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研究生:邱雅靖
研究生(外文):Ya-Ching Chou
論文名稱:軸變形影響下之修整型螺旋齒輪對接觸特性探討
指導教授:蔡錫錚
指導教授(外文):Shyi-Jeng Tsai
學位類別:碩士
校院名稱:國立中央大學
系所名稱:機械工程學系
學門:工程學門
學類:機械工程學類
論文種類:學術論文
論文出版年:2016
畢業學年度:104
語文別:中文
論文頁數:192
中文關鍵詞:平行軸漸開線螺旋齒輪軸彎曲變形螺旋修整雙隆起修整受載齒面接觸分析成形磨隆起修整受載傳動誤差
外文關鍵詞:parrallel-axis involute helical gearshaft deformationhelix modificationdouble crowningLTCAform grindingloaded transmission error
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一般而言,漸開線齒輪有著加工簡單,中心距組裝誤差敏感度低等特性。且和正齒輪相比,螺旋齒輪更有較高的接觸率以及漸進的接觸線變化,使其能承受較大的負載,傳動過程也較為平順;故做為傳動元件,螺旋齒輪的應用範圍相當廣泛。
一般來說,齒輪受載時,與其結合之軸也會因此產生變形。為探討軸變形對螺旋齒輪受載齒面接觸特性的影響,本研究以影響係數法為基礎建立受載齒面接觸分析模型。此分析模型,係納入齒面接觸變形、輪齒彎曲變形、軸扭轉變形以及軸彎曲變形,結合齒對「負載-變形」與力平衡等關係式建立而得。此分析模型亦包括齒面接觸分析模型,除建立螺旋齒輪三維齒面方程式,並藉由齒面漸開線特性建立嚙合條件以及納入齒輪對組裝關係,並以此求得兩嚙合齒面之間距,做為受載齒面接觸分析模型之依據。因此本分析模型可模擬出嚙合齒面之真實接觸斑與應力分佈,特別非赫茲接觸亦可模擬而得。
本論文共探討三種不同螺旋齒輪齒面:無修整齒面,理想雙隆起齒面以及成形磨加工之雙隆起齒面;以及兩種不同軸支撐配置:簡支配置與懸壁支撐配置。在研究中,共探討齒面接觸應力分佈,嚙合過程齒對負載分配與最大應力變化,傳動誤差等接觸特性。
由無修整齒面螺旋齒輪對之分析可知,齒面應力分佈在軸彎曲變形影響下,會明顯往近軸承方向偏移,且偏移狀況會隨著不同軸支撐形式下的彎曲變形傾角而改變:彎曲變形傾角越大,則應力偏移狀況越嚴重,齒面負載(分佈不均)係數也變大;同時接觸線在齒頂或齒面寬之齒面位置之接觸應力亦有應力集中現象,尤其齒頂處更為嚴重。另一方面,嚙合過程的負載分配也因軸變形影響,使得近軸承端的接觸位置承受更大的負載;而受載傳動誤差的平均值和峰值差也隨著軸變形量越大而增加。
為了消除齒面應力集中現象以提高齒輪壽命,本研究選擇對小齒輪進行修整。齒面修整方首先藉由螺旋齒線修整,改善先前因軸變形而造成分佈不均的齒面應力;並利用雙隆起修整確實消除齒面寬兩端以及齒頂附近的應力集中狀況。在考慮理想齒面修整方式下,齒面接觸應力呈現一均勻且略為偏向近軸承端的近似水滴接觸斑形式;螺旋齒輪之受載接觸率則因雙隆起修整從約2.47(線接觸)降到1.76(點接觸)左右,且嚙合過程的負載分配形式非常接近一無任何劇烈變動的梯形形狀。
而成形磨因加工後的齒面精度極佳,也能同時修整兩側齒腹,成為為今日常見的螺旋齒輪修整加工方式。但若是依照預訂隆起修整路線進行齒線修整,齒面會產生明顯的扭曲情形。因此本研究針對在不補償修整路徑的狀況下之修整螺旋齒輪,分析其接觸特性。分析結果可知,經成形磨加工之螺旋齒輪右旋右齒腹其雙隆起修整的齒面扭轉趨勢和接觸線的傾斜形式非常接近,因此右旋右齒腹的接觸應力狀況較佳;反之,右旋左齒腹的應力狀況因齒面扭轉趨勢不同,所以接觸應力情況較差。改變旋向後,左旋右齒腹的分析結果和右旋左齒腹之結果相當接近;而左旋左齒腹的狀況則和右旋右齒腹類似。成形磨修整造成的齒面扭曲使得受載傳動誤差的峰值差大幅增加約2倍左右;但是,平均值則有明顯降低的現象。

Compared with spur gears, helical gears have the advantages of relatively higher contact ratio and load capacity, as well as more smooth and silent operation. As a consequence, helical gears ar the most used elements for power transmission.
