臺灣博碩士論文加值系統

本研究探討單一滑動葉片旋轉式壓縮機的動態模擬和分析策略，並研究間隙對壓縮機的動態反應與性能之影響，同時提出精確、快速的接觸偵測與穿透深度的計算方法。本研究所考量的接觸分為葉片與滑槽的接觸，以及葉片前緣之半圓柱面與定子內壁之包絡線型曲面的接觸。研究方法主要是利用連續赫茲接觸力模型，模擬元件發生接觸時的正向接觸力，並考慮摩擦力的影響，再以多體動力學推導含拘束力的壓縮機系統動力方程式，而數值積分法則使用四階Adams預測-修正法配合四階Runge-Kutta法。為了更加感受其模擬結果的合理性，藉由計算機圖學的輔助，發展一般性之電腦動態模擬軟體，並將模擬結果以二維及三維動畫的方式呈現。由分析結果得知，在壓縮機於啟動狀態的初期或低轉速下，葉片的加速度將產生不穩定的情況。隨著轉速的提高，葉片運動的不穩定現象將會減少，而壓縮機的輸入功率、輸出功率和摩擦功率則會隨著轉速的提高而增大。當間隙存在時，壓縮機於運轉的過程中，葉片的運動會不平順，而此不穩定之現象，在葉片發生碰撞時最為明顯。隨著間隙的增大，壓縮機的輸入功率、輸出功率與機械效率則有明顯降低的趨勢，而葉片與定子及轉子將出現更多不同的接觸模式，且葉片的接觸力之最大值與平均值亦會明顯的變大。本研究中，以定子內壁之葉瓣數為3作為分析的基礎，轉速則設定為1500 rpm。當無間隙設定時，葉片的最大正向接觸力約為333 N，且葉片將出現10種不同的接觸模式，而壓縮機的機械效率約為75.7 %。當間隙設定為0.02 mm時，葉片的最大正向接觸力約為345 N，且葉片將出現16種不同的接觸模式，而壓縮機的機械效率約為74 %。當間隙設定為0.05 mm時，葉片的最大正向接觸力約為362.8 N，且葉片將出現19種不同的接觸模式，而壓縮機的機械效率約為70.8 %。當間隙值為0.1 mm時，壓縮機的機械效率約為65.12 %，當間隙值達0.2 mm時，壓縮機的機械效率則降為約54.2 %。由研究結果得知，本文之方法對於其它存在此二類接觸問題之壓縮機，亦可合理的應用。

This study develops dynamic simulation and analysis strategic of a single sliding-vane rotary compressor and analyzes effects of clearance on dynamic response and performance of the rotary compressor. It also provides accurate and fast solutions for contact detection and computing technology of penetration. Two types of contact are considered. One type is the contact between the vane and the vane slot, and the other type is the contact between the semi-cylindrical surface of the vane and the envelope surface of the stator. The continuous Hertzian contact force model is utilized to calculate contact forces of the compressor components with consideration of frictional forces. The dynamic equations including constrained forces of the compressor system are derived by using multibody dynamics. The numerical integration uses the fourth-order Adams predictor-corrector method incorporated with the fourth-order Runge-Kutta method. With two-dimensional and three-dimensional animations of computer graphics, it is easy to understand simulated results. When the compressor is turned on or it operated with low rotational speed, the unstable acceleration of the vane occurs. As the rotational speed increases, the unstability of the vane is reduced while the input power, the output power and the friction power of the compressor increase. The vane becomes unstable when the clearance exists. The unstability of vane is significant when impact happens. As the clearance increases, the input power, output power and mechanical efficiency of the compressor are decreased, and contact types of the vane with the stator and the rotor increase. The maximum and average contact forces of the vane also increase. A stator with three lobes and the rotational speed of 1500 rpm with and without clearance are studied. If there is no clearance, the vane has 10 kinds of contact modes, the maximum contact force of the vane is 333 N, and the mechanical efficiency of the compressor is 75.7 %. If the clearance is equal to 0.02 mm, the vane has 16 kinds of contact modes, the maximum contact force of the vane is 345 N, and the mechanical efficiency of the compressor is 74 %. If the clearance is equal to 0.05 mm, the vane has 19 kinds of contact modes, the maximum contact force of the vane is 362.8 N, and the mechanical efficiency of the compressor is 70.8 %. If the clearance is equal to 0.1 mm and 0.2 mm, the mechanical efficiency of the compressor is 65.1 % and 54.2 %, respectively. It is expected that the results of this study can also be applied to compressors with these two types of contact problem.

