications of gears in the design phase. In this study, instead of conventional discrete models with equivalent masses and springs, continuum geometry models are adopted in investigating the dynamic responses of the helical gear pairs. An approach proposed to automatically generating high quality meshing elements is applicable to wide varieties of gears. Firstly, profile equations of a transverse section of a rack cutter are derived. Then, using the homogenous transformation matrix and equation of meshing for gears, equations of involutes, fillets, and other curves in the gear teeth are deduced. Next, not CAD models but using a C code, meshing elements of the analyzed gears are generated after calculating nodal coordinates directly from the derived tooth profile equations. Creation of several element examples of the gears displays effectiveness of the proposed approach. Moreover, using the software LS-DYNA dynamic responses which include dynamic fillet stresses, contact forces and contact stresses, and transmission errors of a standard spur pair and helical gear pairs are calculated. The dynamic fillet stress of a spur gear pair using the proposed method is compared with the experimental result in order to verify the numerical correctness. Finally, for the purpose of facilitating the design of helical gear pairs, influences of several factors which are assembly errors, crowning modification, addendum chamfering, and undercutting, on the dynamic characteristics of helical gear pairs are discussed.

中文摘要 .................................................................... I 英文摘要 .................................................................... II 誌謝 ....................................................................... IV 目錄 ........................................................................ V 圖表目錄 .................................................................. VIII 符號表 ................................................................... XII 第一章 序論 ................................................................. 1 1.1 研究背景 ................................................................. 1 1.2 研究動機與目的 .......................................................... 2 1.3 文獻回顧 ................................................................. 3 1.4 內容大綱 ................................................................. 5 第二章 齒輪幾何模式 ............................................................ 7 2.1 齒輪修整 ................................................................. 7 2.1.1 隆齒修整 ................................................................ 7 2.1.2 隆齒之分類 .............................................................. 9 2.2 齒輪誤差 .................................................................. 9 2.3 齒輪輪廓方程式之創成 ...................................................... 15 2.3.1 齒條形刀具截面方程式 .................................................... 15 2.3.2 創成齒輪之齒廓方程式 .................................................... 18 2.4 齒輪過切 ................................................................ 22 2.5 齒頂導角 ................................................................. 23 2.6 齒輪接觸分析 ............................................................. 24 第三章 螺旋齒輪對網格模式建立 .................................................. 25 3.1 產生齒輪網格過程 .......................................................... 25 3.1.1 齒形參數 ............................................................... 25 3.2 齒輪網格模式建立 .......................................................... 29 第四章 螺旋齒輪動態分析 ....................................................... 34 4.1 分析項目 ................................................................. 34 4.2 LS-DYNA分析之條件設定 .................................................... 36 4.2.1 邊界與初始條件 ......................................................... 36 4.2.2 阻尼 .................................................................. 37 4.3 螺旋齒輪對之動態分析 ...................................................... 38 4.4 組合誤差對動態影響 ........................................................ 42 4.4.1 齒根應力 ............................................................... 43 4.4.2 接觸力與接觸應力 ........................................................ 44 4.4.3 傳動誤差 ............................................................... 45 4.5 隆齒修整對動態影響 ........................................................ 47 4.5.1 齒根應力 ............................................................... 47 4.5.2 接觸力與接觸應力 ........................................................ 51 4.5.3 傳動誤差 ............................................................... 52 4.6 齒頂導角對動態影響 ........................................................ 54 4.6.1 齒根應力 ............................................................... 54 4.6.2 接觸力與接觸應力 ........................................................ 56 4.6.3 傳動誤差 ............................................................... 57 4.7 過切對動態影響 ........................................................... 57 4.7.1 齒根應力 ............................................................... 58 4.7.2 接觸力與接觸應力 ........................................................ 58 4.7.3 傳動誤差 ............................................................... 60 第五章 結論與未來展望 ......................................................... 61 5.1 結論 .................................................................... 61 5.2 未來展望 ................................................................. 62 參考文獻 ..................................................................... 63

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