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論文名稱 Title |
凸輪連桿機構設計與分析 Design and Analysis of Cam-Link Mechanisms |
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系所名稱 Department |
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畢業學年期 Year, semester |
語文別 Language |
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學位類別 Degree |
頁數 Number of pages |
311 |
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研究生 Author |
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指導教授 Advisor |
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召集委員 Convenor |
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口試委員 Advisory Committee |
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口試日期 Date of Exam |
2009-06-23 |
繳交日期 Date of Submission |
2009-07-16 |
關鍵字 Keywords |
有理式B仿線、運動學、凸輪連桿機構、平面凸輪機構、靜動力學、最佳化問題、運動合成、平面連桿機構 Cam-link mechanism, Kinetostatics, Planar linkage, Motion synthesis, Kinematics, Rational B-spline, Optimization problem, Planar cam mechanism |
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統計 Statistics |
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中文摘要 |
基本型平面凸輪機構與平面連桿機構常應用於工業自動化機器裡。在決定凸輪連桿機構的設計方法與設計程序上,最基本的運動合成以及運動曲線的產生方式仍需有效的設計程序與最佳化方法以明確定義機構運動構造與運動性能。 為得到凸輪連桿機構的多目標最佳化問題之結果,文中首先介紹現有之基因演算法以作為此類問題的演算法則。探討其參數於演化過程中的影響,並說明經適當的定義其參數與變數條件,則可成功求得多目標問題之最佳解答。 為比較凸輪連桿機構於運動合成時之曲線,先介紹現有凸輪機構運動曲線類型及其運動特性,並提出有理式B仿線。為展現有理式B仿線之靈活度,使用基因演算法近似現有多段組合成的三角函數曲線。此外,選擇相關運動特性目標最小化問題,用有理式B仿線與基因演算法進行搜尋且找到較佳之結果。 為合成出不同構造的凸輪連桿機構,先推導兩種平面連桿機構與四種平面凸輪機構的運動特性,再使用基因演算法以求取具有不同運動曲線、壓力角及曲率半徑等限制下之最小凸輪尺寸。接著探討曲柄滑塊具有函數產生之運動且滑塊於肘節位置為起始點之運動合成問題。經分析發現使用傳統加速度連續之運動曲線合成滑塊運動時,曲柄的角加速度無法連續。為達到曲柄運動角加速度連續的特性,具有位移對時間四次微分為零之邊界條件的運動曲線可符合此一目標。再經由構造合成的設計程序,依照連桿機構與凸輪機構的輸入對輸出關係進行組合,並使用設計限制進行機構構造篩選,以得到符合條件的機構構造。 為應用凸輪連桿機構於實際問題,介紹一種具有曲柄滑塊為肘節機構的機器。經由可用空間之限制,以及運動限制等條件,尋找可行的機構構造,並進行尺寸定義、運動及靜動力特性分析。接著使用基因演算法以求得有理式B仿線所產生之最佳運動特性與最小靜動力特性之多目標最佳化問題,以有效減少與延長凸輪從動件之接觸應力及壽命。 由於曲柄滑塊機構的運動合成於肘節位置存在加速度不連續的問題。為得證理論之正確性,藉由實驗之設立,平面曲柄滑塊機構的運動特性亦與理論結果進行比較,實驗結果確立若合成滑塊在肘節位置的邊界條件具有位移對時間之四次或四次以上微分為零的運動曲線,則可使曲柄具有角加速度連續的特性。 綜合以上所述,本文提供建議給設計者於建立凸輪連桿機構具有一肘節式曲柄滑塊機構時所需考慮的運動特性,亦提供設計方法於簡單型函數產生凸輪連桿機構之合成、分析與實測。 |
Abstract |
The basic planar cam mechanisms and link mechanisms are widely used in industrial automatic machines. In determining the design method and design procedure for the cam-link mechanism, the basic kinematic synthesis and motion curve generation method require effective design procedure and optimization method to determine the kinematic structure of the mechanism and its kinematic performance clearly. In order to determine the result of the multi-objective optimization problem for the cam-link mechanism, the genetic algorithm is defined as the problem solver and begins this dissertation. By considering the influences of the parameters in the evolving procedure and by defining the conditions of the parameters and variables properly, the best solutions of the multi-objective optimization problem can then be solved successfully. By comparing the curves for the motion synthesis of the cam-link mechanism, the existing motion functions with their kinematic characteristics used in cam mechanisms are introduced and the rational B-spline motion function is proposed. By using the genetic algorithm to approximate the motion curves that is combined with trigonometric functions, the flexibility of the rational B-spline is demonstrated. Furthermore, to minimize different kinematic characteristics of the single-objective minimization problems, these problems are also searched by using rational B-splines with genetic algorithm for having better results. For synthesizing different structures of cam-link mechanisms, first of all is to derive the kinematics of the two planar link mechanisms and four planar cam mechanisms and then the genetic algorithm is used here