In general, the shafts are deformed caused by the loads acting in the gears. In order to explore the effects of the gear shaft deformation on the contact characteristics of loaded teeth, an LTCA (loaded tooth contact analysis) model are developed in the thessis. The proposed LTCA model, based on the influence coefficient method, involves the contacted deformation and the bending of the tooth, as well as the torsion and deflection of the shaft. This LTCA model includes also a tooth contact model for gear meshing analysis, in which the mathematical equations of the actual helical gear flank surfaces, the assembly conditions of the gear pair, the contact points and the tooth gaps can be involved. Because the actual three dimensional flank surfaces are considered in the LTCA model, the contact stress distribution and the also the corresponding contact patterns can be simulated, even also the non-hertz contact of the the engaged teeth.
Three different tooth forms of the helical gears are considered in the study: non-modified flanks, ideally double corwned flanks and double crownded flanks by profile-grinding. On the other hands, two support arrangements of the shafts are also involved in the analysis: simple support arrangement and overhand arrangement. The contact characteristics, analyzed in the thessis, include the contact stress distribution, the variation of the load sharing and the max. contact stress during the gear meshing, as well as the transmission errors.
It can be recognized from the analysis results of the non-modified helical gears that the contact stress near the face-end of the bearing-side increases obviously due to the effect of shaft deformation. This phenomenon of uneven stress distribution is more serious with the increasing of shaft deformation (inclination angle). On the other hand, the concentrated stresses can be also found in the tooth conact on the tip and the face-end. The peak-to-peak value and the average amount of the loaded transmission error in this case increases also by the enlarged deformation of the loaded shaft.
In order to increase the load capacity and elimate the stress concentration, the flank modification of helical gears is, an essentail but alos effective method. In the study, double crowning modification is choosen, which consists of the helix modification, the lead crowning modification, and the profile crowning crowning modification. Helix modification can reduce the uneven contact stress distribution on flanks due to the shaft doformation, while lead and profile crowning can elimate stress concentration on the face-end, and tip/root of flanks. The analysis results of the ideally modified helical gears show that an uniform oval-shaped contact pattern is performed. The stress concentration on the flanks during the gear meshing does not occur, but the loaded contact ratio decreases from 2.47 (line contact) to 1.76 (point contact). In addition, the variation of the load sharing of the single tooth pair during the gear meshing becomes smooth.
Profile grinding is now a widely applied finishing method for helical gears, because of its high precision and good productivity. However, twisted tooth flank occurs if there are no correction measures conducted for profile grinding of the lead crowned flnaks. The contact characteristics of the double crowned flanks with nature twist are thus studied by using the proposed LTCA approach. Because the ease-off twisted right flank of gears with right hand helix angle is similar to the diagonal flank modification, the contact characteristics of this flank side are better than those of the left flank. Similarly, this guidline is also valid for the gear with left hand helix angle, where the contact characteristics of the left flank is better than those of the right flank. Besides, the peak-to-paek value of twisted flanks with the modification parameters used in the case is twice larger than that of ideally modified flank, but the averge value is smaller.