目錄

中文摘要 i
英文摘要 ii
目錄 iv
圖目錄 vii
表目錄 xi
符號表 xii

第一章 緒論 1
1.1 研究動機 1
1.2 旋轉式壓縮機 2
1.2.1 旋轉式壓縮機的類型 3
1.2.2 單一滑動葉片旋轉式壓縮機 4
1.3 國內外相關研究 7
1.3.1 旋轉式壓縮機之動態分析相關文獻 7
1.3.2 接頭具間隙之動態分析相關文獻 10
1.4 研究目的 18
1.5 研究方法 19
1.6 本文架構 19

第二章 壓縮機的作用力 21
2.1 葉片的運動模式 21
2.1.1 葉片的接觸區域 22
2.1.2 葉片與轉子滑槽的相對運動模式 23
2.1.3 葉片與定子內壁的相對運動模式 24
2.2 廣義座標 25
2.3 葉片、定子與轉子的作用力 27
2.3.1 系統力 29
2.3.2 接觸力 31
2.3.3 摩擦力 36
2.4 元件的接觸力與摩擦力推導 39
2.4.1 葉片與轉子滑槽的接觸力與摩擦力之推導 39
2.4.2 葉片與定子內壁的接觸力與摩擦力之推導 42

第三章 接觸偵測與穿透深度 45
3.1 接觸偵測 45
3.1.1 葉片與轉子滑槽的接觸偵測 45
3.1.1.1 第一個接觸區域的接觸偵測 46
3.1.1.2 第二個接觸區域的接觸偵測 48
3.1.1.3 第三個接觸區域的接觸偵測 50
3.1.1.4 第四個接觸區域的接觸偵測 51
3.1.2 葉片與定子內壁的接觸偵測 53
3.1.2.1 第五個接觸區域的接觸偵測 53
3.1.2.2 第六個接觸區域的接觸偵測 54
3.2 穿透深度 55
3.2.1 葉片與轉子滑槽的穿透深度 56
3.2.2 葉片與定子內壁的穿透深度 61

第四章 壓縮機多體系統的動力分析 64
4.1 元件的拘束 64
4.2 多體系統的動力模型 69
4.3 動力系統的數值積分 72
4.3.1 四階Adams預測-修正法 72
4.3.2 積分步距的改變 75
4.4 動力分析的流程 77
4.5 分析結果的座標轉換 79

第五章 結果 81
5.1 壓縮機的基本參數、初始狀態與轉速設定 81
5.1.1 基本參數 81
5.1.2 初始狀態 83
5.1.3 轉速設定 83
5.2 理論的驗證 85
5.2.1 葉片沿著徑向的運動分析結果驗證 85
5.2.2 葉片沿著切向的運動分析結果驗證 87
5.3 壓縮機在固定轉速的動態分析結果 88
5.3.1 葉片在固定轉速的運動分析結果 88
5.3.1.1 葉片在固定轉速沿著徑向的運動分析結果 88
5.3.1.2 葉片在固定轉速沿著切向的運動分析結果 90
5.3.2 葉片在固定轉速時的動力分析結果 91
5.3.2.1 葉片在固定轉速時的正向接觸力 91
5.3.2.2 轉子與定子在固定轉速時的拘束力矩 93
5.3.3 葉片在固定轉速時的接觸模式 95
5.4 壓縮機在啟動狀態的運動分析結果 101
5.4.1 葉片在啟動狀態沿著徑向的運動分析結果 101
5.4.2 葉片在啟動狀態沿著切向的運動分析結果 104
5.5 間隙對壓縮機性能的影響 107
5.6 壓縮機運轉模擬動畫之結果展示 110
5.6.1 二維模擬動畫之結果展示 110
5.6.2 三維模擬動畫之結果展示 117