to find out the minimal cam dimension with the limits of the motion curves, the pressure angles, and the radius of curvatures. Then, the kinematic synthesis problem of the function generation slider-crank mechanisms as the slider starts at the toggle position is discussed. Through the analysis finds out that when using the traditional motion functions with acceleration continuity to synthesize the slider motion, the angular acceleration of the crank cannot be continuous. To achieve the acceleration continuity of the crank motion, the curve that contains the fourth derivatives of the displacement with respect to time are set to be zeros can fulfill the continuity requirement. Then using the structural synthesis design procedure, by following the input and output relations of the link mechanisms and cam mechanisms with design constraints to select the proper structures of the mechanisms. To apply the cam-link mechanism in real application, a machine containing a slider-crank mechanism as toggle mechanism is introduced. Through the design constraints of space and motion limits to find out the possible mechanism structure and define the dimensions and then analyze the kinematics and kinetostatics of the machine. By using the genetic algorithm to solve the multi-objective optimization problem, the parameters of the rational B-spline are adjusted to have optimal kinematics and minimal kinetostatics to reduce the contact stress and to improve the fatigue life of the cam follower. Due to the existing problem of the slider-crank mechanism that contains discontinuous acceleration at the toggle position, to prove the correctness of the theoretical results, the experimental tests are measured and verified with the theoretical results in high similarity. The results show that when the slider motion curves begin at the toggle position with boundary motion constraints up to fourth or more than fourth derivatives of the displacement with respect to time that are set to be zeros, the angular accelerations of the cranks are continuous. In summary, this dissertation provides suggestions of the kinematic characteristics for the designer to design cam-link mechanisms that contain a slider-crank mechanism as the toggle mechanism and design methods for the synthesis, analysis and experimental test of the simple function generation cam-link mechanism. |
目次 Table of Contents |
誌 謝 II 摘 要 III ABSTRACT IV TABLE OF CONTENTS VI LIST OF FIGURES X LIST OF TABLES XX CHAPTER I INTRODUCTION 1 1.1 Introduction 1 1.2 Motivation 1 1.3 Structure of the Dissertation 2 CHAPTER II GENETIC ALGORITHM 5 2.1 Introduction and Historical Review 5 2.2 Foundations of Genetic Algorithms 7 2.3 Improving the Simple Genetic Algorithms 9 2.3.1 Encoding 9 2.3.2 Fitness Function Definition 10 2.3.3 Reproduction 12 2.3.4 Crossover 13 2.3.5 Mutation 14 2.3.6 Elitist strategy 15 2.3.7 Extinction and Immigration Strategy 15 2.3.8 Stopping Conditions 16 2.4 Optimization Example 17 2.4.1 Analytical Optimization 19 2.4.2 Influence of the Initial Population Range and the Crossover Rate on the Genetic Algorithm Optimization Problem 22 2.4.3 Influence of Population Size on the Genetic Algorithm Optimization Problem 25 2.5 Concluding Remarks 27 CHAPTER III MOTION CURVES 28 3.1 Introduction and Historical Review 28 3.2 Basic Motion Functions 30 3.2.1 Simple Harmonic Motion (SHM) Curve 31 3.2.2 Cycloidal Displacement Motion (CDM) Curve 31 3.2.3 Skewed Modified Trapezoidal (SMT) Acceleration Curve 31 3.2.4 Polynomial Motion Curve 35 3.2.5 Fourier Series Motion Curve 37 3.2.6 Berzak and Freudenstein 3-4-5-6-7 Polynomials 38 3.2.7 Johnson Polynomial Motion Curve 39 3.3 Spline Motion Functions (Piecewise polynomial) 39 3.3.1 Bézier Motion Function 39 3.3.2 Rational B-Spline Motion