摘要 i
Abstract iii
謝誌 vi
目錄 vii
圖目錄 xi
表目錄 xviii
符號說明 xix
第 1 章 前言 1
1.1 研究背景 1
1.2 文獻回顧 2
1.3 研究目的 4
1.4 論文架構 5
第 2 章 螺旋齒輪齒面數學模型 6
2.1 無修整螺旋齒輪之齒面方程式 6
2.1.1 齒輪座標系定義 6
2.1.2 標準螺旋齒輪漸開線齒面方程式 7
2.2 理想修整齒面之數學模型 11
2.2.1 齒線修整模式 11
2.2.2 齒形修整 14
2.2.3 雙隆起修整螺旋齒輪之齒面方程式 15
2.3 成形磨齒面修整之數學模型 16
2.3.1 成形磨組裝位置座標系設定 16
2.3.2 磨輪標準齒面方程式 18
2.3.3 成形磨修整加工之嚙合方程式 21
2.3.4 修整齒面方程式 24
2.4 修整齒輪齒面偏差量計算 25
第 3 章 螺旋齒輪受載齒面接觸分析模型 27
3.1 螺旋齒輪對嚙合分析 27
3.1.1 接觸條件 27
3.1.2 接觸點關係式建立 28
3.2 受載齒面接觸應力計算基本模型 30
3.2.1 單齒對接觸狀況 30
3.2.2 多齒對接觸負載 32
3.3 嚙合齒對齒面間距計算 33
3.3.1 無修整齒面間距計算 34
3.3.2 修整齒面間隙角計算 37
3.4 不同負載形式之影響係數 38
3.4.1 齒面接觸赫茲變形影響係數 39
3.4.2 螺旋齒輪輪齒彎曲變形影響係數 40
3.4.3 軸扭轉變形影響係數 46
3.4.4 軸彎曲變形影響係數 48
3.5 傳動誤差定義 50
第 4 章 無修整齒對齒面受載接觸分析 52
4.1 分析案例之參數 52
4.1.1 齒輪參數與分析案例 52
4.1.2 軸支撐配置 54
4.2 特定位置的接觸應力分佈 55
4.2.1 不考慮軸影響接觸應力分佈 56
4.2.2 考慮軸支撐形式對接觸應力之影響 60
4.2.3 螺旋角旋向影響接觸應力分佈 65
4.3 無修整齒輪之嚙合過程分析 70
4.3.1 嚙合過程之負載分配 70
4.3.2 嚙合過程之最大接觸應力 78
4.3.3 5嚙合過程之受載傳動誤差 82
第 5 章 理想修整齒面受載接觸分析 86
5.1 齒線修整設計 86
5.1.1 螺旋修整對線接觸齒面接觸應力分佈之影響 87
5.1.2 無限齒面寬下隆起修整對點接觸齒面應力分佈之影響 90
5.1.3 有限齒面寬下的修整量選擇 92
5.1.4 齒線修整後之特定位置接觸應力 95
5.1.5 小結 100
5.2 齒形修整設計 100
5.2.1 不同齒形修整量比較 100
5.2.2 雙隆起修整後特定位置接觸應力 103
5.2.3 修整齒面偏差情形 110
5.3 較佳修整參數之嚙合過程分析 113
5.3.1 嚙合過程之負載分配 113
5.3.2 嚙合過程之齒面負載分佈 118
5.3.3 嚙合過程之傳動誤差 121
第 6 章 成形磨修整齒面受載接觸分析 130
6.1 修整齒面特性 130
6.1.1 成形磨齒線拋物線隆起修整 131
6.1.2 成形磨雙隆起修整 137
6.2 嚙合過程分析 144
6.2.1 嚙合過程負載分配 144
6.2.2 嚙合過程之歯面最大應力 145
6.2.3 嚙合過程傳動誤差 148
第 7 章 結論與展望 151
7.1 結論 151
7.2 未來展望 153
附錄A 154
參考文獻 155


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