近似線接觸型態之歪斜軸漸開線錐形齒輪對齒面接觸強度之研究

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吳思漢(Szu-han Wu) 查詢紙本館藏

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機械工程學系

論文名稱

近似線接觸型態之歪斜軸漸開線錐形齒輪對齒面接觸強度之研究
(Contact Strength of a Skew Conical Involute Gear Drive in Approximate Line Contact)

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摘要(中)

漸開線錐形齒輪在空間齒輪機構之應用具有多種組合可能，尤其可應用於小軸交角場合，並具有加工容易、組裝誤差敏感度低、背隙可以控制等優點，為漸開線特殊齒輪中應用最廣的一種。歪斜軸組合之錐形齒輪對由於點接觸的嚙合特性，因此具有低組裝誤差敏感度的優點，然而卻也因此有接觸應力過大的問題，而仍無法廣泛地應用於重負載動力傳動的場合。本研究從幾何設計著手，透過「移位嚙合設計」與「近似線接觸」兩個方法來達到增加接觸斑的面積、從而降低齒面接觸應力的目的。利用錐形齒輪沿齒面寬連續移位之特性，本研究有系統地建立完整計算流程，可以使錐形齒輪嚙合位置改變，而形成正移位嚙合，即以其具有較小主曲率的齒面來進行傳動。另一方面，則透過齒輪參數的設計，使錐形齒輪之接觸斑形成點接觸，但是形狀為扁長之橢圓形式。因此在受負載情況下，接觸斑會因為齒面彈性變形而形成近似於線接觸的型態。此種接觸即為所謂「近似線接觸」。
針對此「近似線接觸」嚙合型態的歪斜軸錐形齒輪對，本研究系統化地分析不同的齒輪參數組合，探討因組裝誤差或加工誤差而發生齒端邊緣接觸的敏感度，以及齒輪參數與齒面承載能力的關係。並根據此分析結果歸納出一「近似線接觸」嚙合的設計方法，包括錐形齒輪對移位嚙合設計流程，以及建議的設計準則。
為了能確實評斷齒面的承載能力，本研究對此近似線接觸設計進行受負載輪齒接觸分析。由於近似線接觸的接觸斑在負載下可超過齒面寬，係屬於非赫茲接觸問題，以及輪齒的彎曲變形會影響其接觸應力分佈。因此本研究首先建立一以漸開線齒輪對的作用線為基礎的嚙合模型，可適用於嚙合齒面之曲率很接近的近似線接觸情況。其次再利用影響係數法的概念發展一齒面接觸應力計算模型。其中不但考慮齒面接觸變形的效應，亦將輪齒彎曲撓曲的影響納入考慮。此齒面接觸應力計算模型可應用於點接觸、線接觸、單齒對嚙合、多齒對嚙合、乃至於具齒面修整之齒面等各種不同的接觸情況。同時並採用有限元素方法來驗證接觸應力計算結果的可靠度。
本文中提供一設計實例，從幾何設計到受負載輪齒接觸分析作一完整地分析和討論。此設計實例的容許組裝誤差、以及容許加工誤差皆符合一般實務應用需求，而且其承載能力亦足夠應用於重負載傳動。使用接觸應力計算模型所得到的結果與有限元素方法的結果誤差約5%，證明此計算模型的可靠度，而且具有比有限元素方法更有效率的優點。
為了驗證此近似線接觸嚙合型態的歪斜軸齒輪對的齒面承載能力適用於重負載傳動，本研究開發一功率封閉型齒輪試驗裝置，實際進行107負載循環齒輪失效檢驗之實驗。被測齒輪的材料為中碳鋼S45C、且進行調質熱處理，但不作表面硬化。實驗以三組齒輪對，分別進行扭矩為200、300、350Nm負載等級(最大接觸應力分別為892、1021、1074 N/mm2)之試驗。由於本實驗屬於高負載低速、錐形齒輪的齒面粗糙度不佳、以及齒輪僅作調質熱處理而無表面硬化處理等因素，齒面破壞的型態除了屬於疲勞破壞的點蝕之外，更嚴重的破壞形式為冷刮痕(cold scoring)或是磨耗(wear)。
針對「近似線接觸」嚙合型態的歪斜軸錐形齒輪對，本研究提出有效且可靠的設計方法、嚙合模型、以及齒面接觸應力計算模型。經過實驗驗證，近似線接觸設計之錐形齒輪對，確實具有更佳的齒面承載能力。

摘要(英)

