凹面錐形齒輪修整設計模式之研究; A Study on Flank Modification for Concave Conical Gears

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Title:	凹面錐形齒輪修整設計模式之研究;A Study on Flank Modification for Concave Conical Gears
Authors:	鍾禮隆;Li-lung Jhong
Contributors:	機械工程研究所
Keywords:	凹面錐形齒輪;近似線接觸;受負載齒輪接觸分析;物件導向程式;齒面修整;loaded tooth contact analysis;concave conical gear;object-oriented programming
Date:	2009-10-12
Issue Date:	2010-06-10 16:55:03 (UTC+8)
Publisher:	國立中央大學圖書館
Abstract:	錐形齒輪傳動機構具有組裝敏感度低、可調整背隙及組合多樣化等優點，但是由於其接觸的型態為點接觸形式，於重負載的傳動場合容易產生接觸應力過大的問題。為了解決齒面承載能力不足問題，許多學者則針對其接觸形式進行改善。其中以齒面修整成凹面形式降低接觸應力最受重視。此種修整之錐形齒輪設計已實際應用於重負載的傳動。傳統凹面錐形齒輪之設計，採取圓弧修整路徑，但圓弧路徑之曲線特性完全由曲率半徑所決定，在設計上無法對修整齒面進行額外的調整，尤其在較小曲率半徑的情況下，修整後齒面在近兩端面處的曲率變化不同，易導致齒面干涉。本研究從設計觀點出發，以平面的二次曲線做為刀具修整路徑，探討不同曲線型態及參數對修整後之凹面錐形齒輪對接觸特性的影響。修整路徑除了基本的圓弧之外還包含了橢圓、拋物線與雙曲線等不同的曲線類型。為探討各曲線對修整齒面之接觸特性影響，本研究提出二次曲線之曲率半徑、曲率半徑變化率及形狀係數等做為修整路徑之設計參數。在建立流程中，為簡化不同類型的二次曲線刀具路徑推導齒面數學模型，本研究同時以數值方法與物件化程式建立凹面錐形齒輪齒面模型計算流程。針對修整齒面特性的探討，本論文首先經由分析不同修整路徑，整理出各類型路徑之基本特性，並比較不同路徑曲線修整之齒面所產生的齒面偏差量。同時也驗證修整型凹面齒輪對在有組裝誤差下，並無傳遞誤差。藉由齒面受負載接觸分析之數值計算方法，各修整齒輪參數對齒面接觸狀況之影響及其變化趨勢得以進一步分析。由分析數據可以得知，圓弧路徑之曲率半徑有極限值，當路徑節點處曲率半徑過小，在齒面近大端處會產生干涉情況。相對而言，雙曲線修整路徑除了在路徑節點處兩側的曲線變化較為平緩，有利於路徑節點處較小曲率半徑的選取之外，同時也可以透過另外兩個曲線參數進行修整路徑規劃，其中形狀係數增加會使齒面接觸斑往齒面寬大端方向拓展，而形成類似水滴狀的接觸斑；此一非對稱改變可透過曲率半徑變化率來調整接觸斑的形狀。本論文最後依照分析出來的參數特性，提供一簡易的修整齒輪設計流程，提出使用曲率半徑及曲率半徑變化率兩項參數來進行最佳化齒面設計的概念。並以設計實例比較傳統圓弧路徑與雙曲路徑修整齒輪之差異，證明以雙曲線路徑修整凹面錐形齒輪確實是能夠得到較佳的齒面接觸狀態。 Conical gears are suitable to apply in transmission because of advantages like low assembly sensitivity, adjustable backlash control and various possibilities of combination. But they have a problem of less surface durability due to the point contact. The concept of concave conical gear was thus proposed to reduce the higher contact stress, and its practical application was also found in power transmission. The path of the cutter for the conventional concave conical gears is designed as a circle, whereby the radius is the only one parameter for determination of the curve characteristics. Consequently there is no additional possibility to modify the tooth flank for better performance. Especially, the curvatures of modified tooth flank near the toe and the heel of the face-width are quite different with a small curvature radius of the cutting path, and an interference between two mating tooth flanks could occur. From a viewpoint of design, the thesis focuses on designing the cutting path by using planar quadric curves to explore the influence of the different curve types and their corresponding parameters on the contact characteristics of the concave conical gears. Besides the circle the cutting paths discussed in the thesis include also ellipse, parabola and hyperbola. Three kinds of variables of the quadric curves are defined in the study as design parameters for analysis of the contact characteristics: namely, the curvature radius, the gradient of curvature radius, and the form factor. In order to simplify the derivation of mathematical model, the tooth flank of the concave conical gear is calculated numerically by using object-oriented programming based on mathematic relations of gearing theory. The study at first analyzes different types of cutting paths and summarizes their characteristics, and compares the deviations of the tooh flanks generated by different cutting paths. Furthermore, it is also proved that the proposed concave conical gears have no transmission error under assembly error. By using numerical approach for loaded tooth contact analysis, the influences of the dsign parameters of the different cutting paths on the contact characteristics of concave conical gears were analyzed. From the analyzed results, the circular cutting path has a limited curvature radius, whereby the tooth flanks of the mating gears interfere with each other near the heel of the face-width. Comparatively, the tooth flanks generated by the hyperbolic cutting path changes more flat. It is thus advantageous that a smaller curvature radius at the pitch point of the cutting path can be chosen. Another advantage of the hyperbolic cutting path is also that another two parameters can be applied additionally to adjust the cutting path for optimum contact characteristics of gears. For example, a larger form factor can cause the contact pattern to expend to the heel asymmetrically, and thus forming as teardrop-shape. This irregular contact pattern can be adjusted by using the gradient of curvature radius. Finally this thesis offers a set of design guidelines for modifying the concave conical gears with better contact characteristics based on the proposed design parameters of the cutting paths. An example to compare the conventional concave conical gear using circular cutting path with the novel concave conical gear using hyperbolic cutting pather was moreover discussed. It is verified that the hyperbolic cutting path is much better than the circular cutting path for modifying the concave conical gears.
Appears in Collections:	[Graduate Institute of Mechanical Engineering] Electronic Thesis & Dissertation

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