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研究生:石伊蓓
研究生(外文):Yi-pei Shih
論文名稱:面滾式戟齒輪齒面修整數學模式之研究
論文名稱(外文):STUDY ON THE FLANK MODIFICATION OF FACE HOBBED HYPOID GEARS
指導教授:馮展華馮展華引用關係
指導教授(外文):Zhang-Hua Fong
學位類別:博士
校院名稱:國立中正大學
系所名稱:機械工程所
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:2007
畢業學年度:95
語文別:英文
論文頁數:158
中文關鍵詞:六軸戟齒輪創成機齒面相對修形齒面誤差修正面滾式切製法戟齒輪
外文關鍵詞:ease-off flank modificationflank correctionface hobbinghypoid gearsix-axis CNC Cartesian-type hypoid generator
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根據齒輪原理和微分幾何的觀念,本研究推導出泛用型面滾式戟齒輪創成機之數學模式。此數學模式能模擬現行之面滾式(face hobbing)和面銑式(face milling)切齒系統,並能計算和分析戟齒輪(hypoid gear)之齒面幾何和接觸性能。本研究建立戟齒輪的齒面修整(flank modification)和修正(flank correction)方法,以縮短戟齒輪設計和製造的開發時程。本論文內容包括:

(a) 根據齒輪原理、微分幾何以及泛用型搖台式戟齒輪創成機構,推導出泛用的面滾式戟齒輪之齒面數學模式。本研究所建立的數學模式分為三個模組: 銑刀盤、假想產形齒輪以及工件齒輪和假想產形齒輪的相對位置。由於模組化的安排,此數學模式非常適合於戟齒輪設計、製造分析軟體之物件導向程式的開發。除了可以模擬現有使用單刀軸和雙刀軸銑刀盤之面滾式戟齒輪外,它也很容易再加以簡化來模擬面銑式加工和各種動態輔助齒面修整運動。
(b) 根據齒輪原理以及萬用傘齒輪測試機之機構,推導出面滾式戟齒輪之接觸分析的數學模式。此數學模式能模擬戟齒輪對在無負載條件下,齒面接觸齒印的位置、形狀和其運動誤差曲線。
(c) 運用所發展之泛用型面滾式戟齒輪齒面和接觸分析數學模式,即可模擬並分析現有面滾式戟齒輪之齒面位置和接觸性能,其適用之切製法包含克林根貝格(Klingelnberg)的Cyclo-Palloid法、奧里康(Oerlikon)的刀傾全展成法(Spiroflex)和刀傾半展成法(Spirac),以及Gleason的TriAC®法。
(d) 根據本文推導之戟齒輪齒面數學模式和敏感度分析(sensitivity analysis)技術,發展一個基於泛用型戟齒輪創成機的面滾式戟齒輪齒面誤差修正方法。此修正方法之目的在於降低戟齒輪製造時因機器誤差和熱處理變形所造成的齒面誤差。依據量測的齒面誤差量和設計變數的敏感度矩陣以計算設計變數之修正量,再使用此修正量來調整齒輪齒面並分析其與理論齒面的誤差值。
(e) 應用齒面相對修形(ease-off)和敏感度分析技術,發展一個面滾式戟齒輪齒面設計方法。此方法可用於齒輪齒面幾何之最佳化,以改善齒輪對之接觸情形。首先,使用泛用的面滾式戟齒輪齒面方程式,建立齒面相對修形的數學模式。接著,分析各設計變數變化時對齒面相對修形的敏感度。再依據由經驗得到的最佳齒面相對修形誤差值和敏感度矩陣,使用線性迴歸的方法計算出設計變數修整量,來獲得具有與最佳齒面相對修形相仿的齒面幾何。此修整方法亦能運用於其他形式的齒輪齒面設計。
(f) 根據新型的六軸卡氏座標式電腦數控戟齒輪創成機,建立六軸戟齒輪創成機之戟齒輪齒面數學模式。透過從泛用搖台式至六軸卡氏座標式戟齒輪機的轉換關係,即可獲得各種應用於卡氏座標式戟齒輪切齒機的切製法(包含面銑式和面滾式)之運動函式。根據所建立之數學模式,提出一個六軸卡氏座標式創成機所切製之戟齒輪齒面誤差修正方法。此修正法並不需要透過機械設定轉換而能直接調整創成機之六軸運動來達成任意齒面修正的目的。
Based on the theory of gearing and differential geometry, we develop a mathematical model for the universal face-hobbing hypoid gear generator that can (1) simulate both existing face-hobbing and face-milling cutting systems; (2) calculate tooth surface geometry; and (3) analyze contact performance of hypoid gears. Applying the proposed mathematical model to universal flank modification and correction methods for both face milled and face hobbed hypoid gears substantially shortens the time needed for new hypoid gear set development.
Specifically, this dissertation proposes solutions to six research problems:
(a) First, based on the theory of gearing, differential geometry and the universal face-hobbing hypoid gear generator, we derive a mathematical model for face hobbed hypoid gear tooth surfaces. This model contains three modules: a cutter head, an imaginary generating gear, and the relative motion between the imaginary generating gear and the work gear. Given its modular arrangement, the model is suitable for the development of an object-oriented programming (OOP) code. Not only can this mathematical model simulate existing face hobbed hypoid gears with either a single or dual cutter head, but it includes most existing flank modification features and can be easily simplified to simulate the face-milling cutting.
(b) Second, based on the theory of gearing and the universal bevel gear-rolling tester, we derive a mathematical model of no-load tooth contact analysis (TCA) for the face hobbed hypoid gear. This model enables evaluation of the meshing and contact characteristics between the contact surfaces of hypoid gears.
