直齒斜齒輪之數學模式與接觸分析

Abstract

直齒斜齒輪之數學模式與接觸分析 學生：張岐康 指導教授：蔡忠杓 國立交通大學機械工程研究所 摘要 在現今的工業界，直齒斜齒輪在相交軸的傳遞動力系統中，可以說是受到相當廣範的運用，但能找到與其相關的研究文獻卻不多。本文係根據齒輪原理及微分幾何的觀念，推導出具奧托伊德 齒形的直齒斜齒輪之數學模式。此一數學模式能模擬具奧托伊德齒形的直齒斜齒輪之創成機構、齒面特徵及其運動特性，並可迅速且精確的得到直齒斜齒輪齒面幾何、齒輪對之運動誤差、轉位及過切等相關的特性。本論文研究包括： (1) 建立具奧托伊德齒形的直齒斜齒輪之齒面數學模式，並以精簡之參數來描述齒面幾何。所推導之齒面數學模式可以當做具奧托伊德齒形的直齒斜齒輪之檢驗標準及建立有限元素法之網格的運用。 (2) 根據齒面接觸分析的理論來發展出具奧托伊德齒形的直齒斜齒輪之齒輪接觸分析之數學模式，並利用此一數學模式來分析齒輪對在傳動時所造成的運動誤差，並探討其齒面的接觸特徵。 (3) 建立具奧托伊德齒形之轉位直齒斜齒輪之齒面數學模式，並與正常的齒面比較其中的差異，並探討轉位直齒斜齒輪之齒厚及齒高等齒面參數之變化。 (4) 根據微分幾何的觀念，探討具奧托伊德齒形之轉位直齒斜齒輪之過切存在條件式，並依據此條件式來找出避免齒形過切之機器設定及進刀量，並研究過切與轉位係數間的關係及其條件式。 本文所發展之齒面數學模式不僅可以作為具奧托伊德齒形之直齒斜齒輪檢驗的標準、建立有限元素法分析之網格，而且可以做為日後建立具奧托伊德齒形之直齒斜齒輪設計製造之專家系統的基礎。本文並以許多的例子及圖表來說明所發展的齒面數學模式之應用及正確性。相信對從事具奧托伊德齒形的直齒斜齒輪之工程人員在設計與製造上都會有相當的參考價值與助益。

MATHEMATICAL MODEL AND TOOTH CONTACT ANALYSIS OF STRAIGHT BEVEL GEARS Student：Chi-Kang Chang Advisor：Chung-Biau Tsay Institute of Mechanical Engineering National Chaio Tung University ABSTRACT Straight bevel gears are widely used for power transmissions between the intersected shafts. However, only a few of technical literatures related to straight bevel gears can be found. In this thesis, a mathematical model of the straight bevel gear with octoid form has been developed based on the theory of gearing and the concept of differential geometry. The developed mathematical model can be used to simulate the gear manufacture and meshing conditions of the gear set. The tooth undercutting , tooth contact analysis, profile shifting, and kinematic errors have also been investigated. The research subjects are: (a) Development of a general mathematical model to represent the geometry of the straight bevel gear tooth surfaces with the octoid form. The tooth surfaces of straight bevel gear are described by the simplified surface coordinates, and they are termed as the "soft" straight bevel gears. The developed mathematical model can be used to the surface inspection as the standard gear tooth surface profile. (b) Based on the theory of tooth contact analysis, a tooth contact analysis computer simulation program has been developed. According to the developed computer program is used to simulate the kinematic errors of the gear train and to investigate the bearing contact of straight bevel gear. (c) Constructing a mathematical model for the profile-shifted straight bevel gears, and comparing the differences between the ordinary profile and shifted profile. Relations between the tooth surface parameters and its tooth thickness and tooth height have also been investigated. (d) Investigation on the necessary and sufficient conditions for undercutting of the straight bevel gear with octoid form. Machine-tool settings for tooth non-undercutting are also investigated according to the developed tooth undercutting conditions. The relationship between the undercutting and the coefficient of the profile shifting are also studied. The proposed mathematical model of straight bevel gears is the basis for the inspection, finite element analysis and expert system development of straight bevel gears. The applications and correctness of the proposed mathematical model are illustrated by some numerical examples, tables and computer graphs. Those investigations and results shown in this thesis are most helpful to engineers in the design and manufacture of straight bevel gears with octoid form.