臺灣博碩士論文加值系統

並聯式機構具有高剛性、高速、高承載力及複雜的曲面加工能力等優點，使得並聯式機構被應用在工具機產業。為了發揮這些優點、提升性能，運動學、動態特性及剛性分析是有必要深入研究的。在本文中以台灣工業技術研究院機械所開發的並聯式工具機為例子進行研究。
關於3-PRS並聯式機構的運動分析，首先利用幾何關係所建立的耦合非線性三角方程式來描述3-PRS機構的運動學，使用貝祖消去法(Bezout Elimination Method)求得64組可能的解。將這64個解分為6個群組，探討各群組的機構構型以判斷出正確的解。此外，本文中提出利用最佳化方法，訂出正確解的行為限制條件，可有效率的求得順向運動學的解。順向運動學方程式亦可用來推導進給誤差模型，分析進給誤差如滾珠導螺桿背隙對於運動軌跡的影響。
應用牛頓-歐拉法(Newton-Euler Method)及運動分析導出的方程式來推導具剛體接頭之3-PRS並聯式機構的運動方程式，並且導入接頭剛性、間隙及摩擦力於動態模型中。利用線性化的技巧來處理複雜的三角微分方程式，忽略加速度項與速度項則可推導出靜態剛性，改變不同的滑行對位置可建立工作空間中的剛性地圖，並可求得等效剛性與接頭剛性間的比值。應用倫基-庫達數值積分方法(Runge-Kutta Method)可求得系統動態響應，數值結果顯示當接頭間隙越大時，動態響應及接頭傳遞力亦隨著增大。系統自然頻率可利用解線性化的運動方程式之特徵方程式求得，結果顯示自然頻率隨著滑行對位置改變，並利用自然頻率對應之模態形狀來解釋其變化的趨勢。有關具接頭間隙之系統自然頻率，由於接頭間隙與摩擦效應造成系統的不連續性，本文則利用脈衝響應來求得，接頭間隙對於以旋轉運動為主的運動模態的自然頻率的影響最大。最後探討系統幾何尺寸、連桿密度、接頭剛性對系統自然頻率的靈敏度。

Parallel mechanisms have been applied to machine tools due to their advantages of high stiffness, high speed, large load carrying capacity and complicated surface processing ability. In order to take these advantages, comprehensive analyses on kinematics, dynamics and stiffness should be performed. In this dissertation, a parallel typed machine tool developed by the Industry Technology Research Institute (ITRI) in Taiwan is used as the research target for performing analyses.
For the kinematics of the 3-PRS parallel mechanism, the geometric method is applied to formulate three coupled trigonometric equations to describe the direct kinematics. One algorithm for solving the three coupled equations is to use the Bezout’s elimination method which leads to a total of 64 solutions. By categorizing the 64 solutions into 6 groups and further examining the corresponding configuration for each group, the desired solution to the direct kinematics can be obtained. The other algorithm for solving the coupled equations is to apply optimization technique with side and behavior constraints. This approach can efficiently obtain the desired solution without checking all the possible solutions. The direct kinematics model is applied to develop the error model of the feed system. The backlash effects on the motion trajectory are also investigated
Based on the kinematics, the equations of motion of the 3-PRS parallel mechanism with rigid joints is formulated by using the Newton-Euler approach. Furthermore, joint flexibility, clearance and friction effects are included in the dynamics model. The linearization technique is applied to deal with the trigonometric differential equations. The static stiffness at the tool-tip is calculated by neglecting the terms of inertia and velocities from linearized equations of motion. The stiffness map is built within the working space. To obtain the dynamic response, Runge-Kutta method is adopted to solve the nonlinear equations of motion. The results shows that the dynamic responses and the transmitting forces increase as the joint clearances increase. The system natural frequencies are determined by solving the characteristic equations of the linearized equations of motion. The results show that the natural frequencies vary as the sliders positions change, and the cause of the frequencies shift is explained by analyzing their associated mode shapes. The impulse response is also applied to determine the natural frequencies of the system with joint clearances because of its discontinuity. The joint clearances significantly affect the modes which the rotational motions are dominated. Finally, the sensitivity analysis on geometric parameters, link density and joint flexibility to natural frequencies are performed.

摘要 i
ABSTRACT ii
LIST OF TABLES vi
LIST OF FIGURES vii
NOMENCLATURE xi
CHAPTER 1 INTRODUCTION 1
1.1 Motivation 1
1.2 Literature Review 2
1.2.1 Kinematic analysis 2
1.2.2 Dynamic analysis 5
1.2.3 Stiffness analysis 6
1.3 Dissertation Outline 7
CHAPTER 2 KINEMATIC ANALYSIS 10
2.1 Direct Kinematics 10
2.1.1 Direct kinematic formulation 11
2.1.2 Approach I – the Bezout’s elimination method 15
2.1.3 Approach II – optimization technique 19
2.2 Inverse Kinematics 22
2.3 Error Analysis 24
2.3.1 Feed error formulation 24
2.3.2 Ball screw backlash effect 27
2.4 Summary 29
CHAPTER 3 DEVELOPMENT OF DYNAMICS FORMULATION 45
3.1 System Equations of Motion 46
3.1.1 Consideration with rigid joints 47
3.1.2 Consideration with flexible joints 50
3.1.3 Consideration with joint clearance and friction effects 54
3.2 Formulation for Equivalent Stiffness 56
3.2.1 Linearization of equations of motion 57
3.2.2 Equivalent stiffness formulation 58
3.3 Formulation for Natural Frequency 59
CHAPTER 4 STINNFESS AND DYNAMICS ANALYSES 64
4.1 Stiffness Analysis 64
4.2 Dynamic Response and Transmitting Force Analyses 65
4.3 Natural Frequency Analysis 67
4.4 Sensitivity Analysis of Natural Frequency 70
CHAPTER 5 CONCLUSIONS 97
REFERENCES 101
APPENDIX 109
PUBLICATIONS 112

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