Of all the different types of gears, conical gears as the special type of general spatial gearing makes them not only suitable for parallel axis transmission, but also for cases with intersecting or skew axes. However, one of the weak spots of conical gears is less surface durability in spatial applications due to point contact problems, although transmission accuracy is not correspondingly sensitive to assembly error. From the viewpoint of design, this study suggested two concepts to increase the surface durability of conical gears. Skew conical gear drives can be designed as gear pairs, with either “profile-shifted transmission”, or “approximate line contact”.
The profile-shifted transmission can give us the possibility to adjust the contact position of a conical gear pair for better tooth contact bearing just only through changing certain gearing or assembly data. On the other hand, an approximate line contact drive has a contact ellipse with a large major-to-minor-axis ratio, which allows it to overcome the weakness of conical gear drives for application in power transmission. This gearing design approach is characterized by reduced edge contact sensitivity and increased surface durability.
The edge contact sensitivity that can arise with this kind of gear drive due to assembly or manufacturing errors is evaluated by analyzing the value of the shift of the line of action caused by such errors. The surface durability is evaluated by calculating the Hertz stress. Some guidelines are developed based on the analysis of the influence of the gearing parameters on the edge contact sensitivity and the surface durability made possible using this design approach for conical gear drives in the approximate line contact.
An efficient approach for loaded tooth contact analysis (LTCA) of conical gear drives is developed. Two new models are developed for the meshing and contact stress anslyses. This approach differs from conventional TCA methods in that the meshing analysis is based on the line of action characteristic of involute gearing. A numerical method is applied to calculate the contact stress. The non-Hertzian contact problem is solved giving due consideration to the influences of the tooth contact deformation and tooth bending deflection. The approach is not only suitable for application for cases of non-Hertzian contact, but also for practical cases, such as in gear drives with end-relief.
A practical example is given to demonstrate the feasibility of the approximate line contact design. The LTCA results of the example are also compared with those of solved by the finite element method (FEM) to which they are in good consistence. Finally, a back-to-back testing equipment was developed for testing the surface fatigue strength of a skew conical gear drive. The objective of the tests was to determine the limited service life (number of load cycles) of the skew conical gear drive in approximate line contact for tooth pitting fatigue failure under several load levels. Three test gear pairs (S45C, hardening and tempering, but no case hardening) with different applied torques were tested: 200 Nm, 300 Nm, and 350 Nm. The results indicated that the surface durability of the approximate line contact design is indeed increased. Furthermore, the kinds of gear surface failures for the test were pitting, cold scuffing, and wear.

關鍵字(中)

★ 齒面破壞試驗
★ 齒面疲勞強度
★ 錐形齒輪
★ 受負載輪齒接觸分析
★ 齒端邊緣接觸
★ 近似線接觸

關鍵字(英)

★ conical gear
★ approximate line contact
★ edge contact
★ loaded tooth contact analysis
★ surface fatigue strength
★ gear tooth surface failure

論文目次

摘要 i
Abstract iii
誌謝 v
目錄 vi
圖目錄 ix
表目錄 xv
符號說明 xvi
第1章緒論 1
1-1 研究背景 1
1-1-1 漸開線錐形齒輪之特性 2
1-1-2 漸開線錐形齒輪應用上之限制 4
1-2 文獻回顧 4
1-3 研究動機與目的 6
1-4 研究方法 8
1-5 論文架構 10
第2章理論基礎 12
2-1 齒輪運動學 12
2-1-1 創成齒輪與刀具的相對速度螺旋 12
2-1-2 歪斜軸齒輪對的相對速度螺旋 13
2-2 漸開線錐形齒輪數學模式 16
2-2-1 標準齒形 16
2-2-2 具有齒端修整齒面 20
2-3 歪斜軸漸開線錐形齒輪對移位嚙合設計 22
2-3-1 橫向齒形角係數 24
2-3-2 組裝關係 25
2-3-3 運動學基本關係 26
2-3-4 無背隙嚙合關係 28
2-3-5 齒輪加工參數與工作參數關係 29
2-4 歪斜軸漸開線錐形齒輪對嚙合基本關係 31
2-4-1 作用線 31
2-4-2 漸開線錐形齒輪齒面的主曲率與主方向 34
2-4-3 Hertz 理論 36
2-4-4 傳動函數 37
第3章近似線接觸之幾何設計 40
3-1 設計概念 40
3-1-1 基本構想 40
3-1-2 線接觸嚙合條件 40
3-2 齒端邊緣接觸分析 42
3-2-1 軸交角誤差之影響 43
3-2-2 偏位誤差之影響 46
3-2-3 加工誤差之影響 47
3-3 齒面承載能力分析 50
3-4 設計方法與準則 52
第4章歪斜軸錐形齒輪對受負載接觸分析 56
4-1 嚙合模型 56
4-2 齒面接觸應力計算模型 61
4-2-1 非赫茲應力基本計算模型 62
4-2-2 多齒對嚙合計算模型 66
4-2-3 齒面接觸變形影響係數 66
4-2-4 輪齒懸臂梁撓曲影響係數 68
4-2-5 數值計算流程 72
4-3 有限元素模型 75
第5章實例探討 78
5-1 近似線接觸與線接觸型態之歪斜軸錐形齒輪對 78
5-1-1 齒輪數據與容許誤差 78
5-1-2 近似線接觸型態受負載輪齒接觸分析 81
5-1-3 線接觸型態受負載輪齒接觸分析 85
5-1-4 相對速度分析 91
5-2 齒端修整對接觸應力的影響 94
第6章齒面承載能力實驗驗證 97
6-1 齒面破壞的種類 97
6-1-1 點蝕 100
6-1-2 擦損或刮痕 101
6-2 實驗驗證設備設計 102
6-2-1 實驗設備工作原理與架構 102
6-2-2 實驗設備規格與設計重點 104
6-3 實驗基本條件 108
6-3-1 被測齒輪 108
6-3-2 潤滑條件 109
6-3-3 齒面強度試驗判定失效的標準 110
6-4 實驗內容規劃 110
6-4-1 負載規劃 110
6-4-2 實驗步驟 111
6-5 實驗結果與討論 113
6-5-1 齒面點蝕面積率 114
6-5-2 齒面破壞型態 116
第7章結論與未來展望 120
7-1 結論 120
7-2 未來展望 122
參考文獻 123
附錄被測齒輪檢驗報告 132
作者簡介 134

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指導教授

蔡錫錚(Shyi-Jeng Tsai)

審核日期

2009-7-29

推文