Functions 44 3.4 Design Examples 47 3.4.1 Example 3-1 – MS Motion Function Approximated by RBS 47 3.4.2 Example 3-2 – MT Motion Function Approximated by RBS 53 3.4.3 Example 3-3 – Kinematic Constraints Minimized by RBS 55 3.4.4 Example 3-4 – Specific Motion Characteristics Approximated by RBS 59 3.5 Concluding Remarks 61 CHAPTER IV KINEMATICS AND KINEMATIC STRUCTURES OF PLANAR CAM AND LINK MECHANISMS 62 4.1 Introduction and Historical Review 62 4.2 Planar Cam Mechanisms 66 4.2.1 Cam Profile Determination 71 4.2.2 Cam Performance Index 73 4.2.3 Rigid Body Transformation Procedure 75 4.2.4 Normalized Motion Functions to the Follower Motion 76 4.2.5 A Rotating Cam with an Oscillating Roller Follower 77 4.2.6 A Translating Cam with an Oscillating Roller Follower 80 4.2.7 A Rotating Cam with a Translating Roller Follower 82 4.2.8 A Translating Cam with a Translating Roller Follower 85 4.2.9 Cam Size Minimization Procedure 88 4.2.10 Design Example 4-1: A Rotating Cam with a Translating Roller Follower having Symmetric D-R-D-R Motions 90 4.2.11 Design Example 4-2: A Rotating Cam with a Translating Roller Follower having Asymmetric D-R-D-R Motions 94 4.3 Planar Linkage 97 4.3.1 Kinematics of the Four-Bar Linkage 99 4.3.2 Kinematics of the Slider-Crank Mechanism 106 4.3.2.1 When the crank is the input 107 4.3.2.2 When the slider is the input 108 4.3.3 Limiting Positions of the Slider-Crank Mechanism 109 4.3.4 Transmission Angle of the Slider-Crank Mechanism 112 4.3.4.1 Design example 4-3: In-line slider-crank mechanism motion synthesis 112 4.3.4.2 Design example 4-4: Offset slider-crank mechanism motion synthesis 118 4.3.4.3 Design example 4-5: Slider motion synthesis by using Polynomial and B-spline functions 123 4.3.4.4 Design example 4-6: Synthesize the slider motion by using B-spline function to reduce the peak crank angular acceleration value 138 4.4 The Kinematic Structure and the Creation of Function Generation CLMs 142 4.4.1 Graph Representation of the Kinematic Structure of Planar Linkage and Simple Cam Mechanisms 145 4.4.2 Design Procedure for the creation of CLMs according to structure and function 148 4.4.3 Creation of Function Generation CLMs 149 4.5 Design Procedure for Motion Synthesis and Kinematic Analysis of CLMs 156 4.6 Concluding Remarks 160 CHAPTER V KINEMATIC AND KINETOSTATIC ANALYSIS OF CAM-LINK MECHANISM 162 5.1 Design Example Introduction 162 5.2 Current Design of Link Mechanisms 163 5.2.1 Link Mechanism Driven by a Hydraulic Actuator 163 5.2.2 Link Mechanism Driven by a Gear Reduction Servo Motor with a Dwell Mechanism 165 5.3 CLM Selecting Criteria and Dimensional Synthesis 169 5.4 Kinematic Analysis of the CLM 175 5.5 Kinetostatic Analysis of the CLM 184 5.6 Contact Stress Analysis of the Cams and the Bearing Life of the Followers 198 5.7 Minimization Problem of the CLM 201 5.7.1 SMT Motion Function Applied to the Minimization Problem 202 5.7.2 RBS Motion Function Applied to the Minimization Problem 206 5.8 Concluding Remarks 215 CHAPTER VI EXPERIMENTAL TESTS OF THE KINEMATIC CHARACTERISTIC OF THE SIX-LINK CAM-LINK MECHANISM 216 6.1 Introduction 216 6.2 Dimensional Synthesis of the Six-Link CLM 216 6.3 Experimental Setup 225 6.4 Experimental Results 231 6.5 Concluding Remarks 257 CHAPTER VII CONCLUSIONS AND RESEARCH OPPORTUNITIES 258 7.1 Summary 258 7.2 Contributions 259 7.3 Research Opportunities 259 7.4 Conclusion 260 REFERENCES 261 APPENDICES 274 Appendix A Kinematics of Motion Curves 274 Appendix A.1 Kinematic Characteristics of Motion Curves 274 Appendix A.2 Derivatives of the Rational B-Spline Function 277 Appendix B Computer Program Verification Based on the Slider Kinematics of the Slider-Crank Mechanism 280 VITA 284 PUBLICATION LIST 285 |
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