蝸線傘齒輪之齒輪接處分析及其最佳化

Abstract

A mathematical model of the circular-cut spiral bevel gear set has been developed, and simulations of the gear manufacture and working conditions have also been investigated in this thesis. The research subjects are: (a) Constructing a general mathematical model to represent the geometry of the circular-cut spiral bevel gear tooth surfaces. The tooth surfaces of the spiral bevel gear are described by the simplified surface coordinates, and they are termed as the "soft" spiral bevel gear. (b) Developing a mathematical model to simulate the kinematical characteristics of the universal bevel gear roll tester. By employing the "imaginary" roll tester to imitate the working conditions of the "soft" spiral bevel gear set, and using the loaded tooth contact analysis (LTCA) techniques to evaluate the meshing conditions between the contacting tooth surfaces. (c) Applying the proposed mathematical model to simulate the existing Gleason spiral bevel gear cutting machines, and to study the relations between the Gleason machine settings and parameters of the mathematical model. (d) To create a kinematical optimization model for the spiral bevel gear set. The objects of the optimization are: (1) minimizing the rotational kinematic errors, (2)to discard the discontinuous teeth meshing, and (3) to move the contact pattern toward the center region of the tooth surfaces with a proper bias. (e) To investigate the undercutting conditions of the circular-cut spiral bevel gear. Deriving the equations for the limit of the generating tool setting, and the edge of regression on the generated gear tooth surface. The proposed mathematical models are the basis for the expert system of designing the spiral bevel gear set. The applications of the proposed mathematical model are illustrated by numerical examples and computer graphics. 根據齒輪原理及微分幾何，作者推導出圓切式蝸線傘齒輪之數學模式．此數學模式能 模擬現存之大部分圓切式蝸線傘齒輪創成機及測試機的特性，精確且迅速的得到齒面 幾何及齒輪對之運動特性．本論文包括： （１）建立圓切式蝸線傘齒輪之齒面數學模式，並以精簡之參數來描述齒面的幾何． 所推導之齒面數學模式可當作圓切式蝸線傘齒輪之檢驗標準． （２）根據萬用傘齒輪測試機（Universal bevel gear roll tester）之機構，發展 出圓切式蝸線傘齒輪之齒輪接觸分析的數學模式．利用此數學模式來模擬蝸線傘齒輪 對在負載下運轉的情形，同時得到齒面嚙合的齒印及剛體運動誤差． （３）運用所發展之數學模式來模擬格里森（Gleason ）公司之圓切式蝸線傘齒輪創 成機器，並將數學模式所用之參數轉換成實際加工時所用的機器設定．同時發展一系 列之圓切式蝸線傘齒輪電腦輔助設計軟體，只要將機器設定輸入電腦就可得到齒形之 電腦繪圖及齒輪對之運轉情形． （４）根據前所推導之數學模式及最佳化理論，發展出圓切式蝸線傘齒輪之最佳化模 式．以此模式可將運動誤差、間歇性之齒面接觸及齒印位置調整到最佳之折衷點． （５）根據微分幾何，證明圓切式蝸線傘齒輪之過切存在的條件，並據此條件找出避 免齒形過切之機器設定及進刀量的限制． 本文所發展之數學模式不僅可作為圓切式蝸線傘齒輪之檢驗標準，且可作為圓切式蝸 線傘齒輪設計製造之專家系統的基礎．文中並以許多例子來說明所發展之數學模式的 應用及其可靠度．