小齒數比之錐形齒輪對修整設計、分析與疲勞測試

Flank Modified Conical Gear Drives with Small Gear Ratio: Design, Analysis and Fatigue Test

林旻鴻(Min-Hung Lin)

指導教授 ： 蔡錫錚

國立中央大學/工學院/機械工程學系/碩士(2014年)

摘要

相較於傘齒輪，鉋切錐形齒輪對由於為錐形齒輪與正齒輪之組合，可有較低組裝誤差敏感度，同時齒面幾何關係複雜度低、易於設計，若藉由模具生產方式製作，更可降低成本；因此在大軸交角、大量生產的傳動機構應用場合具有取代傘齒輪的潛力。但由於在點接觸設計下，受載齒對接觸情況類似正齒輪對，負載分配形成非連續狀況，此結果可能導致齒對接觸開始、結束或齒對變化發生負載急遽變動以及齒頂邊緣應力集中現象，進而易導致齒面磨耗、破損，甚至造成齒根斷裂。因此為能廣泛應用，必須確認齒輪之承載能力是否滿足設計要求，特別是應用於粉末冶金模造齒輪之場合。
因此本論文之研究目的，即在魏健宇[4]所提出之錐形齒輪對非標準設計的基礎上，提出齒形拋物線隆起方式修整錐形齒輪齒面，藉以改善負載分配，得以漸進變化。為確認修整模型之可行性，本論文分別提出無負載及受載齒面接觸分析模型，並以測試齒輪對實際進行不同組裝誤差下齒印量測，最後則進行疲勞破壞測試實驗，以確認所設計之齒面疲勞強度。
本論文首先在錐形齒輪對設計上，加入最大比滑率之影響做為設計指標以降低齒面磨耗。並利用正齒輪漸開線接觸點法線特性，簡化齒面接觸分析模型中之齒面接觸點之求解，再以此方法分析探討在修整參數、組裝誤差等對接觸點在齒面位置偏移以及傳動誤差之影響。分析結果顯示：(1) 修整量越大，其接觸開始(結束)位置附近之接觸點往大(小端)之偏移量越大；(2) 無負載之各組裝誤差下，無修整錐形齒輪對接觸率與偏移後接觸點所在之壓力角值成反比； (3) 接觸點偏移至大端之組裝條件下，傳動誤差曲線在接觸開始位置即產生不連續狀況，其餘組裝條件下傳動誤差曲線相似，為連續拋物線。
另一方面，嚙合齒對受載齒面接觸分析模型係利用影響係數法建立，以求得各組裝誤差下嚙合過程負載分配、齒面應力分佈及受載傳動誤差。分析結果顯示在理想組裝下：(1) 嚙合過程負載分配變化連續，且接觸開始、結束之負載為零；(2) 接觸結束位置附近之接觸斑受小齒輪齒頂邊緣之影響，會發生小面積之邊緣應力集中現象；(3) 受載傳動誤差曲線為連續拋物線，且隨負載越大，其變化振幅值越低。而在組裝誤差下，接觸點會偏移至大 (小端)，此時：(1) 齒對接觸結束(開始)之負載不為零，且隨接觸點偏移量越大其負載越大；(2) 嚙合過程中之負載變化與受載傳動誤差曲線於單齒對接觸開始(結束)之位置會產生不連續狀況。
各組裝誤差下接觸位置除以齒面接觸分析進行模擬外，並以以齒印量測實驗及CAD干涉模擬分析兩種方式進行驗證，驗證結果顯示數值模擬結果與實驗結果極為相近。
齒輪齒面疲勞測試則以有、無修整之粉末冶金錐形齒輪對於功率封閉型的測試平台上進行過載疲勞破壞測試，以預估其疲勞強度壽命。由實驗結果可知，在相同負載及運轉圈數下，修整錐形齒輪對之齒面破壞程度遠較未修整齒輪對小。而在量測之振動頻譜分析上，由結果可知修整錐形齒輪整體振幅較未修整為低，而齒輪在第一倍嚙合頻率之振幅降低受到負載分配改善之影響尤為明顯。

