摘 要
本文主旨為探討轉位正齒輪對(Profile shifted gear pair)，運轉過程中齒面摩耗(Tooth wear)與動態負載(Dynamic responses)間之交互影響。文中分別就齒對動態負載分析提出了雙剛度(Two-step mesh stiffness)解析模式與變參數(Time-varying parameters)數值解模式。雙剛度解析模式為一簡化之動態模式，其主要假設為齒面摩擦力對動態負載所造成的效應可忽略，且齒對之嚙合剛度在雙齒接觸區(Double tooth contact period)與單齒接觸區(Single tooth contact period)分別視為定值。文中利用雙剛度模式推導得齒對在不同轉速下動態負載之時域與頻域的解析解，經由時域與頻域間交叉分析比對，探討齒面摩耗對動態負載頻譜分佈所可能造成的影響。在變參數數值解模式中，文中利用一擬線性數值解法，逐步模擬各嚙合位置齒對各參數的變化，這些參數包括齒對嚙合剛度、阻尼、摩擦係數與摩擦溫昇效應等參數。配合此具時變參數模式，文中分就不同材質齒對在不同運轉條件下之摩耗與動態負載之交互關係，進行模擬分析。分析實例有鋼齒對、耐隆 66 (Nylon 66)與聚縮荃(POM)等塑膠齒對，以及耐隆/鋼齒對。
雙剛度模式數值結果顯示，齒對間傳遞扭矩(Transmitted torque)的頻譜分佈混合有頻率調變(Frequency modulation)與振幅調變(Amplitude modulation)。頻率調變起因於齒對嚙合運轉過程中單、雙齒接觸區交替變換，而簡諧變化的輸入扭矩則產生振幅調變。頻譜分佈圖上顯示摩耗對傳遞扭矩的影響，主要在嚙合頻率低倍頻處。變參數數值解模式數值結果顯示，隨運轉週期之遞增，摩耗與動態負載互為影響。摩耗量以接近最低單齒接觸點處具最大值，而最大動態負載將隨齒對運轉週期之遞增先降低而後昇高。

Abstract
The interaction among the tooth wear, dynamic loads, and its corresponding frequency spectrum variation in a spur gear pair is investigated in the dissertation. A mathematical model for the dynamic responses of a profile shifting involute gear pair is derived. For simplicity, a two-step mesh stiffness model is proposed to approximate the load sharing and the time-varying stiffness effects in a mating gear pair. The analytic solutions for the dynamic responses of this simplified dynamic model are derived in both time and frequency domains. The effect of the tooth wear on the spectrum variation has been illustrated analytically. The amplitude and frequency modulations introduced from the periodic load fluctuation have also been discussed.
In order to characterize the time-varying mesh stiffness, frictional coefficient, damping ratio, temperature rise and load sharing effects between engaged teeth, a computational algorithm with the quasi-linear iteration procedure is developed to include above instantaneous parameters in the teeth engagement process. The tooth wear equation proposed by Flodin and Andersson (1997) is employed to simulate the mild wear process for an engaging gear pairs. The interactions between the tooth wear and the dynamic loads for the Steel/Steel, POM/POM, Nylon/Nylon and Steel/Nylon gears pair have been studied numerically. Numerical examples indicate that the proposed models are valid for gear pairs with different materials.

Contents
Acknowledgements i
Abstract ii
Contents v
List of Figures viii
List of Tables xiii
Nomenclature xiv
Chapter 1 Introduction 1
1.1 Background and motivation 1
1.2 Literature 4
1.3 Organization of this dissertation 9
Chapter 2 Two-Step Stiffness Model in a Profile Shifted Gear
Pair 12
2.1 Introduction 12
2.2 Dynamic model for an engaged gear pair 12
2.2.1 Mesh period 17
2.2.2 Mesh stiffness 19
2.2.3 Damping ratio 25
2.3 Two-step mesh stiffness model 25
2.4 Spectra of transmitted torque 31
2.4.1 For a constant input torque 31
2.4.2 For a harmonically fluctuated input torque 34
Chapter 3 Time-Varying Tooth Parameters Model in a Profile
Shifted Gear Pair 37
3.1 Introduction 37
3.2 Time-varying tooth parameters model for an engaged
spur gear pair 38
3.2.1 Position dependent tooth pair mesh stiffness 39
3.2.2 Damping ratio 43
3.2.3 Friction coefficient 44
3.2.4 Polar mass moment of inertia 47
3.2.5 Temperature rise effect 47
3.3 Wear model for an engaged spur gear pair 49
3.4 Computational algorithm for the cumulative tooth
profile wear simulation 51
Chapter 4 Numerical Results and Discussions 57
4.1 Introduction 57
4.2 Two-step mesh stiffness model for a Seel/Steel gear
pair 57
4.2.1 Spectra for the different profile shifted gear pairs 62
4.2.2 Input torque fluctuation on the frequency spectrum
variation 67
4.2.3 Spectra variation introduced from the tooth profile
error 70
4.3 Time-varying tooth parameters model for different gear
pairs 70
4.3.1 Steel/Steel gear pair 75
4.3.2 Nylon/Nylon gear pair 85
4.3.3 POM/POM gear pair 90
4.3.4 Nylon/Steel gear pair 97
Chapter 5 Conclusions 108
References 110
Vita 117

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