第六章 討論 120
6.1 理論驗證的探討 120
6.2 壓縮機在固定轉速的分析結果探討 122
6.2.1 葉片在固定轉速的運動分析結果探討 122
6.2.1.1 葉片在固定轉速沿著徑向的運動分析結果探討 122
6.2.1.2 葉片在固定轉速沿著切向的運動分析結果探討 123
6.2.1.3 葉片在固定轉速時的加速度探討 124
6.2.2 葉片在固定轉速時的正向接觸力探討 125
6.2.3 轉子與定子在固定轉速時的拘束力矩探討 126
6.2.4 葉片在固定轉速時的接觸模式探討 128
6.3 壓縮機在啟動狀態的運動分析結果探討 131
6.4 間隙對壓縮機性能影響的探討 133
6.5 壓縮機運轉模擬動畫的結果探討 135
6.5.1 運動軌跡的結果探討 135
6.5.2 二維與三維模擬動畫的結果探討 137

第七章 結論與建議 138

參考文獻 141

附錄A 定子內壁曲線之推導 A.1
附錄B 壓縮室氣體的熱力性質分析 B.1
附錄C 數值積分的介紹 C.1
附錄D 程式的使用說明 D.1

圖目錄

Fig. 1.1 Sliding vane type 3
Fig. 1.2 Rolling piston type 4
Fig. 1.3 Single sliding-vane rotary compressor 5
Fig. 1.4 Components of compressor 5
Fig. 1.5 Working process 6
Fig. 1.6 Forces acting on vane [6] 8
Fig. 1.7 Kinematic modes of vane 8
Fig. 1.8 Forces acting on vane [8] 9
Fig. 1.9 Revolute joint with clearance 10
Fig. 1.10 Slider joint with clearance 10
Fig. 1.11 Revolute clearance joint modeled by spring dashpot approach 11
Fig. 1.12 Kinematic modes of revolute clearance joint 11
Fig. 1.13 Revolute clearance joint modeled by massless link approach 12
Fig. 1.14 Revolute clearance joint with lubricant 13
Fig. 1.15 Kinematic modes of slider clearance joint 15
Fig. 1.16 Slider-crank mechanism with clearance in TBS joint 16
Fig. 1.17 TBS clearance joint 16
Fig. 1.18 Topological structure of mechanism with clearance in TBS joint 16
Fig. 1.19 Rigid-link mechanism with rotating slider joint and clearance 17
Fig. 1.20 Kinematic modes of rotating slider clearance joint 17

Fig. 2.1 Kinematic mode of vane without clearance 21
Fig. 2.2 Contact regions of vane 22
Fig. 2.3 Relative kinematic modes of vane and rotor 23
Fig. 2.4 Relative kinematic modes of vane and stator 25
Fig. 2.5 Generalized coordinate 26
Fig. 2.6 Forces acting on vane 27
Fig. 2.7 Forces acting on stator 28
Fig. 2.8 Forces acting on rotor 28
Fig. 2.9 Pressure distributions acting on vane 29
Fig. 2.10 System forces acting on vane 30
Fig. 2.11 Enlarged view for vane and rotor 32
Fig. 2.12 Enlarged view for vane and stator 32
Fig. 2.13 Contact between two spherical surfaces 34
Fig. 2.14 Contact between a spherical surface and a plane 34
Fig. 2.15 Enlarged view for friction of vane and rotor 36
Fig. 2.16 Enlarged view for friction of vane and stator 36
Fig. 2.17 Contact regions of vane and rotor 39
Fig. 2.18 Contact regions of vane and stator 42

Fig. 3.1 Half-plane method 45
Fig. 3.2 First contact region of contact detection type 1 46
Fig. 3.3 Special case of vane and rotor 47
Fig. 3.4 First contact region of contact detection type 2 48
Fig. 3.5 Second contact region of contact detection type 1 48
Fig. 3.6 Second contact region of contact detection type 2 49
Fig. 3.7 Third contact region of contact detection type 1 50
Fig. 3.8 Third contact region of contact detection type 2 51
Fig. 3.9 Fourth contact region of contact detection type 1 51
Fig. 3.10 Fourth contact region of contact detection type 2 52
Fig. 3.11 Fifth contact region of contact detection 53
Fig. 3.12 Sixth contact region of contact detection 54
Fig. 3.13 Contact forces of object 55
Fig. 3.14 First contact region of penetration cases 57
Fig. 3.15 Differentiated flow chart of penetration cases 57
Fig. 3.16 Fifth contact region of penetration 61
Fig. 3.17 Enlarged view for fifth contact region of penetration 62