(c) Third, using the proposed mathematical models, we investigate the tooth geometries and TCA results of existing face hobbed hypoid gears with cutting methods such as Klingelnberg’s Cyclo-Palloid, Oerlikon’s Spiroflex and Spirac, and Gleason’s TriAC® formate processes.
(d) Fourth, based on the proposed mathematical model of tooth surfaces and the sensitivity analysis, we develop a general flank correction methodology for both face milled and face hobbed hypoid gears that aims to minimize the flank deviations caused by machine error and heat treatment deflection during gear cutting. According to the measured flank deviations and the flank sensitivity matrix produced for the design parameters, the corrections for the design parameters can be obtained by linear regression. These corrective design parameters are then applied to the gear cutting machine simulation program to calculate the modified tooth surfaces, which are then compared with the theoretical tooth surfaces to assess the correction result.
(e) Fifth, based on the ease-off topography of the gear drive and the sensitivity analysis, we develop a flank modification methodology for face hobbed hypoid gears that can be used to optimize the tooth surface geometries of face milled and face hobbed bevel gears and so improve their contact performance. After using the equation of universal face hobbed hypoid gear tooth surfaces to establish a mathematical model for the ease-off topography of the gear drive, we analyze the sensitivity of the ease-off to each design parameters including cutter parameters, machine settings, and auxiliary flank modification (AFM) motion parameters. According to this sensitivity matrix and the deviations between the optimum and initial ease-off, the modifications of pinion design parameters for obtaining tooth surfaces with an optimum ease-off topography can be determined by linear regression. This proposed flank modification methodology can also be used as the basis for a general technique of flank design for similar gear types.
(f) Sixth, based on the six-axis Cartesian-type CNC hypoid gear generator, we establish a mathematical model for hyoid gears. Specifically, we obtain the six-axis movement for both face milling and face hobbing operations by way of the conversion from a universal cradle-type to a six-axis Cartesian-type machine. Using this six-axis mathematical model, we then develop a kinematic flank correction method that can be used directly to modulate the six-axis movement for producing hypoid gear tooth surfaces without machine-setting conversion. The proposed model can also be used to obtain a free-form flank correction.
摘 要 I
ABSTRACT III
致 謝 VI
TABLE OF CONTENTS VII
LIST OF TABLES X
LIST OF FIGURES XII
NOMENCLATURE XVI
CHAPTER 1 INTRODUCTION 1
1.1 Development of the Spiral Bevel and Hypoid Gears Manufacturing Systems 1
1.2 Literature Review 3
1.3 Dissertation Outline 6
CHAPTER 2 MATHEMATICAL MODEL FOR A CRADLE-TYPE UNIVERSAL FACE-HOBBING HYPOID GEAR GENERATOR 8
2.1 Introduction 8
2.2 Equations for the Face-Hobbing Cutter Head 8
2.3 Equations for the Imaginary Generating Gear 12
2.4 Equations for the Gear Tooth 15
2.5 Tooth Contact Analysis 19
2.6 Discussion 22
2.7 Concluding Remarks for Chapter 2 23
CHAPTER 3 SIMULATED RESULTS FOR THE TOOTH SURFACES AND TCA OF FACE HOBBED HYPOID GEARS 34
3.1 Introduction 34
3.2 Gear Blank for the Face Hobbed Hypoid Gear 35
3.3 Klingelnberg’s Cyclo-Palloid Cutting System 35
3.4 Oerlikon’s Spiroflex Cutting System 37
3.5 Oerlikon’s Spirac Cutting System 38
3.6 Gleason’s TriAC® Formate Cutting System 39
3.7 Concluding Remarks for Chapter 3 39
CHAPTER 4 FLANK CORRECTION FOR SPIRAL BEVEL AND HYPOID GEARS ON A CRADLE-TYPE UNIVERSAL HYPOID GEAR GENERATOR 55
4.1 Introduction 55
4.2 Flank Correction for Spiral Bevel and Hypoid Gears Based on a Universal Hypoid Gear Generator 56
4.3 Second-Order Surface Approximation for the Flank Topography Variation 58
4.4 Numerical Example 59
4.5 Concluding Remarks for Chapter 4 60
CHAPTER 5 FLANK MODIFICATION METHODOLOGY FOR SPIRAL BEVEL AND HYPOID GEARS BASED ON THE EASE-OFF TOPOGRAPHY 80
5.1 Introduction 80
5.2 Ease-Off Topography of a Gear Drive 81
5.3 Design Parameters for Spiral Bevel and Hypoid Gears 85
5.4 Flank Modification for Spiral Bevel and Hypoid Gears 86
5.5 Numerical Example 88
5.6 Concluding Remarks for Chapter 5 90
CHAPTER 6 FLANK CORRECTION FOR SPIRAL BEVEL AND HYPOID GEARS ON A SIX-AXIS CNC HYPOID GEAR GENERATOR 103
6.1 Introduction 103
6.2 Mathematical Model of a Universal Face-Hobbing Hypoid Gear Generator 104
6.3 Mathematical Model of a Cartesian-Type Hypoid Gear Generator 106
6.4 Flank Correction for Spiral Bevel and Hypoid Gears Based on a Six-Axis Cartesian CNC Hypoid Gear Generator 112
6.5 Numerical Example and Discussion 114
6.6 Concluding Remarks for Chapter 6 116
CHAPTER 7 CONCLUSIONS AND FURTHER RESEARCH 132
7.1 Conclusions 132
7.2 Future Research 134
REFERENCES 136
APPENDIX A EXPERIMEMT RESULTS OF REAL HYPOID GEAR CUTTING 141
A.1 Face Hobbed Hypoid Gear Cutting 141
A.2 Corrections Based on the Universal and Six-Axis Flank Correction Methods 142
A.3 Acknowledgments 143
PUBLICATIONS LIST 157
作者簡介 158
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