關鍵字

小齒數比 ； 齒面修整 ； 受載齒面接觸分析 ； 組裝誤差分析 ； 齒面疲勞實驗

並列摘要

Comparing with bevel gears, conical gear drives, which are paired with a shaped conical gear and a spur gear, have some significant advantages, such as lower assembly sensitivity, low complexity in geometry of tooth surface for ease of design, and lower manufacturing cost of molded gears (e.g. powder metallurgy gears). The conical gear drives have thus a potential to replace bevel gears in the application of mass production and transmission with a large shaft angle. But the bearing contact during gear meshing is similar to spur gear pairs that the load sharing is discontinuous. This phenomenon can cause an impact on tooth pairs and concentrated contact stresses on flanks at contact begin or end and. Abrasive wear or pitting damage on tooth flanks, and even breakage of teeth could occur. In order to expand the application of conical gears, the load capacity of the gear drives must be able to calculate to fulfill the design requirements, especially for application of powder metallurgy gear pair widely used by confirm modified flanks.
The aim of this thesis is to propose a parabolic profile crowning approach for conical gears based on the developed design approach for nonstandard conical gears haiving a small gear ratio [4], as to improve a smooth variation of the load sharing of the contact tooth pair. A Loaded and an unloaded tooth contact analysis approach for conical gear pairs are also proposed respectively to verify the feasibility of the proposed approach for flank modification. Finally, test gears are manufactured to measure the tooth contact patterns considering assembly errors. A fatigue test is also conducted on a close-loop test rig to validate the surface durability.
The specific sliding is at first introduced in the design approach as an additional design criterion to reduce the wear on flanks. In order to to simplify the solving process for the positions of contact points, the proposed tooth contact analysis (TCA) model is developed based on the geometric characteristics of involute. The influences of the flank modification parameters on the contact positions and the transmission errors of the gear pair under different assembly errors are analyzed by using the TCA model. The results show that：(1) The shift distance of contact positions from the middle of the face-width to the heel (the toe) at the contact begin (end) is larger with increased modification amounts; (2) The unloaded contact ratio of non-modified flanks of conical gear drives under assembly errors is inversely proportional to the working pressure angle due to shifting of contact position; (3) The unloaded transmission error (UTE) performs similar to parabolic curve in the presence of assembly errors. But, the UTE will be discontinuous at contact begin if the contact position is shifted to the heel
On the other hand, a loaded tooth contact analysis (LTCA) model based on influence coefficient method is also developed to confirm the load capacity of the modified conical gear drives. The tooth contact stress, the load sharing and the loaded transmission errors during gear meshing are analyzed. The results for gear drives under ideal condition show that：(1) The load sharing during gear meshing becomes continuous and no load exists at contact begin or end; (2) A concentrated contact stress on a small area occurs at the contact end due to contact with the tip edage of the pinion; (3) The loaded transmission error (LTE) is also similar to parabolic curve. The change of the amplitude value is lower with the increased load. The results for gear drives under assembly errors show that：(1) The load at contact end or begin doesn’t disappear and is higher with the increased shift distance; (2) The variation of the load sharing and the LTE become discontinuous at the position from single to double tooth pair contact, and vice versa.
The TCA model is also validated by measuring the contact patterns of test gears and interference analysis in a CAD software under assembly errors. The results show that the TCA model is in good agreement with the mentioned methods.
A overloaded fatigue test is conducted on a power close-loop test rig with two types of PM conical gears, each with non-modified and modified flank. The experimental results show that the pitting damage on modified flanks is far less than non-modified flanks under the same conditions. Finally, the analysis results of the measured vibration data show that the vibration amplitudes of the gear pairs with flank modification are lower than those of the gear pairs without modification. Especially the reduction of the vibration amplitudes at the 1st gear mesh frequency is affected strongly by the improvement of the load sharing of the gear drives due to flank modification.

並列關鍵字

Small gear ratio ； Flank modification ； Loaded tooth contact analysis ； Assembly error analysis ； Fatigue test

參考文獻

[5]Litvin, F.L., A. Fuentes, M. Howkins, “ Design, generation and TCA of new type of asymmetric face-gear drive with modified geometry Comput ” , Methods Appl. Mech. Engrg., 190 (2001), pp. 5837–5865。

[9]Chakraborty, J., Bhadoria, B.S., “ Design parameters for face gears ” , Journal of Mechanisms 6, pp.435-445, 1971。

[10]Chakraborty, J., Bhadoria, B.S., “ Some studies on hypoid face gears ” , Mechanism and Machine Theory 8, pp. 339-349, 1973。

[13]Yaejima, K., Morosumi, M., “Designing of Generalized Profile-Shifted Involute Face Gear 1st Report” , The Japan Socity of Mechanical Engineers, pp.275-282 , 1996。

[18]Tsuji, I., Gunbara, H., Kawasaki, K., Abe, Y., Suzuki, K., “ Development of High-Performance Face Gear Set ”, The Japan Socity of Mechanical Engineers, pp.6-14 , 2011。