Fig. 4.1 Kinematic constraints of stator 64
Fig. 4.2 Kinematic constraints of rotor 65
Fig. 4.3 Flow chart of change step 76
Fig. 4.4 Flow chart of dynamic analysis 78
Fig. 4.5 Coordinate transformation 79

Fig. 5.1 Initial conditions of compressor 83
Fig. 5.2 Rotor rotational speed ω versus time, t = 0 ~ 1.2 s 84
Fig. 5.3 Calculated vane displacement sr versus rotor angular displacement 86
Fig. 5.4 Calculated vane velocity vr versus rotor angular displacement 86
Fig. 5.5 Calculated vane acceleration ar versus rotor angular displacement 86
Fig. 5.6 Calculated vane velocity vθ versus rotor angular displacement 87
Fig. 5.7 Calculated vane acceleration aθ versus rotor angular displacement 88
Fig. 5.8 Vane displacement sr versus rotor angular displacement at 1500 rpm 88
Fig. 5.9 Vane velocity vr versus rotor angular displacement at 1500 rpm 89
Fig. 5.10 Vane acceleration ar versus rotor angular displacement at 1500 rpm 89
Fig. 5.11 Vane velocity vθ versus rotor angular displacement at 1500 rpm 90
Fig. 5.12 Vane acceleration aθ versus rotor angular displacement at 1500 rpm 90
Fig. 5.13 Vane force FVc1 versus rotor angular displacement at 1500 rpm 91
Fig. 5.14 Vane force FVc2 versus rotor angular displacement at 1500 rpm 91
Fig. 5.15 Vane force FVc3 versus rotor angular displacement at 1500 rpm 92
Fig. 5.16 Vane force FVc4 versus rotor angular displacement at 1500 rpm 92
Fig. 5.17 Vane force FVc5 versus rotor angular displacement at 1500 rpm 92
Fig. 5.18 Vane force FVc6 versus rotor angular displacement at 1500 rpm 92
Fig. 5.19 Bending moment λ3 versus rotor angular displacement at 1500 rpm 93
Fig. 5.20 Bending moment λ6 versus rotor angular displacement at 1500 rpm 94
Fig. 5.21 Vane contact at c1 region and c2 region without clearance 95
Fig. 5.22 Vane contact at c1 region and c2 region with clearance (cR ,cS = 0.02 mm) 95
Fig. 5.23 Vane contact at c1 region and c2 region with clearance (cR ,cS = 0.05 mm) 95
Fig. 5.24 Vane contact at c3 region and c4 region without clearance 96
Fig. 5.25 Vane contact at c3 region and c4 region with clearance (cR ,cS = 0.02 mm) 96
Fig. 5.26 Vane contact at c3 region and c4 region with clearance (cR ,cS = 0.05 mm) 96
Fig. 5.27 Vane contact at c5 region and c6 region without clearance 97
Fig. 5.28 Vane contact at c5 region and c6 region with clearance (cR ,cS = 0.02 mm) 97
Fig. 5.29 Vane contact at c5 region and c6 region with clearance (cR ,cS = 0.05 mm) 97
Fig. 5.30 Vane contact regions without clearance 98
Fig. 5.31 Vane contact regions with clearance
(cR ,cS = 0.02 mm) 99
Fig. 5.32 Vane contact regions with clearance
(cR ,cS = 0.05 mm) 100
Fig. 5.33 Vane displacement sr versus time, t = 0 ~ 0.5 s 101
Fig. 5.34 Vane displacement sr versus time, t = 0 ~ 0.4 s 102
Fig. 5.35 Vane displacement sr versus time, t = 0.4 ~ 0.5 s 102
Fig. 5.36 Vane velocity vr versus time, t = 0 ~ 0.5 s 102
Fig. 5.37 Vane velocity vr versus time, t = 0 ~ 0.4 s 103
Fig. 5.38 Vane velocity vr versus time, t = 0.4 ~ 0.5 s 103
Fig. 5.39 Vane acceleration ar versus time, t = 0 ~ 0.5 s 103
Fig. 5.40 Vane acceleration ar versus time, t = 0 ~ 0.4 s 104
Fig. 5.41 Vane acceleration ar versus time, t = 0.4 ~ 0.5 s 104
Fig. 5.42 Vane velocity vθ versus time, t = 0 ~ 0.5 s 105
Fig. 5.43 Vane velocity vθ versus time, t = 0 ~ 0.4 s 105
Fig. 5.44 Vane velocity vθ versus time, t = 0.4 ~ 0.5 s 105
Fig. 5.45 Vane acceleration aθ versus time, t = 0 ~ 0.5 s 106
Fig. 5.46 Vane acceleration aθ versus time, t = 0 ~ 0.4 s 106
Fig. 5.47 Vane acceleration aθ versus time, t = 0.4 ~ 0.5 s 106
Fig. 5.48 Power versus clearance at 1000 rpm 107
Fig. 5.49 Power versus clearance at 1500 rpm 107
Fig. 5.50 Power versus clearance at 2000 rpm 108
Fig. 5.51 Mechanical efficiency versus clearance at 1000 rpm 108
Fig. 5.52 Mechanical efficiency versus clearance at 1500 rpm 108
Fig. 5.53 Mechanical efficiency versus clearance at 2000 rpm 109
Fig. 5.54 Mechanical efficiency versus clearance 109
Fig. 5.55 Two-dimensional dynamic simulation drawing 110
Fig. 5.56 Trajectory of vane center, z = 3 111
Fig. 5.57 Trajectory of vane center, z = 5 111
Fig. 5.58 Trajectory of vane center, z = 7 111
Fig. 5.59 Trajectory of vane center, z = 9 112
Fig. 5.60 Trajectory of vane tip circle vs. stator inner contour, z = 3 112
Fig. 5.61 Trajectory of vane tip circle vs. stator inner contour, z = 5 112
Fig. 5.62 Trajectory of vane tip circle vs. stator inner contour, z = 7 113
Fig. 5.63 Trajectory of vane tip circle vs. stator inner contour, z = 9 113
Fig. 5.64 Local enlarged view for trajectory of vane tip circle and
stator inner contour, z = 3 114
Fig. 5.65 Local enlarged view for trajectory of vane tip circle and
stator inner contour, z = 5 114
Fig. 5.66 Local enlarged view for trajectory of vane tip circle and
stator inner contour, z = 7 114
Fig. 5.67 Local enlarged view for trajectory of vane tip circle and
stator inner contour, z = 9 115
Fig. 5.68 Two-dimensional dynamic simulation drawing, t = 0 ~ 0.2 s 116
Fig. 5.69 Three-dimensional simulation drawing of stator 117
Fig. 5.70 Three-dimensional simulation drawing of rotor and vane 117
Fig. 5.71 Three-dimensional dynamic simulation drawing 118
Fig. 5.72 Three-dimensional dynamic simulation drawing of different light source 118
Fig. 5.73 Three-dimensional dynamic simulation drawing, t = 0 ~ 0.2 s 119

Fig. 6.1 Penetration of line and stator inner contour 129
Fig. 6.2 Trajectory of vane center versus rotor angular displacement, z = 3 136

表目錄

Table 3.1 First contact region of contact situations 56

Table 4.1 Runge-Kutta parameters 73

Table 5.1 Dimensions of compressor 82
Table 5.2 Simulation parameters of compressor 82
Table 5.3 Comparisons of calculated vane kinematic results with Huang and
Chung’s data 87
Table 5.4 Vane kinematic results without clearance and with clearance 89
Table 5.5 Calculated vane contact force without clearance and with clearance 93
Table 5.6 Calculated constraint bending moment without clearance and with
clearance 94
Table 5.7 Vane contact modes without clearance 98
Table 5.8 Vane contact modes with clearance (cR ,cS = 0.02 mm) 99
Table 5.9 Vane contact modes with clearance (cR ,cS = 0.05 mm) 100
Table 5.10 Vane contact modes of two-dimensional dynamic simulation case 116

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